# ?sptrf

Computes the Bunch-Kaufman factorization of a symmetric matrix using packed storage.

## Syntax

FORTRAN 77:

call ssptrf( uplo, n, ap, ipiv, info )

call dsptrf( uplo, n, ap, ipiv, info )

call csptrf( uplo, n, ap, ipiv, info )

call zsptrf( uplo, n, ap, ipiv, info )

Fortran 95:

call sptrf( a [,uplo] [,ipiv] [,info] )

## Description

This routine forms the Bunch-Kaufman factorization of a symmetric matrix A using packed storage:

if `uplo='U'`, `A = P*U*D*UT*PT`

if `uplo='L'`, `A = P*L*D*LT*PT,`

where P is a permutation matrix, U and L are upper and lower triangular matrices with unit diagonal, and D is a symmetric block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks. U and L have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of D.

## 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 `P*U*D*UT*PT`.

  If `uplo = 'L'`, the array ap stores the lower triangular part of the matrix A, and A is factored as `P*L*D*LT*PT`.

n

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

ap

  REAL for ssptrf

  DOUBLE PRECISION for dsptrf

  COMPLEX for csptrf

  DOUBLE COMPLEX for zsptrf.

  Array, DIMENSION at least max(1, n(n+1)/2). The array ap contains 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 triangle of A (as specified by uplo) is overwritten by details of the block-diagonal matrix D and the multipliers used to obtain the factor U (or L).

ipiv

  INTEGER.

  Array, DIMENSION at least `max(1, n)`. Contains details of the interchanges and the block structure of D. If `ipiv(i) = k > 0`, then `dii` is a 1-by-1 block, and the i-th row and column of A was interchanged with the k-th row and column.

  If uplo = 'U' and ipiv(i) =ipiv(i-1) = -m < 0, then D has a 2-by-2 block in rows/columns i and `i-1`, and the (`i-1`)-th row and column of A was interchanged with the m-th row and column.

  If uplo = 'L' and ipiv(i) =ipiv(i+1) = -m < 0, then D has a 2-by-2 block in rows/columns i and `i+1`, and the (`i+1`)-th row and column of A was interchanged with the m-th row and column.

info

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

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

  If `info = i`, `dii` is 0. The factorization has been completed, but D is exactly singular. Division by 0 will occur if you use D for solving a system of linear equations.

## 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 sptrf interface are as follows:

a

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

ipiv

  holds the vector of length (n).

uplo

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

## Application Notes

The 2-by-2 unit diagonal blocks and the unit diagonal elements of U and L are not stored. The remaining elements of U and L overwrite elements of the corresponding columns of the matrix A, but additional row interchanges are required to recover U or L explicitly (which is seldom necessary).

If ipiv(i) = i for all `i = 1...n`, then all off-diagonal elements of U (L) are stored explicitly in packed form.

If `uplo = 'U'`, the computed factors U and D are the exact factors of a perturbed matrix `A + E`, where
`|E| ≤ c(n)ε P|U||D||UT|PT`

`c(n)` is a modest linear function of n, and ε is the machine precision. A similar estimate holds for the computed L and D when `uplo = 'L'`.

The total number of floating-point operations is approximately `(1/3)n3` for real flavors or `(4/3)n3` for complex flavors.

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

?sptrs

  to solve `A*X = B`

?spcon

  to estimate the condition number of A

?sptri

  to compute the inverse of A.

Parent topic: Routines for Matrix Factorization