NAME

       sorbdb4.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sorbdb4 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1,
           PHANTOM, WORK, LWORK, INFO)
           SORBDB4

Function/Subroutine Documentation

   subroutine sorbdb4 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer
       LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real,
       dimension(*) PHI, real, dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*)
       TAUQ1, real, dimension(*) PHANTOM, real, dimension(*) WORK, integer LWORK, integer INFO)
       SORBDB4

Purpose:

        SORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
        matrix X with orthonomal columns:

                                   [ B11 ]
             [ X11 ]   [ P1 |    ] [  0  ]
             [-----] = [---------] [-----] Q1**T .
             [ X21 ]   [    | P2 ] [ B21 ]
                                   [  0  ]

        X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
        M-P, or Q. Routines SORBDB1, SORBDB2, and SORBDB3 handle cases in
        which M-Q is not the minimum dimension.

        The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
        and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
        Householder vectors.

        B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
        implicitly by angles THETA, PHI..fi

       Parameters:
           M

                     M is INTEGER
                      The number of rows X11 plus the number of rows in X21.

           P

                     P is INTEGER
                      The number of rows in X11. 0 <= P <= M.

           Q

                     Q is INTEGER
                      The number of columns in X11 and X21. 0 <= Q <= M and
                      M-Q <= min(P,M-P,Q).

           X11

                     X11 is REAL array, dimension (LDX11,Q)
                      On entry, the top block of the matrix X to be reduced. On
                      exit, the columns of tril(X11) specify reflectors for P1 and
                      the rows of triu(X11,1) specify reflectors for Q1.

           LDX11

                     LDX11 is INTEGER
                      The leading dimension of X11. LDX11 >= P.

           X21

                     X21 is REAL array, dimension (LDX21,Q)
                      On entry, the bottom block of the matrix X to be reduced. On
                      exit, the columns of tril(X21) specify reflectors for P2.

           LDX21

                     LDX21 is INTEGER
                      The leading dimension of X21. LDX21 >= M-P.

           THETA

                     THETA is REAL array, dimension (Q)
                      The entries of the bidiagonal blocks B11, B21 are defined by
                      THETA and PHI. See Further Details.

           PHI

                     PHI is REAL array, dimension (Q-1)
                      The entries of the bidiagonal blocks B11, B21 are defined by
                      THETA and PHI. See Further Details.

           TAUP1

                     TAUP1 is REAL array, dimension (P)
                      The scalar factors of the elementary reflectors that define
                      P1.

           TAUP2

                     TAUP2 is REAL array, dimension (M-P)
                      The scalar factors of the elementary reflectors that define
                      P2.

           TAUQ1

                     TAUQ1 is REAL array, dimension (Q)
                      The scalar factors of the elementary reflectors that define
                      Q1.

           PHANTOM

                     PHANTOM is REAL array, dimension (M)
                      The routine computes an M-by-1 column vector Y that is
                      orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
                      PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
                      Y(P+1:M), respectively.

           WORK

                     WORK is REAL array, dimension (LWORK)

           LWORK

                     LWORK is INTEGER
                      The dimension of the array WORK. LWORK >= M-Q.

                      If LWORK = -1, then a workspace query is assumed; the routine
                      only calculates the optimal size of the WORK array, returns
                      this value as the first entry of the WORK array, and no error
                      message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                      = 0:  successful exit.
                      < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           July 2012

       Further Details:

             The upper-bidiagonal blocks B11, B21 are represented implicitly by
             angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
             in each bidiagonal band is a product of a sine or cosine of a THETA
             with a sine or cosine of a PHI. See [1] or SORCSD for details.

             P1, P2, and Q1 are represented as products of elementary reflectors.
             See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
             and SORGLQ.

       References:
           [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
           50(1):33-65, 2009.

Author

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