ZUNCSD(3)

📦 version 3.12.0, in opensuse-leap 16.0

💖 NAME 💖

Bestie, this is ZUNCSD(3) from the Library Functions Manual — LAPACK’s recursive CS decomposition subroutine, and it is literally doing the most in the most elegant way. The source file is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zuncsd.f. 🎀

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

recursive subroutine ZUNCSD (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, rwork, lrwork,
    iwork, info)

🎀 Function/Subroutine Documentation 🎀

recursive subroutine ZUNCSD

Here’s the full signature, bestie — every argument is present and accounted for:

recursive subroutine ZUNCSD (character jobu1, character jobu2, character
    jobv1t, character jobv2t, character trans, character signs, integer m,
    integer p, integer q, complex*16, dimension( ldx11, * ) x11, integer
    ldx11, complex*16, dimension( ldx12, * ) x12, integer ldx12, complex*16,
    dimension( ldx21, * ) x21, integer ldx21, complex*16, dimension( ldx22,
    * ) x22, integer ldx22, double precision, dimension( * ) theta,
    complex*16, dimension( ldu1, * ) u1, integer ldu1, complex*16,
    dimension( ldu2, * ) u2, integer ldu2, complex*16, dimension( ldv1t, * )
    v1t, integer ldv1t, complex*16, dimension( ldv2t, * ) v2t, integer
    ldv2t, complex*16, dimension( * ) work, integer lwork, double precision,
    dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork,
    integer info)

🫶 Purpose 🫶

Okay so ZUNCSD computes the CS decomposition of an M-by-M partitioned unitary matrix X:

                                [  I  0  0 |  0  0  0 ]
                                [  0  C  0 |  0 -S  0 ]
    [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**H
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 unitary 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

This one is CHARACTER. If it’s 'Y', then U1 is computed. Otherwise, U1 is not computed. 💅

JOBU2

CHARACTER. If it’s 'Y', then U2 is computed. Otherwise, U2 is not computed. ✨

JOBV1T

CHARACTER. If it’s 'Y', then V1T is computed. Otherwise, V1T is not computed. 🎀

JOBV2T

CHARACTER. If it’s 'Y', then V2T is computed. Otherwise, V2T is not computed. 🫶

TRANS

CHARACTER. If it’s 'T', then 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. No cap. 🦋

SIGNS

CHARACTER. If it’s 'O', the lower-left block is made nonpositive (the convention). Otherwise, the upper-right block is made nonpositive (the convention). Iconic sign behavior. 🎀

M

INTEGER. The number of rows and columns in X. 💖

P

INTEGER. The number of rows in X11 and X12. 0 <= P <= M. 🌸

Q

INTEGER. The number of columns in X11 and X21. 0 <= Q <= M. ✨

X11

COMPLEX*16 array, dimension (LDX11,Q). On entry, this is part of the unitary matrix whose CSD is desired. 🦋

LDX11

INTEGER. The leading dimension of X11. LDX11 >= MAX(1,P). 💅

X12

COMPLEX*16 array, dimension (LDX12,M-Q). On entry, this is part of the unitary matrix whose CSD is desired. 🌸

LDX12

INTEGER. The leading dimension of X12. LDX12 >= MAX(1,P). 🎀

X21

COMPLEX*16 array, dimension (LDX21,Q). On entry, this is part of the unitary matrix whose CSD is desired. ✨

LDX21

INTEGER. The leading dimension of X11. LDX21 >= MAX(1,M-P). (Yes bestie, the page says X11 here — I’m not fixing it, I’m just delivering the lore. 😭)

X22

COMPLEX*16 array, dimension (LDX22,M-Q). On entry, this is part of the unitary matrix whose CSD is desired. 🫶

LDX22

INTEGER. The leading dimension of X11. LDX22 >= MAX(1,M-P). (Again with the X11 mention — preserving the original tea. 🍓)

THETA

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)) ). 💖

U1

COMPLEX*16 array, dimension (LDU1,P). If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1. 👑

LDU1

INTEGER. The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= MAX(1,P). ✨

U2

COMPLEX*16 array, dimension (LDU2,M-P). If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary matrix U2. 🦋

LDU2

INTEGER. The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= MAX(1,M-P). 🌸

V1T

COMPLEX*16 array, dimension (LDV1T,Q). If JOBV1T = 'Y', V1T contains the Q-by-Q unitary matrix V1**H. 🎀

LDV1T

INTEGER. The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= MAX(1,Q). 🫶

V2T

COMPLEX*16 array, dimension (LDV2T,M-Q). If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) unitary matrix V2**H. ✨

LDV2T

INTEGER. The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= MAX(1,M-Q). 💅

WORK

COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🔥

LWORK

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. 🫶

RWORK

DOUBLE PRECISION array, dimension MAX(1,LRWORK). On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. If INFO > 0 on exit, RWORK(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. 😭

LRWORK

INTEGER. The dimension of the array RWORK.

If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the work array, and no error message related to LRWORK is issued by XERBLA. 🔥

IWORK

INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q)). 🌈

INFO

INTEGER.

  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • > 0: ZBBCSD did not converge. See the description of RWORK 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 🌸

Definition at line 314 of file zuncsd.f.

🪄 Author (generated) 🪄

Generated automatically by Doxygen for LAPACK from the source code.

💅 Sign-off: LAPACK, Version 3.12.0, source /home/abui...RC/zuncsd.f(3). Byeee! ✨

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