?poequ

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?poequ

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

Syntax

call spoequ( n, a, lda, s, scond, amax, info )

call dpoequ( n, a, lda, s, scond, amax, info )

call cpoequ( n, a, lda, s, scond, amax, info )

call zpoequ( n, a, lda, s, scond, amax, info )

call poequ( a, s [,scond] [,amax] [,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 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_i,j=s(i)*A_i,j*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 ?laqsy auxiliary function that uses scaling factors computed by ?poequ.

Input Parameters

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

REAL for spoequ

DOUBLE PRECISION for dpoequ

COMPLEX for cpoequ

DOUBLE COMPLEX for zpoequ.

Array: size lda by *.

Contains the n-by-n symmetric or Hermitian positive definite matrix A whose scaling factors are to be computed. Only the diagonal elements of A are referenced.

The second dimension of a must be at least max(1,n).

lda

INTEGER. The leading dimension of a; lda≥ max(1,n).

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 poequ interface are as follows:

a	Holds the matrix `A` of size (`n,n`).
s	Holds the vector of length `n`.

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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?poequ

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