# ?geequ

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

## Syntax

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] )

## Description

This 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.

## 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: 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 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.

## Fortran 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 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).

## 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.

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.

Parent topic: Routines for Matrix Equilibration