BULLETINBULLETIN OF THE OF THE POLISH POLISH ACADEMY ACADEMY OF OF SCIENCES SCIENCES TECHNICALTECHNICAL SCIENCES, SCIENCES, Vol. Vol. XX, 64, No. No. Y, 4, 20162016 DOI: 10.1515/bpasts-2016-00ZZDOI: 10.1515/bpasts-2016-0093

BULLETIN OFInversion THE POLISH ACADEMY of OF selected SCIENCES structures of block matrices of chosen TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 BULLETIN OF THE POLISH ACADEMY OF SCIENCES DOI: 10.1515/bpasts-2016-00ZZInversion of selected structures of block matrices of chosen TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 mechatronic systems DOI: 10.1515/bpasts-2016-00ZZ mechatronic systems 1 1 1 2 InversionTomasz of Trawiselectednski´ ∗, structures Adam Kochan of, Paweł block Kielan matricesand Dariusz of Kurzyk chosen InversionT. of TRAWIŃSKI selected1* structures, A. KOCHAN1 of, P. KIELAN block1 matrices, and D. KURZYK of chosen2 1 Department of Mechatronics,mechatronic Silesian University of Technology, systems Akademicka 10A, 44-100 Gliwice, Poland 12DepartmentInstitute ofof Mathematics,Mechatronics,mechatronic Silesian Silesian University University of ofTechnology, Technology, systems 10A KaszubskaAkademicka 23, St., 44-100 44-100 Gliwice,Gliwice, Poland Poland 2Institute of Mathematics,1 Silesian University 1of Technology, 23 Kaszubska1 St., 44-100 Gliwice, Poland2 Tomasz Trawinski´ ∗, Adam Kochan , Paweł Kielan and Dariusz Kurzyk Tomasz Trawinski´ 1 , Adam Kochan 1, Paweł Kielan1 and Dariusz Kurzyk2 Abstract. This1 Department articleTomasz describes of Trawi Mechatronics, hownski´ to calculate∗, Silesian Adam the University number Kochan of of algebraic Technology,, Paweł operations Kielan Akademicka necessaryand 10A, Dariusz to 44-100 implement KurzykGliwice, block Poland matrix inversion that occur, Abstract. This paper describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occurs, among others,1 Department in2 mathematicalInstitute of of Mechatronics, Mathematics, models of Silesianmodern Silesian University positioningUniversity of of systems Technology, Technology, of mass Kaszubska Akademicka storage 23, devices. 10A, 44-100 44-100 The Gliwice, Gliwice, inversion Poland Poland method of block matrices is among others,Department in mathematical of Mechatronics, models of Silesianmodern positioning University systems of Technology, of mass storage Akademicka devices. 10A, The 44-100inversion Gliwice, method Polandof block matrices is pre- 2 presentedsented as as well. well.Institute PresentedThe presented of formMathematics, form of generalof general Silesian formulas formulas Universitydescribing describing of Technology, the the calculation calculation Kaszubska complexity complexity 23, of 44-100 inverted of inverted Gliwice, formform of Poland block of blockmatrixmatrix were prepared were prepared for threefor differentthree different cases cases of their of division division into into internal internal blocks. blocks. The The obtained obtained results results are compared are compared with a with standard a standard Gaussian Gaussian method methodand the and“inv” the "inv" Abstract. This article describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occur, methodmethod used used in Matlab. in Matlab. The The proposed proposed method method for for matrixmatrix inversion is is much much more more effective effective in comparison in comparison in standard in standard Matlab Matlab matrix matrixinversion inversion among others, in mathematical models of modern positioning systems of mass storage devices. The inversion method of block matrices is Abstract."inv" function“inv”This function article (almost (almost describes two two times how times faster) to faster) calculate and and is the is much much number less less of numerically numerically algebraic operations complex complex than thannecessary standard standard to Gauss implement Gauss method. method. block matrix inversion that occur, presentedamong others, as well. in mathematical Presented form models of general of modern formulas positioning describing systems the calculation of mass storage complexity devices. of inverted The inversion form of method block matrix of block were matrices prepared is forpresentedKey three words:Key different as well.words:arrowhead cases Presented arrowhead of their matrices, form matrices, division of mechatronicgeneral intomechatronic internal formulas systems, systems, blocks. describing matrix Thematrix obtained the inversion, calculation results computational computational are complexity compared complexity. ofcomplexity with inverted a standard form Gaussianof block matrix method were and theprepared "inv" methodfor three used different in Matlab. cases of The their proposed division method into internal for matrix blocks. inversion The obtained is much results more are effective compared in comparison with a standard in standard Gaussian Matlab method matrix and inversion the "inv" "inv"method1. function Introduction used in (almost Matlab. two The times proposed faster) method and is much for matrix less numerically inversion is complex much more than effective standard in Gauss comparison method. in standard Matlab matrix inversion "inv" function1. Introduction (almost two times faster) and is much less numerically complexelementary than standard matrices Gauss (blocks), method. forLs example,Msr ...an electromechan- KeyMathematical words: arrowhead models matrices, of physical mechatronic objects systems, are formulated matrix inversion, using, computationalical system complexity (squirrel cage induction motors) inertia matrix [5] MT L ... Keyamong words:Mathematical others,arrowhead the models Lagrangian matrices, of mechatronicphysical formalism. objects systems, Itare can formulated matrix be represented inversion, using, computationalmay take the complexity form of:D =  sr r  (2) 1. Introduction . . .. by theamong following others, form the Lagrangian of differential formalism. equations It can writtenbe represented in ma-  . . .  1. Introduction Ls Msr ... Mathematicaltrix formby the [1]: following models of form physical of differential objects areequations formulated written using, in matrix   MLsT MLsr ...  Mathematicalform [1]: models of physical objects are formulated using, where Ls, Lr,DM=sr denotessrs matricesrsr of self inductances(2) of stator amongMathematical others, models the Lagrangian of physical formalism. objects are It can formulated be represented using, T = M.sr L.r .... byamong the following others, the form Lagrangian of differential formalism. equations It can writtenbe represented in ma- windings andD rotor=  windings,M.sr L.r matrix..  of mutual stator(2) - rotor in- among others, the LagrangianDq¨ +C formalism.q˙ + Kq+ G It= canτ be represented (1)(1)  . . .  ductances respectively. . The. inertia.. matrices in electromechan- trixby the form following [1]: form of differential equations written in ma-  . . .  trix form [1]: whereicalLs, systemsLr, Msr denotes may have matrices structural of self features inductances allowing of stator them to be trixwhere formwhere [1]:D - denotesD denotes inertial inertial matrix, CC denotes- denotes matrix matrix of centrifugal of cen- where Ls, Lr, Msr denote matrices of self inductances of stator windingswheredividedLs, andLr, intoM rotorsr denotes a windings, number matrices of matrix submatrices, of of self mutual inductances for stator the most - rotorof stator elementary in- trifugaland andCoriolis Coriolis forces,Dq¨ + forces,C Kq˙ denotes+ KqK+ stiffness-G denotes= τ matrix, stiffness G denotes matrix,(1) vectorG windingss r andsr rotor windings, matrix of mutual stator – rotor ductanceswindingsmatrices and respectively. rotor - the windings, blocks The have inertia matrix the matrices of size mutual of in 2 statorelectromechan-2 [2, - 5].rotor Symmetric in- - denotesof gravitational vectorD ofq¨ forces,+ gravitationalCq˙ + τ Kqdenotes+ G forces,=vectorτ ofτ generalized- denotes (1)forces, vector inductances respectively. × Dq¨ +Cq˙ + Kq+ G = τ (1) icalductances systemsinertia respectively. matrices may have encountered structural The inertia features matrices in mathematical allowing in electromechan- them models to be of head whereof generalizedDq -denotes denotes vector forces, inertial of generalizedq matrix,- denotesC -displacements. denotes vector of matrix generalized of cen- dis- The inertia matrices in electromechanical systems may have dividedical systemspositioning into a may number systems have of structural submatrices, of hard features disk for drivesthe allowing most [3, 4],elementary them have to be very dif- trifugalwhereplacements.D and- denotesThe Coriolis Theinertial inertial inertial forces, matrix, matrix, matrix,K present- denotesC present -in denotes (1), stiffness inin (1), matrixpredominant in matrix, predominant of cen- Gcases, structural features allowing them to be divided into a number may be regarded as symmetrical, but for mathematical modelsmatricesdivided offerent submatrices, into - the forms, a number blocks for depending havethe of submatrices,most the onelementary size structures of for2 2 thematrices [2, of most 5]. its – elementarykinematicSymmetric the blocks chains. -trifugal denotescases, and may vector Coriolis be ofregarded gravitational forces, asK symmetrical,- denotes forces, τ stiffness but- denotes for matrix, mathematical vectorG × of wide set of physical objects its elements are indirect functioninertiamatrices have matrices the - the size blocks of encountered 2 2 have [2, 5]. the Symmetric in size mathematical of 2 inertia2 [2, matrices models 5]. Symmetric ofencoun head- of- denotes generalized vector forces, of gravitationalq - denotes forces, vectorτ of- generalized denotes vector dis- Exemplary forms£ of these matrices× are as follows: modelsof time. of wide The set elements of physical of inertial objects matrix its may elements depend are on indirectangularpositioninginertia tered matrices in mathematical systems encountered of hard models disk in mathematicalof drives head positioning [3, 4], models have systems very of head dif- of placements.of generalized The forces, inertialq matrix,- denotes present vector in (1),of generalized in predominant dis- functionor linear of time. displacements. The elements It happens of inertial very often matrix in mathematical may dependferentpositioning hard forms, disk systems drives depending [3, of 4], hard on have structures disk very drives different of [3, its4], kinematicforms, have depending very chains. dif- cases,placements. may be The regarded inertial as matrix, symmetrical, present in but (1), for in mathematical predominant d11 0 00... 0 on angularmodelling or linearof advanced displacements. mechanical Itsystems happens such very as robot often ma in-Exemplaryferenton forms,structures forms depending of ofits thesekinematic on matrices structures chains. are Exemplary of as its follows: kinematic forms of chains. these modelscases, may of wide be regarded set of physical as symmetrical, objects its but elements for mathematical are indirect d d 0... 0 mathematicalnipulators modelling [1, 23, 24]. of In advanced mathematical mechanical models of systems electro-me such-Exemplarymatrices formsare as offollows: these matrices22 are23 as follows:  functionmodels of of wide time. set The of physical elements objects of inertial its elements matrix may are indirect depend as robotchanical manipulators systems matrix [1], [23],parameters [24]. of In self- mathematical and mutual induc mod-- d 0 00d...33 0...0 0 onfunction angular ofor time. linear The displacements. elements of inertial It happens matrix very may often depend in D =11 (3) functionels oftances of electro-mechanical time. of stator The elementsand rotor systems windings of inertial matrix also matrix depend parameters may on dependangular of self-dis- d  0 00... . 0  d11 d022 d 0023 0... .0  mathematicalon angularplacements or modelling linear of displacements.the ofrotor advanced [2]. Similarly, mechanical It happens in mathematical verysystems often such models in   . dk 1,k  and mutual inductances of stator and rotor windings also de-  d22 d23 0... 0 −  asmathematical robotof manipulators head modelling positioning [1], of systems advanced[23], [24]. of mechanicalmodern In mathematical mass systems storage mod- such devices D =   sym22 d3323 0... 0 d  (3) aspend robot on manipulators angular displacements [1], [23], [24]. of the In mathematical rotor [2]. Similarly, mod- in   d 0... 0 k,k  elsas robot of electro-mechanical(hard manipulators disk drives), [1], elements systems [23], [24]. of matrix inertial In mathematical parameters matrices (which of mod- self- are rep- D =   d33 0..... 0   (3) mathematical models of head positioning systems of modern  . dk 1,k  andels of mutual electro-mechanicalresented inductances by mass moments of systems stator of and matrix inertia rotor parametersor windings by masses) also of depend self- de- on  .. −   d11 d12 .d. 13 d... d1,k andmass mutualtemporary storage inductances devices spatial ofconfigurations (hard stator disk and drives), rotor of the windings links elements of its also ofkinematic de- inertial  sym . ddk k,1k,k  pendand mutual on angular inductances displacements of stator of and the rotor rotor windings [2]. Similarly, also de- in  −  matrices (which are represented by mass moments of inertia or  sym d22 0 d...k,k 0 mathematicalpend onchain angular and models therefore displacements of head the angular positioning of the (or rotorlinear) systems [2]. displacement Similarly, of modern of in joints  sym  dk,k   by masses) depend on temporary spatial configurations of the  ...  massmathematical storage[3, 4]. devices modelsA thorough of(hard headanalysis disk positioning drives),of the structures elements systems of of ofthe moderninertial inertia ma-  D =d11 d12 d13 d33 d...1,k 0 (4) linkstrix of its of kinematic mechanical chain systems, and electromechanical therefore the angular systems, (or linear) robot d  d d ... d  matricesmass storage (which devices are represented (hard disk by drives), mass moments elements of of inertia inertial or d11 d2212 d013 ... d01. ,.k  manipulators, head positioning systems of hard disk drives, and   .  0  bymatricesdisplacement masses) (which depend are of jointson represented temporary [3, 4]. by A spatial mass thorough moments configurations analysis of inertia of of the the or struc- d22 d0 ... 0 many others, shows that they have often block structure. The D =   22 33   (4) bytures masses) of the depend inertia on matrixtemporary of mechanical spatial configurations systems, electrome- of the   sym d ,  linksby masses) of its kinematic depend on chain temporary and therefore spatial the configurations angular (or linear) of the =  d33 .... 0 k k  inertia matrices of these systems can be divided internally into D =  d33 .. 0  (4) displacementlinkschanical of its kinematic systems, of joints chain robot [3, 4]. and manipulators, A therefore thorough the analysishead angular positioningof (or the linear) struc- sys-   . 0   A square matrices with entries... equal0 zero except for their turesdisplacementtems of of the hard inertia of disk joints matrix drives, [3, 4]. of and A mechanical thorough many others, analysis systems, shows of electrome- thatthe struc- they have  sym . d0k,k  main diagonal, one row and one column have many applica- chanicalturesoften of* block the systems, inertia structure. robot matrix Themanipulators, of inertia mechanical matrices head systems, positioning of these electrome- systems sys- can  sym dk,k  e-mail: [email protected] Ations square e.g. matrices in wireless with entries communication equal zerok,k except systems for [19], their neural- temschanicalbe of divided hard systems, disk internally drives, robotinto and manipulators, many elementary others, head matrices shows positioning that (blocks), they have sys- for ex-   mainAnetwork square diagonal, matrices models one row with [20] and entries as one well column equal as issues zero have related except many withfor applica- their chemistry oftentemsample, of block hard an structure. disk electromechanical drives, The and inertia many matrices system others, (squirrelshowsof these that systems cage they induction have can A square matrices with entries equal zero except for their tionsmain[21] e.g.diagonal, or in phisics wireless one row [22]. communication and In one this column paper systems have we propose many [19], applica- neural- a method of beoftenmotors) divided blockBull. internally inertia structure. Pol. Ac.: matrix Tech. into The 64(4) elementary [5] inertia may 2016 matrices take matrices the of form these (blocks), of: systems for can ex- 853 networktionsinversion e.g. models in wireless of [20] symetric as communication well matrices as issues containing related systems with [19], non-zero chemistry neural- blocks in ample,be divided an electromechanicalinternally into elementary system matrices (squirrel (blocks), cage induction for ex- Brought to you by | Gdansk University of Technology [21]network or phisics models [22]. [20] as In well this as paper issues we related propose with a method chemistry of motors)ample, an inertia electromechanical matrix [5] may system take the (squirrel form of: cage induction networktheir models main diagonal, [20] as well one as column, issues onerelatedAuthenticated row with and chemistry zeros in remain- inversion[21] or phisics of symetric [22]. matrices In this paper containing we propose non-zero a method blocks ofin motors)∗e-mail: inertia [email protected] matrix [5] may take the form of: [21]ing or phisics entries. [22]. This In kind thisDownload of paper matrices Date we | 1/3/17 propose is significant 11:15 a AM method in modeling of theirinversion main of diagonal, symetric one matrices column, containing one row and non-zero zeros in blocks remain- in their main diagonal, one column, one row and zeros in remain- ∗e-mail: [email protected] ingtheir1 entries. main diagonal, This kind one of column, matrices one is row significant and zeros in in modeling remain- ∗e-mail: [email protected] ing entries. This kind of matrices is significant in modeling 1 1 T. Trawiński, A. Kochan, P. Kielan, and D. Kurzyk

