---
name: "zpbequ"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/zpbequ/3lapack"
---

# zpbequ(3lapack)

computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the two-norm)

## NAME

ZPBEQU - computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the two-norm)

## SYNOPSIS

- SUBROUTINE ZPBEQU(

  UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )

- CHARACTER

  UPLO

- INTEGER

  INFO, KD, LDAB, N

- DOUBLE

  PRECISION AMAX, SCOND

- DOUBLE

  PRECISION S( \* )

- COMPLEX\*16

  AB( LDAB, \* )

## PURPOSE

ZPBEQU computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the two-norm).  S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)\*A(i,j)\*S(j) has ones on the diagonal.  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.\

## ARGUMENTS

- UPLO (input) CHARACTER\*1

  \= 'U':  Upper triangular of A is stored;\
   \= 'L':  Lower triangular of A is stored.

- N (input) INTEGER

  The order of the matrix A.  N \>= 0.

- KD (input) INTEGER

  The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'.  KD \>= 0.

- AB (input) COMPLEX\*16 array, dimension (LDAB,N)

  The upper or lower triangle of the Hermitian 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 (input) INTEGER

  The leading dimension of the array A.  LDAB \>= KD+1.

- S (output) DOUBLE PRECISION array, dimension (N)

  If INFO = 0, S contains the scale factors for A.

- SCOND (output) DOUBLE PRECISION

  If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i).  If SCOND \>= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S.

- AMAX (output) DOUBLE PRECISION

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

- INFO (output) INTEGER

  \= 0:  successful exit\
   \< 0:  if INFO = -i, the i-th argument had an illegal value.\
   \> 0:  if INFO = i, the i-th diagonal element is nonpositive.