SORCSD computes the CS decomposition of an M-by-M partitioned


 orthogonal matrix X:


                                 [  I  0  0 |  0  0  0 ]


                                 [  0  C  0 |  0 -S  0 ]


     [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**T


 X = [-----------] = [---------] [---------------------] [---------]   .


     [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]


                                 [  0  S  0 |  0  C  0 ]


                                 [  0  0  I |  0  0  0 ]


 X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,


 (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are


 R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in


 which R = MIN(P,M-P,Q,M-Q).

JOBU1

          JOBU1 is CHARACTER


          = 'Y':      U1 is computed;


          otherwise:  U1 is not computed.

JOBU2

          JOBU2 is CHARACTER


          = 'Y':      U2 is computed;


          otherwise:  U2 is not computed.

JOBV1T

          JOBV1T is CHARACTER


          = 'Y':      V1T is computed;


          otherwise:  V1T is not computed.

JOBV2T

          JOBV2T is CHARACTER


          = 'Y':      V2T is computed;


          otherwise:  V2T is not computed.

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 <= M.

X11

          X11 is REAL array, dimension (LDX11,Q)


          On entry, part of the orthogonal matrix whose CSD is desired.

LDX11

          LDX11 is INTEGER


          The leading dimension of X11. LDX11 >= MAX(1,P).

X12

          X12 is REAL array, dimension (LDX12,M-Q)


          On entry, part of the orthogonal matrix whose CSD is desired.

LDX12

          LDX12 is INTEGER


          The leading dimension of X12. LDX12 >= MAX(1,P).

X21

          X21 is REAL array, dimension (LDX21,Q)


          On entry, part of the orthogonal matrix whose CSD is desired.

LDX21

          LDX21 is INTEGER


          The leading dimension of X11. LDX21 >= MAX(1,M-P).

X22

          X22 is REAL array, dimension (LDX22,M-Q)


          On entry, part of the orthogonal matrix whose CSD is desired.

LDX22

          LDX22 is INTEGER


          The leading dimension of X11. LDX22 >= MAX(1,M-P).

THETA

          THETA is REAL array, dimension (R), in which R =


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


          C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and


          S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

U1

          U1 is REAL array, dimension (LDU1,P)


          If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.

LDU1

          LDU1 is INTEGER


          The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=


          MAX(1,P).

U2

          U2 is REAL array, dimension (LDU2,M-P)


          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal


          matrix U2.

LDU2

          LDU2 is INTEGER


          The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=


          MAX(1,M-P).

V1T

          V1T is REAL array, dimension (LDV1T,Q)


          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal


          matrix V1**T.

LDV1T

          LDV1T is INTEGER


          The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=


          MAX(1,Q).

V2T

          V2T is REAL array, dimension (LDV2T,M-Q)


          If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal


          matrix V2**T.

LDV2T

          LDV2T is INTEGER


          The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=


          MAX(1,M-Q).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.


          If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),


          ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),


          define the matrix in intermediate bidiagonal-block form


          remaining after nonconvergence. INFO specifies the number


          of nonzero PHI's.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          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.

IWORK

          IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))

INFO

          INFO is INTEGER


          = 0:  successful exit.


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


          > 0:  SBBCSD did not converge. See the description of WORK


                above for details.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2017