?ppequ

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?ppequ

Computes row and column scaling factors intended to equilibrate a symmetric (Hermitian) positive definite matrix in packed storage and reduce its condition number.

Syntax

call sppequ( uplo, n, ap, s, scond, amax, info )

call dppequ( uplo, n, ap, s, scond, amax, info )

call cppequ( uplo, n, ap, s, scond, amax, info )

call zppequ( uplo, n, ap, s, scond, amax, info )

call ppequ( ap, s [,scond] [,amax] [,uplo] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive definite matrix A in packed storage and reduce its condition number (with respect to the two-norm). The output array s returns scale factors such that s(i)s[i + 1] contains

These factors are chosen so that the scaled matrix B with elements b_ij=s(i)*a_ij*s(j) has diagonal elements equal to 1.

This choice of s puts the condition number of B within a factor n of the smallest possible condition number over all possible diagonal scalings.

See ?laqsp auxiliary function that uses scaling factors computed by ?ppequ.

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:

If uplo = 'U', the array ap stores the upper triangular part of the matrix A.

If uplo = 'L', the array ap stores the lower triangular part of the matrix A.

INTEGER. The order of matrix A; n≥ 0.

REAL for sppequ

DOUBLE PRECISION for dppequ

COMPLEX for cppequ

DOUBLE COMPLEX for zppequ.

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

s	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, size (`n`). If `info = 0`, the array `s` contains the scale factors for `A`.
scond	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. If `info = 0`, `scond` contains the ratio of the smallest `s(i)` to the largest `s(i)`.
amax	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Absolute value of the largest element of the matrix `A`.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, the `i`-th diagonal element of `A` is nonpositive.

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 ppequ interface are as follows:

ap	Holds the array `A` of size (`n*(n+1)/2`).
s	Holds the vector of length `n`.
uplo	Must be 'U' or 'L'. The default value is 'U'.

Application Notes

If scond≥ 0.1 and amax is neither too large nor too small, it is not worth scaling by s.

If amax is very close to SMLNUM or very close to BIGNUM, the matrix A should be scaled.

Parent topic: Matrix Equilibration: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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