dpbsv(3)

NAME

DPBSV - compute the solution to a real system of linear

equations A * X = B,

SYNOPSIS

SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
    CHARACTER     UPLO
    INTEGER       INFO, KD, LDAB, LDB, N, NRHS
    DOUBLE        PRECISION AB( LDAB, * ), B( LDB, * )

PURPOSE

DPBSV computes the solution to a real system of linear

equations A * X = B, where A is an N-by-N symmetric positive def

inite band matrix and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as

A = U**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',

where U is an upper triangular band matrix, and L is a

lower triangular band matrix, with the same number of superdiago

nals or subdiagonals as A. The factored form of A is then used

to solve the system of equations A * X = B.

ARGUMENTS

UPLO (input) CHARACTER*1

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N (input) INTEGER

The number of linear equations, i.e., the order of

the matrix A. N >= 0.

KD (input) INTEGER

The number of superdiagonals of the matrix A if

UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >=

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) DOUBLE PRECISION array, dimension

(LDAB,N)

On entry, the upper or lower triangle of the sym

metric band matrix A, stored in the first KD+1 rows of the array.

The j-th column of A is stored in the j-th column of the array AB

as follows: if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j

KD)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for

j<=i<=min(N,j+KD). See below for further details.

On exit, if INFO = 0, the triangular factor U or L

from the Cholesky factorization A = U**T*U or A = L*L**T of the

band matrix A, in the same storage format as A.

LDAB (input) INTEGER

The leading dimension of the array AB. LDAB >=

KD+1.

B (input/output) DOUBLE PRECISION 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, the leading minor of order i of

A is not positive definite, so the factorization could not be

completed, and the solution has not been computed.

FURTHER DETAILS

The band storage scheme is illustrated by the following

example, when N = 6, KD = 2, and UPLO = 'U':

On entry: On exit:

* * a13 a24 a35 a46 * * 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

Similarly, if UPLO = 'L' the format of A is as follows:

On entry: On exit:

a11 a22 a33 a44 a55 a66 l11 l22 l33 l44

l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54

l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 *

*

Array elements marked * are not used by the routine.

LAPACK version 3.0 15 June 2000

DPBSV(3)