# ?pbstf

Computes a split Cholesky factorization of a real symmetric or complex Hermitian positive-definite banded matrix used in ?sbgst/?hbgst .

## Syntax

FORTRAN 77:

call spbstf(uplo, n, kb, bb, ldbb, info)

call dpbstf(uplo, n, kb, bb, ldbb, info)

call cpbstf(uplo, n, kb, bb, ldbb, info)

call zpbstf(uplo, n, kb, bb, ldbb, info)

Fortran 95:

call pbstf(b [, uplo] [,info])

## Description

This routine computes a split Cholesky factorization of a real symmetric or complex Hermitian positive-definite band matrix B. It is to be used in conjunction with ?sbgst/?hbgst.

The factorization has the form `B = ST*S` (or `B = SH*S` for complex flavors), where S is a band matrix of the same bandwidth as B and the following structure: S is upper triangular in the first (n+kb)/2 rows and lower triangular in the remaining rows.

## Input Parameters

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  If `uplo = 'U'`, bb stores the upper triangular part of B.

  If `uplo = 'L'`, bb stores the lower triangular part of B.

n

  INTEGER. The order of the matrix B (`n ≥ 0`).

kb

  INTEGER. The number of super- or sub-diagonals in B

  (`kb ≥ 0`).

bb

  REAL for spbstf

  DOUBLE PRECISION for dpbstf

  COMPLEX for cpbstf

  DOUBLE COMPLEX for zpbstf.

  bb (ldbb,*) is an array containing either upper or lower triangular part of the matrix B (as specified by uplo) in band storage format.

  The second dimension of the array bb must be at least max(1, n).

ldbb

  INTEGER. The first dimension of bb; must be at least kb+1.

## Output Parameters

bb

  On exit, this array is overwritten by the elements of the split Cholesky factor S.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = i`, then the factorization could not be completed, because the updated element bi i would be the square root of a negative number; hence the matrix B is not positive-definite.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine pbstf interface are the following:

b

  Stands for argument bb in Fortan 77 interface. Holds the array B of size (kb+1,n).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

## Application Notes

The computed factor S is the exact factor of a perturbed matrix `B + E`, where

c(n) is a modest linear function of n, and ε is the machine precision.

The total number of floating-point operations for real flavors is approximately n(kb+1)2. The number of operations for complex flavors is 4 times greater. All these estimates assume that kb is much less than n.

After calling this routine, you can call ?sbgst/?hbgst to solve the generalized eigenproblem `Az = λBz`, where A and B are banded and B is positive-definite.

Parent topic: Generalized Symmetric-Definite Eigenvalue Problems