sorbdb4.f man page

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

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.

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:

Definition at line 216 of file sorbdb4.f.