?heequb

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?heequb

Computes row and column scaling factors intended to equilibrate a Hermitian indefinite matrix and reduce its condition number.

Syntax

call cheequb( uplo, n, a, lda, s, scond, amax, work, info )

call zheequb( uplo, n, a, lda, s, scond, amax, work, info )

Include Files

mkl.fi, lapack.f90

Description

The routine computes row and column scalings intended to equilibrate a Hermitian indefinite matrix A and reduce its condition number (with respect to the two-norm).

The array s contains the scale factors, s(i) = 1/sqrt(A(i,i)). These factors are chosen so that the scaled matrix B with elements b(i,j)=s(i)*a(i,j)*s(j) has ones on the diagonal.

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

uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of `A` is stored: If `uplo = 'U'`, the array `a` stores the upper triangular part of the matrix `A`. If `uplo = 'L'`, the array `a` stores the lower triangular part of the matrix `A`.
n	INTEGER. The order of the matrix `A`; `n≥ 0`.
a, work	COMPLEX for cheequb DOUBLE COMPLEX for zheequb. Array a: size `lda` by . Contains the n-by-n symmetric indefinite 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)`. work() is a workspace array. The dimension of work is at least `max(1,3*n)`.
lda	INTEGER. The leading dimension of a; `lda≥ max(1, m)`.

Output Parameters

s	REAL for cheequb DOUBLE PRECISION for zheequb. Array, size (n). If `info = 0`, the array s contains the scale factors for `A`.
scond	REAL for cheequb DOUBLE PRECISION for zheequb. 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 cheequb DOUBLE PRECISION for zheequb. 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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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?heequb

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