# p?poequ

Computes row and column scaling factors intended to equilibrate a symmetric (Hermitian) positive definite distributed matrix and reduce its condition number.

## Syntax

call pspoequ(n, a, ia, ja, desca, sr, sc, scond, amax, info)

call pdpoequ(n, a, ia, ja, desca, sr, sc, scond, amax, info)

call pcpoequ(n, a, ia, ja, desca, sr, sc, scond, amax, info)

call pzpoequ(n, a, ia, ja, desca, sr, sc, scond, amax, info)

## Description

This routine computes row and column scalings intended to equilibrate a real symmetric or complex Hermitian positive definite distributed matrix `sub(A) = A(ia:ia+n-1, ja:ja+n-1)` and reduce its condition number (with respect to the two-norm). The output arrays sr and sc return the row and column scale factors

These factors are chosen so that the scaled distributed matrix B with elements `bij=s(i)*aij*s(j)` has ones on the diagonal.

This choice of sr and sc puts the condition number of B within a factor n of the smallest possible condition number over all possible diagonal scalings.

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

## Input Parameters

n

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

a

  (local)

  REAL for pspoequ

  DOUBLE PRECISION for pdpoequ

  COMPLEX for pcpoequ

  DOUBLE COMPLEX for pzpoequ.

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

  The array a contains the n-by-n symmetric/Hermitian positive definite distributed matrix sub(A) whose scaling factors are to be computed. Only the diagonal elements of sub(A) are referenced.

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

sr, sc

  (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`, the array `sr(ia:ia+n-1)` contains the row scale factors for sub(A). sr is aligned with the distributed matrix A, and replicated across every process column. sr is tied to the distributed matrix A.

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

scond

  (global)

  REAL for single precision flavors;

  DOUBLE PRECISION for double precision flavors.

  If `info = 0`, scond contains the ratio of the smallest sr(i) ( or sc(j)) to the largest sr(i) ( or sc(j)), with

  `ia≤i≤ia+n-1` and `ja≤j≤ja+n-1`.

  If `scond ≥ 0.1` and amax is neither too large nor too small, it is not worth scaling by sr ( or sc ).

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 i-th argument is an array and the j-th 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 = k, the k-th diagonal entry of sub(A) is nonpositive.

Parent topic: Routines for Matrix Equilibration