MB03AD

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

Purpose

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

             [  C1  S1  0 ]   [ 1  0   0  ]
        Q =  [ -S1  C1  0 ] * [ 0  C2  S2 ]
             [  0   0   1 ]   [ 0 -S2  C2 ]

  makes the first column of the real Wilkinson single/double shift
  polynomial of the general product of matrices, stored in the
  array A, parallel to the first unit vector.

      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, S1, C2, S2
C     .. Array Arguments ..
      INTEGER           AMAP(*), S(*)
      DOUBLE PRECISION  A(LDA1,LDA2,*)

Mode Parameters

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

  K       (input)  INTEGER
          The number of factors.  K >= 1.

  N       (input)  INTEGER
          The order of the factors in the array A.  N >= 3.

  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.

  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)
          On entry, the leading N-by-N-by-K part of this array must
          contain a n-by-n product (implicitly represented by its K
          factors) in 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', C2 and S2 contain the parameters
          for the second Givens rotation.

  Two Givens rotations are properly computed and applied.

  None

Program Text

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