Square matrices with entries equal zero except for their of proposed method which exhibits the smallest increase of main diagonal, one row and one column have many applica- number of algebraic operation due to block matrix dimension tions e.g. in wireless communication systems [19], neural-net- increase. work models [20] as well TomaszasTomasz issues Trawi Trawi relatednski,´ nski,´ with Adam Adam chemistry Kochan, Kochan, Paweł Paweł Kielan Kielan and and Dariusz Dariusz Kurzyk Kurzyk Tomasz Trawinski,´ Adam Kochan, Paweł Kielan and Dariusz Kurzyk [21] or phisics [22]. In this paper we propose a method of ofofof mechatronic mechatronic mechatronicinversion systems. systems. of systems. symetric Considered Considered Considered matrices structures structures containing structures of of of blocknon-zero block block matri- matri- matri- blocks2. 2.2.2. Inverting Inverting InvertingInverting the the the block blockblock block matrixmatrix matrix matrix using using using using its its its internal internal internal in their main diagonal, one column, one row and zeros in cescesces can can can be be be used used used to to to describe describe describe an an an inertia inertia inertia matrix matrix matrix in in in mathematical mathematical mathematical structurestructureits internal structure remaining entries. This kind of matrices is significant in structure modelmodelmodel of of of a a heada head head positioning positioning positioning system system system of of of a a harda hard hard disk disk disk drive. drive. drive. It It It seemsseemsmodeling interesting interesting of how mechatronic how fast fast may may systems. a a group group Considered of of symmetric symmetric structures inertia inertia ofSuppose SupposeSupposeSuppose that that thatthat the the thethe inertia inertia inertiainertia matrix matrix matrix of ofof of a aa physical a physicalphysical physical objectobject object cancan can can be be berep rep- rep- rep-- seems interestingblock matrices how can fast be may used a groupto describe of symmetric an inertia inertia matrix in resentedresentedresented in inin the thethe following followingfollowing block block form: form: form: matrices,matrices,matrices, as as as encountered encountered encountered in in in models models models of of of hard hard hard disk disk disk drive drive drive head head head po- po- po- resented in the following block form: sitioningsitioningmathematical system, system, be be model inverted inverted of consideringa considering head positioning their their block blocksystem structure structure of a hard sitioningdisk system, drive. be It inverted seems interesting considering how their fast block may structure a group of a0a0 b1b1 b2b2 ...... bkbk a0 b1 b2 ... bk andand their their reasonable reasonable block block dimension. dimension. It It is is also also worth worth to to in- in- T T and theirsymmetric reasonable inertia block matrices, dimension. as encountered It is also in worth models to of in- hard bTb a1a1 00 ...... 00 vestigatevestigate influence influence an an internal internal structure structure of of block block matrices matrices has has b 1 1 a1 0 ... 0  vestigatedisk influence drive head an internalpositioning structure system, of be block inverted matrices considering has  1T T  onon inversion inversion time time as as well well as as on on numerical numerical complexity complexity of of in- in- DD== bT2b2 00 a2a2 ...... 00 (5)(5) on inversiontheir block time asstructure well as and on their numerical reasonable complexity block dimension. of in- D = b2 0 a2 ... 0   (5)(5) versionversion process. process. The The above above mentioned mentioned issues issues motivated motivated the the  ......  versionIt process. is also worth The to above investigate mentioned influence issues an motivated internal structure the ...... 00 AuthorsAuthors to to investigate investigate the the problem problem of of numerical numerical complexity complexity of of . . . . 0  Authorsof to block investigate matrices the has problem on inversion of numerical time as complexitywell as on numer of -  T T  inversioninversion of of block block matrices matrices with with different different structures. structures. Similar Similar bbTkbk 000000000aakak inversionical of complexity block matrices of inversion with different process. structures. The above Similar mentioned  k k  problemproblemissues was was investigatedmotivated investigated the inin Authors [5], [5], where where to effectiveinvestigate effective method method the problem (based (based of TheThe division division into into elementary elementary matrices matrices of of the the above above matrix matrix is is problemT wasT investigated in [5], where effective method (based The division into elementary matrices of the above matrix is ononLDLLDLnumericalT decomposition)decomposition) complexity of of findingof finding inversion the the inverse inverseof block of of thismatrices this kind kind ofwith of The division into elementary matrices of the above matrix is on LDL decomposition) of finding the inverse of this kind of essentiallyessentiallyessentially arbitrary, arbitrary, arbitrary, but but but there there there may may may occur occur occur practical practical practical reasons reasons reasons [6], [6], [6], matrixmatrixdifferent was was proposed. proposed. structures. However, However, Similar problem approximation approximation was investigated of of complex- complex- in [5], whichwhichessentially define define arbitrary, them. them. It but It happens happens there may in in branchedoccur branched practical head head reasons positioning positioning [6] matrix was proposed. However, approximationT of complex- which define them. It happens in branched head positioning ityity of of thewhere the algorithm algorithm effective was wasmethod very very general.(based general. on We WeLDL propose propose decomposition) a a detailed detailed of systemssystemswhich ofdefine of hard hard them. disk disk drivesIt drives happens [4] [4] andin and branched in in this this case,head case, thispositioning this division division ity of thefinding algorithm the inverse was very of general.this kind We of proposematrix was a detailed proposed.systems systems of hardof hard disk disk drives drives [4] [4] and and in in this this case,case, thisthis division analysisanalysis of of the the number number of of algebraic algebraic operations operations necessary necessary to to im- im- isis correlated correlated with with the the structure structure of of the the kinematic kinematic chain chain of of head head analysisHowever, of the number approximation of algebraic of complexity operations necessaryof the algorithm to im- wasis correlatedis correlated with with the the structure structure of of the the kinematic kinematic chainchain of head plementplementplement inversion inversion inversion of of of considered considered considered block block block matrices. matrices. matrices. In In In chapter chapter chapter 2 2 2 positioningpositioning system. system. Due Due to to the the ability ability to to give give the the physical physical in- in- generalgeneralvery information information general. aboutWe about propose the the internal internal a detailed structure structure analysis of of block blockof the matri- matri-numberpositioning positioning system. system. Due Due to to the the ability ability toto givegive thethe physicalphysical in in-- generalof information algebraic operations about the internalnecessary structure to implement of block inversion matri- of terpretationterpretationterpretation of ofof the the the matricesmatrices matrices of of ofthe the the structure structure structure of an of of arrow an an arrow arrow (arrow (ar- (ar-- ces,ces, further further considered considered in in the the article article is is presented.In presented.In this this chap- chap- terpretation of the matrices of the structure of an arrow (ar- ces, furtherconsidered considered block in matrices. the article is presented.In this chap- rowhead),rowhead),head), they they they became became became the thesubject the subject subject of research. of of research. research. Arrowhead Arrowhead Arrowhead matrix terter the the inverted inverted form form of of block block matrix, matrix, derived derived in in former former works, works, rowhead), they became the subject of research. Arrowhead ter the invertedIn chapter form of2 general block matrix, information derived about in former the internal works, struc- matrixmatrixrepresentation representation representation of the of ofinertia the the inertia inertia matrix matrix matrix describing describing describing the equation the the equa- equa- isis presented. presented. In In former former works, works, the the authors authors have have not not investi- investi- matrix representation of the inertia matrix describing the equa- is presented.ture of Inblock former matrices, works, further the authors considered have in not the investi- article istion tiontionof of ofphysical of physical physical physical object object object object under under under under consideration, consideration, consideration, consideration, may may mayalso may also alsoused also used usedin used the in in in gatedgatedgated mutual mutualpresented.In mutual interactions interactions interactions this chapter between between between the internalinverted internal internal structureform structure structure of block of of of block block blockmatrix, thethedescription description description of ofoperation of operation operation of ofthe of thewireless the wireless wireless links links links[9]. [9].One [9]. Oneof One the of of matricesmatrices (consisting (consisting more more than than 16 16 elements) elements) and and its its numerical numerical the description of operation of the wireless links [9]. One of matricesderived (consisting in former more works, than 16 is elements)presented. and In former its numerical works, thethe thetheproblems problems problems problems associated associated associated associated with with with witharrow arrow arrow arrow matrices matrices matrices matrices is effective is is is effective effective effective determi de- de-- de- complexitycomplexityauthors and and have resultant resultant not investigated computation computation mutual times. times. interactions In In paper paper [18] [18] between the the nation of the eigenvalues of [7–9]. Additionally, considered are complexity and resultant computation times. In paper [18] the terminationterminationtermination of of of the the the eigenvalues eigenvalues eigenvalues of of of [7], [7], [7], [8], [8], [8], [9]. [9]. [9]. Additionally, Additionally, Additionally, numbernumberinternal of of algebraic algebraic structure operation operation of block hashas matrices been been calculated calculated(consisting for for more an an in- in-than parallel matrix inversion methods as described in [10]. In [11] number of algebraic operation has been calculated for an in- consideredconsideredconsidered are are are parallel parallel parallel matrix matrix matrix inversion inversion inversion methods methods methods as as as described described described versionversionversion16 process process processelements) of of of strictly strictly strictlyand its defined defined definednumerical matrix matrix matrix complexity (only (only (only one one one type),and type), type), resultant but but but inin [10].a [10].quick In Inmethod [11] [11] is is presentedof presented solving asystems a quick quick method of method linear of ofequations solving solving systems of systems the computation times. In paper [18] the number of algebraicin [10].arrow In matrix [11] is of presented coefficients a quickis presented. method of solving systems underunderunder different different different partition partition partition of of of input input input block block block matrix, matrix, matrix, i.e. i.e. i.e. into into into 4, 4, 4, 6 6 6 ofof linear linear equations equations of of the the arrow arrow matrix matrix of of coefficients. coefficients. operation has been calculated for an inversion process ofof linearAs equations shown in of[6] the inverted arrow matrixblock matrix of coefficients. (5) can be repre- andand 16 16 elementary elementary matrices. matrices. Other Other papers papers showing showing the the rela- rela- AsAs shown shown in in [6] [6] inverted inverted block block matrix matrix (5) (5) can can be be represented represented and 16strictly elementary defined matrices. matrix Other(only one papers type), showing but under the different rela- Assented shown as: in [6] inverted block matrix (5) can be represented tiontiontion between between between block block block matrix matrix matrix structure structure structure and and and structure structure structure of of of kinematic kinematic kinematic as:as: chainschainspartition of of robots robots of (alternatively (alternativelyinput block matrix, kinematic kinematic i.e. into chains chains 4, 6 of and of head head 16 posi-elemen posi- -as: chains oftary robots matrices. (alternatively Other papers kinematic showing chains the ofrelation head posi-between 1 1 1 1 tioningtioning systems) systems) [3, [3, 4], 4], [6] [6] and and [18] [18] or or structure structure of of winding winding of of c0c0 c0cb01ba1−a1− c0cb02ba2−a1− ...... ξ1ξ1 tioning systems) [3, 4], [6] and [18] or structure of winding of c0 −c−0b1a−1 1 −c−0b2a−2 2 ... ξ1 block matrix structure and structure of kinematic chains of − 1 −1 1T T 2 1 1 electricelectric machines machines [2] [2] and and [5]. [5]. General General formulas, formulas, describing describing a a c1c1 a1−a11−bT1b1c0cb02ba22−a12− ...... ξ2ξ2 electricrobots machines (alternatively [2] and [5]. kinematic General chains formulas, of head describing positioning a  c1 a1− b1 c0b2a2− ... ξ2  relationrelation between between block block matrix’s matrix’s structure structure and and dimension, dimension, and and  cc ...... ξξ relationsystems) between [3, block 4, 6] matrix’s and [18] structure or structure and of dimension, winding of andelectric DDr r== c 2 2 ... 3 3 (6)(6)(6) thethe number number of of algebraic algebraic operations operations have have not not been been presented presented in in Dr =  2 ξ3 (6) the numbermachines of algebraic [2] and operations[5]. General have formulas, not been describing presented a relation in  ......   ......  authors’sauthors’sauthors’sbetween former former former block papers. papers. papers. matrix's In In In chapter chapter chapterstructure 3 3 of3 of ofand this this this dimension, paper paper paper methods methods methods and the  . .    ofof accounting accountingnumber theof the algebraic number number ofoperations of algebraic algebraic have operations, operations, not been necessarypresented necessary in symsym ckck of accounting the number of algebraic operations, necessary sym ck  toto make makeauthors's during during former inversion inversion papers. process process In chapter are are described. described. 3 of this paper Three Three methods dif- dif-   to make during inversion process are described. Three dif-  k k 1 1T T 1 1 T T 1 1 ferentferent casesof cases accounting of of block block the matrices matrices number internal internal of algebraic portioning portioning operations, are are consid- consid- neces- wherewherewherec0c0 =(=(a0a0 ∑k∑j bj bjaj−aj1−j bTbj )j −)1−,c,ici =(=(aiai bTibi(c(0−c10− ++ ferent cases of block matrices internal portioning are consid- where c0 =(a0 −−∑ j b ja−j b j )− ,ci =(ai −−bi (c0− + 1 1T T= (1 1 1 1k− –1 T)–1 = ( T( –1 + − –1 T)–1 )–11 1 ered.ered. Inversionsary Inversion to make times times during for for all allinversion chosen chosen structuresprocess structures are of ofdescribed. block block matri- matri- Three bibai−aj1−j bcTb0j )j −)1 −ab0ib )i−)1 −Σj bforjaforj bji i , ci 1 ,...,1a,...,i k bki, ,c0 ξ 1ξ1 b=i=aj bjc0cb0bkiakk−a1k− , , ered. Inversion times for all chosen structures of block matri- bia−j b j )− bi)−¡ for i ∈∈ {1{,...,¡k–1},} ξ1 =–1 −Tc−0bkak−–1 , cesces are aredifferent compared compared cases with with of times times block of of inversionmatrices inversion usinginternal using standard standard portioning inver- inver- are for1 1iT T 1, …, 1 1k , ξ∈1 =1{ 1TcT0bk ak} 1, 1ξ2 = a1 b−1 c0bk ak , ces are compared with times of inversion using standard inver- ξ2ξ2==a−a1−bTb2c0cfb0 kbak−a1−, ξ, 3gξ3==a−a1−bT¡b c0cb0kbak−a1−. . considered. Inversion times for all chosen structures ofξ 2 = a−ξ1 1b= 1ac10–1bbkTac−kbk,aξ–13 .= a−2 2b 2c20bka−k k. sionsionsion method method method in in in Matlab Matlab Matlab (“inv” (“inv” (“inv” function). function). function). The The The presented presented presented method method method TheThe1 3 inverted inverted1 2 2 blockk0 blockk k matrix matrix2 of2 of inertia inertiak (6) (6) consist consist of of (as (as can can block matrices are compared with times of inversion using The inverted block matrix of inertia (6) consist of (as can ofofof block block block matrix matrix matrix inversion inversion inversion is is ismuch much much more more more effective effective effective than than than the the the one one one bebe observed) observed) elementary elementary matrices, matrices, which which are are calculated calculated on on the the usedused in instandard Matlab. Matlab. inversion Also, Also, in in method this this article article in Matlab the the number number (“inv” offunction). of algebraic algebraic Thebe observed)The inverted elementary block matrices,matrix of whichinertia are(6) consist calculated of (as on can the used inpresented Matlab. Also,method in of this block article matrix the numberinversion of is algebraic much morebasis basisbasisbe ofobserved) of of the the the block block block elementary matrix matrix matrix elements elements elementsmatrices, (5) (5) which (5) before before before are the thecalculated the inversion. inversion. inversion. on Itthe It isIt is is operationsoperationsoperations (necessary (necessary (necessary to to to invert invert invert the the the matrices) matrices) matrices) has has has been been been calcu- calcu- calcu- possiblepossible to to calculate calculate chosen chosen elementary elementary submatrix submatrix (6) (6) without without latedlated and andeffective compared compared than with withthe Gaussianone Gaussian used methodin method Matlab. of of matrixAlso, matrix in inversion, inversion,this articlepossible basis of to the calculate block matrix chosen elements elementary (5) before submatrix the inversion. (6) without It is lated andthe compared number of with algebraic Gaussian operations method of(necessary matrix inversion, to invert the thethepossible need need of ofto calculation calculationcalculate chosen of of the the remainingelementary remaining elementssubmatrix elements of (6) of the thewithout blocks blocks andand it it has has shown shown the the advantages advantages of of proposed proposed method method which which the need of calculation of the remaining elements of the blocks and it hasmatrices) shown has the been advantages calculated of and proposed compared method with which Gaussianmatrix. matrix.matrix.the need of calculation of the remaining elements of the blocks exhibitsexhibitsexhibitsmethod the the the smallest smallest smallest of matrix increase increase increase inversion, of of of number number number and it ofhas of of algebraic algebraic shown algebraic the operation operation operationadvantages matrix. dueduedue to to to block block block matrix matrix matrix dimension dimension dimension increase. increase. increase. 3.3.3. The The The number number number of of of algebraic algebraic algebraic operations operations operations during during during 854 thethethe inversion inversion inversion of of of the the the block block blockBull. matrix matrix matrix Pol. Ac.: Tech. 64(4) 2016 Brought to you by | Gdansk University of Technology ForForFor the the the sake sake sake of of of consequence consequence consequence of of of further further further course course course of of of the the the discussion, discussion, discussion, definitiondefinition of of block block dimension dimension is is formulated. formulated.Authenticated definition of block dimensionDownload is formulated. Date | 1/3/17 11:15 AM DEFDEF1.1. IfIf the the symmetric symmetric block block matrix matrix D D has has been been di- di- DEF 1. If the symmetric block matrix D has been di- videdvidedvided into into into elementary elementary elementary matrices matrices matrices using using usingkkverticalkverticalvertical lines lines lines (into (into (intokk+1+1k+1 columns)columns)columns) and and andkkhorizontalkhorizontalhorizontal lines lines lines (into (into (intokk+1+1k+1 rows) rows) rows) the the the block block block size size size

22 Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. XX(Y) XX(Y) 2016 2016 2 Bull. Pol. Ac.: Tech. XX(Y) 2016 Inversion of selected structures...

n of the block matrix is defined as an ordered pair of num- Table 1 bers (k+1, k+1). Block size of the block matrix is written as: Number of algebraic operations necessary to calculate the leading element c0. n=(k+1) (k+1), or briefly by n=(k+1). Block dimension n 2 3 4 5 × Sum of algebraic operations loI 4 7 10 13 In order to demonstrate effectiveness of the computing algo- rithm of matrix inversion, the number of algebraic operations needed to be performed will be calculated for block matrices Table 2 with different internal divisions into elementary blocks - sub- Number of algebraic operations necessary to calculate the negative feedback matrices. All of the analyzed cases of the block matrix will matrix di. have a structure such as matrix (5), it will differ only in size Block dimension n 2 3 4 5 of elementary matrices. As mentioned above, each of the in- Sum of algebraic operations loII 3 6 9 12 verse elementary matrices (6) can be calculated individually. A detailed analysis of the structure of the matrix (6) reveals that there are four different types of items - elementary submatrices 3.1. Case 1 - one-piece elementary matrices - a partition of inverted block matrix requiring the calculation. These are as in the 1-1-1-1 order If the block matrix can be divided into follows: one-piece elementary matrices (it will take the form of a ma- trix (5)), then the number of algebraic operations related to 1. first elementary matrix, which later will be called the leading the calculation of the leading element c0 (7) is determined by: element, has the following form: block size n of the block matrix, the inversion process, opera- 1 T tions of addition (subtraction) in triples of matrices b ja−j b j , k 1 T 1 T 1 multiplication in triples of matrices b ja−j b j , and a j matrix in- c0 =(a0 ∑b ja−j b j )− (7) − j version. By convention, such division of the block matrix for one-piece elementary matrices will be referred as division in 2. elementary matrices, that in physical interpretation can be 1-1-1-1 order. Calculated number of algebraic operations for responsible for negative feedback, have following forms: various dimensions of the block matrices is shown in Table 1.

