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.

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.

SIGNS

          SIGNS is CHARACTER


          = 'O':      The lower-left block is made nonpositive (the


                      "other" convention);


          otherwise:  The upper-right block is made nonpositive (the


                      "default" convention).

M

          M is INTEGER


          The number of rows and columns in X.

P

          P is INTEGER


          The number of rows in X11 and X12. 0 <= P <= M.

Q

          Q is INTEGER


          The number of columns in X11 and X21. 0 <= Q <=


          MIN(P,M-P,M-Q).

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.

LDX11

          LDX11 is INTEGER


          The leading dimension of X11. If TRANS = 'N', then LDX11 >=


          P; else LDX11 >= Q.

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.

LDX12

          LDX12 is INTEGER


          The leading dimension of X12. If TRANS = 'N', then LDX12 >=


          P; else LDX11 >= M-Q.

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.

LDX21

          LDX21 is INTEGER


          The leading dimension of X21. If TRANS = 'N', then LDX21 >=


          M-P; else LDX21 >= Q.

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.

LDX22

          LDX22 is INTEGER


          The leading dimension of X22. If TRANS = 'N', then LDX22 >=


          M-P; else LDX22 >= M-Q.

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.

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.

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.

TAUQ2

          TAUQ2 is REAL array, dimension (M-Q)


          The scalar factors of the elementary reflectors that define


          Q2.

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.

November 2011

  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.

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