MB03AD

Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector

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

Purpose

  To compute two Givens rotations (C1,S1) and (C2,S2) such that the
  orthogonal matrix

            [ Q  0 ]        [  C1  S1  0 ]   [ 1  0   0  ]
        Z = [      ],  Q := [ -S1  C1  0 ] * [ 0  C2  S2 ],
            [ 0  I ]        [  0   0   1 ]   [ 0 -S2  C2 ]

  makes the first column of the real Wilkinson double shift
  polynomial of the product of matrices in periodic upper Hessenberg
  form, stored in the array A, parallel to the first unit vector.
  Only the rotation defined by C1 and S1 is used for the real
  Wilkinson single shift polynomial (see SLICOT Library routine
  MB03BE).

Specification
      SUBROUTINE MB03AD( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
     $                   S1, C2, S2 )
C     .. Scalar Arguments ..
      CHARACTER         SHFT
      INTEGER           K, LDA1, LDA2, N, SINV
      DOUBLE PRECISION  C1, C2, S1, S2
C     .. Array Arguments ..
      INTEGER           AMAP(*), S(*)
      DOUBLE PRECISION  A(LDA1,LDA2,*)

Arguments

Mode Parameters

  SHFT    CHARACTER*1
          Specifies the number of shifts employed by the shift
          polynomial, as follows:
          = 'D':  two shifts (assumes N > 2);
          = 'S':  one real shift.

Input/Output Parameters
  K       (input)  INTEGER
          The number of factors.  K >= 1.

  N       (input)  INTEGER
          The order of the factors in the array A.
          N >= 2, for a single shift polynomial;
          N >= 3, for a double shift polynomial.

  AMAP    (input)  INTEGER array, dimension (K)
          The map for accessing the factors, i.e., if AMAP(I) = J,
          then the factor A_I is stored at the J-th position in A.
          AMAP(1) is the pointer to the Hessenberg matrix.

  S       (input)  INTEGER array, dimension (K)
          The signature array. Each entry of S must be 1 or -1.

  SINV    (input)  INTEGER
          Signature multiplier. Entries of S are virtually
          multiplied by SINV.

  A       (input)  DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
          The leading N-by-N-by-K part of this array must contain
          the product (implicitly represented by its K factors)
          in periodic upper Hessenberg form.

  LDA1    INTEGER
          The first leading dimension of the array A.  LDA1 >= N.

  LDA2    INTEGER
          The second leading dimension of the array A.  LDA2 >= N.

  C1      (output)  DOUBLE PRECISION
  S1      (output)  DOUBLE PRECISION
          On exit, C1 and S1 contain the parameters for the first
          Givens rotation.

  C2      (output)  DOUBLE PRECISION
  S2      (output)  DOUBLE PRECISION
          On exit, if SHFT = 'D' and N > 2, C2 and S2 contain the
          parameters for the second Givens rotation. Otherwise,
          C2 = 1, S2 = 0.

Method
  Two Givens rotations are properly computed and applied.

Further Comments
  None
Example

Program Text

  None
Program Data
  None
Program Results
  None

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