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spbtrf

 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 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