Computers and Mathematics with Applications 58 (2009) 1699–1710 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa An inversion algorithm for a banded matrixI Rui-Sheng Ran a, Ting-Zhu Huang b,∗ a Automation Department, CISDI Engineering CO., LTD. Chongqing, 400013, PR China b School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, PR China article info a b s t r a c t Article history: In this paper, an inversion algorithm for a banded matrix is presented. The n twisted Received 26 July 2007 decompositions of a banded matrix are given first; then the inverse of the matrix is Received in revised form 4 June 2009 obtained, one column at time. The method is about two times faster than the standard Accepted 8 July 2009 method based on the LU decomposition, as is shown with the analysis of computing complexity and the numerical experiments. Keywords: Banded matrix ' 2009 Elsevier Ltd. All rights reserved. Inverse Twisted decomposition The computing complexity Tridiagonal matrix Pentadiagonal matrix 1. Introduction The inverses of tridiagonal and banded matrices are necessary in solving many problems, such as computing the condition number, investigating the decay rate of the inverse and solving a linear system whose coefficient matrix is banded or tridiagonal. For banded matrices, there exist some investigations on the inverses of the matrices and the solution of the banded linear system, see [1–4]. For tridiagonal matrices, many efficient algorithms and explicit expressions of the elements of the inverse in some special cases for such inverse are presented, see [5–10]. In Meurant [6], the twisted decompositions of symmetric tridiagonal or block tridiagonal matrices are proposed to give the inverses of the matrices. In this paper, for a banded matrix subject to some conditions, a method for obtaining the inverse is presented. The results are related with n twisted decompositions of the banded matrix, which are first given in this paper. According to the jth twisted decomposition, the formulae for computing the jth column elements of the inverse are obtained. When we let j D 1; 2;:::; n, a method is presented which inverts the banded matrix in order starting with the first column. The standard method solves n linear systems for all the columns of the inverse, by using the LU decomposition. For convenience, it is denoted with LU method. From the analysis of the computing complexity, the method proposed is about two times faster than the LU method. Let p denote the bandwidth. Numerical experiments show that for p n, the error of the proposed method is considerable smaller and the time complexity is likewise smaller than that for the LU method. 2. The inverse of banded matrices Let A D .aij/ be a nonsingular equal bandwidth matrix of order n such that aij D 0 if ji − jj > p, and aij 6D 0 if ji − jj ≤ p, where p is a fixed positive integer such that 2p < n. Obviously, the bandwidth of A is p. In general, many technical problems have a coefficient matrix with bandwidth p which is much less than n. I This research was supported by NSFC (10771030), the Key Project of Chinese Ministry of Education (107098), the Specialized Research Fund for the Doctoral Program of Higher Education (20070614001), Sichuan Province Project for Applied Basic Research (2008JY0052) and the Project of National Defense Key Lab. (9140C6902030906). ∗ Corresponding author. E-mail addresses: [email protected] (R.-S. Ran), [email protected], [email protected] (T.-Z. Huang). 0898-1221/$ – see front matter ' 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2009.07.069 1700 R.-S. Ran, T.