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### NAME

ZGBSV - compute the solution to a complex 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

### SYNOPSIS

SUBROUTINE ZGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO ) INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS INTEGER IPIV( * ) COMPLEX*16 AB( LDAB, * ), B( LDB, * )

### PURPOSE

ZGBSV computes the solution to a complex 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 decomposi‐ tion 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 triangu‐ lar matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals. The factored form of A is then used to solve the sys‐ tem 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) COMPLEX*16 array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+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 superdiag‐ onals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= 2*KL+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) COMPLEX*16 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. 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 ele‐ ments of U because of fill-in resulting from the row interchanges.