spbtrf(3P)

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, ∗ )

#include <sunperf.h>

void spbtrf(char uplo, int n, int kd, float ∗sab, int ldab, int ∗info) ;

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, 27 Jun 23, 1989