-Z. Huang / Computers and Mathematics with Applications 58 (2009) 1699–1710 Throughout the paper, it is assumed that all principal minors of A are nonzero. In [11, p. 96], the LU decomposition of A is given, where the matrix L is a lower triangular matrix whose lower semi bandwidth is p and the matrix U is a upper triangular matrix whose upper semi bandwidth is p. Similarly, the UL decomposition can be given. Based on the two decompositions, we first give n twisted decompositions of the equal bandwidth matrix A. With the results in hand, we obtain simple and direct formulae for finding the inverse of A in order by column, starting with the first. Theorem 2.1. Let A D .aij/ be a nonsingular equal bandwidth matrix of order n. For j D 1; 2;:::; n, .i/ when j D 1,A D UL; when j D n, A D LU; .ii/ when j D 2;:::; n − 1,there exists a twisted decomposition such that A D LjUj, where 0 1 1 l 1 B 21 C B : : : C B : :: :: C B C B : C B :: C BlpC1;1 1 C B : C B :: ··· C D B lj;j−1 1 ljC1;j ljCp;j C Lj B : C ; B :: : :: :: :: C B : : : : : C B C B :: :: C B ljCp−1;j−1 : : ln;n−pC B C B :: :: : C B : : : C @ A 1 ln:n−1 1 0u11 ············ u1;pC1 1 : : B : : C B : : C B C B :: :: C B : : C B ······ C B uj−1;j−1 uj−1;jCp−1 C B C B uj;j C Uj D B u C u C C C ; B j;j 1 j 1;j 1 C B : : : C B : :: :: C B C B : : C B :: :: C B uj;jCp C B : : : C @ :: :: :: A un−p;n ··· un−1;n un;n and the sequences lij and uij may be obtained with the formulae: 1: when r D 1; 2;:::; j − 1 .1/ r D 1 air uri D ari .i D 1;:::; p C 1/; lir D .i D 2;:::; p C 1/I urr .2/ r D 2;:::; j − 1 .a/ computing the rth row of U − Xr 1 uri D ari − lrkuki .i D r;:::; r C p/I (1) kDmax.1;i−p;r−p/ .b/ computing the rth column of L − rP1 air − likukr kDmax.1;i−p;r−p/ lir D .i D r C 1;:::; r C p/I (2) urr 2: when r D n; n − 1;:::; j C p .1/ r D n air uir D ari .i D n − p;:::; n/; lri D .i D n − p;:::; n − 1/I urr R.-S. Ran, T.-Z. Huang / Computers and Mathematics with Applications 58 (2009) 1699–1710 1701 .2/ r D n − 1;:::; j C p .a/ computing the rth row of U min.n;iCp;rCp/ X uir D ari − lkr uik .i D r;:::; r − p/I (3) kDrC1 .b/ computing the rth column of L min.n;iCp;rCp/ P air − lkiurk kDrC1 lri D .i D r − 1;:::; r − p/I (4) urr 3: when r D j C p − 1;:::; j C 1; j .1/ computing the rth row of U min.n;iCp;rCp/ j−1 X X uir D ari − lkr uik − lrkuki .i D r; r − 1;:::; j/I (5) kDrC1 kDmax.1;r−p;i−p/ .2/ computing the rth column of L min.n;iCp;rCp/ j−1 P P air − lkiurk − likukr kDrC1 kDmax.1;r−p;i−p/ lri D .i D r − 1;:::; j/: (6) urr Proof. The three cases regarding the formulae for computing lij and uij may be similarly solved. 1. For the first case, when r D 1, the initial values are derived: ai1 u1i D a1i.i D 1;:::; p C 1/ and li1 D .i D 2;:::; p C 1/: u11 When r D 2;:::; j − 1, multiplying the rth row of Lj with the ith column of Uj, we have − Xr 1 uri C lrkuki D ari.i D r;:::; r C p/; kDmax.1;i−p;r−p/ so we have Eq. (1), i.e. the rth row of Uj is obtained; multiplying the rth column of Uj with ith row of Lj, we have j−1 X lriurr C likukr D air .i D r C 1;:::; r C p/; kDmax.1;r−p;i−p/ so we have Eq. (2), i.e. the rth column of Lj is obtained. Performing the above process, each column of Lj and each row of Uj may be obtained in turn respectively. Note that each step is always based on the result obtained in the former step. 2. The second case may be similarly obtained. Different from the first case, the process begins with r D n and iterates forward. From r D n − 1 to j C p, the rth row of Uj and the rth column of Lj are given in turn. When r D n, the initial value are: ain uin D ani.i D n − p;:::; n/ and lni D .i D n − 1;:::; n − p/: unn 3. The third case is about the central rows and columns Lj and Uj. By using the results of the above two cases, it may also be obtained when iterating forward from r D j C p − 1 to r D j. For the rth row of Uj, multiplying the rth row of Lj with ith .i D r; r − 1;:::; j/ column of Uj, we have min.n;iCp;rCp/ j−1 X X uir C lkr uik C lrkuki D ari: kDrC1 kDmax.1;r−p;i−p/ For the rth column of Lj, multiplying the rth column of Uj with ith .i D r − 1;:::; j/ row of Lj, we have min.n;iCp;rCp/ j−1 X X lriurr C lkiurk C likukr D air : kDrC1 kDmax.1;r−p;i−p/ So we have Eq.