Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?geequ
Computes row and column scaling factors intended to equilibrate a general matrix and reduce its condition number.
Syntax
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 )
call geequ ( a , r , c [ , rowcnd ] [ , colcnd ] [ , amax ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
sgeequ dgeequ cgeequ zgeequ geequ
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 b_{ij}=r(i)*a_{ij}*c(j)b_{ij}=r[i-1]*a_{ij}*c[j-1] have absolute value 1.
See ?laqge (Scales a general rectangular matrix, using row and column scaling factors computed by ?geequ.) auxiliary function that uses scaling factors computed by ?geequ .
Input Parameters
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: size lda by * .
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 (size m ), c (size 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)r[i] to the largest r(i)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)c[i] to the largest c(i)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 , i > 0 , and
i≤m , the i -th row of A is exactly zero;
i>m , the ( i - m )th column of A is exactly zero.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
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 LAPACK 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 .
Application Notes
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.
?lamch (Determines machine parameters for floating-point arithmetic.)
SMLNUM = slamch ('s')
BIGNUM = 1 / SMLNUMIf 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 SMLNUM or very close to BIGNUM , the matrix \(A\) should be scaled.