Advances in Social Science, Education and Humanities Research, volume 550 Proceedings of the 1st International Conference on Mathematics and Mathematics Education (ICMMED 2020)

The Determinant of Pentadiagonal Centrosymmetric Matrix Based on Sparse Hessenberg’s Algorithm N Khasanah 1,*

1 Mathematics Department, Faculty of Science and Technology, UIN Walisongo Semarang, Indonesia *Corresponding author. Email: [email protected]

ABSTRACT The algorithm of general pentadiagonal matrix has been evaluated before for computational purpose. The properties of this matrix on sparse structure are exploited to compute an efficient algorithm. This article propose a new construction of pentadiagonal matrix having centrosymmetric structure called pentadiagonal centrosymmetric matrix. Moreover, by applying the algorithm of determinant sparse Hessenberg matrix, an explicit formula of pentadiagonal centrosymmetric matrix’s determinant is developed.

Keywords: general pentadiagonal, centrosymmetric structure, sparse Hessenberg, algorithm, determinant.

1. INTRODUCTION The aim of this paper is to construct a new form of pentadiagonal matrix with centrosymmetric structure. Pentadiagonal matrix is the one construction of sparse Due to application both matrices and evaluation the matrix widely applied in areas of science and structure, the algorithm of determinant of this matrix is engineering, such in numerical solution of ordinary and proposed by using Hessenberg rule. partial differential equations (ODE and PDE), interpolation problems and boundary value problems (BVP) [1]. The rule of this matrix is necessary at many areas, then the evaluation of this matrix are needed, 2. PRELIMINARIES particularly at determinant process. Based on the special First of all, some definitions and properties are given structure of this matrix, some researchers focus on for clear discussion at determinant process of determinant problem based on computation stand point. pentadiagonal centrosymmetrix matrix, as follows. The main basic research about pentadiagonal matrix is started by [2] using two-term recurrence for evaluating Definition 1 [11]. The matrix is called as nn general determinant general matrix. Based on previous research pentadiagonal matrix which has definition such [3] at computing the determinant of a tridiagonal matrix, D  d , where the entry d  0 for ji  2 or generalization of the DETGTRI algorithm are obtained.   ij ,1nji ij This algorithm as the major concept for the next can be written as discussion on constructing determinant of pentadiagonal  ddd  matrix [4-11].  11 12 13   dddd  On the other side, centrosymmetric matrix also has a 21 22 23 24  ddddd  special structure arise at some applications, for instance  31 32 33 34 35  (1) pattern recognition process [12]. By using analytical D       process this special entries of centrosymmetric matrix,   d  ,2 nn    the computation of determinant matrix is important to be nn  3,1 nn  2,1 nn  1,1 dddd  ,1 nn evaluated. Studies about this topics are given at some    nn 2, nn 1, ddd ,nn  papers [13-16] for a number of fast algorithm for computing determinant process by applying Hessenberg algorithm of determinant. This kind of matrix having rules on numerical analysis and arise at determinant centrosymmetric matrix [17-18].

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Definition 2 [13]. The entries of n -by- n lower discussion, we focus on general pentadiagonal matrix, Hessenberg matrix is the form as follows where is even number as its matrix ordo only. Based on the definition before about general pentadiagonal (1) and  h h 0 0 0   11 1,2  centrosymmetric matrix (4), the formationpentadiagonal  h21 h2,2 h2,3 0 0  centrosymmetric matrix is formed by the step:. H       0  (2)  

hn1,1 hn1,2  hn1,n1 hn1,n  3.1. Construct Sparse Hessenberg Matrix   h h  h h  n,1 n,2 n,n1 n,n  Based on definition of pentadiagonal and centrosymmetric matrix, then the pentadiagonal Definition 3 [15]. The n order of sparse Hessenberg centrosymmetric matrix is written as (1) where the entries matrix is the matrix with the contruction as have centrosymmetric structure dij  d ni1,n j1 for 1 i  n ,1 j  n . For a deeper understanding, let we  h11 h1,2 h1,3    show an example the form of 88 pentadiagonal matrix h21 h2,2 h2,3 h2,4  to be the specific matrix called pentadiagonal  h h h h   3,2 3,3 3,4 3,5  centrosymmetric matrix such : S        (3)    d d d     h 11 12 13  n2,n    d d d d  h h h   21 22 23 24   n1,n2 n1,n1 n1,n      d 31 d 32 d 33 d 34 d 35  hn,n1 hn,n     d d d d d  42 43 44 45 56 is D8     d 53 d 54 d 55 d 56 d 57  Definition 4 [14]. The centrosymmetric matrix is the  d d d d d  matrix where A  a  R nn and has the entries as  64 65 66 67 68   ij nn  d 75 d 76 d 77 d 78  aij  ani1,n j1 for 1 i  n ,1 j  n or    d 86 d 87 d 88  pentadiagonal form matrix. Then it constructed to be  a11 a12  a1,n    pentadiagonal having centrosymmetric structure such  a 21 a 22  a 2,n    A      (4)  d d d     11 12 13  a 2,n  a 22 a 21  d d d d    21 22 23 24 a  a a  d d d d d   1,n 12 11   31 32 33 34 35   d d d d d  ~ 42 43 44 45 46 (5) D8     d 46 d 45 d 44 d 43 d 42  3. RESULTS AND DISCUSSION  d d d d d   35 34 33 32 31  d d d d This section discuss about the main result of this  24 23 22 21    paper on constructing a new form of pentadiagonal  d 13 d 12 d 11  centrosymmetric matrix. Then, the algorithm to compute After getting this form matrix, the next work is determinant of this matrix is proposed also. Based on the transforming pentadiagonal into sperse Hessenberg rule of sparse Hesssenberg matrix, the steps of this matrix. This step is usefull for applying Hesssenberg algorithm is explained for deeper understanding algorithm at determinant computation. Based on the The construction of general pentadiagonal matrix and definition of pentadiagonal form (1), then constructing its algorithm for computing determinant have been found with centrosymetric entries, it continues by constructing sparse Hessenberg matrix H  d written as [4] by [4]. As the different point of view from general  ij 1i, jn structure of pentadigonal matrix, this article shows a new construction pentadiagonal matrix having  d d d   11 12 13  centrosymmetric structure. The formation of this matrix d d d d  we called as pentadiagonal centrosymmetric matrix. 21 22 23 24  d d d d  After constructing new form of the matrix, next we derive  32 33 34 35  H        (6) numerical algorithm for computing this new matrix.   Based on the algorithm of sparse Hessenberg, the     d n2,n  explanations of this algorithm are written as follows.  d d d   n1,n2 n1,n1 n1,n   d d  First, let consider transforming the general  n,n1 n,n  pentadagonal matrix (1) into the formation of pentadiagonal centrosymmetric matrix. In this

