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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

July 2012

  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.

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