Computes row and column scaling factors intended to equilibrate a general matrix and reduce its condition number.
FORTRAN 77:
call sgeequ( m, n, a, lda, r, c, rowcnd, colcnd, amax, info )
call dgeequ( m, n, a, lda, r, c, rowcnd, colcnd, amax, info )
call cgeequ( m, n, a, lda, r, c, rowcnd, colcnd, amax, info )
call zgeequ( m, n, a, lda, r, c, rowcnd, colcnd, amax, info )
FORTRAN 95:
call geequ( a, r, c [,rowcnd] [,colcnd] [,amax] [,info] )
C:
lapack_int LAPACKE_sgeequ( 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_dgeequ( 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_cgeequ( 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_zgeequ( 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 );
The routine computes row and column scalings intended to equilibrate an m-by-n 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 bij=r(i)*aij*c(j) have absolute value 1.
See ?laqge auxiliary function that uses scaling factors computed by ?geequ.
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 sgeequ DOUBLE PRECISION for dgeequ COMPLEX for cgeequ DOUBLE COMPLEX for zgeequ. 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). |
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 of the matrix A. If info = 0, the array c contains the column scale factors of 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). |
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). |
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 and i ≤ m, the i-th row of A is exactly zero; i > m, the (i-m)th column of A is exactly zero. |
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 Fortran 95 Interface Conventions.
Specific details for the routine geequ interface are as follows:
a |
Holds the matrix A of size (m, n). |
r |
Holds the vector of length (m). |
c |
Holds the vector of length n. |
All the components of r and c are restricted to be 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.
SMLNUM and
BIGNUM are parameters representing machine precision. You can use the
?lamch routines to compute them. For example, compute single precision (real and complex) values of
SMLNUM and
BIGNUM as follows:
SMLNUM = slamch ('s')
BIGNUM = 1 / SMLNUM
If rowcnd ≥ 0.1 and amax is neither too large nor too small, it is not worth scaling by r.
If colcnd ≥ 0.1, it is not worth scaling by c.
If amax is very close to overflow or very close to underflow, the matrix A should be scaled.