SRC/dorbdb1.f(3)

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SRC/dorbdb1.f(3)

from the Library Functions Manual — okay bestie, this is the LAPACK page for SRC/dorbdb1.f, and it is literally giving tall-skinny-matrix main character energy 💅✨

✨ NAME ✨

SRC/dorbdb1.f

🌸 SYNOPSIS 🌸

Functions/Subroutines

       subroutine dorbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
           taup2, tauq1, work, lwork, info)
           DORBDB1

🦋 Function/Subroutine Documentation 🦋

   subroutine dorbdb1 (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(*) work,
       integer lwork, integer info)
       DORBDB1

🎀 Purpose 🎀

okay so DORBDB1 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 DORBDB2, DORBDB3, and DORBDB4 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. 🌈

💅 Parameters 💅

  • M — M is INTEGER. It’s the number of rows X11 plus the number of rows in X21. 💖
  • P — P is INTEGER. It’s the number of rows in X11. 0 <= P <= M. ✨
  • Q — Q is INTEGER. It’s 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, it’s 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 DOUBLE PRECISION array, dimension (LDX21,Q). On entry, it’s 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 DOUBLE PRECISION array, dimension (Q). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details. 🔥
  • PHI — PHI is DOUBLE PRECISION array, dimension (Q-1). The entries of the bidiagonal blocks B11, B21 are defined by 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. ⭐
  • 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 👑

shoutout to the author squad, 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 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 201 of file dorbdb1.f. ⭐

yep, that’s exactly where it lives, bestie. 📍

🍓 Author 🍓

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