?geequb

Computes row and column scaling factors restricted to a power of radix to equilibrate a general matrix and reduce its condition number.

Syntax

FORTRAN 77:

call sgeequb( m, n, a, lda, r, c, rowcnd, colcnd, amax, info )

call dgeequb( m, n, a, lda, r, c, rowcnd, colcnd, amax, info )

call cgeequb( m, n, a, lda, r, c, rowcnd, colcnd, amax, info )

call zgeequb( m, n, a, lda, r, c, rowcnd, colcnd, amax, info )

lapack_int LAPACKE_sgeequb( int matrix_order, lapack_int m, lapack_int n, const float* a, lapack_int lda, float* r, float* c, float* rowcnd, float* colcnd, float* amax );

lapack_int LAPACKE_dgeequb( int matrix_order, lapack_int m, lapack_int n, const double* a, lapack_int lda, double* r, double* c, double* rowcnd, double* colcnd, double* amax );

lapack_int LAPACKE_cgeequb( int matrix_order, lapack_int m, lapack_int n, const lapack_complex_float* a, lapack_int lda, float* r, float* c, float* rowcnd, float* colcnd, float* amax );

lapack_int LAPACKE_zgeequb( int matrix_order, lapack_int m, lapack_int n, const lapack_complex_double* a, lapack_int lda, double* r, double* c, double* rowcnd, double* colcnd, double* amax );

Include Files

The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.

Description

The routine computes row and column scalings intended to equilibrate an m-by-n general matrix A and reduce its condition number. The output array r returns the row scale factors and the array c - the column scale factors. These factors are chosen to try to make the largest element in each row and column of the matrix B with elements b(ij)=r(i)*a(ij)*c(j) have an absolute value of at most the radix.

r(i) and c(j) are restricted to be a power of the radix between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of a but works well in practice.

This routine differs from ?geequ by restricting the scaling factors to a power of the radix. Except for over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries' magnitudes are no longer equal to approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

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.

m	INTEGER. The number of rows of the matrix `A`; `m ≥ 0`.
n	INTEGER. The number of columns of the matrix `A`; `n ≥ 0`.
a	REAL for sgeequb DOUBLE PRECISION for dgeequb COMPLEX for cgeequb DOUBLE COMPLEX for zgeequb. Array: DIMENSION `(lda,*)`. Contains the m-by-n matrix `A` whose equilibration factors are to be computed. 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

r, c	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Arrays: `r(m)`, `c(n)`. If `info = 0`, or info > m, the array r contains the row scale factors for the matrix `A`. If `info = 0`, the array c contains the column scale factors for the matrix `A`.
rowcnd	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. If `info = 0` or `info > m`, rowcnd contains the ratio of the smallest `r(i)` to the largest `r(i)`. If `rowcnd ≥ 0.1`, and amax is neither too large nor too small, it is not worth scaling by r.
colcnd	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. If `info = 0`, colcnd contains the ratio of the smallest `c(i)` to the largest `c(i)`. If `colcnd ≥ 0.1`, it is not worth scaling by c.
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 very close to 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` and `i ≤ m`, the `i`-th row of `A` is exactly zero; `i > m`, the (`i`-m)-th column of `A` is exactly zero.