?sptrf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?sptrf

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

Syntax

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 )

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

Include Files

mkl.fi, lapack.f90

Description

The routine computes the factorization of a real/complex symmetric matrix A stored in the packed format using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is:

if uplo='U', A = U*D*U^T
if uplo='L', A = L*D*L^T,

where U and L are products of permutation and triangular matrices with unit diagonal (upper triangular for U and lower triangular for L), 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.

NOTE:

This routine supports the Progress Routine feature. See Progress Function for details.

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 U*D*U^T.

If uplo = 'L', the array ap stores the lower triangular part of the matrix A, and A is factored as L*D*L^T.

INTEGER. The order of matrix A; n≥ 0.

REAL for ssptrf

DOUBLE PRECISION for dsptrf

COMPLEX for csptrf

DOUBLE COMPLEX for zsptrf.

Array, size 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

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, size at least max(1, n). Contains details of the interchanges and the block structure of D. If ipiv(i) = k >0, then d_ii 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, d_ii 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.

LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine sptrf interface are as follows:

ap	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 array ap, 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||U^T|P^T

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: Matrix Factorization: LAPACK Computational Routines

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