1 General relationship between the dimension of the block d = c b a− (8) i − 0 i i matrix block dimension n and the number of algebraic opera- for i 1,...,k , auxiliary matrices having forms: tions that are needed for the calculation of the leading element ∈ { } c 1 T 0 (7), is as follows: e = a− b (9) i − i i for i 1,...,k 1 . loI = 3(n 1)+1 (13) ∈ { − } − 3. elementary matrices, which in physical interpretation can be The calculations effort necessary to perform the designation responsible for positive couplings, are forms of the products of negative feedback matrix d (8), is associated with the im- of the matrices (8) and (9) i plementation of algebraic operations necessary to: calculate eid j (10) the matrix product c0 and bi, calculate the inverse form of the matrix ai (in present case - dividing by an element of the ma- for i 1,...,k 1 , j 1,...,k . ∈ { − } ∈ { } trix). Calculated numbers of algebraic operations for different 4. block matrices, which can be called in physical interpreta- block dimension n of the block matrix, the negative feedback tion as self inertia matrices, have following forms: matrix di (8), are shown in Table 2. It should be noted, that in- T 1 1 T 1 1 creasing size of the block matrix by one, results in occurrence ci =(ai b (c− + bia− b )− bi)− (11) − i 0 j j of an additional negative feedback matrix di - in the first row Inversion of selected structures of block formatricesi of1 ,...,chosenk mechatronic. systems and first column of the inverted block matrix. ∈ { } General relationship between block dimension of the block By introducing the above indications of elementary matri- matrix and the number of algebraic operations that are needed 3. The number of algebraic operations during By introducing the above indications of elementary ma- ces, inverted block matrix (6) takes the following form: for the calculation of the negative feedback matrix di (8) is as trices, inverted block matrix (6) takes the following form: the inversion of the block matrix follows: c d d ... d For the sake of consequence of further course of the discussion, 0 1 2 k c e d ... e d l = 3(n 1) (14) definition of block dimension is formulated.  1 1 2 1 k  oII − c ... e d Inversion of selected structures...Dr = 2 2 k (12) for n = 2,3,... . Definition 1. If the symmetric block matrixInversion D Inversionhas been of divided selected of selected structures... structures... . .   .. .  The number of algebraic operations needed in order to deter- n into elementary matrices using k vertical lines (into k+1 col-  Table 1  n of the block matrix is defined as an orderedInversion pair of num- of selected structures... Table 1  mine the matrix of positive feedback eid j (10), is related with n of theumns) block and matrix k horizontal is defined lines as (into an ordered k+1 rows) pair the of block num- size Number of algebraic sym operationsTable necessaryTable 1 1 to calculatec the leading element c0. nbersof (thek+1, blockk+1). matrix Block is size defined of the as block an ordered matrix pairis written of num- as: Number of algebraic operations necessary to calculatek the leading element c0. bers (nk +1,of thek+1). block Block matrix size is ofdefined the block as an matrixorderedInversion is pair written of ofnum as: selected-NumberNumber structures... of algebraic of algebraic operations operations necessary necessary to calculate to calculate the leading the leading element elementc0. c .the calculation of the auxiliary matrix ei, and its product with bersbers (k+1, (k+1,k+1).k+1). Block Block size size of theof the block block matrix matrix is written is written as: as:  Block dimensionTable 1n 2 3 4 5 0 n=(ofk+1) the block(k+1), matrix or briefly is defined by n=(k as+1). an ordered pair of num-Later in this article the numberBlock dimension of algebraicn 2 operations3 4 required5 negative feedback matrix d j. It should be emphasized that this n=(k+1)bers(k× +1),(k+1, or k briefly+1). Block by n =(sizek+1). of the block matrix is written as: NumberLater in of this algebraic articleBlock operations theBlock dimensionnumber dimension necessary ofn algebraicn2 to calculate23 operations34 the4 leading5 required5 element c . n=(k+1)×(k+1), or briefly by n=(k+1). Sum of algebraic operations loI 4 7 10 13 0 bersn of (n× thek =+1,× ( blockkk++1).1) matrix( Blockk+1), isor size definedbriefly of the by as blockn an = ( orderedk matrix+1). pair is written of num- as:to implementto implementSum of in algebraic in order order tooperations to calculate calculateTablel 1 theoI 4the inverted inverted7 10 blockblock13 matrixmatrix type of matrices occurs for block dimension n 3. These cal- In order to demonstrate effectiveness of the computing algo- Sum of algebraic operationsoI loI 4 7 10 13 ≥ Inn=( orderInk+1) order to( demonstratek to+1), demonstrate£ or briefly effectiveness effectiveness by n=(k+1). of the of the computing computing algo- algo- Number of algebraic operationsBlock dimension necessary ton calculate2 3 the leading4 5 element c0. rithmbersIn order(k of+1,× matrix tok+1). demonstrate inversion, Block size effectiveness the of number the block of ofmatrix algebraic the computing is written operations algo- as: (5)(5) for for three three different different cases cases of of the the internal internal structure structure ofof input culations require: inversion of the matrix ai, multiplication of rithm of matrix inversion, the number of algebraic operations Sum of algebraicBlock operations dimensionlnoI 24 37 410 513 1 T rithmrithmn=( ofk+1) matrixofIn matrix( orderk+1), inversion, inversion,to or demonstrate briefly the bythe numbern number=(effectivenessk+1). of algebraic of algebraic of the operations computing operations al-matricesmatrices will will be be calculated. calculated. matrices ai− bi , multiplication by ( 1) and multiplication by neededIn order to× be to demonstrate performed will effectiveness be calculated of the for computing block matrices algo- Table 2 − neededneeded togorithm beto be performed of performed matrix inversion, will will be becalculated the calculated number for of for blockalgebraic block matrices operations matrices Sum of algebraic operationsTable 2 loI 4 7 10 13 withrithmIndifferent order of matrix to demonstrate internal inversion, divisions effectiveness the number into elementary of of algebraic the computing blocks operations - algo- sub- Number of algebraic operations necessaryTable 2 to calculate the negative feedback with differentneeded internal to be performed divisions will into be elementary calculated for blocks block - sub-matricesNumber 3.1.Number ofCase algebraic of algebraic1 – operationsone-piece operations necessary elementary necessarymatrix to calculated . to matrices calculate the negative the – negativea partition feedback feedback with different internal divisions into elementary blocks - sub- Bull. Pol. Ac.: Tech. XX(Y) 2016 matrix dii. 3 matrices.neededrithm of to matrix beAll performed of inversion, the analyzed will the be number cases calculated of of the algebraic for block block matrix operations matrices will matrixTabledi. 2 matrices.with All different of the internal analyzed divisions cases ofinto the elementary block matrix blocks will – sub- in the 1‒1‒1‒1 order. If the matrixblock dmatrixi. can be divided into matrices. All of the analyzed cases of the block matrix will Number of algebraic operationsBlock dimension necessaryn to calculate2 3 the4 negative5 feedback havewithneededmatrices. different a structure to be performedAll internal such of the as divisions analyzed will matrix be into(5),calculated cases itelementary willof the for differ block block blocks only matrix matrices in - sub- sizewill one-piece elementaryBlock Blockmatrices dimension dimensionTable (itn will 2 n2 take23 the34 form45 of a5 matrix have a structure such as matrix (5), it will differ only in size Sum of algebraicBlock operations dimensionmatrix ldin. 23 36 49 125 haveof elementary a structure matrices. such as Asmatrix mentioned (5), it will above, differ each only of inthe size in- Number of algebraicSum of algebraic operations operations necessaryloII to calculate3 6 the9 negative12 feedback ofmatrices.with elementaryhave different a All structure internalmatrices.of the such analyzed divisions Asas matrix mentioned cases into (5), elementary ofit will above, the differ block each blocks only matrix of in the- size sub- will in- of (5)), thenSum the of algebraicnumber operationsof algebraicloII operations3 6 related9 12 to the cal- ofof elementary elementary matrices. matrices. As As mentioned mentioned above, above, each each of the of the in- in- Sum of algebraicBlock operations dimensionmatrix dloII.n 32 63 94 125 versehavematrices.elementary a elementary structure All of matrices. such matrices the analyzed as As matrix (6) mentioned can cases (5), be calculated it of above, will the differ block each individually. of only matrix the ininverse will size A culation of the leading element c0 (7)i is determined by: block verseverse elementary elementary matrices matrices (6) (6) can can be calculated be calculated individually. individually. A A Sum of algebraic operations l 3 6 9 12 detailedofhave elementaryelementary a structure analysis matrices. matrices such of the as structure (6) Asmatrix can mentioned be (5), of calculated the it will matrixabove, differ individually. (6) each only reveals of in the A size that in-de- size n of the block matrix,Block dimensionthe inversionoIIn 2process,3 4operations5 of detaileddetailed analysis analysis of theof the structure structure of the of the matrix matrix (6) (6) reveals reveals that that 3.1. Case 1 - one-piece elementary matrices–1 T - a partition thereverseof elementarytailed are elementary four analysis different matrices. matrices of the types Asstructure (6) ofmentioned can items be of calculated -the elementary above, matrix eachindividually. (6) submatrices reveals of the in-that A3.1. 3.1.addition Case Case 1 (subtraction)Sum - 1 one-piece - of one-piece algebraic in elementary operationstriples elementary ofl oIImatrices matrices3 matrices 6bjaj -9b aj , -partition 12multipli a partition- there are four different types of items - elementary submatrices 3.1. Case 1 - one-pieceIf theelementary block–1 T matrix matrices can be - divided a partition into theredetailedofverse invertedthere are elementary analysis four are block four different matrixofmatricesdifferent the types structure requiring (6)types of can itemsof of beitems the the calculated - calculation. elementary – matrix elementary individually.(6) Thesesubmatrices reveals submatrices are that Aas incation the 1-1-1-1in triples order of matricesIf the bja blockj bj , and matrix aj matrix can be inversion. divided into of invertedof inverted block block matrix matrix requiring requiring the the calculation. calculation. These These are areas asin thein the 1-1-1-1 1-1-1-1 order orderIf theIf theblock block matrix matrix can can be divided be divided into into oftherefollows: invertedof are inverted four block different block matrix matrix types requiring ofrequiring items the - calculation.the elementary calculation. These submatrices These are are as one-piece3.1.By convention, Case elementary 1 - such one-piece division matrices elementary of the (it block will matricesm take atrix the for form -one-piece a partition of a ma- follows:follows:detailed analysis of the structure of the matrix (6) reveals that one-pieceone-piece elementary elementary matrices matrices (it will (it will take take the theform form of a of ma- a ma- follows:of invertedas follows: block matrix requiring the calculation. These are as 3.1.trixinelementary the (5)),Case 1-1-1-1 then 1 matrices - one-piece orderthe number willIf be elementary the ofreferred block algebraic matrixas matricesdivision operations can bein - a1‒1–1‒1 divided partition related into to there are four different types of items - elementary submatrices trixtrix (5)), (5)), then then the the number number of algebraic of algebraic operations operations related related to to 1. first1. elementary first elementary matrix, matrix, which which later later will will be be called called the the leading leading inone-piecetheorder. the calculation 1-1-1-1Calculated elementary order of number the leading matricesIf theof algebraic block element (it will matrix operationsc take0 (7) can the is be determinedfor form divided various of a into ma- by: 1. 1.firstfollows:offirst inverted elementary elementary block matrix, matrix matrix, which requiring which later later the will willcalculation. be called be called the These the leading leading are as thethe calculation calculation of the of theleading leading element elementc0 (7)c0 is(7) determined is determined by: by: 1. firstelement, elementaryelement, has the has matrix, following the following which form: laterform: will be called the leading theblockdimensions calculation size n ofof the of the block the block leading matrices matrix, element is theshown inversionc0 in(7) Table is determinedprocess, 1. opera- by: element,follows:element, has has the the following following form: form: blockone-pieceblocktrix size (5)), sizen of thenelementaryn theof the theblock numberblock matricesmatrix, matrix, of the algebraic(it inversionthe will inversion take operations process, the process,form opera- related of aopera- ma- to element, has the following form: block size n of the block matrix, the inversion process, opera-1 T 1. first elementary matrix, which later will be called the leading trixthetions calculation (5)), of addition then the of (subtraction) the number leading of element algebraic in triplesc0 operations of(7) matrices is determined relatedb1jaT−j b by: toj , tionstions of addition of addition (subtraction) (subtraction) in triples in triples of matrices of matricesb ja−b jba−j ,1bTj , k tions of addition (subtraction)Table in1 triples1 ofT matrices jb ja−j b , 1. element,first elementary has the matrix, following whichk form: later will1 T be1 called the leading theblockmultiplication calculation size n ofin of the triples the block leading of matrix, matrices element theb1 inversionjca0T−1(7)bT is, and determined process,a j matrix opera-j by: in-j c =(a k b j1a−T1bT1)−1 (7)(7) multiplicationNumber of algebraic in triples operations of matrices necessaryb jtoa− jcalculateb j , and thea leadingj matrix in- c0 =(a0 ∑b ja−j b j )− (7)multiplication in triples of matrices b ja−j b j ,1 andT a j matrix in-1 T element, has thec0 =( followinga0 − form:b∑ja−j bjj1)−T 1 (7) blockmultiplicationtionsversion. ofsize addition Byn of convention, in the triples (subtraction) block of matrix,such matrices division in the triplesb inversionja− ofj ofb thej matrices, and block process,a j matrixb j opera-a−j b forin-j , c0 =(−a0∑− j b ja−j b j )− (7) version. By convention,element such division c0 of the block matrix for −j ∑k version. By convention, such division of the1 blockT matrix for1 T j 1 T 1 tionsversion.multiplicationone-piece of addition By elementary convention, in (subtraction)triples matrices of such matrices divisionin will triplesb bea of− referred ofb the matrices, and block asa division matrixbmatrixja−j b forin-j in, c =(a b a− b )− (7)one-pieceone-pieceBlock elementary elementarydimension matrices n matrices will will be2 referred bej j referred3 j as4 division asj5 division in in 2. elementary matrices,0 that0 in∑k physicalj j j interpretation can be one-piece elementary matrices will be referred1 T as division in 2. 2.elementaryelementary2. elementary matrices, matrices, matrices, that that in −that physical in physicalin physical interpretation1 T interpretation interpretation1 can can becan bebe multiplicationversion.1-1-1-1 order. By convention, in Calculated triples of such matrices number divisionb ofa algebraic− ofb the, and block operationsa matrix matrix in- for 2. elementary matrices,c =( thata inj physicalb a− b interpretation)− can(7) be1-1-1-11-1-1-1 order. order. Calculated Calculated number number of algebraic ofj algebraicj j operations operationsj for for responsible for negative0 0 feedback,∑ j j havej following forms: 1-1-1-1Sum order. of algebraic Calculated operations number loI of4 algebraic7 10 operations13 for responsibleresponsible for negative for negative feedback,− feedback,j have following have following forms: forms: version.variousone-piece dimensions By elementary convention, of matricesthe such block division will matrices be of referred the is shown block as in matrixdivision Table for 1. in 2. responsibleelementary formatrices, negative that feedback, in physical have interpretation following forms: can bevarious dimensions of the block matrices is shown in Table 1. one-piecevarious1-1-1-1General dimensions order. elementary relationship Calculated of matrices the between block number will matrices the of be dimensionalgebraic referred is shown as operations of division in the Table block forin 1. 1 GeneralGeneral relationship relationship between between the the dimension dimension of the of theblock block 2. responsibleelementary formatrices, negative thatdi feedback,= inc physical0bi1a havei− interpretation following forms: can(8) be(8) GeneralGeneral relationshiprelationship between between the the dimension dimension of the of theblock block di =i c0−bia0i−i i−1 (8) 1-1-1-1variousmatrix block order.dimensions dimension Calculated of then numberblockand the matrices of number algebraic is of shown algebraic operations in Table opera- for 1. di−= −c0bia− (8)matrixmatrix block block dimension dimensionn andn and the thenumber number of algebraic of algebraic opera- opera- responsible for negative feedback,− havei following forms: matrixtionsmatrix that block are dimension dimensionneeded for n n theandand calculationthe the number number ofof the ofalgebraic algebraic leading opera element opera-- for i 1,...,k , auxiliary matrices1 having forms: tionsvarioustions thatGeneral that are dimensions areneeded relationship needed for of thefor the between calculationthe block calculation matrices the of dimension the of is leading the shown leading ofin element theTable element block 1. for i ∈1for,...,{ i k ,1, …, auxiliary} k , dauxiliaryi matrices= c0 bmatricesi havinga− having forms: forms: (8) tionstions that are are needed needed for for the the calculation calculation of ofthe the leading leading element element for∈i{∈ {1,...,2}f k}, auxiliaryg matrices− i having forms: cmatrix0General(7), is block as relationship follows: dimension betweenn and the the number dimension of algebraic of the blockopera- ∈ { } 1 T1 c0 (7),c0 (7), is as is follows:as follows: deii== c1a0i−bTiabi−i (8)(9) c00 (7), is as follows: for i 1,...,k , auxiliaryei =i a matrices−i− bi−i 1 Ti having forms: (9) matrixtions that block are dimension needed forn theand calculation the number of of the algebraic leading element opera- ei−= −a− b (9)(9) ∈ { } − i i tionsc (7), that isas are follows: needed forl the= calculation3(n 1)+1 of the leading element(13) forfor ii 1,...,,...,k , auxiliary1 . matrices1 T having forms: 0 l =loI3=(n3(n1)+11)+ 1 (13)(13)(13) forfori i ∈1,...,{1 k k1} .1 . e = a− b (9) oI − for∈i{∈ {1,...,−k −} 1}. i i i c0 (7), is as follows: loI = 3−(n −1)+1 (13) 3. 3.elementaryelementaryformatrices, i matrices, 1, …, whichk which 1 in. physical in− physical1 T interpretation interpretation can can be be − ∈ { 2 f − } ¡ eig= ai− bi (9) The calculations effort necessary to perform the designation 3. elementaryforresponsiblei 1,..., matrices, fork positive1 . which couplings, in− physical are forms interpretation of the products can be The calculationsThe calculations effort effort necessaryloI =necessary3(n to1 perform )+to perform1 the the designation designa-(13) responsibleresponsible for for positive positive couplings, couplings, are are forms forms of the of the products products ofThe negative calculations feedback effort matrix necessarydi −(8), to is perform associated the with designation the im- responsible∈ { for positive− } couplings, are forms of the productsof negativeof negative feedback feedback matrixl matrix=di3(8),(dni (8), is1)+ associated is1 associated with with the im-the(13) im- 3.of theofelementaryfor the3. matricesi elementary matrices1,..., matrices, (8)k (8) andmatrices,1 and (9). which (9) which inphysical in physical interpretation interpretation can can be be oftion negative of negative feedback feedback matrixoI matrixd d(8),i (8), is is associated associated with with thethe im- of the∈ matrices{ (8)− and} (9) plementationplementationThe calculations of algebraic of algebraic effort operations necessary operationsi− necessary to performnecessary to: the to:calculate designation calculate 3. responsibleelementaryresponsiblematrices, for positive for positive which couplings, couplings, in physical are are forms interpretation forms of of the the products products can be plementationimplementation of of algebraic algebraic operations necessary necessary to:to: calculate calculate eid j (10) oftheThe negative matrix calculations product feedback effortc0 matrixand necessarybi,d calculatei (8), to is perform associated the inverse the withdesignation form the of im- the ofresponsible theof matrices the matrices for (8)positive and (8) (9)and couplings,eid (9)ej id j are forms of the products(10)(10)thethe matrixthe matrixmatrix product product productc0 candc0 0andandbi ,b calculateib, icalculate, calculate the the inversethe inverse inverse formform form ofof thethe of the eid j (10) oftheplementationmatrix negative matrixai (in product feedback present of algebraicc0 matrixcaseand -b operationsi dividing,d calculatei (8), is by associated necessary the an inverse element with to: form of calculate the the of im- ma- the forof thei matrices1,...,k (8)1 and, j (9) 1,...,k . matrixmatrixmatrixai (in aii (in present(in present present case case case - dividing– -dividing dividing by by an by an element anelement element ofof thethe of ma- thema- ma- forfori i 1,...,1,...,k k1 ,1j , j 1,...,1,...,k .k . plementationmatrixtrix). Calculatedai (in present of algebraic numbersc case -b ofoperations dividing algebraic by necessary operations an element to: for of calculate different the ma- for∈ i{∈ {1,...,−k −} 1}∈, j{∈ {1e,...,id}j k}. (10)(10)trix).trix).thetrix). Calculated matrix Calculated product numbers numbers numbers0 and of of algebraic of algebraici, algebraic calculate operations operations operations the inverse forfor different fordifferent form different of the 4. 4.blockblock matrices,∈ matrices,{ which− which} can can∈ be{ becalled called} in physical in physical interpreta- interpreta- thetrix).block matrix Calculated dimension product numbersncofand thebblock of, calculatealgebraic matrix, the operations the inverse negative formfor feedback different of the 4. block matrices, which can beeid calledj in physical interpreta-(10) blockblockmatrixblock dimension dimension ai (in presentn of nn of theof0 the case theblock block block -i dividing matrix, matrix, matrix, the by the annegative the negative element negative feedbackfeedback of feedback the ma- tiontionfor asi self as self1 inertia,..., inertiak matrices,1 matrices,, j have1 have,..., followingk following. forms: forms: block dimension n of the block matrix, the negative feedback ∈for{ i 1, …, − }k 1∈ {, j 1, …, } k . matrixmatrixmatrixtrix).matrixd Calculated(8), adiii (8),(in(8), are presentare areshown shown numbers shown casein in Table in Table - of Table dividing algebraic 2. 2. It 2.It should should It by should operations an be elementbe noted, noted, be noted, for of thatthat differentthe that in-in- ma- in- 4. tionblockfor i as matrices, self1,...,2 inertiaf k which1 matrices,,¡j cang 1 be have,...,2 f calledk following. ing physical forms: interpreta- matrixi di (8), are shown in Table 2. It should be noted, that in- ∈ { − } ∈ { } creasingtrix).creasingblock Calculatedsize dimensioni size of the of theblock numbersn of block thematrix of blockmatrix algebraic by matrix, one, by one, results operations the results negative in occurrence in for occurrence different feedback 4. block matrices,=( whichT T can(1 be1 + called1 T1 inT1) physical1 1) 1 interpreta- tion4. as block selfci =(matrices,c inertiai =(ai ai matrices,b which(bci−(c can+−0 haveb +beia−b calledia followingb−j )b− inj ) bphysical−i)−b forms:i)− interpretation(11)(11) blockcreasing dimension size of then of block the block matrixTable 2 matrix, by one, the results negative in occurrence feedback −i Ti0 0 1 j jj1 Tj 1 1 of anofmatrix additional an additionaldi (8), negative are negative shown feedback in feedback Table matrix 2. matrix It shoulddi -d ini be- the in noted, firstthe first row that row in- tion as selfc inertiai =(−ai matrices,−bi (c0− have+ bia following−j b j )− b forms:i)− (11) Number of algebraic operations necessary to calculate the negative as self inertia matrices,− have following forms: matrixofcreasingand an first additionaldi columnsize(8), areof the negative ofshown the block inverted in feedback matrix Table block 2. by It matrix one, should matrix. resultsdi be- in noted, in the occurrence first that row in- forfori i 1,...,1,...,k .k . T 1 1 T 1 1 andand first first column column of the of theinvertedfeedback inverted block matrix block matrix. d matrix. ∈ { ci =(} ai bi (c− + bia−j b j )− bi)− (11) creasingand first sizecolumn of the of blockthe inverted matrix block by one,i matrix. results in occurrence for∈i{∈ {1,...,} k}. − T 0 1 1 T 1 1 GeneralofGeneral an additional relationship relationship negative between between feedback block block dimension matrix dimensiondi - of in the theof blockthe first block row ∈ { ci =(} ai bi (c0− + bia−j b j )− bi)− (11)(11) GeneralBlock dimension relationship n between block2 dimension3 4 of5 the block ByforBy introducingi introducing1,..., thek the. above− above indications indications of elementary of elementary matri- matri-matrixofmatrixand an and first additional and the column thenumber number negative of theof algebraic invertedof feedback algebraic block operations matrix operations matrix.d thati - in that are the neededare first needed row By introducing∈ { } the above indications of elementary matri- andmatrix first and column the number of the inverted of algebraic block operations matrix. that are needed ces,ces, invertedfor invertedi block1,..., block matrixk . matrix (6) (6) takes takes the the following following form: form: forfor theGeneral the calculationSum calculation of relationship algebraic of the of operations thenegative between negative loII feedback block feedback3 dimension matrix6 matrix9 di of(8)12di the is(8) as block is as ces, inverted∈for{ i block 1, …, } matrixk . (6) takes the following form: for the calculation of the negative feedback matrix di (8) is as By introducing2 f the aboveg indications of elementary matri-follows:follows:matrixGeneral and relationship the number between of algebraic block operations dimension that of are thei needed block By introducing thec aboved indicationsd ... of elementaryd matri- matrixfollows: and the number of algebraic operations that are needed ces, inverted blockc0 matrixc0 d1 (6)d1 d takes2 d2 ... the... followingdk dk form: for the calculation of the negative feedback matrix di (8) is as ces, inverted block matrixc0 d (6)1 takesd2 the... followingdk form: forfollows: the calculation of the negative= ( feedback) matrix di (8) is as Bull. Pol. Ac.: Tech. 64(4)c1 c 20161e1de21d2...... e1dek1dk loII l=oII3=(n3(n1) 1) (14)855(14)   c1 e1d2 ... e1dk  follows: l = 3−(n −1) (14)  c0 d1 1dc22 ... e1ddkk  Brought to you by | GdanskoII University of Technology D D=r = c2 c2 ...... e2dek2dk (12)(12) for n = 2,3,... . − r Dr  c0 d1 dc2 ... eddk  for forn =n2=,3,...2,3,.... . l = 3(n 1) Authenticated (14) Dr =  c1 e1d2 2. . . e21.dk  (12) for n = 2,3,... . oII   c e d ...... e .d  TheThe number number of algebraic of algebraicDownloadl operations= operations3 (Daten − needed 1| 1/3/17) needed in11:15 order in AM order to deter- to deter-(14)   1 1c 2 ...... e1.dk  oII Dr=  2 . 