SRC/sorcsd.f(3)

💖 NAME 💖

SRC/sorcsd.f is the name, bestie. The manual name is Library Functions Manual, and the page title is SRC/sorcsd.f(3). This is the LAPACK Fortran subroutine sorcsd, living its recursive subroutine life. 🫶

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

recursive subroutine sorcsd (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)
    SORCSD

🎀 Function/Subroutine Documentation 🎀

Full signature, no cap:

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

🎯 Purpose: 🎯

Okay so SORCSD computes the CS decomposition of an M-by-M partitioned orthogonal matrix X. The source diagrams it like this, comment markers included because Fortran is gonna Fortran:

!>                                 [  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 ]

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. C and S are R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which R = MIN(P,M-P,Q,M-Q). 💅

🌸 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).
• 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 REAL 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 REAL 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 REAL 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, it literally says X11 there; I’m preserving the source’s exact energy. 🙈
• X22 — X22 is REAL 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).
• THETA — THETA is REAL 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)) ).
• U1 — U1 is REAL 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 REAL 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 REAL array, dimension (LDV1T,Q). If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal matrix V1**T.
• LDV1T — LDV1T is INTEGER. The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= MAX(1,Q).
• V2T — V2T is REAL 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 REAL 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: SBBCSD did not converge. See the description of WORK above for details.

📚 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 📍

Definition at line 295 of file sorcsd.f.

📝 Author 📝

Generated automatically by Doxygen for LAPACK from the source code.

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Okay bestie, that’s the whole page. Footer vibe: LAPACK Version 3.12.0 SRC/sorcsd.f(3) — mwah! 💖✨