## MB03AB

### Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant with explicit shifts)

[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 needed for the real
      Wilkinson single shift polynomial (see the SLICOT Library routines
      MB03BE or MB03BF). The shifts are defined based on the eigenvalues
      (computed externally by the SLICOT Library routine MB03BB) of the
      trailing 2-by-2 submatrix of the matrix product. See the
      definitions of the arguments W1 and W2.

Specification

          SUBROUTINE MB03AB( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, W1,
         $                   W2, C1, S1, C2, S2 )
    C     .. Scalar Arguments ..
          CHARACTER         SHFT
          INTEGER           K, LDA1, LDA2, N, SINV
          DOUBLE PRECISION  C1, C2, S1, S2, W1, W2
    C     .. Array Arguments ..
          INTEGER           AMAP(*), S(*)
          DOUBLE PRECISION  A(LDA1,LDA2,*)

Arguments

Mode Parameters

      SHFT    CHARACTER*1
              Specifies the number and type of shifts employed by the
              shift polynomial, as follows:
              = 'C':  two complex conjugate shifts;
              = 'D':  two real identical shifts;
              = 'R':  two real shifts;
              = 'S':  one real shift.
              When the eigenvalues are complex conjugate, this argument
              must be set to 'C'.

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, defined
              by A(:,:,AMAP(1)).

      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 an
              n-by-n 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.

      W1      (input)  DOUBLE PRECISION
              The real part of the first eigenvalue.
              If SHFT = 'S', this argument is not used.

      W2      (input)  DOUBLE PRECISION
              The second eigenvalue, if both eigenvalues are real, else
              the imaginary part of the complex conjugate pair.

      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, C2 and S2 contain the parameters for the second
              Givens rotation. If SHFT = 'S', C2 = 1, S2 = 0.

Method

      Givens rotations are properly computed and applied.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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