2.k  (12) formineThen = the number2, matrix3,... of. of algebraic positive operations feedback− e neededid j (10), in isorder related to deter- with   sym c ... ecd  minemine the the matrix matrix of positive of positive feedback feedbackeid je(10),id j (10), is related is related with with Dr =sym sym 2 . ck 2c.kk  (12) for n = 2,3,... . e d   . .  theminethe calculationThe calculation the number matrix of the of of of algebraic theauxiliary positive auxiliary operations matrixfeedback matrixe , andneedediei,j and(10), its product in its is order product related withto deter- with   sym . . c..k  the calculation of the auxiliary matrixi ei, and its product with Later in this article the number of algebraic.. operations.  required thenegativeThe calculation number feedback of of algebraic the matrix auxiliaryd operations. It matrixshould needede bei, emphasizedand in its order product to that deter- with this LaterLater in this in this article article the the number number of algebraic of algebraic operations operations.  required requirednegativenegativemine feedback the feedback matrix matrix of matrix positived j. Itd j should.feedback It should bee emphasized beid j emphasized(10), is relatedthat that this with this Later in this article thesym number of algebraic operationsck  required minenegative the feedbackmatrix of matrix positived feedback. It shoulde bed (10), emphasized is related that with this to implementto implement in order in order to calculate to calculate the the inverted inverted block block matrix matrixtypethetype of calculation matrices of matrices occurs of occurs the for auxiliary block forj block dimension matrix dimensioni eij,n andn3. its These3. product These cal- with cal- to implement in ordersym to calculate the invertedck  block matrix type of matrices occurs for block dimension n ≥ 3. These cal- to(5) implement for three different in order cases to calculate of theinternal the inverted structure block of matrix input thetypeculations calculation of matrices require: of occurs the inversion auxiliary for block of matrixthe dimension matrixei, anda≥,n multiplicationits≥ product3. These with cal- of (5)(5)Later for for three in three this different article different the cases number cases of theof of the internal algebraic internal structure operations structure of inputrequired of inputculationsculationsnegative require: feedback require: inversion inversionmatrix ofd j the. of It thematrix should matrixa bei, multiplication emphasizedai, multiplication≥ that of this of Later in this article the number of algebraic operations required negative1 feedbackT 1 T matrix d . It should be emphasizeda that this matrices(5)tomatrices implement for will three will be differentcalculated. be in calculated. order cases to calculate of the internal the inverted structure block of matrix inputmatricesculationstypematrices ofa− matricesab require:−1,b multiplicationT , occurs multiplication inversion forj block by of( the by dimension1 matrix() and1) and multiplicationi,n multiplication multiplication3. These by cal- by of matrices will be calculated. matricesi ai−i 1bTi , multiplication by ( 1) and multiplication by matricesto implement will be in calculated. order to calculate the inverted block matrix typematrices of matricesa− b , occurs multiplication for block by− dimension(−1) andn multiplication≥3. These cal- by (5) for three different cases of the internal structure of input culations require:i i inversion of the matrix− ai, multiplication≥ of (5) for three different cases of the internal structure of input culations require:1 T inversion of the matrix a , multiplication of matricesBull. Pol. Ac.: will Tech. be calculated. XX(Y) 2016 matrices a− b , multiplication by ( 1) andi multiplication by Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. XX(Y) XX(Y) 2016 2016 i 1 Ti 3 3 matrices will be calculated. matrices a− b , multiplication by ( −1) and multiplication by Bull. Pol. Ac.: Tech. XX(Y) 2016 i i − 3 Bull. Pol. Ac.: Tech. XX(Y) 2016 3 Bull. Pol. Ac.: Tech. XX(Y) 2016 3 Inversion of selected structures... n of the block matrix is defined as an ordered pair of num- Table 1 Tomasz Trawinski,´ Adam Kochan, Paweł Kielan and Dariusz Kurzyk bers (k+1, k+1). Block size of the block matrix is written as: Number of algebraic operations necessary to calculate the leading element c0. Tomasz Trawinski,´ Adam Kochan, Paweł Kielan and Dariusz Kurzyk Block dimension n 2 3 4 5 n=(k+1) (k+1), or briefly by n=(k+1). Table 3 × Table 3 Sum of algebraic operations loI 4 7 10 13 Number of algebraic operations necessary to calculate the positive feedback Number of algebraic operations necessary to calculate the positive feedback In order to demonstrate effectiveness of the computing algo- matrix eid j. rithm of matrix inversion, the number of algebraic operations matrix eid j. Block dimension n 3 4 5 6 needed to be performed will be calculated for block matrices Table 2 Block dimension n 3 4 5 6 Sum of algebraic operations loIII 3 9 18 30 with different internal divisions into elementary blocks - sub- Number of algebraic operations necessary to calculate the negative feedback Sum of algebraic operations loIII 3 9 18 30 matrices. All of the analyzed cases of the block matrix will matrix di. have a structure such as matrix (5), it will differ only in size Block dimension n 2 3 4 5 Table 4 Table 4 Fig. 1. The number of algebraic operations required to be imple- Sum of algebraic operations loII 3 6 9 12 Number of algebraic operations necessary to calculate the self inertia Fig. 1. The number of algebraic operations required to be imple- of elementary matrices. As mentioned above, each of the in- Number of algebraic operations necessary to calculate the self inertia mented in order to calculate the matrix inverse: a) - for a single- matrices c . mented in order to calculate the matrix inverse: a) - for a single- verse elementary matrices (6) can be calculated individually. A matrices ci . element matrix (special case) the number of operations - 1, b) - for i element matrix (special case) the number of operations - 1, b) - for detailed analysis of the structure of the matrix (6) reveals that Block dimension n 2 3 4 5 block matrices with block dimension 2 the number of operations - 15, Block dimension n 2 3 4 5 block matrices with block dimension 2 the number of operations - 15, there are four different types of items - elementary submatrices 3.1. Case 1 - one-piece elementary matrices - a partition Sum of algebraic operations loIV 8 16 24 32 c) - for matrices with block dimension 3 the number of operations - Sum of algebraic operations loIV 8 16 24 32 c) - for matrices with block dimension 3 the number of operations - in the 1-1-1-1 order If the block matrix can be divided into 32, d) - for matrices with block dimension 4 the number of operations of inverted block matrix requiring the calculation. These are as 32, d) - for matrices with block dimension 4 the number of operations follows: one-piece elementary matrices (it will take the form of a ma- - 52 matrix d . Calculated numbers of algebraic operations for var- - 52 trix (5)), then the number of algebraic operations related tomatrix dj . Calculated numbers of algebraic operations for var- ious dimensionsj of the block matrix, for the positive feedback 1. first elementary matrix, which later will be called the leading the calculation of the leading element c0 (7) is determined by:ious dimensions of the block matrix, for the positive feedback matrix e d are shown in Table 3. element, has the following form: block size n of the block matrix, the inversion process, opera-matrix ei dj are shown in Table 3. 1 T Generali j relationship between block dimension of the block tions of addition (subtraction) in triples of matrices b ja−j b j , General relationship between block dimension of the block k 1 T matrix and the number of algebraic operations that are needed 1 T 1 multiplication in triples of matrices b ja− b , and a j matrix in- c0 =(a0 b ja− b )− (7) j j matrix and the number of algebraic operations that are needed ∑ j j for the calculation of the positive feedback matrix eid j (10) is − j version. By convention, such division of the block matrix forfor the calculation of the positive feedback matrix e d (10) is as follows (assuming that matrix d was calculated ini previousj one-piece elementary matrices will be referred as division inas follows (assuming that matrix di was calculated in previous 2. elementary matrices, that in physical interpretation can be step): i 1-1-1-1 order. Calculated number of algebraic operations forstep): responsible for negative feedback, have following forms: (n 2)(n 1) various dimensions of the block matrices is shown in Table 1. l = 3 (n− 2)(n− 1) (15) oIII = Fig. 2. Matrix partitioning into an 1-2-2-2 order: a) - block dimen- loIII 3 − 2 − (15) Fig. 2. Matrix partitioning into an 1-2-2-2 order: a) - block dimen- 1 General relationship between the dimension of the block 2 sions of submatrices, b) - signs assignment to the submatrices di = c0bia− (8) for n = 3,4,.... − i matrix block dimension n and the number of algebraic opera-for n = 3,4,.... sions of submatrices, b) - signs assignment to the submatrices The number of algebraic operations needed in order to de- for i 1,...,k , auxiliary matrices having forms: tions that are needed for the calculation of the leading element The number of algebraic operations needed in order to de- ∈ { } Table 5 c0 (7), is as follows: termine self inertia matrices (11) is associated with following Table 5 1 T termine self inertia matrices (11) is associated1 T with following Number of algebraic operations necessary to calculate the leading element c0. ei = ai− bi (9) calculations: the product of matrices bia−1biT ; matrix ai inver- Number of algebraic operations necessary to calculate the leading element c0. − calculations: the product of matrices b ai− b ; matrix a inver- sion (in this case by division by the elementi i i of this matrix);i Block dimension n 2 3 4 5 for i 1,...,k 1 . loI = 3(n 1)+1 (13)sion (in this case by division by the element of this matrix); Block dimension n1 2 3 4 5 − Inversion of 4-elementary submatrices a− 15 30 45 60 ∈ { − } the summation in the inner brackets; inversion of the internal Inversion of 4-elementary submatrices aj 1 15 30 45 60 3. elementary matrices, which in physical interpretation can be the summation in the inner brackets; inversion of the internal 1 −jT The calculations effort necessary to perform the designationexpression (in parentheses) and multiplying it by the matri- Multiplication of triple matrices b ja−j 1bTj 9 18 27 36 responsible for positive couplings, are forms of the products expression (in parentheses) and multiplying it by the matri- Multiplication of triple matrices b ja−j b j 9 18 27 36 of negative feedback matrix d (8), is associated with the im-ces b and aT , subtraction of expressions contained in exter- Additions / Subtractions 1 2 3 4 of the matrices (8) and (9) i ces bi and aiT , subtraction of expressions contained in exter- Additions / Subtractions 1 2 3 4 plementation of algebraic operations necessary to: calculatenal parenthesesi i and its inversions. Calculated numbers of al- Inversions / Divisions 1 1 1 1 nal parentheses and its inversions. Calculated numbers of al- Inversions / Divisions 1 1 1 1 e d (10) the matrix product c0 and bi, calculate the inverse form of thegebraic operations for different block dimensions n of block Sum of algebraic operations loI 26 51 76 101 i j gebraic operations for different block dimensions n of block Sum of algebraic operations loI 26 51 76 101 matrix ai (in present case - dividing by an element of the ma-matrix, necessary for calculation of self inertia matrices, are i ,...,k j ,...,k matrix, necessary for calculation of self inertia matrices, are for 1 1 , 1 . trix). Calculated numbers of algebraic operations for differentshown in Table 4. ∈ { − } ∈ { } shown in Table 4. 4. block matrices, which can be called in physical interpreta- block dimension n of the block matrix, the negative feedback General relationship between block dimension of the block elementary matrices, so could be present at the block main di- T. Trawiński, A. Kochan, GeneralP. Kielan, and relationship D. Kurzyk between block dimension of the block elementary matrices, so could be present at the block main di- tion as self inertia matrices, have following forms: matrix d (8), are shown in Table 2. It should be noted, that in-matrix and the number of algebraic operations that are needed agonal as to: one-piece matrix, 2 2 dimensional square and i matrix and the number of algebraic operations that are needed agonal as to: one-piece matrix, 2× 2 dimensional square and creasing size of the block matrix by one, results in occurrencefor the calculation of the self inertia matrices ci (11) is as fol- symmetric matrices. The effect of× such division of input ma- T 1 1 T 1 1 for the calculation of the self inertia matrices ci (11) is as fol- symmetric matrices. The effect of such division of input ma- ci =(ai bi (c0− + bia−j b j )− bi)− (11) of an additional negative feedback matrix d - in the first rowlows: trix into elementary matrices, is that in the first line rectangular − creasing size of the block matrix by one, resultsi in occurrencelows: for the calculation of the self inertia matrices ci (11) is as fol- trix into elementary matrices, is that in the first line rectangular and first column of the inverted block matrix. matrices (1 2 dimensional) and in the first column rectangu- for i 1,...,k . of an additional negative feedback matrix di – in the first row lows: matrices (1× 2 dimensional) and in the first column rectangu- ∈ { } Generaland first relationship column of the between inverted block block dimension matrix. of the block loIV = 8(n 1) (16) lar matrices× (with dimension 2 1) are located. Such a divi- loIV = 8(n− 1) (16)(16) lar matrices (with dimension 2× 1) are located. Such a divi- By introducing the above indications of elementary matri- matrix andGeneral the relationship number of algebraicbetween block operations dimension that of are the needed block − sion of the block matrix into one-piece× elementary and 2 by 2 for n = 1,2,... . sion of the block matrix into one-piece elementary and 2 by 2 ces, inverted block matrix (6) takes the following form: for thematrix calculation and the number of the negativeof algebraic feedback operations matrix that dare(8) needed is asfor nfor= n1 =,2 1, 2, …,... . dimensional symmetric matrices will be called the division in i The total amount of algebraic operations needed to calculate dimensional symmetric matrices will be called the division in follows:for the calculation of the negative feedback matrix di (8) is as The totalThe total amount amount of algebraic of algebraic operations operations needed needed to calculateto calcu- the 1-2-2-2 order. The block matrix (3) fulfils the above men- the inverted form of block matrix of the block dimension n the 1-2-2-2 order. The block matrix (3) fulfils the above men- c d d ... d follows: thelate inverted the inverted form of form block of block matrix matrix of the of the block block dimension dimensionn tioned conditions, and its partition into elementary matrices in 0 1 2 k and assuming that all matrices are one-piece, is given by the tioned conditions, and its partition into elementary matrices in andn assuming and assuming that that all matricesall matrices are are one-piece, one-piece, is is given given byby the 1-2-2-2 order is shown in Figure 2. c1 e1d2 ... e1dk loII = 3(n 1) (14)(14)following formula: 1-2-2-2 order is shown in Figure 2.   − followingfollowing formula: formula: The number of algebraic operation necessary to calculate c ... e d The number of algebraic operation necessary to calculate Dr = 2 2 k (12) for nfor= n2 =,3 2, 3, …,... . Tomasz Trawinski,´ Adam Kochan, Paweł Kielan and Dariusz Kurzyk2   n2 n the leading element c0 (7) is determined by block dimension n  .. .  The number of algebraic operations needed in order to de- lo = 3 n + 19 n 10 (17)(17) the leading element c0 (7) is determined by block dimension n  . .  The number of algebraic operations needed in order to deter- lo = 3 2 + 192 − 10 (17) of input matrices, but also by dimensions of matrices in triple   termine the matrix of positiveTable3 feedback ei dj (10), is related 2 2 − of input1 T matrices, but also by dimensions of matrices in triple mine the matrix of positive feedback eid j (10), is related with b ja−j 1bTj . The number of algebraic operations necessary to cal-  sym ck  Numberwith ofthe algebraic calculation operations of the necessary auxiliary to calculate matrix thee , positiveand its feedbackproductfor nfor= n1 =,2 1, 2, …,... . b ja− b . The number of algebraic operations necessary to cal-   i for n = 1,2,... . j j 1 T the calculation of the auxiliarymatrix matrixeid j. ei, and its product with b a b   with negative feedback matrix dj. It should be emphasized that RelationshipRelationship showing showing the the number number of of algebraic algebraic operationsoperations culate matrices j −j 1 Tj results from two rectangular matrices Later in this article the number of algebraic operations required negative feedback matrix d . It should be emphasized that this Relationship showing the number of algebraic operations culate matrices b ja−j b j results from two rectangular matrices this type of matricesBlock occurs dimensionj for blockn 3 dimension4 5 n 6 3. Theseneeded needed to beto donebe done to to calculate calculate the the elementary elementary matricesmatrices (with(with multiplications (with dimensions 1 2 and 2 1) with square to implement in order to calculate the inverted block matrix type of matrices occurs for block dimension n 3. These¸ cal-needed to be done to calculate the elementary matrices (with multiplications (with dimensions 1× 2 and 2× 1) with square calculationsSum of algebraicrequire: operationsinversionl oIIIof the3 matrix9 a18, multiplication30 the blockthe block dimension dimension growth growth of of the the block block matrix), matrix), can can bebe repre-repre- matrix (with dimension 2 2). Furthermore× it× is necessary to ≥i the block dimension growth of the block matrix), can be repre- matrix (with dimension 2× 2). Furthermore it is necessary to (5) for three different cases of the internal structure of input culationsof matrices require: a–1b inversionT, multiplication of the by matrix ( 1) aandi, multiplication multiplication by of sented graphically in Fig. 1. calculate the inverse form of 4 elementary a 1 matrix. The 1 T i i sented graphically in Figure 1. × −j 1 matrices will be calculated. matrices a− b , multiplication by ( 1¡) and multiplication bysented graphically in Figure 1. calculate the inverse form of 4 elementary a−j matrix. The matrix idj. Calculatedi numbers of algebraic− operations for var- calculation effort which should be carried out for leading ele- Table 4 calculation effort which should be carried out for leading ele- ious dimensions of the block matrix, for the positive feedback3.2. Fig. Case 1. 2 The - elementary number of algebraic matrices operations in partition required in the to be 1-2- imple-ments c0 calculations is summarized in Table 5. Number of algebraic operations necessary to calculate the self inertia 3.2. Case 2 - elementary matrices in partition in the 1-2- ments c0 calculations is summarized in Table 5. Bull. Pol. Ac.: Tech. XX(Y) 2016 matrix eidj are shown in Table 3. 3 mented in order to calculate the matrix inverse: a) - for a single- matrices c . 2-2 order Assume that the block matrix can be divided into Generally, relationship between block dimension n of block i 2-2 orderelementAssume matrix (special that the case) block the matrixnumber can of operations be divided - 1, into b) - for Generally, relationship between block dimension n of block Block dimensionTablen 3 2 3 4 5 block matrices with block dimension 2 the number of operations - 15, 4 Bull. Pol. Ac.: Tech. XX(Y) 2016 NumberSum of of algebraic algebraic operations l oIVnecessary8 to16 calculate24 the32 positive 4 c) - for matrices with block dimension 3 the number of operations - Bull. Pol. Ac.: Tech. XX(Y) 2016 feedback matrix eidj 32, d) - for matrices with block dimension 4 the number of operations Block dimension n 3 4 5 6 - 52 matrix d j. Calculated numbers of algebraic operations for var- Fig. 1. The number of algebraic operations required to be implemented ious dimensionsSum of algebraic of the blockoperations matrix, loIII for3 the positive9 18 feedback30 in order to calculate the matrix inverse: a) – for a single-element matrix matrix e d are shown in Table 3. (special case) the number of operations – 1, b) – for block matrices i j with block dimension 2 the number of operations – 15, c) – for ma- GeneralGeneral relationship relationship between between block block dimension dimension of of thethe blockblock trices with block dimension 3 the number of operations – 32, d) – for matrixmatrix and and the the number number of of algebraic algebraic operations operations that that areare neededneeded matrices with block dimension 4 the number of operations – 52 for the calculation of the positive feedback matrix e d (10) is for the calculation of the positive feedback matrix eidi jj (10) is as follows (assuming that matrix d was calculated in previous as follows (assuming that matrix di wasi calculated in previous step):step): 3.2. Case 2 – elementary matrices in partition in the 1‒2–2‒2 (n 2)(n 1) order. Assume that the block matrix can be divided into elemen- = loIII 3 − − (15)(15) taryFig. matrices, 2. Matrix so partitioning could be present into an at 1-2-2-2 the block order: main a) diagonal - block dimen-as 2 to: one-piece matrix, 2 2 dimensional square and symmetric ma- sions of submatrices, b)£ - signs assignment to the submatrices for nfor= n3 =,4 3, 4, …,.... trices. The effect of such division of input matrix into elementary The numberThe number of algebraicof algebraic operations operations needed needed in in order order toto de-de- matrices, is that in the first line rectangular matrices (1 2 dimen- £ terminetermine self self inertia inertia matrices matrices (11) (11) is is associated associatedwith with followingfollowing sional) and in the first column rectangularTable 5 matrices (with dimen- calculations:calculations: the the product product of of matrices matrices bb a–11bbT;T matrix; matrix a inversiona inver- sionNumber 2 of1) algebraicare located. operations Such necessarya division to of calculate the block the leading matrix element into c0. i ii− i i i i £ sion(in (in this this case case by division by division by the by element the element of this matrix); of this matrix);the sum- one-piece elementary and Block2 by 2 dimension dimensionaln symmetric2 3 matrices4 5 1 the summationmation in the ininner the brackets; inner brackets; inversion inversion of the internal of the expression internal will Inversionbe called ofthe 4-elementary division in submatricesthe 1‒2–2‒2a−j order.15 The30 block45 matrix60 T 1 T expression(in parentheses) (in parentheses) and multiplying and multiplying it by the matrices it by thebi and matri- ai , 3) fulfilsMultiplication the above of mentioned triple matrices conditions,b ja−j b j and9 its 18partition27 into36 subtractionT of expressions contained in external parentheses elementary matrices Additionsin 1‒2–2‒2 / Subtractions order is shown1 in Fig.2 2.3 4 ces bi and ai , subtraction of expressions contained in exter- nal parenthesesand its inversions. and itsCalculated inversions. numbers Calculated of algebraic numbers operations of al- Inversions / Divisions 1 1 1 1 for different block dimensions n of block matrix, necessary for gebraic operations for different block dimensions n of block Sum of algebraic operations loI 26 51 76 101 calculation of self inertia matrices, are shown in Table 4. a) b) matrix, necessary for calculation of self inertia matrices, are shown in Table 4. Table 4 elementary matrices, so could be present at the block main di- GeneralNumber relationship of algebraic between operations block necessary dimension to calculate of the self block agonal as to: one-piece matrix, 2 2 dimensional square and matrix and the number ofinertia algebraic matrices operations ci that are needed × symmetric matrices. The effect of such division of input ma- for the calculationBlock dimension of the n self inertia matrices2 3ci (11)4 is5 as fol- lows: trix into elementary matrices, is that in the first line rectangular Sum of algebraic operations loIII 8 16 24 32 matrices (1 2 dimensional) and in the first column rectangu- × l = 8(n 1) (16) lar matrices (with dimension 2 1) are located. Such a divi- oIV − × General relationship between block dimension of the block Fig.sion 2. of Matrix the blockpartitioning matrix into into an 1‒2–2‒2 one-piece order: elementary a) – block anddimen 2- by 2 for nmatrix= 1, 2and,... the. number of algebraic operations that are needed dimensionalsions of submatrices, symmetric b) – matricessigns assignment will be to called the submatrices the division in The total amount of algebraic operations needed to calculate the 1-2-2-2 order. The block matrix (3) fulfils the above men- the inverted form of block matrix of the block dimension n tioned conditions, and its partition into elementary matrices in and856 assuming that all matrices are one-piece, is given by the 1-2-2-2 order is shown in FigureBull. 2. Pol. Ac.: Tech. 64(4) 2016 following formula: TheBrought number to you of by algebraic | Gdansk operationUniversity of necessaryTechnology to calculate 2 Authenticated n n the leading element c0 (7) is determined by block dimension n l = 3 + 19 10 (17) Download Date | 1/3/17 11:15 AM o 2 2 − of input matrices, but also by dimensions of matrices in triple 1 T for n = 1,2,... . b ja−j b j . The number of algebraic operations necessary to cal- 1 T Relationship showing the number of algebraic operations culate matrices b ja−j b j results from two rectangular matrices needed to be done to calculate the elementary matrices (with multiplications (with dimensions 1 2 and 2 1) with square × × the block dimension growth of the block matrix), can be repre- matrix (with dimension 2 2). Furthermore it is necessary to × 1 sented graphically in Figure 1. calculate the inverse form of 4 elementary a−j matrix. The calculation effort which should be carried out for leading ele- 3.2. Case 2 - elementary matrices in partition in the 1-2- ments c0 calculations is summarized in Table 5. 2-2 order Assume that the block matrix can be divided into Generally, relationship between block dimension n of block

