?syequb

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

Syntax

FORTRAN 77:

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

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

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

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

lapack_int LAPACKE_ssyequb( int matrix_order, char uplo, lapack_int n, const float* a, lapack_int lda, float* s, float* scond, float* amax );

lapack_int LAPACKE_dsyequb( int matrix_order, char uplo, lapack_int n, const double* a, lapack_int lda, double* s, double* scond, double* amax );

lapack_int LAPACKE_csyequb( int matrix_order, char uplo, lapack_int n, const lapack_complex_float* a, lapack_int lda, float* s, float* scond, float* amax );

lapack_int LAPACKE_zsyequb( int matrix_order, char uplo, lapack_int n, const lapack_complex_double* a, lapack_int lda, double* s, double* scond, double* amax );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.h

Description

The routine computes row and column scalings intended to equilibrate a symmetric 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

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

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	REAL for ssyequb DOUBLE PRECISION for dsyequb COMPLEX for csyequb DOUBLE COMPLEX for zsyequb. Array a: DIMENSION `(lda,)`. 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 single precision flavors DOUBLE PRECISION for double precision flavors. Array, DIMENSION (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 overflow or underflow, 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.