MB03AI

Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant with evaluation, Hessenberg factor is the last one)

[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 routines
  MB03BE or MB03BF). All factors whose exponents differ from that of 
  the Hessenberg factor are assumed nonsingular. The matrix product
  is evaluated.

Specification
      SUBROUTINE MB03AI( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
     $                   S1, C2, S2, DWORK )
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,*), DWORK(*)

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.  N >= 2.

  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(K) 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', C2 and S2 contain the parameters
          for the second Givens rotation. Otherwise, C2 = 1, S2 = 0.

Workspace
  DWORK   DOUBLE PRECISION array, dimension (N*(N+2))

Method
  The necessary elements of the real Wilkinson double shift
  polynomial are computed, and suitable Givens rotations are 
  found. For numerical reasons, this routine should be called
  when convergence difficulties are encountered for small order
  matrices and small K, e.g., N, K <= 6. 

Further Comments
  None
Example

Program Text

  None
Program Data
  None
Program Results
  None

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