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zpbequ

 NAME
      ZPBEQU - compute row and column scalings intended to equili-
      brate 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 equili-
      brate a Hermitian positive definite band matrix A and reduce
      its condition number (with respect to the two-norm).  S con-
      tains 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 diago-
      nal 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 under-
              flow, 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 entry is nonpo-
              sitive.