TESTING/EIG/dcsdts.f(3)

💖 TESTING/EIG/dcsdts.f 💖

okay bestie, this page is from the Library Functions Manual and it's all about TESTING/EIG/dcsdts.f — yes, manual name folded in right here at the top, no cap. It's giving very specific numerical-testing main character energy, and honestly? Iconic. 💅✨

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✨ NAME ✨

TESTING/EIG/dcsdts.f

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine dcsdts (m, p, q, x, xf, ldx, u1, ldu1, u2, ldu2, v1t, ldv1t,
           v2t, ldv2t, theta, iwork, work, lwork, rwork, result)
           DCSDTS

🦋 Function/Subroutine Documentation 🦋

subroutine dcsdts

   subroutine dcsdts (integer m, integer p, integer q, double precision,
       dimension( ldx, * ) x, double precision, dimension( ldx, * ) xf, integer
       ldx, 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( * ) theta,
       integer, dimension( * ) iwork, double precision, dimension( lwork )
       work, integer lwork, double precision, dimension( * ) rwork, double
       precision, dimension( 15 ) result)
       DCSDTS

Purpose 💫

bestie, here's the exact math tea — source comment markers and all — because every symbol matters:

           !>
           !> DCSDTS tests DORCSD, which, given an M-by-M partitioned orthogonal
           !> matrix X,
           !>              Q  M-Q
           !>       X = [ X11 X12 ] P   ,
           !>           [ X21 X22 ] M-P
           !>
           !> computes the CSD
           !>
           !>       [ U1    ]**T * [ X11 X12 ] * [ V1    ]
           !>       [    U2 ]      [ X21 X22 ]   [    V2 ]
           !>
           !>                             [  I  0  0 |  0  0  0 ]
           !>                             [  0  C  0 |  0 -S  0 ]
           !>                             [  0  0  0 |  0  0 -I ]
           !>                           = [---------------------] = [ D11 D12 ] ,
           !>                             [  0  0  0 |  I  0  0 ]   [ D21 D22 ]
           !>                             [  0  S  0 |  0  C  0 ]
           !>                             [  0  0  I |  0  0  0 ]
           !>
           !> and also DORCSD2BY1, which, given
           !>          Q
           !>       [ X11 ] P   ,
           !>       [ X21 ] M-P
           !>
           !> computes the 2-by-1 CSD
           !>
           !>                                     [  I  0  0 ]
           !>                                     [  0  C  0 ]
           !>                                     [  0  0  0 ]
           !>       [ U1    ]**T * [ X11 ] * V1 = [----------] = [ D11 ] ,
           !>       [    U2 ]      [ X21 ]        [  0  0  0 ]   [ D21 ]
           !>                                     [  0  S  0 ]
           !>                                     [  0  0  I ]
           !>

Parameters 🎀

• M — M is INTEGER — The number of rows of the matrix X. M >= 0.
• P — P is INTEGER — The number of rows of the matrix X11. P >= 0.
• Q — Q is INTEGER — The number of columns of the matrix X11. Q >= 0.
• X — X is DOUBLE PRECISION array, dimension (LDX,M) — The M-by-M matrix X.
• XF — XF is DOUBLE PRECISION array, dimension (LDX,M) — Details of the CSD of X, as returned by DORCSD; see DORCSD for further details.
• LDX — LDX is INTEGER — The leading dimension of the arrays X and XF. LDX >= max( 1,M ).
• U1 — U1 is DOUBLE PRECISION array, dimension(LDU1,P) — The P-by-P orthogonal matrix U1.
• LDU1 — LDU1 is INTEGER — The leading dimension of the array U1. LDU >= max(1,P).
• U2 — U2 is DOUBLE PRECISION array, dimension(LDU2,M-P) — The (M-P)-by-(M-P) orthogonal matrix U2.
• LDU2 — LDU2 is INTEGER — The leading dimension of the array U2. LDU >= max(1,M-P).
• V1T — V1T is DOUBLE PRECISION array, dimension(LDV1T,Q) — The Q-by-Q orthogonal matrix V1T.
• LDV1T — LDV1T is INTEGER — The leading dimension of the array V1T. LDV1T >= max(1,Q).
• V2T — V2T is DOUBLE PRECISION array, dimension(LDV2T,M-Q) — The (M-Q)-by-(M-Q) orthogonal matrix V2T.
• LDV2T — LDV2T is INTEGER — The leading dimension of the array V2T. LDV2T >= max(1,M-Q).
• THETA — THETA is DOUBLE PRECISION array, dimension MIN(P,M-P,Q,M-Q) — The CS values of X; the essentially diagonal matrices C and S are constructed from THETA; see subroutine DORCSD for details.
• IWORK — IWORK is INTEGER array, dimension (M)
• WORK — WORK is DOUBLE PRECISION array, dimension (LWORK)
• LWORK — LWORK is INTEGER — The dimension of the array WORK
• RWORK — RWORK is DOUBLE PRECISION array
• RESULT — the test-ratio queen 👑, and here is every single formula exactly:

RESULT is DOUBLE PRECISION array, dimension (15)
The test ratios:
First, the 2-by-2 CSD:
RESULT(1) = norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 )
RESULT(2) = norm( U1'*X12*V2 - D12 ) / ( MAX(1,P,M-Q)*EPS2 )
RESULT(3) = norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 )
RESULT(4) = norm( U2'*X22*V2 - D22 ) / ( MAX(1,M-P,M-Q)*EPS2 )
RESULT(5) = norm( I - U1'*U1 ) / ( MAX(1,P)*ULP )
RESULT(6) = norm( I - U2'*U2 ) / ( MAX(1,M-P)*ULP )
RESULT(7) = norm( I - V1T'*V1T ) / ( MAX(1,Q)*ULP )
RESULT(8) = norm( I - V2T'*V2T ) / ( MAX(1,M-Q)*ULP )
RESULT(9) = 0        if THETA is in increasing order and
                     all angles are in [0,pi/2]

          = ULPINV   otherwise.
Then, the 2-by-1 CSD:
RESULT(10) = norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 )
RESULT(11) = norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 )
RESULT(12) = norm( I - U1'*U1 ) / ( MAX(1,P)*ULP )
RESULT(13) = norm( I - U2'*U2 ) / ( MAX(1,M-P)*ULP )
RESULT(14) = norm( I - V1T'*V1T ) / ( MAX(1,Q)*ULP )
RESULT(15) = 0        if THETA is in increasing order and
                      all angles are in [0,pi/2]

           = ULPINV   otherwise.
( EPS2 = MAX( norm( I - X'*X ) / M, ULP ). )

Author

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

Definition at line 226 of file dcsdts.f.

👑 Author 👑

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okay, that's the whole page, bestie! Final sign-off: this footer energy is LAPACK Version 3.12.0, with the source/version tag TESTING/EIG/dcsdts.f(3) — and no separate date is shown here, so I'm not inventing one. Periodt. 💖✨🫶