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

NAME

SPBSV - compute the solution to a real system of linear equations A ∗ X = B,

SYNOPSIS

SUBROUTINE SPBSV(
UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )

CHARACTER UPLO

INTEGER INFO, KD, LDAB, LDB, N, NRHS

REAL AB( LDAB, ∗ ), B( LDB, ∗ )

PURPOSE

SPBSV computes the solution to a real system of linear equations
   A ∗ X = B, where A is an N-by-N symmetric positive definite 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 superdiagonals 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) REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric 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) 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, 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.