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From the above matrix, we choose λ from the  d d d  i1,i  11 12 13  d d d d  matrix of 21 ~22 ~23 ~24  d d d d   32 ~33 ~34 ~35  (8)  d 43 d 44 d 45 d 56   I i1  λ λ λ λ D  ~ ~ ~   8 7 6 3    d 45 d 44 d 43 d 42   1   ~ ~ ~  λ  (7) d 34 d 33 d 32 d 31 i1    ~ ~ ~  i1,i 1  d 23 d 22 d 21    ~ ~    d d   I ni1   12 11 

As the ilustration, for 88 pentadiagonal For a general formula, the number of i1,i is defined centrosymmetrix matrix (5) then by choosing d ~ as    i1,i1 where d  d . By the concept of d 3,1 i1,i ~ 21 21 3,2   to transform matrix where : d i,i1 d 2,1 V  n n1 3 , it has the form of VD  H [14]. 1  Therefore,   1   detD detV detD detVD detH, (9)  d3,1    1  with the d e tV1  d2,1  λ D   1  3   Based on the element of D which integer 1   numbers or d Z , then it can use the following  1  ij   matrix  1     1  I i1   d d d     11 12 13   1  λ i1  ~ (10) d 21 d 22 d 23 d 24     d i1,i1 d i,i1  d d d d d     31 32 33 34 35     I ni1   d d d d d  42 43 44 45 46   Moreover, the computation of determinant of d d d d d  46 45 44 43 42  Hessenberg matrix is equal for computing the  d d d d d   35 34 33 32 31  determinant of pentadigonal centrosymmetric matrix. It  d 24 d 23 d 22 d 21  means that the next step is how to compute Hessenberg   matrix.  d 13 d 12 d 11 

 d d d  3.2. Construct the Algorithm of Determinant  11 12 13  d d d d  Pentadiagonal Centrosymmetric Matrix 21 ~22 23 24  d d d d   32 33 34 35  Based on the study before about algorithm of  d d d d d  determinant matrix, this determinant is constructed by  42 43 44 45 46 .   using the two-term recuurence takes the following  d 46 d 45 d 44 d 43 d 42  explanation.  d d d d d   35 34 33 32 31   d 24 d 23 d 22 d 21   P q  First, take n n matrix Z   n1 n1  , where  d d d  n  T   13 12 11  r n1 s n  Z has n 1  n 1 , q ,s are the scalar and Repeating the same process, by taking n1     n1 n T n1 d 42 d13 r has 1 n 1 as size of this block matrix. Next, 4,3   ,,12   and the end step resulting d 32 d 23 the determinant of Zn recursivelly is written as [14] : the following form of matrix.

f0 1 T 1 f i  α i f i1 , where α1  d1 , α i  si  ri1 Zi1 qi1

Generally, the determinant of Z i matrix is written by

detZi  fi , where i 1,2,,n .

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Futheremore, by applying previous algorithm at 4. CONCLUSION sparse Hessenberg matrix having the following recursive. The algorithm of sparse Hessenberg matrix is applied on constructing the algorithm of determinant general T 1 α n  dn,n  hn,n1e n1H n1 dn2,nen2  dn1,nen1  pentadiagonal centrosymmetric matrix. This algorithm is

T 1 T 1 used caused by same structure of main matrix is sparse α  d  d d e n1H n1e  d e n1H n1e n n,n n,n1  n2,n n2 n1,n n1  Hessenberg matrix, therefore it will be applicable. , where e is vector unit.

By subtitute the equations

T 1 f n3 e n1 H n1e n2   d n1,n2 and ACKNOWLEDGMENTS f n1 The author grateefully acknowledge Miss Annisa Nur T 1 f n2 e n1 H n1e   then become Latifah for her help in editing the paper and the technical n1 f n1 assistance.  f f  α  d  d  n3 d d  n2 d  . n n,n n,n1  n1,n2 n2,n n1,n   f n1 f n1  REFERENCES Moreover, by multiply the equation with fn1 we have

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