## MB02TZ

### Estimation of the reciprocal condition number of a complex upper Hessenberg matrix

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To estimate the reciprocal of the condition number of a complex
      upper Hessenberg matrix H, in either the 1-norm or the
      infinity-norm, using the LU factorization computed by MB02SZ.

Specification

          SUBROUTINE MB02TZ( NORM, N, HNORM, H, LDH, IPIV, RCOND, DWORK,
         $                   ZWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          NORM
          INTEGER            INFO, LDH, N
          DOUBLE PRECISION   HNORM, RCOND
    C     .. Array Arguments ..
          INTEGER            IPIV(*)
          DOUBLE PRECISION   DWORK( * )
          COMPLEX*16         H( LDH, * ), ZWORK( * )

Arguments

Mode Parameters

      NORM    CHARACTER*1
              Specifies whether the 1-norm condition number or the
              infinity-norm condition number is required:
              = '1' or 'O':  1-norm;
              = 'I':         Infinity-norm.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrix H.  N >= 0.

      HNORM   (input) DOUBLE PRECISION
              If NORM = '1' or 'O', the 1-norm of the original matrix H.
              If NORM = 'I', the infinity-norm of the original matrix H.

      H       (input) COMPLEX*16 array, dimension (LDH,N)
              The factors L and U from the factorization H = P*L*U
              as computed by MB02SZ.

      LDH     INTEGER
              The leading dimension of the array H.  LDH >= max(1,N).

      IPIV    (input) INTEGER array, dimension (N)
              The pivot indices; for 1 <= i <= N, row i of the matrix
              was interchanged with row IPIV(i).

      RCOND   (output) DOUBLE PRECISION
              The reciprocal of the condition number of the matrix H,
              computed as RCOND = 1/(norm(H) * norm(inv(H))).

Workspace

      DWORK   DOUBLE PRECISION array, dimension (N)

      ZWORK   COMPLEX*16 array, dimension (2*N)

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      An estimate is obtained for norm(inv(H)), and the reciprocal of
      the condition number is computed as
         RCOND = 1 / ( norm(H) * norm(inv(H)) ).

References

      -

Numerical Aspects

                                 2
      The algorithm requires 0( N ) complex operations.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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