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zbbcsd.f(3) LAPACK zbbcsd.f(3)

zbbcsd.f -


subroutine zbbcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E, B22D, B22E, RWORK, LRWORK, INFO)
 
ZBBCSD

ZBBCSD
Purpose:
 ZBBCSD computes the CS decomposition of a unitary 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 | ]**H = [---------] [---------------] [---------] . [ | 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 ZUNCSD for details.)
The bidiagonal matrices B11, B12, B21, and B22 are represented implicitly by angles THETA(1:Q) and PHI(1:Q-1).
The unitary matrices U1, U2, V1T, and V2T are input/output. The input matrices are pre- or post-multiplied by the appropriate singular vector matrices.
Parameters:
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 unitary 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 COMPLEX*16 array, dimension (LDU1,P)
          On entry, an LDU1-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.
U2
          U2 is COMPLEX*16 array, dimension (LDU2,M-P)
          On entry, an LDU2-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.
V1T
          V1T is COMPLEX*16 array, dimension (LDV1T,Q)
          On entry, a LDV1T-by-Q matrix. On exit, V1T is premultiplied
          by the conjugate 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.
V2T
          V2T is COMPLEX*16 array, dimenison (LDV2T,M-Q)
          On entry, a LDV2T-by-(M-Q) matrix. On exit, V2T is
          premultiplied by the conjugate 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.
B11D
          B11D is DOUBLE PRECISION array, dimension (Q)
          When ZBBCSD converges, B11D contains the cosines of THETA(1),
          ..., THETA(Q). If ZBBCSD fails to converge, then B11D
          contains the diagonal of the partially reduced top-left
          block.
B11E
          B11E is DOUBLE PRECISION array, dimension (Q-1)
          When ZBBCSD converges, B11E contains zeros. If ZBBCSD fails
          to converge, then B11E contains the superdiagonal of the
          partially reduced top-left block.
B12D
          B12D is DOUBLE PRECISION array, dimension (Q)
          When ZBBCSD converges, B12D contains the negative sines of
          THETA(1), ..., THETA(Q). If ZBBCSD fails to converge, then
          B12D contains the diagonal of the partially reduced top-right
          block.
B12E
          B12E is DOUBLE PRECISION array, dimension (Q-1)
          When ZBBCSD converges, B12E contains zeros. If ZBBCSD fails
          to converge, then B12E contains the subdiagonal of the
          partially reduced top-right block.
B21D
          B21D is DOUBLE PRECISION array, dimension (Q)
          When CBBCSD converges, B21D contains the negative sines of
          THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
          B21D contains the diagonal of the partially reduced bottom-left
          block.
B21E
          B21E is DOUBLE PRECISION array, dimension (Q-1)
          When CBBCSD converges, B21E contains zeros. If CBBCSD fails
          to converge, then B21E contains the subdiagonal of the
          partially reduced bottom-left block.
B22D
          B22D is DOUBLE PRECISION array, dimension (Q)
          When CBBCSD converges, B22D contains the negative sines of
          THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
          B22D contains the diagonal of the partially reduced bottom-right
          block.
B22E
          B22E is DOUBLE PRECISION array, dimension (Q-1)
          When CBBCSD converges, B22E contains zeros. If CBBCSD fails
          to converge, then B22E contains the subdiagonal of the
          partially reduced bottom-right block.
RWORK
          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LRWORK
          LRWORK is INTEGER
          The dimension of the array RWORK. LRWORK >= MAX(1,8*Q).
If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the work array, and no error message related to LRWORK 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 ZBBCSD did not converge, INFO specifies the number
                of nonzero entries in PHI, and B11D, B11E, etc.,
                contain the partially reduced matrix.
Internal Parameters:
  TOLMUL  DOUBLE PRECISION, 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.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2013
Definition at line 330 of file zbbcsd.f.

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