SORBDB(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 SORBDB(3) — like, the manual page, but make it make sense 🎀

okay so before we even start: this is sorbdb.f from the LAPACK SRC directory (the file path is literally /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sorbdb.f, and the original header calls it /home/abui...RC/sorbdb.f(3) in the Library Functions Manual). And it lives under Functions/Subroutines, bestie. We're in Fortran land. Buckle up. 🦋

✨ SYNOPSIS ✨

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

SORBDB — that's the whole vibe. 💜

🌸 Function/Subroutine Documentation 🌸

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

SORBDB (yes, said twice, it's that iconic 👑)

🫶 Purpose:

so what does it DO. okay. SORBDB simultaneously bidiagonalizes the blocks of an M-by-M partitioned orthogonal matrix X, like this:

                                [ 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 here's the rule, no cap: 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 SORCSD 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.

🧚 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 REAL 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,
      • the rows of triu(X11,1) specify reflectors for Q1;
    • else TRANS = 'T', and
      • the rows of triu(X11) specify reflectors for P1,
      • 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 REAL 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. 👀 (yes, that's literally what it says — LDX11 in the LDX12 description. i'm just the messenger, bestie. not me fixing it, the facts stay facts.)
  • X21 — X21 is REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (P). The scalar factors of the elementary reflectors that define P1.
  • TAUP2 — TAUP2 is REAL array, dimension (M-P). The scalar factors of the elementary reflectors that define P2.
  • TAUQ1 — TAUQ1 is REAL array, dimension (Q). The scalar factors of the elementary reflectors that define Q1.
  • TAUQ2 — TAUQ2 is REAL array, dimension (M-Q). The scalar factors of the elementary reflectors that define Q2.
  • WORK — WORK is REAL 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. 🔥 (classic move — ask it how much space it wants and it just tells you.)

  • 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. 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 SORCSD for details. 📐

P1, P2, Q1, and Q2 are represented as products of elementary reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2 using SORGQR and SORGLQ. ✨

🎀 References:

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

Definition at line 284 of file sorbdb.f. 📍

🦋 Author

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and that's the whole page, bestie — LAPACK, Version 3.12.0, dated /home/abui...RC/sorbdb.f(3). go forth and bidiagonalize, ilysm 💖🌸🫶
📦 where this page comes from