SRC/dorbdb4.f(3)

🦋 SRC/dorbdb4.f — Library Functions Manual 🦋

okay bestie, this is the manual page for SRC/dorbdb4.f. It’s giving LAPACK Fortran routine, and we’re keeping every single parameter, condition, and footnote. no cap. 💅

✨ NAME ✨

SRC/dorbdb4.f — the source file for the dorbdb4 routine. The original NAME section is literally just SRC/dorbdb4.f; we’re not gonna pretend otherwise. 🫶

🌈 SYNOPSIS 🌈

Functions/Subroutines

subroutine dorbdb4 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
    taup2, tauq1, phantom, work, lwork, info)
    DORBDB4

💖 Function/Subroutine Documentation 💖

subroutine dorbdb4 (integer m, integer p, integer q, double precision,
    dimension(ldx11,*) x11, integer ldx11, double precision,
    dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*)
    theta, double precision, dimension(*) phi, double precision,
    dimension(*) taup1, double precision, dimension(*) taup2, double
    precision, dimension(*) tauq1, double precision, dimension(*) phantom,
    double precision, dimension(*) work, integer lwork, integer info)
    DORBDB4

Purpose

okay so DORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonormal columns. The source draws this exact diagram:

           !>
           !>                            [ B11 ]
           !>      [ X11 ]   [ P1 |    ] [  0  ]
           !>      [-----] = [---------] [-----] Q1**T .
           !>      [ X21 ]   [    | P2 ] [ B21 ]
           !>                            [  0  ]
           !>

X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P, M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in which M-Q is not the minimum dimension. iykyk. 🦋

The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors.

B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented implicitly by angles THETA, PHI. Yes, the source says B12 right there — we’re keeping it exactly as written, bestie. 🙈

Parameters

• M — INTEGER. The number of rows in X11 plus the number of rows in X21.
• P — INTEGER. The number of rows in X11. 0 <= P <= M.
• Q — INTEGER. The number of columns in X11 and X21. 0 <= Q <= M and M-Q <= min(P,M-P,Q).
• X11 — DOUBLE PRECISION array, dimension (LDX11,Q). On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1.
• LDX11 — INTEGER. The leading dimension of X11. LDX11 >= P.
• X21 — DOUBLE PRECISION array, dimension (LDX21,Q). On entry, the bottom block of the matrix X to be reduced. On exit, the columns of tril(X21) specify reflectors for P2.
• LDX21 — INTEGER. The leading dimension of X21. LDX21 >= M-P.
• THETA — DOUBLE PRECISION array, dimension (Q). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
• PHI — DOUBLE PRECISION array, dimension (Q-1). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
• TAUP1 — DOUBLE PRECISION array, dimension (M-Q). The scalar factors of the elementary reflectors that define P1.
• TAUP2 — DOUBLE PRECISION array, dimension (M-Q). The scalar factors of the elementary reflectors that define P2.
• TAUQ1 — DOUBLE PRECISION array, dimension (Q). The scalar factors of the elementary reflectors that define Q1.
• PHANTOM — DOUBLE PRECISION array, dimension (M). The routine computes an M-by-1 column vector Y that is orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and Y(P+1:M), respectively.
• WORK — DOUBLE PRECISION array, dimension (LWORK).
• LWORK — INTEGER. The dimension of the array WORK. LWORK >= M-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 — INTEGER.
  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.

Author

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

Further Details

The upper-bidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry in each bidiagonal band is a product of a sine or cosine of a THETA with a sine or cosine of a PHI. See [1] or DORCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors. See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR and DORGLQ.

References

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

Definition at line 210 of file dorbdb4.f.

Definition at line 210 of file dorbdb4.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

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That’s the whole tea, bestie: LAPACK Version 3.12.0, SRC/dorbdb4.f(3) — and the footer didn’t hand us a date, so no fake date energy here. 💅✨🎀