4 Bull. Pol. Ac.: Tech. XX(Y) 2016 Inversion of selected structures...

Table 6 Table 8 Number of algebraic operations necessary to calculate the negative feedback Number of algebraic operations necessary to calculate the self inertia matrix di. matrices ci. Block dimension n 2 3 4 5 Block dimension n 2 3 4 5 1 1 T Inversion of 4-elementary submatrices a−j 15 30 45 60 Multiplication of triple matrices biai− bi 24 48 72 96 1 1 Multiplication of triple matrices c0biai− 9 18 27 36 Addition with c0− 1 2 3 4 − 1 1 T Sum of algebraic operations loII 23 46 69 92 Inverting of matrix (c0− + biai− bi ) 1 2 3 4 T 1 1 T 1 Calculation of matrix bi (c0− + biai− bi )− bi 8 16 24 32 Subtractions 4 8 12 16 Table 7 Inversion of 4-elementary submatrices 15 30 45 60 Number of algebraic operations necessary to calculate the positive feedback Sum of algebraic operations loIV 53 106 159 212 matrix eid j. Block dimension n 3 4 5 6 1 Inversion of 4-elementary submatrices a−j 15 45 90 150 1 T Matrix multiplication a− b 8 24 48 80 − i i Matrix multiplication eid j 4 12 24 40 Sum of algebraic operations loIII 27 81 162 270

matrix with its partition into elementary matrices with dimen- sions according to 1-2-2-2 order, and numbers of algebraic op- erations necessary to calculate the leading elements c0 (7), is as follows:

l = 25(n 1)+1 (18) oI − for n = 1,2,... . Fig. 3. The number of algebraic operations required to calculate the Calculated numbers of algebraic operations needed for a de- inverse matrix: a) - for a single-element matrix (special case) the num- termination of the negative feedback matrix d (8), in relation Inversion of selected structures of block matrices of chosen mechatronic systems i ber of operations - 1, b) - for block matrices with block dimension 2 to different block dimensions of input matrices is presented in the number of operations - 102, c) - for matrices with block dimen- Table 6. It is worth to underline, that increase of block dimen- sion 3 the number of operations - 230, d) - for matrices with block The number of algebraic operation necessary to calculatesion byThe one calculated results in number appearing of algebraic of additional operations negative needed feed- to dimension 4 the number of operations - 385 back matrices d (1 2 and 2 1 dimensional) in the first row the leading element c0 (7) is determined by block dimension n obtain the positivei × couplings× matrices ei dj (10), upon different of input matrices, but also by dimensions of matrices in tripleand block first column dimensions of the of invertedinput block block matrix, matrix. is shown in Table 7. and the number of algebraic operations needed to calculate ci –1 T Inversion of selected structures... bjaj bj . The number of algebraic operations necessary to cal- The overall relationship between the block dimension n of a matrices (11), is as follows: –1 T culate matrices bjaj bj results from two rectangular matricesblock matrix (partitioned into submatricesTable 7 in the 1-2-2-2 order) multiplications (with dimensionsTable 6 1 2 and 2 1) with square Number of algebraic operations necessaryTable 8 to calculate the positive Number of algebraic operations necessary to calculate£ the£ negative feedbackand theNumber number of of algebraic algebraic operations operations necessary needed to calculate to calculate the self inertia the loIV = 53(n 1) (21) matrix (with dimension 2 2). Furthermore it is necessary to feedback matrix ei dj − matrix£ di. negative feedback matrix di (8),matrices is as follows:ci. calculate the inverse form of 4 elementary a–1 matrix. The cal- for n = 1,2,... . Block dimension n 2 j 3 4 5 Block dimension n Block dimension n 3 2 4 3 5 4 6 5 culation effort which should be carried out for leading elements The sum of algebraic operations needed to calculate an in- 1 l = 23(n 1) 1–1 T (19) Inversion of 4-elementary submatrices a−j 15 30 45 60 InversionMultiplication of 4-elementary of tripleoII submatrices matrices bia i−aj bi 1524 45 4890 72150 96 c0 calculations is summarized in Table 5. − verse block matrix with block dimension n (partitioned in 1-2- Multiplication of triple matrices c b a 1 9 Inversion18 27 of selected36 structures... Addition with c 1 1 2 3 4 0 i i− Inversion of selectedfor structures...n Matrix= 2,3 ,...multiplication. a–1bT 0− 8 24 48 80 − i i 1 1 T 2-2 order) can be obtained by the following expression: Sum of algebraic operations loII 23 46 69 92 Inverting of matrix¡ (c− + bia− b ) 1 2 3 4 Table 6Table 5 The calculated number ofTable algebraic0 8 i operationsi needed to ob- Table 6 Matrix multiplication eTi dj Table1 8 1 T 1 4 12 24 40 Number of algebraic operations necessary to calculate the leading Calculation of matrix bi (c0− + biai− bi )− bi 8 16 24 32 2 Number of algebraic operations necessary to calculate the negativeInversion feedback of selectedtainNumber the structures... positive of algebraic couplings operations matrices necessarye toid calculatej (10), the upon self inertiadifferent n n element c Subtractions 4 8 12 16 lo = 27 + 121 73 (22) matrix di. 0 blockSum dimensions of algebraic of operations inputmatrices block loIII matrix,ci. is shown27 81 in162 Table270 7. 2 2 − Tablei 7 i Block dimension n 2 3 4 5 Inversion of 4-elementaryBlock dimension submatricesn 2 153 30 4 45 5 60 NumberBlock of algebraic dimension operationsBlock n dimension necessaryTable 6n to calculate2 3 the2 positive4 3 5 4 feedback5 Overall relationshipBlock between dimensionTable dimensionn 8 2 of3 a block4 matrix5 Relations showing the number of algebraic operations Number of algebraic operations necessary1 to calculate the negative feedback OverallNumber ofrelationshipSum algebraic of algebraic operations between operations necessary 1dimensionT loIV to calculate 53of a block106 the self matrix159 inertia 212 Inversion of 4-elementary submatricesmatrixa−ej 1d . 15 –1 30 45 60 Multiplication of triple matrices biai−1biT 24 48 72 96 InversionInversion of 4-elementary of 4-elementary submatrices submatricesa−j i j 15 aj 3015 4530 6045 60 (partitionedMultiplication into offollowing triple matrices 1-2-2-2biai− b order)i 24 and48 the number72 96 of needed to be done to calculate the elementary matrices of in- matrix d1i. (partitioned into following 1‒2–2‒2matrices1 corder)i. and the number of Multiplication of triple matrices c0biai−1 9 18 27 36 Addition with c0−1 1 2 3 e4d Multiplication of triple matricesBlock− dimensionc0biai− –1n 9T 3 18 4 27 536 6 algebraic operations neededAddition to with calculatec0− the1 submatrices2 3 4i j verted block matrix (with an increase of block dimension n of Multiplication of tripleBlock matrices− dimension bjaj bnj 2 9 3 18 4 27 5 36 algebraic operations neededBlock1 to dimension calculate1 T n the2 submatrices3 4 ei dj 5 Sum of algebraic operations loII 123 46 69 92 Inverting of matrix (c0−1 + biai−1biT ) 1 2 3 4 InversionSum of of 4-elementary algebraic operations submatricesloII a− 231 1546 4569 9092 150 (10), is asInverting follows: of matrix (c0− + biai− bi ) 1 T1 2 3 4 the input block matrix) can be represented graphically, as pre- Inversion of 4-elementary submatrices aj − 15 30 45 60 (10), isMultiplication as follows:T of triple1 matrices1 T b1 a− b 24 48 72 96 j Calculation of matrix bT (c−1 + bia−1bT )−1bi ii i8 16 24 32 Additions / Subtractions 1 T 1 2 3 4 Calculation of matrix bi (c0− + b ai− bi ) b 8 16 24 32 Matrix multiplication ai− bi 1 8 24 48 80 i 0 i i i − i 1 sented in Figure 3. It should be considered that the input block Multiplication of triple matrices −c0biai− 9 18 27 36 AdditionSubtractions(n 2)( withn c0−14) 1 8 2 12 3 16 4 Inversions / DivisionsMatrix multiplication− e d 4 112 1 24 1 401 Subtractions 4 8 12 16 matrix has been partitioned into elementary matrices with di- i j InvertingloIII of matrix= 27 (c −1 + b a −1bT ) 1 2 3(20)(20) 4 Sum of algebraicTable operations 7 loII 23 46 69 92 Inversion of 4-elementary submatrices0− 2i i− i 15 30 45 60 Sum of algebraic operations loIII 27 81 162 270 Inversion of 4-elementaryT 1 submatrices1 T 1 15 30 45 60 mensions resulting from following 1-2-2-2 order. Number ofSum algebraic of algebraic operations operations necessary loI to calculate the positive26 51 feedback76 101 Calculation of matrix b (c + b a b ) b 8 16 24 32 Number of algebraic operations necessary to calculate the positive feedback Sum of algebraici operations0− i i−loIVi − 53i 106 159 212 matrix eid j. for nfor= n3 =,4 3, 4, …,...Sum. of algebraic operations loIV 53 106 159 212 matrix eid j. Subtractions 4 8 12 16 Table 7 TheThe number number of algebraicof algebraic operations operations for for different different blockblock di-di- 3.3. Case 3 - elementary matrices in partition in the 1-2-1-2 Block dimension n 3 4 5 6 Inversion of 4-elementary submatrices 15 30 45 60 matrixNumber withGenerally, of algebraic its partition relationship operations into necessary between elementary1 to calculateblock matrices dimension the positive with n feedbackof dimen- blockmensions mensionsn of n blockof block matrix matrix (resulting (resulting from from the thepartitioning partitioning in order Assume that the block matrix can be divided into ele- Inversion of 4-elementary submatrices a−j 1 15 45 90 150 Sum of algebraic operations loIV 53 106 159 212 Inversion of 4-elementary submatrices a−j 15 45 90 150 sionsmatrix according with its to partition 1-2-2-2 matrix order,into1 elementaryTei andd j. numbers matrices of algebraic with dimen op--1-2-2-21‒2–2‒2 order) order) of the of selfthe inertiaself inertia matrices matricesci (11), ci (11), are are shown shown in mentary matrices in such a way that 1-element matrices (1 1) Matrix multiplication a−1bT 8 24 48 80 Matrix multiplication ai− bi 8 24 48 80 × erationssions necessaryaccording toto calculateBlock1‒2–2‒2− dimensioni order, thei leadingn and3 numbers elements4 of5 calgebraic(7),6 isTable in 8.Table 8. and square matrices (2 2)occur alternately on the main diag- Matrix multiplication eid j 4 12 24 400 operationsMatrix necessary multiplication to calculateeid j the14 leading12 24elements40 c0 (7), × as follows:Inversion of 4-elementary submatrices a− 15 45 90 150 The overall relationship between the block dimension n of onal. All matrices appearing on the main diagonal are symmet- Sum of algebraic operations loIII j27 81 162 270 is asSum follows: of algebraic operations loIII 1 T27 81 162 270 Table 8 Matrix multiplication a− b 8 24 48 80 the block matrix (partitioned in the following 1-2-2-2 order) rical. The result of this partition is that the elementary matrices − i i Number of algebraic operations necessary to calculate the self Matrixl multiplicationoI = 25(n e1id)+j 1 4 12 24 40(18)(18) − inertia matrices ci Sum of algebraic operations l 27 81 162 270 matrix withfor n its = 1, 2, … partition into elementaryoIII matrices with dimen- Bull. Pol. Ac.: Tech. XX(Y) 2016 5 matrixfor withn = 1 its,2 partition,... . into elementary matrices with dimen- Fig.Block 3. dimension The number n of algebraic operations2 required3 to4 calculate5 the sions accordingCalculated to 1-2-2-2 numbers order, of algebraic and numbers operations of algebraic needed for op- a de- Calculated numbers of algebraic operations needed for a de- inverseMultiplication matrix: of a) triple - for amatrices single-element b a–1bT matrix24 (special48 72 case)96 the num- erations necessary to calculate the leading elements c0 (7), is i i i erationsterminationtermination necessary of totheof calculatethe negative negative the feedback feedback leading matrix matrix elementsd di i(8), (8),c0 in(7),in relationrelation is matrix with its partition into elementary matrices with dimen- ber of operations–1 - 1, b) - for block matrices with block dimension 2 as follows:to different block dimensions of input matrices is presented in Addition with c0 1 2 3 4 sionsto different according block to dimensions 1-2-2-2 order, of input and numbers matrices of is algebraic presented op- in the number of operations - 102, c) - for matrices with block dimen- TableTable 6. It 6. is It worthis worth to underline,to underline, that that increase increase of of block block dimen-dimen- sion 3 the number of–1 operations–1 T - 230, d) - for matrices with block erations necessary to calculate the leading elements c (7), is Inverting of matrix (c0 +biai bi ) 1 2 3 4 sion by one resultsloI = in25 appearing(n 1)+ of1 additional negative (18)feedback0 dimension 4 the number of operations - 385 sion by one resultsloI in= 25 appearing(n− 1)+ of1 additional negative(18) feed- T –1 –1 T –1 as follows:matrices d (1 2 and 2 1 −dimensional) in the first row and first Calculation of matrix bi (c0 +biai bi ) bi 8 16 24 32 back matricesi d £(1 2 and£ 2 1 dimensional) in the first row for n = 1column,2,... .of thei inverted block matrix. for n = 1,2,... . × × Fig. 3.Subtractions The number of algebraic operations required4 to8 calculate12 16 the Calculatedand first column numbers of of the algebraic inverted operations block matrix. needed for a de- and the number of algebraic operations needed to calculate ci Calculated numbers of algebraicloI = 25(n operations1)+1 needed for a de-(18)inverse matrix: a) - for a single-element matrix (special case) the num- terminationThe overall of therelationship negative feedback betweenTable− matrix 6 the blockd (8), dimension in relationn of a matricesInversion (11),of 4-elementary is as follows: submatrices 15 30 45 60 termination of the negative feedback matrix di (8), in relation ber of operations - 1, b) - for block matrices with block dimension 2 forblocknNumber= matrix1,2,... of (partitioned algebraic. operations into submatrices necessary to in calculate the 1-2-2-2 the negative order) to different block dimensions of input matrices is presented in the numberFig.Sum 3. of ofThealgebraic operations number operations of - 102, algebraic l c) - for operations matrices53 required with106 block to159 calculate dimen-212 the andCalculated the number numbers of algebraic offeedback algebraic operations matrix operations d needed needed to calculate for a de-the oIV Table 6. It is worth to underline, that increasei of block dimen- sioninverse 3 the number matrix: of a)operations - for a single-elementloIV -= 230,53( d)n - matrix for1) matrices (special with case) block the num-(21) terminationnegative feedback of the matrix negatived feedback(8), is as follows: matrix di (8), in relation − sion by oneBlock results dimension in appearing n i of additional negative2 3 feed-4 5 dimensionber of 4 operations the number - 1, of b) operations - for block - 385 matrices with block dimension 2 to different block dimensions of input matrices is presented in for n = 1,2,... . back matrices di (1 2 and 2 1 dimensional)–1 in the first row theThe number overall of operationsrelationship - 102, between c) - forthe matrices block dimension with block n dimen-of back matricesInversiondi (1 of× 4-elementary2 and 2× 1submatrices dimensional) aj in15 the30 first45 row 60 The sum of algebraic operations needed to calculate an in- andTable first column 6. It is of worth× the inverted to underline,loII×= block23(n that matrix.1 increase) of block dimen-(19)andthesion the block number 3 the matrix number ofalgebraic (partitioned of operations operations in the - 230, following needed d) - for 1‒2–2‒2 tomatrices calculate order) withc block and first column of the inverted block matrix.− –1 anddimensionverse the number block 4 the matrix of number algebraic with of block operations operations dimension - 385neededn (partitioned to calculate inci 1-2- Thesion overall byMultiplication one relationship results of triple in between appearing matrices the ofc block0b additionaliai dimension9 negative18 n27of afeed-36 and the number of algebraic operations needed to calculate ci Thefor overalln = 2,3 relationship,... . between the¡ block dimension n of a matrices2-2 order) (11), is can as be follows: obtained by the following expression: back matrices di (1 2 and 2 1 dimensional) in the first row matrices (11), is as follows: block matrixTheSum calculated (partitioned of algebraic number× operations into submatrices of algebraic ×loII inoperations the 1-2-2-223 46 needed order)69 to92 ob- and theand number first column of algebraic of the inverted operations block needed matrix. to calculate the and the number ofl algebraic= 53(2n operations1) needed to calculate(21) ci andtain the number the positive of algebraic couplings operations matrices neededeid j (10), to calculate upon different the loIV = 53n(n 1) n (21)(21) negativeThe feedback overall matrixrelationshipd (8), between is as follows: the block dimension n of a matrices (11), is asl follows:o = 27 +−121 73 (22) negativeblock feedback dimensionsThe overall matrix ofrelationship inputdi (8), block is between as matrix, follows: the is block shown dimension in Table n 7. of 2 2 − block matrix (partitioned into submatrices in the 1-2-2-2 order)for n = 1,2,... . Overalla block relationshipmatrix (partitioned between into dimension submatrices of ain blockthe 1‒2–2‒2 matrix Thefor Relations sumn = 1, 2, … of algebraic showing operations the number needed of to algebraic calculate operations an in- and the number of algebraicl = 23( operationsn 1) needed to calculate(19) the The sum of algebraic operationsloIV = 53 needed(n 1) to calculate an in-(21) (partitionedorder) and into the followingnumberloII = of23 algebraic( 1-2-2-2n 1) order)operations and needed the number (19)to calcu of- neededThe sum to be of done algebraic to calculate operations the− elementaryneeded to calculate matrices an of in- negative feedback matrix di (8),− is as follows: verse block matrix with block dimension n (partitioned in 1-2- algebraiclate the operations negative feedback needed matrix to calculate di (8), theis as submatrices follows: eid j inverseforvertedn = blockblock1,2,... matrixmatrix. (withwith block an increase dimension of block n (partitioned dimension in n of for n = 2,3,... . 2-2 order) can be obtained by the following expression: (10), is as follows: 1‒2–2‒2theThe input sum order) block of can algebraic matrix) be obtained can operations be by represented the neededfollowing graphically, to expression: calculate as an pre- in- The calculated number ofloII algebraic= 23(n operations1) needed to ob-(19)(19) − 2 n tain the positive couplings matrices eid j (10), upon different versesented block in Figure matrix 3. withIt shouldn block be dimension consideredn that(partitioned the input in block 1-2- tain thefor positive n = 2, 3, … couplings matrices( eid)(j (10),) upon different lo = 27 + 121 73 (22)(22) for n = 2,3,... . n 2 n 1 2-2matrix order) has can been belo partitioned= obtained27 + by121 into the elementary following73 expression: matrices(22) with di- block dimensions of inputloIII block= 27 matrix,− is− shown in Table 7. (20) 2 2 − The calculated number of algebraic2 operations needed to ob- mensions resulting from following 1-2-2-2 order. Overall relationship between dimension of a block matrix Relations showing the number2 of algebraic operations tainfor n the= 3 positive,4,... . couplings matrices eid j (10), upon different n n (partitionedBull. intoPol. Ac.: following Tech. 64(4) 1-2-2-2 2016 order) and the number of needed to be done to calculatelo = 27 the+ elementary121 73 matrices of857 in-(22) algebraicblockThe operations dimensions number of needed of algebraic input to block calculate operations matrix, the for is submatrices shown different in Table blocke d 7. di- 3.3. Case 3 - elementary matrices2 in2 − partition in the 1-2-1-2 algebraic operations needed to calculate the submatrices eidj verted blockBrought matrix to you (with by | anGdansk increase University of block of Technology dimension n of mensionsOveralln relationshipof block matrix between (resulting dimension from theof a partitioning block matrix in orderRelationsAssume showing that the the block number matrix of can algebraic be divided operations into ele- (10), is as follows: the input block matrix) can be representedAuthenticated graphically, as pre- (partitioned1-2-2-2 order) into of following the self inertia 1-2-2-2 matrices order)c and(11), the are number shown ofin neededmentary to matrices be done in to suchDownload calculate a way Date thethat | elementary 1-element1/3/17 11:15 matricesAM matrices (1 of in-1) i sented in Figure 3. It should be considered that the input block× algebraicTable 8. operations needed(n to2)( calculaten 1) the submatrices eid j vertedand square block matrices matrix (with(2 2)occur an increase alternately of block on dimension the main diag-n of loIII = 27 − − (20) matrix has been partitioned× into elementary matrices with di- (10),The is overall as follows:l relationshipoIII = 27 − between2 − the block dimension(20)n of theonal. input All matricesblock matrix) appearing can be on represented the main diagonal graphically, are symmet- as pre- 2 mensions resulting from following 1-2-2-2 order. the block matrix (partitioned in the following 1-2-2-2 order) sentedrical. The in Figure result of 3. this It should partition be considered is that the elementary that the input matrices block for n = 3,4,... . (n 2)(n 1) matrix has been partitioned into elementary matrices with di- The number of algebraicloIII = operations27 − for− different block di-(20)3.3. Case 3 - elementary matrices in partition in the 1-2-1-2 2 mensions resulting from following 1-2-2-2 order. mensionsBull. Pol.n of Ac.: block Tech. matrix XX(Y) 2016 (resulting from the partitioning in order Assume that the block matrix can be divided into ele- 5 for n = 3,4,... . order 1-2-2-2 order) of the self inertia matrices ci (11), are shown in mentary matrices in such a way that 1-element matrices (1 1) The number of algebraic operationsi for different block di- 3.3. Case 3 - elementary matrices in partition in the× 1-2-1-2 Table 8. and square matrices (2 2)occur alternately on the main diag- mensions n of block matrix (resulting from the partitioning in order Assume that× the block matrix can be divided into ele- The overall relationship between the block dimension n of onal. All matrices appearing on the main diagonal are symmet- 1-2-2-2 order) of the self inertia matrices c (11), are shown in mentary matrices in such a way that 1-element matrices (1 1) the block matrix (partitioned in the followingi 1-2-2-2 order) rical. The result of this partition is that the elementary matrices× Table 8. and square matrices (2 2)occur alternately on the main diag- × The overall relationship between the block dimension n of onal. All matrices appearing on the main diagonal are symmet- Bull. Pol. Ac.: Tech. XX(Y) 2016 5 Bull.the Pol. block Ac.: Tech. matrix XX(Y) (partitioned 2016 in the following 1-2-2-2 order) rical. The result of this partition is that the elementary matrices5

