# ?pptrf

Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.

## Syntax

FORTRAN 77:

call spptrf( uplo, n, ap, info )

call dpptrf( uplo, n, ap, info )

call cpptrf( uplo, n, ap, info )

call zpptrf( uplo, n, ap, info )

Fortran 95:

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

## Description

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

`A = UH*U`

  if `uplo='U'`

`A = L*LH`

  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 packed in the array ap, and how A is factored:

  If uplo = 'U', the array ap stores the upper triangular part of the matrix A, and A is factored as UH*U.

  If uplo = 'L', the array ap stores the lower triangular part of the matrix A; A is factored as L*LH.

n

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

ap

  REAL for spptrf

  DOUBLE PRECISION for dpptrf

  COMPLEX for cpptrf

  DOUBLE COMPLEX for zpptrf.

  Array, DIMENSION at least max(1, n(n+1)/2). The array ap contains either the upper or the lower triangular part of the matrix A (as specified by uplo) in packed storage (see Matrix Storage Schemes).

## Output Parameters

ap

  The upper or lower triangular part of A in packed 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 pptrf interface are as follows:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size `(n*(n+1)/2)`.

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 is approximately `(1/3)n3` for real flavors and `(4/3)n3` for complex flavors.

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

?pptrs

  to solve `A*X = B`

?ppcon

  to estimate the condition number of A

?pptri

  to compute the inverse of A.

Parent topic: Routines for Matrix Factorization