# ?sytrf

Computes the Bunch-Kaufman factorization of a symmetric matrix.

## Syntax

FORTRAN 77:

call ssytrf( uplo, n, a, lda, ipiv, work, lwork, info )

call dsytrf( uplo, n, a, lda, ipiv, work, lwork, info )

call csytrf( uplo, n, a, lda, ipiv, work, lwork, info )

call zsytrf( uplo, n, a, lda, ipiv, work, lwork, info )

Fortran 95:

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

## Description

This routine forms the Bunch-Kaufman factorization of a symmetric matrix:

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

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

where A is the input matrix, 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 stored and how A is factored:

  If `uplo = 'U'`, the array a stores the upper triangular part of the matrix A, and A is factored as `P*U*D*UT*PT`.

  If `uplo = 'L'`, the array a 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.

a

  REAL for ssytrf

  DOUBLE PRECISION for dsytrf

  COMPLEX for csytrf

  DOUBLE COMPLEX for zsytrf.

  Array, DIMENSION `(lda,*)`. The array a contains either the upper or the lower triangular part of the matrix A (see uplo). The second dimension of a must be at least `max(1, n)`.

lda

  INTEGER. The first dimension of a; at least `max(1, n)`.

work

  Same type as a. A workspace array, dimension at least `max(1,lwork)`.

lwork

  INTEGER. The size of the work array `(lwork ≥ n)`.

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

  See Application Notes for the suggested value of lwork.

## Output Parameters

a

  The upper or lower triangular part of a is overwritten by details of the block-diagonal matrix D and the multipliers used to obtain the factor U (or L).

`work(1)`

  If `info=0`, on exit `work(1)` contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

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 (`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 (`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 sytrf interface are as follows:

a

  holds the matrix A of size (n, n)

ipiv

  holds the vector of length (n)

uplo

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

## Application Notes

For better performance, try using `lwork = n*blocksize`, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

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 are stored in the corresponding columns of the array 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 the corresponding elements of the array a.

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:

?sytrs

  to solve `A*X = B`

?sycon

  to estimate the condition number of A

?sytri

  to compute the inverse of A.

Parent topic: Routines for Matrix Factorization