Bull. Pol. Ac.: Tech. XX(Y) 2016 5 Tomasz Trawinski,´ Adam Kochan, Paweł Kielan and Dariusz Kurzyk

Table 10 Number of algebraic operations necessary to calculate the negative feedback matrix di. Block dimension n 2 3 4 5 1 Inversion of 4-elementary submatrices a−j 15 16 31 32 1 Multiplication of triple matrices c b a− 8 10 18 20 − 0 i i Sum of algebraic operations loII 23 26 49 52

Table 11 Relationship between dimensions - numbers of elements of positive feedback matrices e d and dimensions of input matrices. Fig. 4. Matrix partitioning into 1-2-1-2 order: a) - block dimensions i j of submatrices, b) - marks assignment to the submatrices Block dimension n 2 3 4 5 1 Inversion of 4-elementary submatrices a−j 15 16 31 32 1 T. Trawiński, A. Kochan, P. Kielan, and D. Kurzyk Multiplication of triple matrices c0bia− 8 10 18 20 Table 9 − i l Number of algebraic operations necessary to calculate the leading element c0. Sum of algebraic operations oII 23 26 49 52 Block dimension n 2 3 4 5 a) 1 b) Inversion of 4-elementary submatrices a−j 15 16 31 32 1 T culated numbers of algebraic operations needed to obtain the Multiplication of triple matrices b ja−j b j 9 10 19 20 Additions / Subtractions 1 2 3 4 negative feedback matrices di (8), are shown in Table 10. Inversions / Divisions 1 1 1 1 The relationship between the dimension of the block matrix (partitioned into blocks in the following 1-2-1-2 order) and the Sum of algebraic operations loI 26 29 54 57 number of algebraic operations needed to calculate the nega- tive feedback matrices di (8), is as follows: in the first row are, alternately, rectangular matrices (of dimen- sion 1 2) and 1-element matrices (1 1), and the first column loII = 23d + 3(n d 1) (24) × × − − is the block transposition of the first row. Such partitioning of for d < n n = 2,3,... . where d - numbers of 4-element sub- input block matrix will be referred to the partition in the 1-2- ∧ matrices on the input block matrix diagonal. The number of 1-2 order. Such conditions correspond to an exemplary block algebraic operations required to determine the positive feed- matrixFig. shown4. Matrix in partitioning Figure 4 and into the1‒2–1‒2 matrix order: given a) – by block (3). dimensions Number of submatrices, b) – marks assignment to the submatrices back matrices e d (10) depend on the block dimension n of of algebraic operations necessary to be implemented in order i j the input block matrix and the numbers of 1- and 4-element Fig. 3. The number of algebraic operations required to calculate the in-to calculate the form of the leading element c (7), using the 0 submatrices lying on the main diagonal. As a result, we ob- verse matrix: a) – for a single-element matrix (special case) the numberpartitioning of input matrices into elementary matrices in the Table 9 tain three types of calculated matrices, namely: 1-element, of operations – 1, b) – for block matrices with block dimension 2 the1-2-1-2 order, depend on the block dimension n of input ma- number of operations – 102, c) – for matrices with block dimension 3 Number of algebraic operations necessary to calculate the leading 2-element (rectangular), 4-element (square and symmetrical) trix. If a block dimension is n = 2, the input block matrix is the number of operations – 230, d) – for matrices with block dimension element c0 matrices. Analyzing the relationship (10) and the block matrix 4 the number of operations – 385 composed of four matrices (Fig.4.a) and the number of alge- Block dimension n 2 3 4 5 form shown in Figure 4, it can be concluded that in the calcu- braic operations necessary to calculate the leading element c0 –1 lation of the 1-element matrices only 1-element matrices take (7) isInversion the same ofas 4-elementary in the case submatrices analyzed a ini Chapter15 16 3.2 -31 formula32 part, while in the calculation of the 2-element matrices two Relations showing the number of algebraic operations(18). Multiplication of triple matrices b a–1bT 9 10 19 20 i i i kind of elementary matrices: 1-element and 4-element subma- needed to be done to calculate the elementary matrices of in- Total algebraic operations will be different for larger block Additions / Subtractions 1 2 3 4 trices are involved. In the calculation of the 4-element matrices verted block matrix (with an increase of block dimension ndimensions n of the input block matrix, and will depend on the two 4-element submatrices, one 1-element and two 2-element of the input block matrix) can be represented graphically, asnumber Inversions of 1-and / Divisions 4-element submatrices, lying1 on1 the1 main1 di- presented in Fig. 3. It should be considered that the input block submatrices are involved. The relationships between the di- agonalSum of of the algebraic input blockoperations matrix. l Algebraic26 computation29 54 effort57 matrix has been partitioned into elementary matrices with di- oI mensions of the matrices creating the positive feedback matri- associated with the leading element c0 (7) is shown in Table 9. mensions resulting from following 1‒2–2‒2 order. ces eid j (10), and a number of its internal elements generated In general, the relationship between the block dimension n In general, the relationship between the block dimension are shown in Table 11. of the block matrix and the number of algebraic operations, 3.3. Case 3 – elementary matrices in partition in the 1‒2–1‒2 n of the block matrix and the number of algebraic operations, The number of algebraic operations, with a different block needed to calculate the leading element c0, using the partition- order. Assume that the block matrix can be divided into ele- needed to calculate the leading element c0, using the parti- dimension n of block matrix (resulting from the partitioning in mentary matrices in such a way that 1-element matrices (1 1)ing intotioning elementary into elementary matrices matrices following following the 1-2-1-2 the 1‒2–1‒2 order, order, is as £ following 1-2-1-2 order), are shown in Figure 5. and square matrices (2 2) occur alternately on the main di-follows:is as follows: £ The sum of all 2-element matrices (in rows and columns, agonal. All matrices appearing on the main diagonal are sym- occurring above the main diagonal of inverted block matrix) metrical. The result of this partition is that the elementary l = n + 24d + 2(n d 1) (23)(23) oI under increasing block dimension n of the input block ma- matrices in the first row are, alternately, rectangular matrices − − (of dimension 1 2) and 1-element matrices (1 1), and thefor dfor< dn < n = n2 =,3 2, 3, …;,... . where whered -d numbers– numbers of of 4-element 4-element sub-sub- trix (partitioning into elementary matrices in the 1-2-1-2 or- £ £ ∧ ^ first column is the block transposition of the first row. Suchmatrices matrices on on input input block block matrix matrix diagonal. diagonal. The calculation ef- der), increases accordingly with sequence described by the partitioning of input block matrix will be referred to the par-fort necessaryThe calculation to be performed effort necessary in order to to be determine performed the in nega-order so-called third diagonal of Lozanic triangle [15]. Successive tition in the 1‒2–1‒2 order. Such conditions correspond to antive to feedback determine matrices the negativedi (8), feedback also in matrices this case di will (8), dependalso in this on numbers which represents block dimension n (of input block exemplary block matrix shown in Fig. 4 and the matrix giventhe blockcase will dimension depend on of the the block input dimension matrix andof the the input number matrix of and 1- matrix) correspond a number, representing the sum of all 2- by (3). Number of algebraic operations necessary to be imple-andthe 4-element number of matrices 1- and 4-element located on matrices its main located diagonal. on its Themain cal- di- element positive feedback matrices eid j (according to the func- mented in order to calculate the form of the leading element agonal. The calculated numbers of algebraic operations needed c0 (7), using the partitioning of input matrices into elementary6 to obtain the negative feedback matrices di (8), are shown in Bull. Pol. Ac.: Tech. XX(Y) 2016 matrices in the 1‒2–1‒2 order, depend on the block dimension Table 10. n of input matrix. If a block dimension is n = 2, the input block matrix is composed of four matrices (Fig. 4a) and the Table 10 number of algebraic operations necessary to calculate the Number of algebraic operations necessary to calculate the negative leading element c0 (7) is the same as in the case analyzed in feedback matrix di Chapter 3.2 – formula (18). Block dimension n 2 3 4 5 Total algebraic operations will be different for larger block –1 dimensions n of the input block matrix, and will depend on the Inversion of 4-elementary submatrices aj 15 16 31 32 –1 number of 1- and 4-element submatrices, lying on the main Multiplication of triple matrices c0biai 8 10 18 20 diagonal of the input block matrix. Algebraic computation effort ¡ Sum of algebraic operations loII 23 26 49 52 associated with the leading element c0 (7) is shown in Table 9.

858 Bull. Pol. Ac.: Tech. 64(4) 2016 Brought to you by | Gdansk University of Technology Authenticated Download Date | 1/3/17 11:15 AM Tomasz Trawinski,´ Adam Kochan, Paweł Kielan and Dariusz Kurzyk

Table 10 Number of algebraic operations necessary to calculate the negative feedback matrix di. Block dimension n 2 3 4 5 1 Inversion of 4-elementary submatrices a−j 15 16 31 32 1 Multiplication of triple matrices c b a− 8 10 18 20 − 0 i i Sum of algebraic operations loII 23 26 49 52

Table 11 Relationship between dimensions - numbers of elements of positive feedback matrices e d and dimensions of input matrices. Fig. 4. Matrix partitioning into 1-2-1-2 order: a) - block dimensions i j of submatrices, b) - marks assignment to the submatrices Block dimension n 2 3 4 5 1 Inversion of 4-elementary submatrices a−j 15 16 31 32 1 Multiplication of triple matrices c b a− 8 10 18 20 Table 9 − 0 i i l Number of algebraic operations necessary to calculate the leading element c0. Sum of algebraic operations oII 23 26 49 52 Block dimension n 2 3 4 5 Inversion of 4-elementary submatrices a 1 15 16 31 32 −j Inversion of selected structures of block matrices of chosen mechatronic systems 1 T culated numbers of algebraic operations needed to obtain the Multiplication of triple matrices b ja−j b j 9 10 19 20 Additions / Subtractions 1 2 3 4 negative feedback matrices di (8), are shown in Table 10. The relationship between the dimension of the block matrix Inversions / Divisions 1 1 1 1 The relationship between the dimension of the block matrix under increasing block dimension n of the input block matrix (partitioned into blocks in the following 1-2-1-2 order) and the Sum of algebraic operations loI 26 29 54 57 (partitioned into blocks in the following 1‒2–1‒2 order) and the (partitioning into elementary matrices in the 1‒2–1‒2 order), numbernumber of algebraicof algebraic operations operations neededneeded to to calculate calculate the the negative nega- increases accordingly with sequence described by the so-called tive feedback matrices di (8), is as follows: feedback matrices di (8), is as follows: third diagonal of Lozanic triangle [15]. Successive numbers in the first row are, alternately, rectangular matrices (of dimen- which represents block dimension n (of input block matrix) sion 1 2) and 1-element matrices (1 1), and the first column loII = 23d + 3(n d 1) (24)(24) correspond a number, representing the sum of all 2-element × × − − is the block transposition of the first row. Such partitioning of positive feedback matrices ei dj (according to the function f2(n)) for d < n n = 2,3,... . where d - numbers of 4-element sub- input block matrix will be referred to the partition in the 1-2- for d <∧ n n = 2, 3, …; where d – numbers of 4-element sub- are shown in Table 12. matrices on the^ input block matrix diagonal. The number of 1-2 order. Such conditions correspond to an exemplary block matrices on the input block matrix diagonal. algebraic operations required to determine the positive feed- matrix shown in Figure 4 and the matrix given by (3). Number The number of algebraic operations required to determine Table 12 back matrices e d (10) depend on the block dimension n of Number of algebraic operations necessary to calculate the positive of algebraic operations necessary to be implemented in order the positive feedbacki j matrices ei dj (10) depend on the block thedimension input block n of matrix the input and block the matrix numbers and ofthe 1- numbers and 4-element of 1- and feedback matrix ei dj to calculate the form of the leading element c (7), using the 0 submatrices4-element lying submatrices on the lying main on diagonal. the main Asdiagonal. a result, As a we result, ob- partitioning of input matrices into elementary matrices in the Block dimension n 3 4 5 6 tainwe three obtain types three of types calculated of calculated matrices, matrices, namely: namely: 1-element, 1-element, –1 T –1 1-2-1-2 order, depend on the block dimension n of input ma- Fig. 5, case 1 (1-element) ai bi c0bjaj 0 0 6 6 2-element2-element (rectangular), (rectangular), 4-element 4-element (square (square and and symmetrical)symmetrical) trix. If a block dimension is n = 2, the input block matrix is –1 T –1 matrices.matrices. Analyzing Analyzing the the relationship relationship (10) (10) and and the the blockblock matrixmatrix Fig. 5, case 2 or 3 (2-element) ai bi c0bjaj 27 54 108 162 composed of four matrices (Fig.4.a) and the number of alge- formform shown shown in Figurein Fig. 4, 4, it it can can be be concluded concluded that that in the in calculation the calcu- –1 T –1 braic operations necessary to calculate the leading element c Fig. 5, case 4 (4-element) ai bi c0bjaj 0 51 51 153 0 lationof ofthe the 1-element 1-element matrices matrices only only 1-element 1-element matrices matrices take part, take (7) is the same as in the case analyzed in Chapter 3.2 - formula while in the calculation of the 2-element matrices two kind Sum of algebraic operations loIII 27 105 165 321 part, while in the calculation of the 2-element matrices two (18). of elementary matrices: 1-element and 4-element submatrices kind of elementary matrices: 1-element and 4-element subma- Total algebraic operations will be different for larger block are involved. In the calculation of the 4-element matrices two trices are involved. In the calculation of the 4-element matrices dimensions n of the input block matrix, and will depend on the 4-element submatrices, one 1-element and two 2-element sub- The sum of all 2-element matrices (in rows and columns, two 4-element submatrices, one 1-element and two 2-element number of 1- and 4-element submatrices, lying on the main di- matrices are involved. The relationships between the dimen- occurring above the main diagonal of inverted block matrix) submatrices are involved. The relationships between the di- agonal of the input block matrix. Algebraic computation effort sions of the matrices creating the positive feedback matrices under increasing block dimension n of the input block matrix mensions of the matrices creating the positive feedback matri- associated with the leading element c (7) is shown in Table 9. ei dj (10), and a number of its internal elements generated are (partitioning into elementary matrices in the 1‒2–1‒2 order), 0 ces showne d (10), in Table and a11. number of its internal elements generated increases accordingly with sequence described by the so-called In general, the relationship between the block dimension n i j are shown in Table 11. third diagonal of Lozanic triangle [15]. Successive numbers of the block matrix and the number of algebraic operations, The number of algebraic operations,Table 11 with a different block which represents block dimension n (of input block matrix) needed to calculate the leading element c , using the partition- 0 dimensionRelationshipn of between block matrix dimensions (resulting – numbers from of theelements partitioning of positive in correspond to a number representing the sum of all 2-element ing into elementary matrices following the 1-2-1-2 order, is as feedback matrices e d and dimensions of input matrices following 1-2-1-2 order), arei j shown in Figure 5. positive feedback matrices ei dj (according to the function f2(n)) follows: are shown in Table 13. TheBlock sum dimension of all 2-element n matrices (in rows2 and3 columns,4 5 –1 occurringInversion above of 4-elementary the main diagonal submatrices of a invertedj 15 block16 31 matrix)32 loI = n + 24d + 2(n d 1) (23) Table 13 − − under increasing block dimension n of–1 the input block ma- Multiplication of triple matrices c0bia 8 10 18 20 Number of 2-element positive feedback matrices eidj for d < n n = 2,3,... . where d - numbers of 4-element sub- trix (partitioning into elementary¡ matricesi in the 1-2-1-2 or- ∧ matrices on input block matrix diagonal. The calculation ef- der),Sum increases of algebraic accordingly operations withloII sequence23 described26 49 by52 the n 3 4 5 6 7 8 9 10 11 12 13 … fort necessary to be performed in order to determine the nega- so-called third diagonal of Lozanic triangle [15]. Successive f2(n) 1 2 4 6 9 12 16 20 25 30 36 … tive feedback matrices di (8), also in this case will depend on numbersThe which number represents of algebraic block operations, dimension withn a(of different input blockblock the block dimension of the input matrix and the number of 1- matrix)dimension correspond n of block a number, matrix (resulting representing fromthe the sumpartitioning of all 2-in and 4-element matrices located on its main diagonal. The cal- elementfollowing positive 1‒2–1‒2 feedback order), matrices are showneid j in(according Fig. 5. to the func- The sum of all 4-element submatrices (lying above the main The sum of all 2-element matrices (in rows and columns, diagonal) due to a increase of the block dimension n of the input occurring above the main diagonal of inverted block matrix) block matrix (partitioning in 1-2-1-2 order), increases following 6 Bull. Pol. Ac.: Tech. XX(Y) 2016 a numerical sequence described by repeated triangular numbers [12-14]. Successive numbers which represents block dimension n corresponds a number, representing the sum of all 4-element positive feedback matrices ei dj, according to function f4(n) shown in Table 14.

