BandDiagonalMod Module

$ SUBROUTINE SGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO ) $ $ -- LAPACK driver routine (version 3.1) -- $ Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. $ November 2006 $ $ .. Scalar Arguments .. $ INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS $ .. $ .. Array Arguments .. $ INTEGER IPIV( * ) $ REAL AB( LDAB, * ), B( LDB, * ) $ .. $ $ Purpose $ ======= $ $ SGBSV computes the solution to a real system of linear equations $ A * X = B, where A is a band matrix of order N with KL subdiagonals $ and KU superdiagonals, and X and B are N-by-NRHS matrices. $ $ The LU decomposition with partial pivoting and row interchanges is $ used to factor A as A = L * U, where L is a product of permutation $ and unit lower triangular matrices with KL subdiagonals, and U is $ upper triangular with KL+KU superdiagonals. The factored form of A $ is then used to solve the system of equations A * X = B. $ $ Arguments $ ========= $ $ N (input) INTEGER $ The number of linear equations, i.e., the order of the $ matrix A. N >= 0. $ $ KL (input) INTEGER $ The number of subdiagonals within the band of A. KL >= 0. $ $ KU (input) INTEGER $ The number of superdiagonals within the band of A. KU >= 0. $ $ NRHS (input) INTEGER $ The number of right hand sides, i.e., the number of columns $ of the matrix B. NRHS >= 0. $ $ AB (input/output) REAL array, dimension (LDAB,N) $ On entry, the matrix A in band storage, in rows KL+1 to $ 2KL+KU+1; rows 1 to KL of the array need not be set. $ The j-th column of A is stored in the j-th column of the $ array AB as follows: $ AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL) $ On exit, details of the factorization: U is stored as an $ upper triangular band matrix with KL+KU superdiagonals in $ rows 1 to KL+KU+1, and the multipliers used during the $ factorization are stored in rows KL+KU+2 to 2KL+KU+1. $ See below for further details. $ $ LDAB (input) INTEGER $ The leading dimension of the array AB. LDAB >= 2KL+KU+1. $ $ IPIV (output) INTEGER array, dimension (N) $ The pivot indices that define the permutation matrix P; $ row i of the matrix was interchanged with row IPIV(i). $ $ B (input/output) REAL array, dimension (LDB,NRHS) $ On entry, the N-by-NRHS right hand side matrix B. $ On exit, if INFO = 0, the N-by-NRHS solution matrix X. $ $ LDB (input) INTEGER $ The leading dimension of the array B. LDB >= max(1,N). $ $ INFO (output) INTEGER $ = 0: successful exit $ < 0: if INFO = -i, the i-th argument had an illegal value $ > 0: if INFO = i, U(i,i) is exactly zero. The factorization $ has been completed, but the factor U is exactly $ singular, and the solution has not been computed. $ $ Further Details $ =============== $ $ The band storage scheme is illustrated by the following example, when $ M = N = 6, KL = 2, KU = 1: $ $ On entry: On exit: $ $ * * * + + + * * * u14 u25 u36 $ * * + + + + * * u13 u24 u35 u46 $ * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 $ a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 $ a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * $ a31 a42 a53 a64 * * m31 m42 m53 m64 * * $ $ Array elements marked * are not used by the routine; elements marked $ + need not be set on entry, but are required by the routine to store $ elements of U because of fill-in resulting from the row interchanges.

Arguments