ZGBSV(l) — LAPACK driver routine (version 2.0)

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 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) 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 superdiagonals 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.

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.