Table 14 Number of 4-element positive feedback matrices eidj

n 3 4 5 6 7 8 9 10 11 12 13 …

f4(n) 0 1 1 3 3 6 6 10 10 15 15 …

Fig. 5. Relationship between dimensions and number of elements of The sum of all 1-element matrices due to increase of the positive feedback matrices ei dj and dimensions of input matrices block dimension n of the input block matrix (partitioning in

Bull. Pol. Ac.: Tech. 64(4) 2016 859 Brought to you by | Gdansk University of Technology Authenticated Download Date | 1/3/17 11:15 AM Inversion of selected structures...

Table 15 Number of 1-element positive feedback matrices eid j. n 3 4 5 6 7 8 9 10 11 12 13 ...

f1(n) 0 0 1 1 3 3 6 6 T.10 Trawiński,10 15 A. Kochan,... P. Kielan, and D. Kurzyk

1‒2–1‒2 order), increasesTable accordingly 16 with a numerical se- a) b) c) Numberquence ofdescribed algebraic operations by repeated necessary triangular to calculate numbers the self inertia[12‒14]. Successive numbers representingmatrices ci .the block dimension n corre- sponds a number,Block dimension representingn 2 the sum3 of4 all 1-element5 6 positive7 Inversion of selectedfeedback structures... matrices e d , according to function f (n) are shown Fig. 5. Relationship between dimensions and numberInversion ofInversionInversion elements of of of selectedof selected selected structures... structures... structures...4-element matricesi j 53 53 106 1061 159 159 in Table 15. positive feedback matrices eid j and dimensions of input matrices 1-element matrices Table- 158 8 16 16 24 Sum of algebraicNumber operations of 1-elementloIV TableTable53 positiveTable 15 1561 15 feedback114 matrices122 e175id j. 183 NumberNumberNumber of of 1-element of 1-element 1-element positive positiveTable positive feedback15 feedback feedback matrices matrices matricese ed de. d. . Table 12 n 3 4 5 6 7 8 9 10 11 i 12ij ij j 13 ... Number of 1-element positive feedback matrices eidj d) Number of algebraic operations necessary to calculate the positive feedback f1n(nn)n 3 303 4 404 5 5 15 6 6 16 7 7 738 8 839 9 961010106 111111101212121013131315...... matrix e d . sive numbers representing the block dimension n corresponds i j f1f(1nf()1n()nn) 0 030 0 0 40 1 1 151 1 163 3 373 3 386 6 6 96 6 6101010101110101210 15151315 ...…...... Block dimension n 3 4 5 6 a number, representing the sum of all 1-element positive feed- f (n) 0 0 1 1 3 3 6 6 10 10 15 … 1 T 1 back matrices4 eid j, according toTable function 16 f1(n) are shown in Fig.5, case 1 (1-element) ai− bi c0b ja−j 0 0 6 6 1 T 1 Number of algebraic operationsTableTableTable necessary 16 16 16 to calculate the self inertia Fig.5, case 2 or 3 (2-element) a− b c b a− 27 54 108 162 Table 15. i i 0 j j matrices c . 1 T 1 TheNumberNumber overallNumber of of algebraic of algebraicrelationship algebraic operations operations operations between necessary necessary necessary thei to to calculate block to calculate calculate dimension the the theself self self inertia inertia inertian of Tomasz Trawinski,´ Adam Kochan, Paweł Kielan and Dariusz Kurzyk Fig.5, case 4 (4-element) ai− bi c0b ja−j 0 51 51 153 The overall relationship between the block dimension n of the input blockBlock matrix dimension (partitioningmatricesmatricesmatricesn c2ic. i in.ci. 3 1-2-1-24 order)5 and6 the 7 Sum of algebraic operations loIII 27 105 165 321 the input block matrix (partitioning in 1‒2–1‒2 order) and the Fig. 5. Relationship between dimensions and number of elements ofnumber of algebraicBlockBlock4-elementBlock dimension dimension operations dimension matricesn n n needed2 2532 3 353 to3 calculate4 41064 5 51065 the6 positive61596 7 71597 Table 17 number of algebraic operations needed to calculate the positive Inversion of the matrices of block dimension n = 3. Fig.positiveFig.Fig. 5. 5. Relationship5. Relationship feedback Relationship matrices between between betweene dimensionsid dimensionsj dimensionsand dimensions and and and number number number of input of of elementsof matriceselements elements of of of 4-element4-element1-element4-element matrices matrices matrices535353- 5353538 1061061068 10610610616 1591591591615915915924 feedbackfeedback matrices matriceseid eji d(10),j (10), is is as as follows: follows: positivepositivepositive feedback feedback feedback matrices matrices matriceseededanddandand dimensions dimensions dimensions of of input of input input matrices matrices matrices 1-element1-element1-element matrices matrices matrices - - - 8 8 8 8 8 8 161616 161616 242424 Block inversion method - "inv" iTablei j ij j 13 Sum of algebraic operations loIV 53 61 114 122 175 183 SumSumSum of of algebraic of algebraic algebraic operations operations operationsloIVloIVloIV 535353 616161 114114114 122122122 175175175 183183183 time of internal loop [s] 0.0416(37) 0.0791(86) Number of 2-element positiveTable feedback 12 matrices eid j. loIII = 51 f4(n)+27 f2(n)+6 f1(n) (25)(25) Fig. 6. The number of algebraic operations required to calculate the time of outer loop [s] 41.637 79.186 Numbern 3 of4 algebraic5 6 operations7 8Table necessaryTableTable 129 12 12 to10 calculate11 the12 positive13 feedback... inverted matrix: a) – for an input block matrix with block dimension NumberNumberNumber of of algebraic of algebraic algebraic operations operations operations necessarymatrix necessary necessaryei tod toj calculate. to calculate calculate the the thepositive positive positive feedback feedback feedback for siven = 3 numbers,4,... . representing the block dimension n correspondsn = 2 the number of algebraic operations – 102, b) – for block matrices f (n) 1 2 4 6 9 12 16 20 25 30 36 ... for n = 3, 4, … 2 matrixmatrixmatrixe ed de. d. . sivesiveCalculatedsive numbers numbers numbers number representing representing representing of algebraic the the the block block block dimensionoperations dimension dimensionn forncorrespondsncorrespondscorresponds different with n = 3 the number of operations – 143, c) – for matrices with n = 4 Block dimensioni ij ij j n 3 4 5 6 a number,Calculated representing number of thealgebraic sum ofoperations all 1-element for different positive block feed- the number of operations – 322, d) – for matrices n = 5 the number Table 18 BlockBlockBlock dimension dimension1 dimensionT n n1n 3 3 3 4 4 4 5 5 5 6 6 6 blockaa number,backadimensions number, number, dimensions matrices representing representing representing n ofeni dinputofj, input according the theblock the sum block sum sum matrix of of to matrix ofall all function all 1-element(partitioning 1-element 1-element (partitioningf1( positiven positive )in positiveare 1‒2–1‒2 in shown feed- 1-2- feed- feed- in Fig.5, case 1 (1-element) ai− bi c0b ja−j 0 0 6 6 of operations – 396 Inversion of the matrices of block dimension n = 5. 1 1T1T T 1 1 1 1-2backbackback order), matrices matrices matrices requiredeeded,d to, according, according obtain according the to to selftofunction function function inertiaf f( matricesfn()n()nare)areare shown shownc shown(11), in in in Fig.5,Fig.5,Fig.5, case case case 1 1 (1-element) 1(1-element) (1-element)a−a−ab1−bcTbcbcjbajb−aj−a−1 0 0 0 0 0 0 6 6 6 6 6 6 Tableorder), 15. requiredi i jitoj jobtain the self inertia matrices1 1 1 ci (11),i are Fig.5, case 2 or 3 (2-element)Tableiaii−iib 14ii0ic00b0 jjaj−j j 27 54 108 162 Block inversion method - "inv" 1 1T1T T 1 1 1 areTableTable shownTable 15. 15. 15. in Table 16. Fig.5,Fig.5,Fig.5, caseNumber case case 2 2 or 2or 3of or 3 (2-element) 4-element 3(2-element) (2-element) positivea−a−ab1−bcTb feedbackcbcjbajb−aj−a−1 matrices272727 5454ei54d j. 108108108 162162162 shownThe in overall Table relationship16. between the block dimension n of Fig.5, case 4 (4-element)iaii−iibii0ic00b0 jjaj−j j 0 51 51 153 time of internal loop [s] 0.0663(32) 0.131(38) 1 1T1T T 1 1 1 ThetheTheTheThe overallinput overall overall overall block relationship relationship relationship relationship matrix (partitioning between between between between the the the the in block block block 1-2-1-2 block dimension dimension dimension dimension order)n andnofnof theof n Fig.5,Fig.5,3Fig.5, case4 caseSum case 45 4 (4-element) of 4(4-element) (4-element) algebraic6 7 ai−a operations8i−abi−ibcib90cib0cjba0jb−ja10loIIIj−ja−j 110 270 0 51125110551 13515116551...153153321153 thethethethe input input input input block block block block matrix matrix matrix matrix (partitioned (partitioning (partitioning (partitioningTable 16 in in in 1-2-1-2 in1-2-1-2 1-2-1-2 1-2-1-2 order) order) order) order) and and and and the the the verted input block matrix (according to the growth of block time of outer loop [s] 66.332 131.38 f (n) 0 Sum1SumSum1 of of algebraic of algebraic3 algebraic3 operations operations6 operations6 loIIIl10oIIIloIII 27102727 1051510510515165165165...321321321 number of algebraic operations needed to calculate the positive 4 Number of algebraic operations necessary to calculate the self dimension) can be represented graphically in Fig. 6. It should numbernumbernumberfeedbacknumber of of of algebraic algebraicof algebraicmatrices algebraic operations operationse operationsid operationsj (10), needed is needed asneeded follows: to to calculate tocalculate calculate calculate the self the the positive positiveinertia positive inertia matrices ci be considered that the input block matrix is partitioned into Table 13 matricesfeedbackfeedbackfeedbackci matrices(11), matrices matrices is aseiedi follows:edjidj(10),(10),j (10), is is as is as follows:as follows: follows: elementary matrices with dimensions consistent with the result Table 19 Number of 2-elementTable positiveTableTable 13 13 13 feedback matrices eid j. Block dimensionnl n =n51 f (n)+272 f (3n)+46 f (n5) 6 7 Fig.(25) 6. The number of algebraic operations required to calculate the tion f2(n)) are shown in Table 12. 53 oIII+ 8( 14) for n =2 2,4,6,81,... of 1‒2–1‒2 partition order. Inversion of the matrices of block dimension n = 7. NumberNumberNumber of of 2-element of 2-element 2-element positive positive positive feedback feedback feedback matrices matrices matriceseiedijde.ijd. j. l4-element= matricesloIIIloIII2loIII==51=51251f−4f(4fn(4)+n()+n)+27272753f2f(2fn(2)+n(53)+n)+66106f16f(1fn(1)n(106)n) 159 (26)159(25)(25)(25)inverted matrix: a) - for an input block matrix with block dimension The sumn 3 of all4 2-element5 6 7 matrices8 9 (in10 rows11 and12 columns,13 ... foroIVn = 3,4,...n .1 n 1 Block inversion method - "inv" 53 −2 + 8 −2 for n = 1,3,5,7,... n = 2 the number of algebraic operations - 102, b) - for block matrices f2(nnn) n 313 3 424 4 5 54 5 6 6 6 7 797 8128 8 9169 9 10102010 11112511 12123012 13133613 ...... time of internal loop [s] 0.0932(57) 0.1648(0) occurring above the main diagonal of inverted block matrix) forforforn1-elementCalculatedn=n=3=,34,3,...4, ,...4matrices,.... . number. of algebraic– 8 operations8 16 for16 different24 with n = 3 the number of operations - 143, c) - for matrices with n = 4 f2f(2nf()2n()n) 1 1 1 2 2 2 4 4 4 6 6 6 9 9 9 121212 161616 202020 252525 303030 363636 ...... under increasing block dimension n of the input block ma- TheblockCalculatedCalculatedCalculated sum dimensions of algebraic number number numbern of of operations of input of algebraic algebraic algebraic block needed operations matrix operations operations to (partitioning calculate for for for different different the different in in-the 1-2- number4. The of number operations of - 322, algebraic d) - for matrices operationsn = 5 the number of time of outer loop [s] 93.257 164.8 Sum of algebraic operations loIV 53 61 114 122 175 183 trix (partitioning into elementary matrices in the 1-2-1-2 or- verseblockblock1-2block input dimensionsorder), dimensions dimensions block required matrixnnofnofof input to input(partitioned input obtain block block block the matrix matrix inself matrix 1-2-1-2 inertia (partitioning (partitioning (partitioning order) matrices may in in 1-2-inc 1-2- be(11), 1-2-operationsand - 396 computation times Table 14 i der), increases accordingly with sequence described by the given1-21-21-2 order), by order), order), the required following required required to to expression: obtainto obtain obtain the the the self self self inertia inertia inertia matrices matrices matricescici(11),c(11),i (11), Number of 4-elementTable positiveTableTable 14 14 14 feedback matrices eid j. are shown in Table 16. so-called third diagonal of Lozanic triangle [15]. Successive areareare shown shownThe shown overall in in Tablein Table Table relationship 16. 16. 16. between the block dimension n of Numerical experiment was carried out in two stages. The Gauss Table 20 NumberNumberNumber of of 4-element of 4-element 4-element positive positive positive feedback feedback feedback matrices matrices matriceseiedijde.ijd. j. The102 overall relationship between thefor blockn = dimension2 n of = numbersn which3 represents4 5 6 block7 8 dimension9 10 n11(of input12 13 block... the input block matrix (partitioned in 1‒2–1‒2 order) and theit ismethod also known is well forknown its computationaland described in complexity, the literature, but it is there also Inversion of the matrices of block dimension n 10. theTheTheThe input overall overall overalln block relationship relationshipn relationship matrix (partitioned between between between the the thein block block 1-2-1-2 block dimension dimension dimension order) andnnofnof theof f4(nnn) n 303 3 4 14 4 5 51 5 6 63 6 7 737 8 868 9 969 10101010 11111011 12121512 13131513 ...... lo =number53 of+ algebraic8( 1)+ operationsl + l needed+ l to calculatefor n = 2 self,6... inertia(27)is noknown information for its computational on the method complexity, "inv" (Matlab but there function) is no in in- Block inversion method - "inv" matrix) correspond to a number representing the sum of all 2- thethenumberthe input input input block2 of block block algebraic matrix2 matrix− matrix operations (partitioned (partitionedoIII (partitionedoII needed in inoI in1-2-1-2 1-2-1-2 1-2-1-2 to calculate order) order) order) andself and and inertiathe the the f4f(4nf()4n()n) 0 0 0 1 1 1 1 1 1 3 3 3 3 3 3 6 6 6 6 6 6 101010 101010 151515 151515 ...... matrices n ci1 (11),n is 1as follows: termsformation of the computational on the method complexity."inv" (Matlab So function) it was in decided terms of to time of internal loop [s] 0.1621(6) 0.2317(4) element positive feedback matrices eid j (according to the func- numbernumbermatricesnumber 53 of of algebraicof−2c algebraic algebraic(11),+ 8 −2 is operations as+ operations operationsl follows:oIII + loII needed needed needed+ loI to to calculatetofor calculate calculaten = 1 self, self3... self inertia inertia inertia i presentthe computational the effectiveness complexity. of shown So algorithms it was decided in the to lightpresent of the time of outer loop [s] 162.16 231.74 tion f2(n)) are shown in Table 13. matricesmatricesmatricescc(11),c(11),(11), is is as is as follows:as follows: follows: ( ) Relationships i i i showingn then number of algebraic operations effectiveness of shown algorithms in the light of the compu- Thetion sumf2 n of) all are 4-element shown in Tablesubmatrices 12. (lying above the main  53 + 8( 1) for n = 2,4,6,8,... computational complexity compared to Gasuss method, and in = n n 2n n n 2n tiontiontionThef2f(2fn(2 sum)n())n) are)) are of are shown shown all shown 2-element in in Tablein Table Table 12. matrices 12. 12. (in rows and columns,needed l tooIV be done535353 to+n+8 calculate+1(8(8( 1n−)1)1 the)forforfor elementarynn=n=2=,24,2,4,6,4,6,8,6 matrices,8,...,8...,... of(26) in-the(26) lighttational of timecomplexity consumed compared during to the Gasuss direct method, calculation and in using the diagonal) due to a increase of the block dimension n of the in- l l l === 532 2 2− 2+2−82−−− for n = 1,3,5,7,... (26)(26)(26) occurringTheTheThe sum sum sum above of of of all all all the 2-element 2-element 2-element main diagonal matrices matrices matrices of (in (in inverted (in rows rows rows and and block and columns, columns, columns, matrix)vertedoIV inputoIVoIV blockn matrixn1n12 1 n (accordingn1n121 to the growth of block di- light of time consumed during the direct calculation using "inv" carrying out numerical experiment, using a double loop, alows put block matrix (partitioning in 1-2-1-2 order), increases fol- 535353−−−++8+8−8−− forforfornn=n=1=,13,1,3,5,3,5,7,5,7,...,7...,... "inv" method and block inversion method. occurringoccurringoccurring above above above the the the main main main diagonal diagonal diagonal of of of inverted inverted inverted block block block matrix) matrix) matrix)mension)The can sum be of represented algebraic2 2 2 2 2 operations graphically2 needed in Figure to calculate 6. It should the in-method and block inversion method. to determine if indeed calculations are carried out in a single lowingunder a numerical increasing sequence block dimension describedn byof repeated the input triangular block ma- The number of algebraic operations that must be performed undertrixunderunder (partitioning increasing increasing increasing block intoblock block elementary dimension dimension dimensionn matricesnofnofof the the the input in input input the block 1-2-1-2block block ma- ma- ma- or-be consideredverseTheTheTheThe sum inputsum sumsum of of that block ofalgebraicof algebraic algebraic thealgebraic matrix input operations operations operationsblock (partitionedoperations matrix needed needed neededneeded in is 1-2-1-2 partitionedto to calculatetoto calculate calculatecalculate order) into the the maythethe in-el- in- in- be The number of algebraic operations that must be per- stream of processor unit. Linear relationship between the num- numbers [12-14]. Successive numbers which represents block inverse input block matrix (partitioned in 1‒2–1‒2 order) mayin orderformed to calculatein order to the calculate inverse the of inverse an input of block an input matrix block using ma- trixder),trixtrix (partitioning (partitioning increases(partitioning accordingly into into into elementary elementary elementary with matrices matrices sequence matrices in indescribedin the the the 1-2-1-2 1-2-1-2 1-2-1-2 by or- or- the or-ementaryverseversegivenverse input input byinput matrices theblock block block following matrix with matrix matrix dimensions (partitioned expression: (partitioned (partitioned consistent in in 1-2-1-2in 1-2-1-2 1-2-1-2 with order) order) order) the may may result may be be be ber of passes the internal loop and the outer loop and the times dimension n corresponds a number, representing the sum of all be given by the following expression: its partitioningtrix using its into partitioning blocks, hasinto beenblocks, compared has been withcompared the num- with der),so-calledder),der), increases increases increases third accordingly diagonal accordingly accordingly of with Lozanic with with sequence sequence sequence triangle described described [15]. described Successive by by by the the theofgivengiven 1-2-1-2given by by by the partition the the following following following order. expression: expression: expression: of performed calculation indicates properly conducted numeri- 4-element positive feedback matrices eid j, according to func- 102 for n = 2 ber ofthe algebraicnumber ofoperations algebraic operations performed performed using standard using standard Gauss so-callednumbersso-calledso-called third which third third diagonal diagonal represents diagonal of of of Lozanic block Lozanic Lozanic dimension triangle triangle triangle [15]. [15].n [15].(of Successive Successiveinput Successive block methods. The number of algebraic operations performed by cal experiment. In other words, in the course of the experiment tion f4(n) shown in Table 14. = 102102102n n forforfornn=n=2=22 Gauss methods. The number of algebraic operations performed numbersmatrix)numbersnumbers correspondwhich which which represents represents represents to a number block block block dimension representing dimension dimensionnn(of then(of(of input sum input input of block block all block 2-4.l Theo  number53 2 + 8( of2 algebraic1)+loIII + l operationsoII + loI for n and= 2, com-6... (27) no events inside the operating system does not interfere with The sum of all 1-element matrices due to increase of the === n n n n n n − Gaussby method,Gauss method, can be can represented be represented as a formulaas a formula [16]: [16]: matrix)matrix)matrix) correspond correspond correspond to to ato a number a number number representing representing representing the the the sum sum sum of of ofall all all 2- 2- 2- lololo 5353532 2+n2+8+1(82(82( 21n)+1)+1 )+loIIIloIIIloIII++l+oIIloIIloII++l+oIloIloIforforfornn=n=2=,26...,26...,6... (27)(27)(27)(27) element positive feedback matrices eid j (according to the func- putation 53 times−2 +−8−−−2 + loIII + loII + loI for n = 1,3... the calculations. For the block inversion we have used matrix block dimension n of the input block matrix (partitioning in 1- 535353n n1n1+1+8+8n 8n1n1+1+l+l l ++l+l l++l+l l forforfornn=n=1=,13...,13...,3... elementtionelementelementf2( positiven positive)) positive are shown feedback feedback feedback in Tablematrices matrices matrices 13.eiediedjid(accordingj (accordingj (according to to the to the the func- func- func-  −2−2 −2 −2−2 −2 oIIIoIIIoIII oIIoIIoII oIoIoI 2 3 7 partitioned in the 1-1-1-1 order. 2-1-2 order), increases accordingly with a numerical sequence NumericalRelationships experiment showing was carried the number out in of two algebraic stages. operations The l = n3 + n2 n (28) tiontiontionThef2f(2fn( sum2)n())n) are)) are of are shown all shown shown 4-element in in Tablein Table Table submatrices 13. 13. 13. (lying above the main  o (28) The calculation results of the matrix inversion times of the described by repeated triangular numbers [12-14]. Succes- GaussneededRelationshipsRelationshipsRelationships method to be is done showingwell showing showing to known calculate the the the andnumber number number the described elementary of of of algebraic algebraic algebraic in the matrices operations literature, operations operations of in- 3 2 − 6 diagonal)TheTheThe sum sum sum due of of allof toall all 4-element a 4-element increase4-element submatrices of submatrices submatrices the block (lying (lyingdimension (lying above above aboven the theof the main the main main in- Relationships showing the number of algebraic operations procedures are summarized in Tables 17-20, linear increase neededneededvertedneeded to toinput beto be be done doneblock done to to matrix calculateto calculate calculate (according the the the elementary elementary elementary to the growth matrices matrices matrices of blockof of ofin- in-where in- di- n - block dimension of the input matrix. diagonal)putdiagonal)diagonal) block due matrixdue due to to ato a increase(partitioning a increase increase of of ofthe the in the block 1-2-1-2 block block dimension dimension dimension order), increasesnnofnofof the the the in- in-fol- in- needed to be done to calculate the elementary matrices of in- where n – block dimension of the input matrix. of calculation time indicates properly conducted experiments. Bull. Pol. Ac.: Tech. XX(Y) 2016 vertedvertedmension)verted input input input canblock block block be matrix represented matrix matrix (according (according (according graphically to to theto the the growth in growth growth Figure of of blockof 6. block block It should di- di-7 di-Results of algebraic calculation, showing the comparison putlowingputput block block block a matrix numerical matrix matrix (partitioning (partitioning (partitioning sequence in described in 1-2-1-2in 1-2-1-2 1-2-1-2 order), by order), order), repeated increases increases increases triangular fol- fol- fol- The results presented times of calculations testify in favor of mension)mension)bemension) considered can can can be be bethat represented represented represented the input graphically graphically block graphically matrix in in Figurein is Figure Figure partitioned 6. 6. It6. It should It should shouldintobetween el- the methods employing the Gauss methods and of the lowingnumberslowinglowing a a numerical a[12-14]. numerical numerical Successive sequence sequence sequence described describednumbers described by which by by repeated repeated repeated represents triangular triangular triangular block bebebe considered considered considered that that that the the the input input input block block block matrix matrix matrix is is partitioned is partitioned partitioned into into into el- el- el- the method of the block matrix inversion compared to the ementary860 matrices with dimensions consistent with the resultinversion using block matrices, are shownBull. Pol. in Ac.: Figures Tech. 7-9.64(4) 2016 numbersdimensionnumbersnumbers [12-14]. [12-14]. [12-14].n corresponds Successive Successive Successive a number, numbers numbers numbers representing which which which represents represents represents the sum block block ofblock all ementaryementaryementary matrices matrices matrices with with with dimensions dimensions dimensions consistent consistent consistent with with with the the the result result result method of "inv" used in Matlab. Computation times are shorter of 1-2-1-2 partition order. In order toBrought compare to you the by | calculation Gdansk University times of of Technology matrix inversion dimension4-elementdimensiondimensionn positivencorrespondsncorrespondscorresponds feedback a a number, a number, number, matrices representing representing representingeid j, according the the the sum sum sum to of of func- allof all all ofof 1-2-1-2of 1-2-1-2 1-2-1-2 partition partition partition order. order. order. than 1.5 to 2 times. performed by block inversion methods andAuthenticated the methods used 4-elementtion4-element4-elementf4(n) positiveshown positive positive in feedback feedback Table feedback 14. matrices matrices matriceseiediedji,dj, accordingj according, according to to func-to func- func- Download Date | 1/3/17 11:15 AM tiontiontionf f(fn()n()nshown)shownshown in in Tablein Table Table 14. 14. 14. 4. The number of algebraic operations and com-in commercial software, the implementation of above men- The4 4 4 sum of all 1-element matrices due to increase of the 4.4.4. The The The number number number of of of algebraic algebraic algebraic operations operations operations and and and com- com- com- 5. Summary blockTheTheThe dimensionsum sum sum of of of all all alln 1-element 1-elementof 1-element the input matrices matrices matricesblock matrix due due due to to (partitioning toincrease increase increase of of of the in the the 1- putation times tioned algorithm using MATLAB software has been done. In block2-1-2blockblock dimension order), dimension dimension increasesnnofnof theof the theaccordingly input input input block block block withmatrix matrix matrix a numerical(partitioning (partitioning (partitioning sequence in in 1-in 1- 1- Numericalputationputationputation experiment times times times was carried out in two stages.the The programming environment the comparison with the "inv" Number of elementary operations required to determine the 2-1-2described2-1-22-1-2 order), order), order), by increases increases repeated increases accordingly accordingly triangular accordingly with numbers with with a a numerical a numerical numerical [12-14]. sequence sequence sequence Succes- NumericalNumericalGaussNumerical method experiment experiment experiment is well was was known was carried carried carried and out describedout out in in in two two two in stages. stages. the stages. literature, The The Thea standard function of the matrix inversion in MATLAB (un- inverse matrix using the described algorithm depend on the describeddescribeddescribed by by by repeated repeated repeated triangular triangular triangular numbers numbers numbers [12-14]. [12-14]. [12-14]. Succes- Succes- Succes- GaussGaussGauss method method method is is wellis well well known known known and and and described described described in in inthe the the literature, literature, literature,der the following conditions: forced CPU affinity for MAT- structure of the matrix and block size of block matrix. The LAB with single core processor) has been made. The priority described algorithm is most effective for matrices with large Bull. Pol. Ac.: Tech. XX(Y) 2016 7 Bull.Bull.Bull. Pol. Pol. Pol. Ac.: Ac.: Ac.: Tech. Tech. Tech. XX(Y) XX(Y) XX(Y) 2016 2016 2016 7of77 performed tasks had been set to high. Calculations were block size and divided into blocks of small size. This is partic- made on computer equipped with CPU AMD Phenom (tm) II ularly noticeable if you compare arrays block divided in the X4 850 3.30 GHz frequency on operating system Windows 7 order 1-1-1-1 and 1-2-2-2 in order, the computational com- Professional 64-bit. Calculations has been performed in two plexity of the array divided into blocks of one-piece require loops: an internal loop performing 10,000 times the calcula- a minimum number of algebraic operations during the matrix tion of matrix inversion and calculating the average time value inversion. Computational complexity in the above two cases, of single pass through the loop; and outer loop repeated 1000 which is obvious, is of the order O(n2). The computational times and calculating the total computation time. This way of complexity of matrix inversion divided in order 1-2-1-2 may at

