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Technische Universit at Chemnitz-Zwickau Technische Universitat ChemnitzZwickau Sonderforschungsbereich Numerische Simulation auf massiv parallelen Rechnern Brahim Benhammouda RANKREVEALING TOPDOWN ULV FACTORIZATIONS Preprint SFB Key words U LV and U RV factorizations Orthogonal factorizations Rankrevealing factorizations Numerical rank Dierentialalgebraic equations AMS sub ject classications F A F Authors address Brahim Benhammouda Fakultatf ur Mathematik TU ChemnitzZwickau D Chemnitz FRG email brahimmathematiktuchemnitzde This work has b een supp orted by the Deutsche Forschungsgemeinschaft under the grant no Me Dierentiellalgebraische Gleichungen PreprintReihe des Chemnitzer SFB SFB January Abstract Rankrevealing U LV and U RV factorizations are useful to ols to determine the rank and to compute bases for nullspaces of a ma trix However in the practical U LV resp U RV factorization each left resp right null vector is recomputed from its corresp onding right resp left null vector via triangular solves Triangular solves are required at initial factorization renement and up dating As a result algorithms based on these factorizations may b e exp ensive es p ecially on parallel computers where triangular solves are exp ensive In this pap er we prop ose an alternative approach Our new rank revealing U LV factorization which we call topdown U LV factor ization T D U LV factorization is based on right null vectors of lower triangular matrices and therefore no triangular solves are required Right null vectors are easy to estimate accurately using condition esti mators such as incremental condition estimator ICE The T D U LV factorization is shown to b e equivalent to the U RV factorization with the advantage of circumventing triangular solves Introduction Recent numerical integration metho ds for dierential algebraic equations D AE s require at each time integration step the computation of the numerical rank and bases for nullspaces of very large matrices These matrices are obtained by a recursive dierentiation algorithm which app ends new rows to the previous matrices The pro cess of incorp orating a new row or column in a matrix is called up dating Other applications are the solution of underdetermined rankdecient least squares problems subset selection problems and information re trieval The singular value factorization SVD p is known to b e an extremely reliable to ol for computing the numerical rank and bases for the nullspaces of a matrix However the SVD is to o exp ensive when it comes to recursive algorithms or realtime applications since its computation re 3 1 quires O n ops and the SVD is dicult to up date Therefore alternative algorithms that are nearly as accurate as the SVD cheaper and easier to up date are desired Recently Stewart prop osed two rankrevealing factoriza tions called U LV and U RV factorizations These two factorizations are 1 Here a op is either an addition or a multiplication eective in exhibiting the numerical rank and bases for the nullspaces The 2 U LV and the U RV factorizations can b e up dated in O n ops sequentially and in O n ops on an array of n pro cessors Recent work related to the U RV and U LV factorization b oth in theory and implementation may b e found in The rankrevealing U LV and the U RV al gorithms are iterative and require estimates of the condition number of some triangular submatrices at every iteration step of initial factorization rene ment and up dating In the U RV and the U LV factorizations small singular values and asso ciated null vectors are estimated by means of conditions es timators A survey of condition estimators is given in In the practical U LV resp U RV factorization however each left resp right null vector is recomputed from its corresp onding right resp left null vector via triangular solves Triangular solves are required for the initial factorization the renement and up dating For some applications triangular solves have to b e p erformed many times in order to achieve a required accu racy Therefore algorithms based on the usual U LV and U RV factorizations may b e very exp ensive on parallel computers where triangular solves are exp ensive For this reason we introduce an alternative rankrevealing U LV factor ization called topdown U LV factorization T D U LV factorization This new factorization relies on right null vectors of lower triangular matrices which are accurately estimated using condition estimators This results in circumventing triangular solves required in the usual rankrevealing U LV and U RV factorizations Our T D U LV factorization is essentially equivalent to the U RV with the advantage of avoiding triangular solves thus it is more suitable