MB03BC

Product singular value decomposition of K-1 triangular factors of order 2

Purpose

  To compute the product singular value decomposition of the K-1
  triangular factors corresponding to a 2-by-2 product of K
  factors in upper Hessenberg-triangular form.
  For a general product of 2-by-2 triangular matrices

                     S(2)        S(3)            S(K)
         A = A(:,:,2)    A(:,:,3)    ... A(:,:,K),

  Givens rotations are computed so that
                                                       S(i)
    [  CV(i-1) SV(i-1) ] [ A(1,1,i)(in)  A(1,2,i)(in) ]
    [ -SV(i-1) CV(i-1) ] [     0         A(2,2,i)(in) ]
                                   S(i)
    [ A(1,1,i)(out) A(1,2,i)(out) ]    [  CV(i) SV(i) ]
  = [     0         A(2,2,i)(out) ]    [ -SV(i) CV(i) ]

  stays upper triangular and

    [  CV(1) SV(1) ]       [ CV(K) -SV(K) ]
    [ -SV(1) CV(1) ] * A * [ SV(K)  CV(K) ]

  is diagonal.

      SUBROUTINE MB03BC( K, AMAP, S, SINV, A, LDA1, LDA2, MACPAR, CV,
     $                   SV, DWORK )
C     .. Scalar Arguments ..
      INTEGER           K, LDA1, LDA2, SINV
C     .. Array Arguments ..
      INTEGER           AMAP(*), S(*)
      DOUBLE PRECISION  A(LDA1,LDA2,*), CV(*), DWORK(*), MACPAR(*),
     $                  SV(*)

Input/Output Parameters

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

  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/output)  DOUBLE PRECISION array, dimension
                          (LDA1,LDA2,K)
          On entry, the leading 2-by-2-by-K part of this array must
          contain a 2-by-2 product (implicitly represented by its K
          factors) in upper Hessenberg-triangular form.
          On exit, the leading 2-by-2-by-K part of this array
          contains modified triangular factors such that their
          product is diagonal.

  LDA1    INTEGER
          The first leading dimension of the array A.  LDA1 >= 2.

  LDA2    INTEGER
          The second leading dimension of the array A.  LDA2 >= 2.

  MACPAR  (input)  DOUBLE PRECISION array, dimension (5)
          Machine parameters:
          MACPAR(1)  overflow threshold,         DLAMCH( 'O' );
          MACPAR(2)  underflow threshold,        DLAMCH( 'U' );
          MACPAR(3)  safe minimum,               DLAMCH( 'S' );
          MACPAR(4)  relative machine precision, DLAMCH( 'E' );
          MACPAR(5)  base of the machine,        DLAMCH( 'B' ).

  CV      (output)  DOUBLE PRECISION array, dimension (K)
          On exit, the first K elements of this array contain the
          cosines of the Givens rotations.

  SV      (output)  DOUBLE PRECISION array, dimension (K)
          On exit, the first K elements of this array contain the
          sines of the Givens rotations.

  DWORK   DOUBLE PRECISION array, dimension (3*(K-1))

  The product singular value decomposition of the K-1
  triangular factors are computed as described in [1].

  [1] Bojanczyk, A. and Van Dooren, P.
      On propagating orthogonal transformations in a product of 2x2
      triangular matrices.
      In Reichel, Ruttan and Varga: 'Numerical Linear Algebra',
      pp. 1-9, 1993.

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Program Text

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