DORBDB(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 DORBDB(3) — Library Functions Manual

okay so bestie, this is DORBDB(3) from the Library Functions Manual, and it’s literally the glow-up page for a LAPACK subroutine. No cap, it does a whole simultaneous bidiagonalization and the math is giving main-character energy. 💖

✨ NAME ✨

The NAME here is just the source path, and yes, it’s exactly this:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dorbdb.f

🦋 SYNOPSIS 🦋

🧚 Functions/Subroutines

subroutine DORBDB (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21,
    ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work,
    lwork, info)
    DORBDB

📚 Function/Subroutine Documentation 📚

Here’s the full typed-out signature, bestie — copied exactly so nobody comes for me:

subroutine DORBDB (character trans, character signs, integer m, integer p,
    integer q, double precision, dimension( ldx11, * ) x11, integer ldx11,
    double precision, dimension( ldx12, * ) x12, integer ldx12, double
    precision, dimension( ldx21, * ) x21, integer ldx21, double precision,
    dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( *
    ) theta, double precision, dimension( * ) phi, double precision,
    dimension( * ) taup1, double precision, dimension( * ) taup2, double
    precision, dimension( * ) tauq1, double precision, dimension( * ) tauq2,
    double precision, dimension( * ) work, integer lwork, integer info)
    DORBDB

🎀 Purpose:

DORBDB simultaneously bidiagonalizes the blocks of an M-by-M partitioned orthogonal matrix X. It’s giving “one routine, many blocks, all handled.” ✨

Here’s the exact diagram vibes:

                                 [ B11 | B12 0  0 ]
     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
 X = [-----------] = [---------] [----------------] [---------]   .
     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
                                 [  0  |  0  0  I ]

X11 is P-by-Q. And this is important: Q must be no larger than P, M-P, or M-Q. If that’s not the case, 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 orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors. 🫶

B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI. Yes, angles doing the invisible work. Iconic.

🌸 Parameters

  • 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. 💅
  • SIGNS — SIGNS is CHARACTER. = 'O': The lower-left block is made nonpositive (the convention); otherwise: The upper-right block is made nonpositive (the convention). 🎀
  • M — M is INTEGER. The number of rows and columns in X. 🌈
  • P — P is INTEGER. The number of rows in X11 and X12. 0 <= P <= M. ✨
  • Q — Q is INTEGER. The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q). 🦋
  • X11 — X11 is DOUBLE PRECISION array, dimension (LDX11,Q). On entry, the top-left block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the columns of tril(X11) specify reflectors for P1, and the rows of triu(X11,1) specify reflectors for Q1; else TRANS = 'T', and the rows of triu(X11) specify reflectors for P1, and the columns of tril(X11,-1) specify reflectors for Q1. 💖
  • LDX11 — LDX11 is INTEGER. The leading dimension of X11. If TRANS = 'N', then LDX11 >= P; else LDX11 >= Q. 🌸
  • X12 — X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q). On entry, the top-right block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the rows of triu(X12) specify the first P reflectors for Q2; else TRANS = 'T', and the columns of tril(X12) specify the first P reflectors for Q2. ✨
  • LDX12 — LDX12 is INTEGER. The leading dimension of X12. If TRANS = 'N', then LDX12 >= P; else LDX11 >= M-Q. (Yep, that’s exactly what the original says — I’m not fixing it, bestie. 🫶)
  • X21 — X21 is DOUBLE PRECISION array, dimension (LDX21,Q). On entry, the bottom-left block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the columns of tril(X21) specify reflectors for P2; else TRANS = 'T', and the rows of triu(X21) specify reflectors for P2. 💜
  • LDX21 — LDX21 is INTEGER. The leading dimension of X21. If TRANS = 'N', then LDX21 >= M-P; else LDX21 >= Q. 🎀
  • X22 — X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q). On entry, the bottom-right block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last M-P-Q reflectors for Q2, else TRANS = 'T', and the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last M-P-Q reflectors for P2. 👑
  • LDX22 — LDX22 is INTEGER. The leading dimension of X22. If TRANS = 'N', then LDX22 >= M-P; else LDX22 >= M-Q. 🌈
  • THETA — THETA is DOUBLE PRECISION array, dimension (Q). The entries of the bidiagonal blocks B11, B12, B21, B22 can be computed from the angles THETA and PHI. See Further Details. ⭐
  • PHI — PHI is DOUBLE PRECISION array, dimension (Q-1). The entries of the bidiagonal blocks B11, B12, B21, B22 can be computed from the angles THETA and PHI. See Further Details. 💫
  • TAUP1 — TAUP1 is DOUBLE PRECISION array, dimension (P). The scalar factors of the elementary reflectors that define P1. 🧚
  • TAUP2 — TAUP2 is DOUBLE PRECISION array, dimension (M-P). The scalar factors of the elementary reflectors that define P2. 🦋
  • TAUQ1 — TAUQ1 is DOUBLE PRECISION array, dimension (Q). The scalar factors of the elementary reflectors that define Q1. 🌸
  • TAUQ2 — TAUQ2 is DOUBLE PRECISION array, dimension (M-Q). The scalar factors of the elementary reflectors that define Q2. 🎀
  • WORK — WORK is DOUBLE PRECISION array, dimension (LWORK). ✨
  • LWORK — LWORK is 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 — INFO is 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 bidiagonal blocks B11, B12, B21, and B22 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are lower bidiagonal. (The original really says B21 in both spots — I’m keeping it exactly as-is. 💀) 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, Q1, and Q2 are represented as products of elementary reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2 using DORGQR and DORGLQ. 🫶

🌈 References:

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

Definition at line 284 of file dorbdb.f.

💜 Author

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