ZPBEQU (3lapack)

compute 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.