cgbsv(3)

NAME

CGBSV - 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 ma

trices

SYNOPSIS

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

PURPOSE

CGBSV 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 ma

trices. The LU decomposition with partial pivoting and row in

terchanges is used to factor A as A = L * U, where L is a product

of permutation and unit lower triangular matrices with KL subdi

agonals, and U is upper triangular with KL+KU superdiagonals.

The factored form of A is then used to solve the system of equa

tions 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 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 superdiago

nals 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 ma

trix P; row i of the matrix was interchanged with row IPIV(i).

B (input/output) COMPLEX 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 ille

gal 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; ele

ments 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.

LAPACK version 3.0 15 June 2000

CGBSV(3)