?hptrf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?hptrf

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

Syntax

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

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

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

Include Files

mkl.fi, lapack.f90

Description

The routine computes the factorization of a complex Hermitian packed matrix A using the Bunch-Kaufman diagonal pivoting method:

if uplo='U', A = U*D*U^H
if uplo='L', A = L*D*L^H,

where A is the input matrix, 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 Hermitian 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 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^H.

If uplo = 'L', the array ap stores the lower triangular part of the matrix A, and A is factored as L*D*L^H.

INTEGER. The order of matrix A; n≥ 0.

COMPLEX for chptrf

DOUBLE COMPLEX for zhptrf.

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 hptrf 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 are stored in 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 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||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 (4/3)n³.

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

?hptrs	to solve `A*X = B`
?hpcon	to estimate the condition number of `A`
?hptri	to compute the inverse of `A`.

Parent topic: Matrix Factorization: LAPACK Computational Routines

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