?poequb

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?poequb

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

Syntax

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

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

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

call zpoequb( n, a, lda, 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).

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. s(i) is a power of two nearest to, but not exceeding 1/sqrt(A_i,i).

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.

Input Parameters

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

REAL for spoequb

DOUBLE PRECISION for dpoequb

COMPLEX for cpoequb

DOUBLE COMPLEX for zpoequb.

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, m).

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)`. If `scond≥ 0.1`, and amax is neither too large nor too small, it is not worth scaling by s.
amax	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Absolute value of the largest element of the matrix `A`. If amax is very close to SMLNUM or BIGNUM, the matrix should be scaled.
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.

Parent topic: Matrix Equilibration: LAPACK Computational Routines

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?poequb

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