SPBSTF(l) — LAPACK routine (version 2.0)

NAME

SPBSTF - compute a split Cholesky factorization of a real symmetric positive definite band matrix A

SYNOPSIS

SUBROUTINE SPBSTF(
UPLO, N, KD, AB, LDAB, INFO )

CHARACTER UPLO

INTEGER INFO, KD, LDAB, N

REAL AB( LDAB, ∗ )

PURPOSE

SPBSTF computes a split Cholesky factorization of a real symmetric positive definite band matrix A.

This routine is designed to be used in conjunction with SSBGST.

The factorization has the form A = S∗∗T∗S where S is a band matrix of the same bandwidth as A and the following structure:

S = ( U )
( M L )

where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.

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 factor S from the split Cholesky factorization A = S∗∗T∗S. See Further Details. 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 factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.

FURTHER DETAILS

The band storage scheme is illustrated by the following example, when N = 7, KD = 2:

S = ( s11 s12 s13                     )
    (      s22 s23 s24                )
    (           s33 s34                )
    (                s44                )
    (           s53 s54 s55           )
    (                s64 s65 s66      )
    (                     s75 s76 s77 )

If UPLO = ’U’, the array AB holds:

on entry:                          on exit:

∗    ∗   a13 a24 a35 a46 a57   ∗    ∗   s13 s24 s53 s64 s75
∗   a12 a23 a34 a45 a56 a67   ∗   s12 s23 s34 s54 s65 s76 a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77

If UPLO = ’L’, the array AB holds:

on entry:                          on exit:

a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 a21 a32 a43 a54 a65 a76   ∗   s12 s23 s34 s54 s65 s76   ∗ a31 a42 a53 a64 a64   ∗    ∗   s13 s24 s53 s64 s75   ∗    ∗

Array elements marked ∗ are not used by the routine.