SRC/sorbdb1.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💖 NAME 💖

SRC/sorbdb1.f — that’s the page title, bestie, and this one lives in the Library Functions Manual (SRC/sorbdb1.f(3) energy). We’re folding the manual name in right here instead of copy-pasting the header line, because we’re cute AND efficient. 💅

🦋 SYNOPSIS 🦋

🌈 Functions/Subroutines 🌈

subroutine sorbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
           taup2, tauq1, work, lwork, info)
           SORBDB1

💅 Function/Subroutine Documentation 💅

🌟 Purpose 🌟

Okay so SORBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonormal columns:

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

X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P, M-P, or M-Q. Routines SORBDB2, SORBDB3, and SORBDB4 handle cases in which Q is not the minimum dimension.

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 Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI.

Not me circling “B12” in my notes — that’s literally what the original says, so it stays. 🫶

🎀 Parameters 🎀

  • M — M is INTEGER. The number of rows X11 plus the number of rows in X21.
  • P — P is INTEGER. The number of rows in X11. 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 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 — LDX11 is INTEGER. The leading dimension of X11. LDX11 >= P.
  • X21 — X21 is REAL 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 — LDX21 is INTEGER. The leading dimension of X21. LDX21 >= M-P.
  • THETA — THETA is REAL array, dimension (Q). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
  • PHI — PHI is REAL array, dimension (Q-1). The entries of the bidiagonal blocks B11, B21 are defined by 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.
  • 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.
  • INFO — INFO is INTEGER.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.

👑 Author 👑

The academic dream team, no cap:

  • 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 SORCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors. See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR and SORGLQ.

🌈 References 🌈

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

📌 Definition at line 201 of file sorbdb1.f 📌

Definition at line 201 of file sorbdb1.f.

💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole page, bestie! Source/version sign-off: LAPACK, Version 3.12.0, page SRC/sorbdb1.f(3) — and there’s no date line in this footer, so we’re not inventing one, periodt. 💅✨🎀

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