DORCSD(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 NAME 💅

Okay so bestie, buckle up: this is the manual page for /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dorcsd.f — and yes the whole path is literally the name, that's just how it is, no cap. It's living in the Library Functions Manual section (section 3, the "functions" girlies 🫶).

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

recursive subroutine DORCSD (jobu1, jobu2, jobv1t, jobv2t, trans, signs,
    m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, u1,
    ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, work, lwork, iwork, info)
    DORCSD

🎀 Function/Subroutine Documentation 🎀

🌈 DORCSD

recursive subroutine DORCSD (character jobu1, character jobu2, character
    jobv1t, character jobv2t, 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( ldu1, * ) u1, integer ldu1, double precision, dimension(
    ldu2, * ) u2, integer ldu2, double precision, dimension( ldv1t, * ) v1t,
    integer ldv1t, double precision, dimension( ldv2t, * ) v2t, integer
    ldv2t, double precision, dimension( * ) work, integer lwork, integer,
    dimension( * ) iwork, integer info)

DORCSD — she's recursive, she's a subroutine, she's the main character 💅

🦋 Purpose

Okay so what does she do? DORCSD computes the CS decomposition of an M-by-M partitioned orthogonal matrix X, and honestly the way she does it is iconic:

                                [  I  0  0 |  0  0  0 ]
                                [  0  C  0 |  0 -S  0 ]
    [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**T
X = [-----------] = [---------] [---------------------] [---------]   .
    [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
                                [  0  S  0 |  0  C  0 ]
                                [  0  0  I |  0  0  0 ]

So X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P, (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively — respectively, bestie, keep up 🫶. And C and S are R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, where R = MIN(P,M-P,Q,M-Q). Periodt.

🌸 Parameters

  • JOBU1 — JOBU1 is CHARACTER. = 'Y': U1 is computed; otherwise: U1 is not computed. 🙈
  • JOBU2 — JOBU2 is CHARACTER. = 'Y': U2 is computed; otherwise: U2 is not computed.
  • JOBV1T — JOBV1T is CHARACTER. = 'Y': V1T is computed; otherwise: V1T is not computed.
  • JOBV2T — JOBV2T is CHARACTER. = 'Y': V2T is computed; otherwise: V2T is not computed.
  • 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). Notice both of those say "the convention" — that's literally what the page says, I'm not editing the lore. 😭
  • 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 <= M.
  • X11 — X11 is DOUBLE PRECISION array, dimension (LDX11,Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX11 — LDX11 is INTEGER. The leading dimension of X11. LDX11 >= MAX(1,P).
  • X12 — X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX12 — LDX12 is INTEGER. The leading dimension of X12. LDX12 >= MAX(1,P).
  • X21 — X21 is DOUBLE PRECISION array, dimension (LDX21,Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX21 — LDX21 is INTEGER. The leading dimension of X11. LDX21 >= MAX(1,M-P). (Yes, the page says "X11" here — that's what it says, so that's what you get 🫠.)
  • X22 — X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX22 — LDX22 is INTEGER. The leading dimension of X11. LDX22 >= MAX(1,M-P). (Same deal — X11 is what the page says, I don't make the rules 💀.)
  • THETA — THETA is DOUBLE PRECISION array, dimension (R), in which R = MIN(P,M-P,Q,M-Q). C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ). Trig ate, honestly. 🌸
  • U1 — U1 is DOUBLE PRECISION array, dimension (LDU1,P). If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
  • LDU1 — LDU1 is INTEGER. The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= MAX(1,P).
  • U2 — U2 is DOUBLE PRECISION array, dimension (LDU2,M-P). If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal matrix U2.
  • LDU2 — LDU2 is INTEGER. The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= MAX(1,M-P).
  • V1T — V1T is DOUBLE PRECISION array, dimension (LDV1T,Q). If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal matrix V1**T. ("Q-by-Q matrix orthogonal matrix" — she said it twice, she meant it ✨.)
  • LDV1T — LDV1T is INTEGER. The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= MAX(1,Q).
  • V2T — V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q). If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal matrix V2**T.
  • LDV2T — LDV2T is INTEGER. The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= MAX(1,M-Q).
  • WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. If INFO > 0 on exit, WORK(2:R) contains the values PHI(1), ..., PHI(R-1) that, together with THETA(1), ..., THETA(R), define the matrix in intermediate bidiagonal-block form remaining after nonconvergence. INFO specifies the number of nonzero PHI's. 🔥
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. 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. 🎀
  • IWORK — IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q)).
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: DBBCSD did not converge. See the description of WORK above for details. 😭
Real talk bestie: if INFO > 0 you got a DBBCSD nonconvergence situation, and the PHI values chilling in WORK(2:R) are your forensic evidence. Go read the WORK paragraph, fr fr.

💜 References

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

👑 Author

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

🌟 Definition at line 295 of file dorcsd.f.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code. (Not me, I'm just the hype woman 💅)

Okay that's the whole thing, bestie — LAPACK, Version 3.12.0, source file /home/abui...RC/dorcsd.f(3), straight from the Library Functions Manual. Slay, go compute your CS decomposition, and remember: C^2 + S^2 = I and so is your worth. 🦋✨🎀

📦 where this page comes from