# ?pbequ

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

## Syntax

FORTRAN 77:

call spbequ( uplo, n, kd, ab, ldab, s, scond, amax, info )

call dpbequ( uplo, n, kd, ab, ldab, s, scond, amax, info )

call cpbequ( uplo, n, kd, ab, ldab, s, scond, amax, info )

call zpbequ( uplo, n, kd, ab, ldab, s, scond, amax, info )

Fortran 95:

call pbequ( a, s [,scond] [,amax] [,uplo] [,info] )

## Description

This routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive definite matrix A in packed storage and reduce its condition number (with respect to the two-norm). The output array s returns scale factors computed as

These factors are chosen so that the scaled matrix B with elements `bij=s(i)*aij*s(j)` has diagonal elements equal to 1. This choice of s puts the condition number of B within a factor n of the smallest possible condition number over all possible diagonal scalings.

See ?laqsb auxiliary function that uses scaling factors computed by ?pbequ.

## Input Parameters

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  Indicates whether the upper or lower triangular part of A is packed in the array ab:

  If `uplo = 'U'`, the array ab stores the upper triangular part of the matrix A.

  If `uplo = 'L'`, the array ab stores the lower triangular part of the matrix A.

n

  INTEGER. The order of matrix A; `n ≥ 0`.

kd

  INTEGER. The number of superdiagonals or subdiagonals in the matrix A; kd ≥ 0.

ab

  REAL for spbequ

  DOUBLE PRECISION for dpbequ

  COMPLEX for cpbequ

  DOUBLE COMPLEX for zpbequ.

  Array, DIMENSION `(ldab,*)`.

  The array ap contains either the upper or the lower triangular part of the matrix A (as specified by uplo) in band storage (see Matrix Storage Schemes).

  The second dimension of ab must be at least `max(1, n)`.

ldab

  INTEGER. The leading dimension of the array ab; ldab ≥ kd +1.

## Output Parameters

s

  REAL for single precision flavors

  DOUBLE PRECISION for double precision flavors.

  Array, DIMENSION (n).

  If `info = 0`, the array s contains the scale factors for A.

scond

  REAL for single precision flavors

  DOUBLE PRECISION for double precision flavors.

  If `info = 0`, scond contains the ratio of the smallest `s(i)` to the largest `s(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`, the i-th diagonal element of A is nonpositive.

## 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 pbequ interface are as follows:

a

  Stands for argument ab in Fortan 77 interface. Holds the array A of size (`kd+1,n`).

s

  Holds the vector of length (n).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

## Application Notes

If `scond ≥ 0.1` and amax is neither too large nor too small, it is not worth scaling by s.

If amax is very close to overflow or very close to underflow, the matrix A should be scaled.

Parent topic: Routines for Matrix Equilibration