---
name: "dpbstf"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/dpbstf/3lapack"
---

# dpbstf(3lapack)

computes a split Cholesky factorization of a real symmetric positive definite band matrix A

## NAME

DPBSTF - computes a split Cholesky factorization of a real symmetric positive definite band matrix A

## SYNOPSIS

- SUBROUTINE DPBSTF(

  UPLO, N, KD, AB, LDAB, INFO )

- CHARACTER

  UPLO

- INTEGER

  INFO, KD, LDAB, N

- DOUBLE

  PRECISION AB( LDAB, \* )

## PURPOSE

DPBSTF computes a split Cholesky factorization of a real symmetric positive definite band matrix A. This routine is designed to be used in conjunction with DSBGST. The factorization has the form  A = S\*\*T\*S  where S is a band matrix of the same bandwidth as A and the following structure:\
\
   S = ( U    )\
\
       ( M  L )\
 where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.\

## 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) DOUBLE PRECISION 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 factor S from the split Cholesky factorization A = S\*\*T\*S. See Further Details. 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 factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.

## FURTHER DETAILS

The band storage scheme is illustrated by the following example, when N = 7, KD = 2:\
 S = ( s11  s12  s13                     )\
\
     (      s22  s23  s24                )\
\
     (           s33  s34                )\
\
     (                s44                )\
\
     (           s53  s54  s55           )\
\
     (                s64  s65  s66      )\
\
     (                     s75  s76  s77 )\
 If UPLO = 'U', the array AB holds:\
 on entry:                          on exit:\
\
  \*    \*   a13  a24  a35  a46  a57   \*    \*   s13  s24  s53  s64  s75\
  \*   a12  a23  a34  a45  a56  a67   \*   s12  s23  s34  s54  s65  s76 a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77 If UPLO = 'L', the array AB holds:\
 on entry:                          on exit:\
 a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77 a21  a32  a43  a54  a65  a76   \*   s12  s23  s34  s54  s65  s76   \* a31  a42  a53  a64  a64   \*    \*   s13  s24  s53  s64  s75   \*    \* Array elements marked \* are not used by the routine.\