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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, * )



     #include <sunperf.h>

     void spbtrf(char uplo, int n, int kd, float *sab, int  ldab,
               int *info) ;

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 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, 27 Jun  23,
     1989

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