NAME

spbtf2 - compute the Cholesky factorization of a real symmetric positive definite band matrix A

SYNOPSIS

  SUBROUTINE SPBTF2( UPLO, N, KD, AB, LDAB, INFO)
  CHARACTER * 1 UPLO
  INTEGER N, KD, LDAB, INFO
  REAL AB(LDAB,*)

  SUBROUTINE SPBTF2_64( UPLO, N, KD, AB, LDAB, INFO)
  CHARACTER * 1 UPLO
  INTEGER*8 N, KD, LDAB, INFO
  REAL AB(LDAB,*)

F95 INTERFACE

  SUBROUTINE PBTF2( UPLO, [N], KD, AB, [LDAB], [INFO])
  CHARACTER(LEN=1) :: UPLO
  INTEGER :: N, KD, LDAB, INFO
  REAL, DIMENSION(:,:) :: AB

  SUBROUTINE PBTF2_64( UPLO, [N], KD, AB, [LDAB], [INFO])
  CHARACTER(LEN=1) :: UPLO
  INTEGER(8) :: N, KD, LDAB, INFO
  REAL, DIMENSION(:,:) :: AB

C INTERFACE

#include <sunperf.h>

void spbtf2(char uplo, int n, int kd, float *ab, int ldab, int *info);

void spbtf2_64(char uplo, long n, long kd, float *ab, long ldab, long *info);

PURPOSE

spbtf2 computes the Cholesky factorization of a real symmetric positive definite band matrix A.

The factorization has the form

   A = U' * U ,  if UPLO = 'U', or

   A = L  * L',  if UPLO = 'L',

where U is an upper triangular matrix, U' is the transpose of U, and L is lower triangular.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

ARGUMENTS

UPLO (input)
Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
```
 = 'U':  Upper triangular
```
```
 = 'L':  Lower triangular
```
N (input)
The order of the matrix A. N > = 0.
KD (input)
The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD > = 0.
AB (input/output)
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'*U or A = L*L' of the band matrix A, in the same storage format as A.
LDAB (input)
The leading dimension of the array AB. LDAB > = KD+1.

INFO (output)

 = 0: successful exit

 < 0: if INFO  = -k, the k-th argument had an illegal value

 > 0: if INFO  = k, the leading minor of order k 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.