# p?geequ

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

## Syntax

call psgeequ(m, n, a, ia, ja, desca, r, c, rowcnd, colcnd, amax, info)

call pdgeequ(m, n, a, ia, ja, desca, r, c, rowcnd, colcnd, amax, info)

call pcgeequ(m, n, a, ia, ja, desca, r, c, rowcnd, colcnd, amax, info)

call pzgeequ(m, n, a, ia, ja, desca, r, c, rowcnd, colcnd, amax, info)

## Description

This routine computes row and column scalings intended to equilibrate an m-by-n distributed matrix `sub(A) = A(ia:ia+m-1, ja:ja+n-1)` 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.

r(i) and c(j) 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 sub(A) but works well in practice.

The auxiliary function p?laqge uses scaling factors computed by p?geequ to scale a general rectangular matrix.

## Input Parameters

m

  (global) INTEGER. The number of rows to be operated on, that is, the number of rows of the distributed submatrix sub(A) `(m ≥ 0)`.

n

  (global) INTEGER. The number of columns to be operated on, that is, the number of columns of the distributed submatrix sub(A) `(n ≥ 0)`.

a

  (local)

  REAL for psgeequ

  DOUBLE PRECISION for pdgeequ

  COMPLEX for pcgeequ

  DOUBLE COMPLEX for pzgeequ .

  Pointer into the local memory to an array of local dimension `a(lld_a,LOCc(ja+n-1))`.

  The array a contains the local pieces of the m-by-n distributed matrix whose equilibration factors are to be computed.

ia, ja

  (global) INTEGER. The row and column indices in the global array A indicating the first row and the first column of the submatrix sub(A), respectively.

desca

  (global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix A.

## Output Parameters

r, c

  (local) REAL for single precision flavors;

  DOUBLE PRECISION for double precision flavors.

  Arrays, dimension `LOCr(m_a)` and `LOCc(n_a)`, respectively.

  If `info = 0`, or `info > ia+m-1`, the array `r (ia:ia+m-1)` contains the row scale factors for sub(A). r is aligned with the distributed matrix A, and replicated across every process column. r is tied to the distributed matrix A.

  If `info = 0`, the array c (ja:ja+n-1) contains the column scale factors for sub(A). c is aligned with the distributed matrix A, and replicated down every process row. c is tied to the distributed matrix A.

rowcnd, colcnd

  (global) REAL for single precision flavors;

  DOUBLE PRECISION for double precision flavors.

  If `info = 0` or `info > ia+m-1`, rowcnd contains the ratio of the smallest r(i) to the largest `r(i) (ia ≤ i ≤ ia+m-1)`. If `rowcnd ≥ 0.1` and amax is neither too large nor too small, it is not worth scaling by `r (ia:ia+m-1)`.

  If `info = 0`, colcnd contains the ratio of the smallest c(j) to the largest `c(j) (ja ≤ j ≤ ja+n-1)`.

  If `colcnd ≥ 0.1`, it is not worth scaling by `c(ja:ja+n-1)`.

amax

  (global) REAL for single precision flavors;

  DOUBLE PRECISION for double precision flavors.

  Absolute value of the largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

info

  (global) INTEGER. If `info=0`, the execution is successful.

  `info < 0`:

  If the ith argument is an array and the jth entry had an illegal value, then `info = -(i*100+j)`; if the i-th argument is a scalar and had an illegal value, then `info = -i`.

  `info > 0`:

  If `info = i` and

  `i≤ m`, the ith row of the distributed matrix

  `sub(A)` is exactly zero;

  `i > m`, the (i-m)th column of the distributed

  matrix sub(A) is exactly zero.

Parent topic: Routines for Matrix Equilibration