spbtrf(3)

NAME

SPBTRF - compute the Cholesky factorization of a real sym

metric 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 sym

metric 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 tri

angular.

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 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).

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 ille

gal value
> 0: if INFO = i, the leading minor of order i is

not positive definite, and the factorization could not be com

pleted.

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

LAPACK version 3.0 15 June 2000

SPBTRF(3)