SBBCSD computes the CS decomposition of an orthogonal matrix in


 bidiagonal-block form,


     [ B11 | B12 0  0 ]


     [  0  |  0 -I  0 ]


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


     [ B21 | B22 0  0 ]


     [  0  |  0  0  I ]


                               [  C | -S  0  0 ]


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


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


                   [    | U2 ] [  S |  C  0  0 ] [    | V2 ]


                               [  0 |  0  0  I ]


 X is M-by-M, its top-left block is P-by-Q, and Q must be no larger


 than P, M-P, or M-Q. (If Q is not the smallest index, 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 bidiagonal matrices B11, B12, B21, and B22 are represented


 implicitly by angles THETA(1:Q) and PHI(1:Q-1).


 The orthogonal matrices U1, U2, V1T, and V2T are input/output.


 The input matrices are pre- or post-multiplied by the appropriate


 singular vector matrices.

JOBU1

          JOBU1 is CHARACTER


          = 'Y':      U1 is updated;


          otherwise:  U1 is not updated.

JOBU2

          JOBU2 is CHARACTER


          = 'Y':      U2 is updated;


          otherwise:  U2 is not updated.

JOBV1T

          JOBV1T is CHARACTER


          = 'Y':      V1T is updated;


          otherwise:  V1T is not updated.

JOBV2T

          JOBV2T is CHARACTER


          = 'Y':      V2T is updated;


          otherwise:  V2T is not updated.

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.

M

          M is INTEGER


          The number of rows and columns in X, the orthogonal matrix in


          bidiagonal-block form.

P

          P is INTEGER


          The number of rows in the top-left block of X. 0 <= P <= M.

Q

          Q is INTEGER


          The number of columns in the top-left block of X.


          0 <= Q <= MIN(P,M-P,M-Q).

THETA

          THETA is REAL array, dimension (Q)


          On entry, the angles THETA(1),...,THETA(Q) that, along with


          PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block


          form. On exit, the angles whose cosines and sines define the


          diagonal blocks in the CS decomposition.

PHI

          PHI is REAL array, dimension (Q-1)


          The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,


          THETA(Q), define the matrix in bidiagonal-block form.

U1

          U1 is REAL array, dimension (LDU1,P)


          On entry, a P-by-P matrix. On exit, U1 is postmultiplied


          by the left singular vector matrix common to [ B11 ; 0 ] and


          [ B12 0 0 ; 0 -I 0 0 ].

LDU1

          LDU1 is INTEGER


          The leading dimension of the array U1, LDU1 >= MAX(1,P).

U2

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


          On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is


          postmultiplied by the left singular vector matrix common to


          [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].

LDU2

          LDU2 is INTEGER


          The leading dimension of the array U2, LDU2 >= MAX(1,M-P).

V1T

          V1T is REAL array, dimension (LDV1T,Q)


          On entry, a Q-by-Q matrix. On exit, V1T is premultiplied


          by the transpose of the right singular vector


          matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].

LDV1T

          LDV1T is INTEGER


          The leading dimension of the array V1T, LDV1T >= MAX(1,Q).

V2T

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


          On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is


          premultiplied by the transpose of the right


          singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and


          [ B22 0 0 ; 0 0 I ].

LDV2T

          LDV2T is INTEGER


          The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).

B11D

          B11D is REAL array, dimension (Q)


          When SBBCSD converges, B11D contains the cosines of THETA(1),


          ..., THETA(Q). If SBBCSD fails to converge, then B11D


          contains the diagonal of the partially reduced top-left


          block.

B11E

          B11E is REAL array, dimension (Q-1)


          When SBBCSD converges, B11E contains zeros. If SBBCSD fails


          to converge, then B11E contains the superdiagonal of the


          partially reduced top-left block.

B12D

          B12D is REAL array, dimension (Q)


          When SBBCSD converges, B12D contains the negative sines of


          THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then


          B12D contains the diagonal of the partially reduced top-right


          block.

B12E

          B12E is REAL array, dimension (Q-1)


          When SBBCSD converges, B12E contains zeros. If SBBCSD fails


          to converge, then B12E contains the subdiagonal of the


          partially reduced top-right block.

B21D

          B21D is REAL array, dimension (Q)


          When SBBCSD converges, B21D contains the negative sines of


          THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then


          B21D contains the diagonal of the partially reduced bottom-left


          block.

B21E

          B21E is REAL array, dimension (Q-1)


          When SBBCSD converges, B21E contains zeros. If SBBCSD fails


          to converge, then B21E contains the subdiagonal of the


          partially reduced bottom-left block.

B22D

          B22D is REAL array, dimension (Q)


          When SBBCSD converges, B22D contains the negative sines of


          THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then


          B22D contains the diagonal of the partially reduced bottom-right


          block.

B22E

          B22E is REAL array, dimension (Q-1)


          When SBBCSD converges, B22E contains zeros. If SBBCSD fails


          to converge, then B22E contains the subdiagonal of the


          partially reduced bottom-right block.

WORK

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


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

LWORK

          LWORK is INTEGER


          The dimension of the array WORK. LWORK >= MAX(1,8*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.


          > 0:  if SBBCSD did not converge, INFO specifies the number


                of nonzero entries in PHI, and B11D, B11E, etc.,


                contain the partially reduced matrix.

  TOLMUL  REAL, default = MAX(10,MIN(100,EPS**(-1/8)))


          TOLMUL controls the convergence criterion of the QR loop.


          Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they


          are within TOLMUL*EPS of either bound.

[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 2016