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## sorbdb()

subroutine sorbdb | ( | character | trans,
|  | character | signs,
|  | integer | m,
|  | integer | p,
|  | integer | q,
|  | real, dimension( ldx11, * ) | x11,
|  | integer | ldx11,
|  | real, dimension( ldx12, * ) | x12,
|  | integer | ldx12,
|  | real, dimension( ldx21, * ) | x21,
|  | integer | ldx21,
|  | real, dimension( ldx22, * ) | x22,
|  | integer | ldx22,
|  | real, dimension( * ) | theta,
|  | real, dimension( * ) | phi,
|  | real, dimension( * ) | taup1,
|  | real, dimension( * ) | taup2,
|  | real, dimension( * ) | tauq1,
|  | real, dimension( * ) | tauq2,
|  | real, dimension( * ) | work,
|  | integer | lwork,
|  | integer | info )

SORBDB

Download SORBDB + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> SORBDB simultaneously bidiagonalizes the blocks of an M-by-M
      !> partitioned orthogonal matrix X:
      !>
      !>                                 [ B11 | B12 0  0 ]
      !>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
      !> X = [-----------] = [---------] [----------------] [---------]   .
      !>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
      !>                                 [  0  |  0  0  I ]
      !>
      !> X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
      !> not the case, then X must be transposed and/or permuted. This can be
      !> done in constant time using the TRANS and SIGNS options. See SORCSD
      !> for details.)
      !>
      !> The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
      !> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
      !> represented implicitly by Householder vectors.
      !>
      !> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
      !> implicitly by angles THETA, PHI.
      !>

Parameters

  [in]

  TRANS

      !>          TRANS is CHARACTER
      !>          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
      !>                      order;
      !>          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
      !>                      major order.
      !>

  [in]

  SIGNS

      !>          SIGNS is CHARACTER
      !>          = 'O':      The lower-left block is made nonpositive (the
      !>                       convention);
      !>          otherwise:  The upper-right block is made nonpositive (the
      !>                       convention).
      !>

  [in]

  M

      !>          M is INTEGER
      !>          The number of rows and columns in X.
      !>

  [in]

  P

      !>          P is INTEGER
      !>          The number of rows in X11 and X12. 0 <= P <= M.
      !>

  [in]

  Q

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

  [in,out]

  X11

      !>          X11 is REAL array, dimension (LDX11,Q)
      !>          On entry, the top-left block of the orthogonal matrix to be
      !>          reduced. On exit, the form depends on TRANS:
      !>          If TRANS = 'N', then
      !>             the columns of tril(X11) specify reflectors for P1,
      !>             the rows of triu(X11,1) specify reflectors for Q1;
      !>          else TRANS = 'T', and
      !>             the rows of triu(X11) specify reflectors for P1,
      !>             the columns of tril(X11,-1) specify reflectors for Q1.
      !>

  [in]

  LDX11

      !>          LDX11 is INTEGER
      !>          The leading dimension of X11. If TRANS = 'N', then LDX11 >=
      !>          P; else LDX11 >= Q.
      !>

  [in,out]

  X12

      !>          X12 is REAL array, dimension (LDX12,M-Q)
      !>          On entry, the top-right block of the orthogonal matrix to
      !>          be reduced. On exit, the form depends on TRANS:
      !>          If TRANS = 'N', then
      !>             the rows of triu(X12) specify the first P reflectors for
      !>             Q2;
      !>          else TRANS = 'T', and
      !>             the columns of tril(X12) specify the first P reflectors
      !>             for Q2.
      !>

  [in]

  LDX12

      !>          LDX12 is INTEGER
      !>          The leading dimension of X12. If TRANS = 'N', then LDX12 >=
      !>          P; else LDX11 >= M-Q.
      !>

  [in,out]

  X21

      !>          X21 is REAL array, dimension (LDX21,Q)
      !>          On entry, the bottom-left block of the orthogonal matrix to
      !>          be reduced. On exit, the form depends on TRANS:
      !>          If TRANS = 'N', then
      !>             the columns of tril(X21) specify reflectors for P2;
      !>          else TRANS = 'T', and
      !>             the rows of triu(X21) specify reflectors for P2.
      !>

  [in]

  LDX21

      !>          LDX21 is INTEGER
      !>          The leading dimension of X21. If TRANS = 'N', then LDX21 >=
      !>          M-P; else LDX21 >= Q.
      !>

  [in,out]

  X22

      !>          X22 is REAL array, dimension (LDX22,M-Q)
      !>          On entry, the bottom-right block of the orthogonal matrix to
      !>          be reduced. On exit, the form depends on TRANS:
      !>          If TRANS = 'N', then
      !>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
      !>             M-P-Q reflectors for Q2,
      !>          else TRANS = 'T', and
      !>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
      !>             M-P-Q reflectors for P2.
      !>

  [in]

  LDX22

      !>          LDX22 is INTEGER
      !>          The leading dimension of X22. If TRANS = 'N', then LDX22 >=
      !>          M-P; else LDX22 >= M-Q.
      !>

  [out]

  THETA

      !>          THETA is REAL array, dimension (Q)
      !>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
      !>          be computed from the angles THETA and PHI. See Further
      !>          Details.
      !>

  [out]

  PHI

      !>          PHI is REAL array, dimension (Q-1)
      !>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
      !>          be computed from the angles THETA and PHI. See Further
      !>          Details.
      !>

  [out]

  TAUP1

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

  [out]

  TAUP2

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

  [out]

  TAUQ1

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

  [out]

  TAUQ2

      !>          TAUQ2 is REAL array, dimension (M-Q)
      !>          The scalar factors of the elementary reflectors that define
      !>          Q2.
      !>

  [out]

  WORK

      !>          WORK is REAL array, dimension (LWORK)
      !>

  [in]

  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.
      !>

  [out]

  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.

Further Details:

      !>
      !>  The bidiagonal blocks B11, B12, B21, and B22 are represented
      !>  implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
      !>  PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
      !>  lower bidiagonal. 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, Q1, and Q2 are represented as products of elementary
      !>  reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2
      !>  using SORGQR and SORGLQ.
      !>

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