for parallel implementations Furthermore the T D U LV uses the null vectors obtained from condition estimators in a straithforward way In this pap er we describ e the T D U LV factorization give an algorithm to compute it and show how this algorithm can b e implemented rened and up dated eciently The remainder of this pap er is organised as follows In section we briey review the usual rankrevealing U LV and U RV factor izations Our new T D U LV factorization metho d is prop osed in section In section we give details of the T D U LV factorization algorithm The new algorithm is presented in section Finally we draw a conclusion in section ULV and URV factorizations In this section we review the rank revealing U LV and U RV factorizations introduced by Stewart We rst introduce the concept of numerical rank of a matrix Given a mn matrix A R m n a singular value factorization SVD see x of A has the form T A U V where U u u and V v v are orthogonal matrices and 1 m 1 m diag is an m n diagonal matrix whose entries the singular 1 n values of A are ordered such that Then the 1 2 n numerical rank of A with resp ect to a threshold is dened as the number of singular values of A strictly larger than ie 1 k k +1 n is a threshold b elow which a singular value of the matrix A is declared to b e numerically null or negligeable The ratio estimates the gap r +1 k b etween large and small singular values of A The numerical rank is well dened whenever the gap is suciently large Ways for choosing the threshold may b e found in For i k n the columns v of V satisfy kAv k and therefore i i are called numerical right nul l vectors k k denotes the matrix norm In the same way columns u u of U are called numerical left nul l vectors k +1 n T since they satisfy ku Ak for i k n i The numerical right nul lspace of A is dened by r span fv v g N k +1 n k in the same way we dene the numerical left nul lspace of A by l span fu u g N k +1 n k mn Given a matrix A R a U LV factorization of A has the form L k T A U V H E mm nn k k with orthogonal matrices U R V R and L R E k (mk )(mk ) (mk )k R lower triangular matrices H R Such a factorization is said to b e rankrevealing if kH E k O and k +1 L is wellconditioned ie L L c where c is some given k k k 1 k tolerance Similarly a U RV factorization of A has the form R F k T V A U G k k (mk )(nk ) where R R G R are upp er triangular matrices and where k k (nk ) F R Such a factorization is said to b e a rankrevealing if R is wellconditioned k T T T and kF G k O k +1 In factorizations and the numerical rank of A is revealed by the dimension of the submatrices L and R resp ectively Orthonormal left and k k right bases for the nullspaces of A are revealed by the matrix U and V resp ectively More precisely columns k through n of U and V span the left and right nullspaces of A resp ectively Factorization and are based on estimating small singular values of the middle factors L and R and the asso ciated left and right null vectors resp ectively Then deating small singular values from the b ottom of ma trices L and R factorizations and are obtained Adaptive versions of the U LV and U RV algorithms and results concerning the eect of esti mated null vectors on the size of odiagonal blo cks H and F are discussed in There it is shown that the sizes of H and F dep end strongly on approximations of the null vectors The norms of H and F in turn aect the accuracy of the approximated nullspaces A renement metho d for the U RV factorization was presented and analysed in Stewart The usual way to compute a U LV factorization of a matrix A is rst to compute an ordinary QL factorization of A p and then to p eel o small singular values one by one from the b ottom of the matrix L This requires approximations of left null vectors of the triangular ma trix L at each iteration step of factorization renement and up dating In the practical rankrevealing U LV factorizations left null vectors are usually obtained from the corresp onding right null vectors via triangular solves For very large problems however this results in an extra cost and may lead to loss of accuracy in the subspaces In the next section we present a more e cient U LV factorization that avoids triangular solves by working with right null vectors of lower triangular matrices This reduces the computational work needed for the triangular solves TDULV factorization In this section we present the rankrevealing T D U LV factorization The idea of our factorization is to compute rst any QL factorization of A for example by using the LAPACK routine xGE 2 QLF then to p eelo small singular values of L one after the other from the top of the matrix L in a sequence of deation steps until
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