man spbtrf (Fonctions bibliothèques) - compute the Cholesky factorization of a real symmetric positive definite band matrix A

NAME

SPBTRF - compute the Cholesky factorization of a real symmetric positive definite band matrix A

SYNOPSIS

SUBROUTINE SPBTRF(
UPLO, N, KD, AB, LDAB, INFO )
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
REAL AB( LDAB, * )

PURPOSE

SPBTRF computes the Cholesky factorization of a real symmetric positive definite band matrix A. The factorization has the form

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

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

where U is an upper triangular matrix and L is lower triangular.

ARGUMENTS

UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.
N (input) INTEGER
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.
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).

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.
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 is not positive definite, and the factorization could not be completed.

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.

Contributed by

Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989