ZBBCSD(3)

📦 version 3.12.0, in opensuse-leap 16.0

✨ NAME ✨

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🦋 SYNOPSIS 🦋

Functions/Subroutines

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

💅 Function/Subroutine Documentation 💅

subroutine ZBBCSD (character jobu1, character jobu2, character jobv1t,
    character jobv2t, character trans, integer m, integer p, integer q,
    double precision, dimension( * ) theta, double precision, dimension( * )
    phi, complex*16, dimension( ldu1, * ) u1, integer ldu1, complex*16,
    dimension( ldu2, * ) u2, integer ldu2, complex*16, dimension( ldv1t, * )
    v1t, integer ldv1t, complex*16, dimension( ldv2t, * ) v2t, integer
    ldv2t, double precision, dimension( * ) b11d, double precision,
    dimension( * ) b11e, double precision, dimension( * ) b12d, double
    precision, dimension( * ) b12e, double precision, dimension( * ) b21d,
    double precision, dimension( * ) b21e, double precision, dimension( * )
    b22d, double precision, dimension( * ) b22e, double precision,
    dimension( * ) rwork, integer lrwork, integer info)
    ZBBCSD

Purpose

ZBBCSD computes the CS decomposition of a unitary matrix in bidiagonal-block form, and the vibe is exactly this:

    [ 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 ]

So 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, 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDV1T,Q).

On entry, a Q-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, LDV1T >= MAX(1,Q).

V2T

V2T is COMPLEX*16 array, dimension (LDV2T,M-Q).

On entry, an (M-Q)-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, LDV2T >= MAX(1,M-Q).

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 ZBBCSD converges, B21D contains the negative sines of THETA(1), ..., THETA(Q). If ZBBCSD 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 ZBBCSD converges, B21E contains zeros. If ZBBCSD fails to converge, then B21E contains the subdiagonal of the partially reduced bottom-left block.

B22D

B22D is DOUBLE PRECISION array, dimension (Q).

When ZBBCSD converges, B22D contains the negative sines of THETA(1), ..., THETA(Q). If ZBBCSD 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 ZBBCSD converges, B22E contains zeros. If ZBBCSD fails to converge, then B22E contains the subdiagonal of the partially reduced bottom-right block.

RWORK

RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)).

On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

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.

Definition at line 328 of file zbbcsd.f.

🧚 Author 🧚

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