# ?laqhb

Scales a Hermetian band matrix, using scaling factors computed by ?pbequ.

## Syntax

call claqhb( uplo, n, kd, ab, ldab, s, scond, amax, equed )

call zlaqhb( uplo, n, kd, ab, ldab, s, scond, amax, equed )

## Description

The routine equilibrates a Hermetian band matrix A using the scaling factors in the vector s.

## Input Parameters

uplo

  CHARACTER*1.

  Specifies whether the upper or lower triangular part of the band matrix A is stored.

  If `uplo = 'U'`: upper triangular.

  If `uplo = 'L'`: lower triangular.

n

  INTEGER. The order of the matrix A.

  `n ≥ 0`.

kd

  INTEGER. The number of super-diagonals of the matrix A if `uplo = 'U'`, or the number of sub-diagonals if `uplo = 'L'`.

  `kd ≥ 0`.

ab

  COMPLEX for claqhb

  COMPLEX*16 for zlaqhb

  Array, DIMENSION (ldab,n). On entry, the upper or lower triangle of the band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array ab as follows:

  if `uplo = 'U'`, `ab(kd+1+i-j,j) = A(i,j)` for `max(1,j-kd) ≤ i ≤ j`;

  if `uplo = 'L'`, `ab(1+i-j,j) = A(i,j)` for `j ≤ i ≤ min(n,j+kd)`.

ldab

  INTEGER. The leading dimension of the array ab.

  `ldab ≥ kd+1`.

scond

  REAL for claqsb

  DOUBLE PRECISION for zlaqsb

  Ratio of the smallest s(i) to the largest s(i).

amax

  REAL for claqsb

  DOUBLE PRECISION for zlaqsb

  Absolute value of largest matrix entry.

## Output Parameters

ab

  On exit, if `info = 0`, the triangular factor U or L from the Cholesky factorization `A = U'*U` or `A = L*L'` of the band matrix A, in the same storage format as A.

s

  REAL for claqsb

  DOUBLE PRECISION for zlaqsb

  Array, DIMENSION (n). The scale factors for A.

equed

  CHARACTER*1.

  Specifies whether or not equilibration was done.

  If `equed = 'N'`: No equilibration.

  If equed = 'Y': Equilibration was done, that is, A has been replaced by `diag(s)*A*diag(s)`.

## Application Notes

The routine uses internal parameters thresh, large, and small, which have the following meaning. thresh is a threshold value used to decide if scaling should be based on the ratio of the scaling factors. If `scond < thresh`, scaling is done.

The values large and small are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If `amax > large` or `amax < small`, scaling is done.

Parent topic: Auxiliary Routines