# ?pbtrf

Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band matrix.

## Syntax

FORTRAN 77:

call spbtrf( uplo, n, kd, ab, ldab, info )

call dpbtrf( uplo, n, kd, ab, ldab, info )

call cpbtrf( uplo, n, kd, ab, ldab, info )

call zpbtrf( uplo, n, kd, ab, ldab, info )

Fortran 95:

call pbtrf( a [, uplo] [,info] )

## Description

This routine forms the Cholesky factorization of a symmetric positive- definite or, for complex data, Hermitian positive-definite band matrix A:

`A = UT*U` for real data, `A = UH*U` for complex data

  if `uplo='U'`

`A = L*LT` for real data, `A = L*LH` for complex data

  if `uplo='L',`

where L is a lower triangular matrix and U is upper triangular.

## Input Parameters

uplo

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

  Indicates whether the upper or lower triangular part of A is stored in the array ab, and how A is factored:

  If `uplo = 'U'`, the upper triangle of A is stored.

  If `uplo = 'L'`, the lower triangle of A is stored.

n

  INTEGER. The order of matrix A; n ≥ 0.

kd

  INTEGER. The number of superdiagonals or subdiagonals in the matrix `A(kd ≥ 0)`.

ab

  REAL for spbtrf

  DOUBLE PRECISION for dpbtrf

  COMPLEX for cpbtrf

  DOUBLE COMPLEX for zpbtrf.

  Array, DIMENSION `(,*)`. The array ap contains either the upper or the lower triangular part of the matrix A (as specified by uplo) in band storage (see Matrix Storage Schemes). The second dimension of ab must be at least `max(1, n)`.

ldab

  INTEGER. The first dimension of the array ab. `(ldab ≥ kd + 1)`

## Output Parameters

ap

  The upper or lower triangular part of A (in band storage) is overwritten by the Cholesky factor U or L, as specified by uplo.

info

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

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

  If `info = i`, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the factorization could not be completed. This may indicate an error in forming the matrix A.

## 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 reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine pbtrf interface are as follows:

a

  Stands for argument ab in Fortan 77 interface. Holds the array A of size `(kd+1,n)`.

uplo

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

## Application Notes

If `uplo = 'U'`, the computed factor U is the exact factor of a perturbed matrix `A + E`, where

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

A similar estimate holds for `uplo = 'L'`.

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

After calling this routine, you can call the following routines:

?pbtrs

  to solve `A*X = B`

?pbcon

  to estimate the condition number of A.

Parent topic: Routines for Matrix Factorization