spbtrf(3F)

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