SRC/dbbcsd.f(3)

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💅 `SRC/dbbcsd.f(3)` 💅

Manual name: Library Functions Manual — folded in right here, bestie, because we’re not copying the header line like a boring NPC. ✨

✨ NAME ✨

SRC/dbbcsd.f — the DBBCSD routine’s page. 💖

✨ SYNOPSIS ✨

🦋 Functions/Subroutines

subroutine dbbcsd (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, work, lwork, info)
    DBBCSD

🦋 Function/Subroutine Documentation 🦋

subroutine dbbcsd (character jobu1, character jobu2, character jobv1t,
    character jobv2t, character trans, integer m, integer p, integer q,
    double precision, dimension( * ) theta, double precision, dimension( * )
    phi, double precision, dimension( ldu1, * ) u1, integer ldu1, double
    precision, dimension( ldu2, * ) u2, integer ldu2, double precision,
    dimension( ldv1t, * ) v1t, integer ldv1t, double precision, 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( * ) work, integer lwork, integer info)
    DBBCSD

🌸 Purpose 🌸

okay so DBBCSD computes the CS decomposition of an orthogonal matrix in bidiagonal-block form. It’s giving very structured, very main-character linear algebra. 💅

Here’s the exact bidiagonal-block glow-up:

    [ 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. Periodt. 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 DORCSD 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. ✨

💖 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 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 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (Q). When DBBCSD converges, B11D contains the cosines of THETA(1), ..., THETA(Q). If DBBCSD 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 DBBCSD converges, B11E contains zeros. If DBBCSD fails to converge, then B11E contains the superdiagonal of the partially reduced top-left block.
B12D — B12D is DOUBLE PRECISION array, dimension (Q). When DBBCSD converges, B12D contains the negative sines of THETA(1), ..., THETA(Q). If DBBCSD 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 DBBCSD converges, B12E contains zeros. If DBBCSD fails to converge, then B12E contains the subdiagonal of the partially reduced top-right block.
B21D — B21D is DOUBLE PRECISION array, dimension (Q). When DBBCSD converges, B21D contains the negative sines of THETA(1), ..., THETA(Q). If DBBCSD 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 DBBCSD converges, B21E contains zeros. If DBBCSD fails to converge, then B21E contains the subdiagonal of the partially reduced bottom-left block.
B22D — B22D is DOUBLE PRECISION array, dimension (Q). When DBBCSD converges, B22D contains the negative sines of THETA(1), ..., THETA(Q). If DBBCSD 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 DBBCSD converges, B22E contains zeros. If DBBCSD fails to converge, then B22E contains the subdiagonal of the partially reduced bottom-right block.
WORK — WORK is DOUBLE PRECISION 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 DBBCSD 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 ⭐

Definition at line 328 of file dbbcsd.f.

🎀 Author 🎀

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