8 Bull. Pol. Ac.: Tech. XX(Y) 2016 Inversion of selected structures of block matrices of chosen mechatronic systems

Results of algebraic calculation, showing the comparison between the methods employing the Gauss methods and of the inversion using block matrices, are shown in Figs. 7‒9.

Fig. 9. Number of algebraic operations performed during matrix inversion versus block dimensions: squares – numbers of algebraic operations for Gauss method, circles – numbers of algebraic operations for inversion using block matrices (partitioned in 1‒2–1‒2 order) Fig. 7. Number of algebraic operations performed during matrix inversion versus block dimensions: squares – numbers of algebraic operations for Gauss method, circles – numbers of algebraic operations for inversion using block matrices (partitioned in 1‒1–1‒1 order) In order to compare the calculation times of matrix inver- sion performed by block inversion methods and the methods used in commercial software, the implementation of above mentioned algorithm using MATLAB software has been done. In the programming environment the comparison with the "inv" a standard function of the matrix inversion in MATLAB (under the following conditions: forced CPU affinity for MATLAB with single core processor) has been made. The priority of performed tasks had been set to high.Calculations were made on computer equipped with CPU AMD Phenom (tm) II X4 850 3.30 GHz frequency on operating system Windows 7 Profes- sional 64-bit. Calculations has been performed in two loops: an internal loop performing 10,000 times the calculation of ma- trix inversion and calculating the average time value of single pass through the loop; and outer loop repeated 1000 times and calculating the total computation time. This way of carrying out numerical experiment, using a double loop, alows to deter- mine if indeed calculations are carried out in a single stream of processor unit. Linear relationship between the number of passes the internal loop and the outer loop and the times of performed calculation indicates properly conducted numerical experiment. In other words, in the course of the experiment no events inside the operating system does not interfere with the calculations. For the block inversion we have used matrix Fig. 8. Number of algebraic operations performed during matrix partitioned in the 1‒1–1‒1 order. inversion versus block dimensions: squares – numbers of algebraic The calculation results of the matrix inversion times of the operations for Gauss method, circles – numbers of algebraic operations procedures are summarized in Tables 17‒20, linear increase for inversion using block matrices (partitioned in 1‒2–2‒2 order) of calculation time indicates properly conducted experiments.

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Table 17 However, in the case of matrix inversion divided in order Inversion of the matrices of block dimension n = 3 1‒2–1‒2 computational complexity is between 1‒1‒1‒1 and 1‒2‒2‒2. In all of the analyzed cases, the elements causing Block inversion method – "inv" the most demanding calculations are the positive feedback 2 time of internal loop [s] 0.0416(37) 0.0791(86) matrices eidj, here the complexity is of the order of O(n ). As shown by formulas (15), (20) and in Table 12, the computa- time of outer loop [s] 41.637 79.186 tional complexity O(n) are linked with the other elements of the matrix like ci, c0 and di. Interesting results were obtained Table 18 by juxtaposing formulas describing the computational com- Inversion of the matrices of block dimension n = 5 plexity of the block matrix inversion and the computational complexity resulting from the Gauss method. Inverting block Block inversion method – "inv" matrix of small size block near n = 5 (and in this case smaller time of internal loop [s] 0.0663(32) 0.131(38) then 4, 5 or 7) have a slightly more complex computation than the Gauss method. However, the block matrix size greater time of outer loop [s] 66.332 131.38 than three (in the case of partitioning in the order 1‒1–1‒1) for presented method is more effective, as well in the case Table 19 of the matrix partitioned in order 1‒2–2‒2, this method is Inversion of the matrices of block dimension n = 7 more effective for matrices with greater block dimension then n = 7. The calculation of the matrix inverse, divided in the Block inversion method – "inv" order 1‒2–1‒2 become more efficient than the Gauss method time of internal loop [s] 0.0932(57) 0.1648(0) on the block dimension greater then n = 5. The exact values of computational complexity of the algorithm for block matrices time of outer loop [s] 93.257 164.8 of the presented structures are given by formulae (17, 22) and (27). Presented method is also characterized by shorter cal- Table 20 culation times compared to the standard method of inverting Inversion of the matrices of block dimension n = 10 used in MATLAB/Simulink, computation times are shorter than 1.5 to 2 times. Block inversion method – "inv" time of internal loop [s] 0.1621(6) 0.2317(4) References time of outer loop [s] 162.16 231.74 [1] J. de Jesus Rubio, C. Torres, C. Aguilar, “Optimal control based in a mathematical model applied to robotic arms”, International Journal of Innovative Computing, Information and Control 7(8), 5045‒5062, (2011). The results presented times of calculations testify in favor [2] T. Trawiński, K. Kluszczyński, W. Kolton, “Lumped parameter of the method of the block matrix inversion compared to the model of double armature VCM motor for head positioning method of “inv” used in Matlab. Computation times are shorter system of mass storage devices”, Przegląd Elektrotechniczny than 1.5 to 2 times. 87(12b), 184‒187, [in Polish] (2011). [3] D. Słota, T. Trawiński, R. Wituła, “Inversion of dynamic matrices of HDD head positioning system”, Appl. Math. Model. 35(3), 1497‒1505, (2011). 5. Summary [4] T. Trawiński, “Kinematic chains of branched head positioning system of hard disk drives”, Przegląd Elektrotechniczny 87(3), Number of elementary operations required to determine the 204‒207,(2011). inverse matrix using the described algorithm depend on the [5] T. Trawiński, “Block form of inverse inductance matrix for poli- structure of the matrix and block size of block matrix. The harmonic model of an induction machine”, XXVIII International described algorithm is most effective for matrices with large Conference on Fundamentals of Electrotechnics and Circuit block size and divided into blocks of small size. This is par- Theory, IC-SPETO’2005, II, p.371, (2005). ticularly noticeable if you compare arrays block divided in the [6] T. Trawiński, Modeling of Driving Lay-out of Mass Storage order 1‒1–1‒1 and 1‒2–2‒2 in order, the computational com- Head Positioning System, Wydawnictwa Politechniki Śląskiej, plexity of the array divided into blocks of one-piece require Gliwice, [in Polish] (2010). a minimum number of algebraic operations during the matrix [7] N. Jakovcevic Stor, I. Slapnicar, J. L. Barlow, “Accurate eigen- value decomposition of arrowhead matrices and applications”, inversion. Computational complexity in the above two cases, 2 Linear Algebra and its Applications, 464, 62–89, (2015). which is obvious, is of the order O(n ). The computational [8] F. Diele, N. Mastronardi, M. Van Barel, E. Van Camp, “On complexity of matrix inversion divided in order 1‒2–1‒2 may computing the spectral decomposition of symmetric arrowhead at first glance appear to be O(n), but if we make a thorough matrices”, Computational Science and Its Applications – ICCSA analysis of the increase in the complexity of the component 2004 Lecture Notes in Computer Science, 3044, 932–941, 2 ei dj, see Table 12, it turns out that it grows as well O(n ). (2004).

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[9] L. Shen, B. W. Suter, “Bounds for eigenvalues of arrowhead ma- [17] W. Hołubowski, D. Kurzyk, T. Trawiński, “A fast method for trices and their applications to hub matrices and wireless commu- computing the inverse of symmetric block arrowhead matrices”, nications”, EURASIP Journal on Advances in Signal Processing, Appl. Math. Inf. Sci. 9(2L), 1‒5, (2015). 2009, (2009). [18] T. Trawiński, “Inversion method of matrices with chosen struc- [10] G. A. Gravvanis, K. M. Giannoutakis, “Parallel exact and ap- ture with help of block matrices”, Przegląd Elektrotechniczny proximate arrow-type inverses on symmetric multiprocessor 85(6), 98‒101, [in Polish] (2009). systems”, Computational Science – ICCS 2006 Lecture Notes [19] H. T. Kung, B. Wilsey Suter, “A hub matrix theory and appli- in Computer Science, 3991, 506‒513, (2006). cations to wireless communications”. EURASIP Journal on Ad- [11] G. A. Gravvanis, “Solving symmetric arrowhead and special vances in Signal Processing 2007(1), 1‒8, (2007). tridiagonal linear systems by fast approximate inverse precon- [20] E. Mizutani, J. W. Demmel, “On structure-exploiting trustregion ditioning”, Journal of Mathematical Modelling and Algorithms regularized nonlinear least squares algorithms for neuralnetwork 1(4), 269‒282, (2002). learning”, Neural Networks 16(5), 745‒753, (2003). [12] J. Tanton, “A dozen questions about triangular numbers”, Math [21] M. Bixon, J. Jortner, “Intramolecular radiationless transitions”, Horizons 13(2), 6‒7, 21‒23 (2005). The Journal of Chemical Physics 48(2), 715‒726, (1968). [13] P. A. Piza, “On the squares of some triangular numbers”, Math- [22] J. W. Gadzuk, “Localized vibrational modes in Fermi liquids. ematics magazine 23(1), 15‒16, (1949). General theory”, Physical Review B 24(4), 1651, (1981). [14] J. A. Ewell, “A trio of triangular number theorems”, American [23] A. Ratajczak, “Trajectory reproduction and trajectory tracking Mathematical Monthly 105(9), 848‒849, (1998). problem for the nonholonomic systems”, Bull. Pol. Ac.: Tech. [15] “Lozanic triangle”, The On-Line Encyclopedia of Integer Se- 64(1), 63‒70 (2016). quences, (2014). [24] A. Mazur, M. Cholewiński, “Implementation of factitious force [16] K. A. Atkinson, An Introduction to Numerical Analysis, John method for control of 5R manipulator with skid-steering platform Wiley and Sons (2nd ed.), New York, 1989. REX”, Bull. Pol. Ac.: Tech. 64(1), 71‒80 (2016).

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