uncsd(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

uncsd(3) — Library Functions Manual 🎀✨

okay bestie, welcome to the Library Functions Manual page for uncsd(3) — yes, the whole thing, and we're doing it with the sparkle it deserves. 💅

✨ NAME ✨

uncsd - {un,or}csd: ??

🌈 SYNOPSIS 🌈

okay so these are the four recursive subroutines this page is serving, and yes, every arg is getting its moment:

  • cuncsd:
recursive subroutine cuncsd (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)

CUNCSD 👑

  • dorcsd:
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 🔥

  • sorcsd:
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 🌸

  • zuncsd:
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)

ZUNCSD 💜

🦋 Detailed Description 🦋

the original page has this heading and then slides straight into the function docs, so no extra lore here — just vibes and then the real tea. 🫶

🔥 Function Documentation 🔥

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

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

CUNCSD 💅

✨ Purpose ✨

okay so CUNCSD computes the CS decomposition of an M-by-M partitioned unitary matrix X, and the math is literally this:

                                [  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 — JOBU1 is CHARACTER. If it's 'Y', U1 is computed; otherwise, U1 is not computed.
  • JOBU2 — JOBU2 is CHARACTER. If it's 'Y', U2 is computed; otherwise, U2 is not computed.
  • JOBV1T — JOBV1T is CHARACTER. If it's 'Y', V1T is computed; otherwise, V1T is not computed.
  • JOBV2T — JOBV2T is CHARACTER. If it's 'Y', V2T is computed; otherwise, V2T is not computed.
  • TRANS — TRANS is CHARACTER. If it's '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. If it's '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 COMPLEX array, dimension (LDX11,Q). On entry, part of the unitary matrix whose CSD is desired.
  • LDX11 — LDX11 is INTEGER. The leading dimension of X11. LDX11 >= MAX(1,P).
  • X12 — X12 is COMPLEX array, dimension (LDX12,M-Q). On entry, part of the unitary matrix whose CSD is desired.
  • LDX12 — LDX12 is INTEGER. The leading dimension of X12. LDX12 >= MAX(1,P).
  • X21 — X21 is COMPLEX array, dimension (LDX21,Q). On entry, part of the unitary matrix whose CSD is desired.
  • LDX21 — LDX21 is INTEGER. The leading dimension of X11. LDX21 >= MAX(1,M-P).
  • X22 — X22 is COMPLEX array, dimension (LDX22,M-Q). On entry, part of the unitary 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 COMPLEX array, dimension (LDU1,P). If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1.
  • LDU1 — LDU1 is INTEGER. The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= MAX(1,P).
  • U2 — U2 is COMPLEX array, dimension (LDU2,M-P). If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary matrix U2.
  • LDU2 — LDU2 is INTEGER. The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= MAX(1,M-P).
  • V1T — V1T is COMPLEX array, dimension (LDV1T,Q). If JOBV1T = 'Y', V1T contains the Q-by-Q unitary matrix V1**H.
  • LDV1T — LDV1T is INTEGER. The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= MAX(1,Q).
  • V2T — V2T is COMPLEX array, dimension (LDV2T,M-Q). If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) unitary matrix V2**H.
  • LDV2T — LDV2T is INTEGER. The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= MAX(1,M-Q).
  • WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • 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.
  • RWORK — RWORK is REAL 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 — LRWORK is 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 — 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: CBBCSD 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 at line 314 of file cuncsd.f. 💖

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)

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 🔥

✨ Purpose

🦋 DORCSD 🦋

✨ Purpose ✨

Okay bestie, so DORCSD computes the CS decomposition of an M-by-M partitioned orthogonal matrix X. Here's the whole lewk, and yes it's giving main character:

                                [  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).

Like, that's the whole vibe. C and S literally can't both be big at the same time because they'd blow past the identity — trigonometry said so 💅

🎀 Parameters 🎀

  • JOBU1 — CHARACTER. Tells us if U1 is invited to the party:
    • = 'Y': U1 is computed;
    • otherwise: U1 is not computed.
  • JOBU2 — CHARACTER. Same energy, but for U2:
    • = 'Y': U2 is computed;
    • otherwise: U2 is not computed.
  • JOBV1T — CHARACTER. And now for V1T:
    • = 'Y': V1T is computed;
    • otherwise: V1T is not computed.
  • JOBV2T — CHARACTER. Last of the job squad, for V2T:
    • = 'Y': V2T is computed;
    • otherwise: V2T is not computed.
  • TRANS — CHARACTER. Storage order check, iykyk:
    • = '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 — CHARACTER. Which block gets the nonpositive treatment:
    • = 'O': The lower-left block is made nonpositive (the convention);
    • otherwise: The upper-right block is made nonpositive (the convention).
  • M — INTEGER. The number of rows and columns in X. It's a square situation, periodt.
  • 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 — DOUBLE PRECISION array, dimension (LDX11,Q). On entry, part of the orthogonal matrix whose CSD is desired. She's the top-left girly 🫶
  • LDX11 — INTEGER. The leading dimension of X11. LDX11 >= MAX(1,P).
  • X12 — DOUBLE PRECISION array, dimension (LDX12,M-Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX12 — INTEGER. The leading dimension of X12. LDX12 >= MAX(1,P).
  • X21 — DOUBLE PRECISION array, dimension (LDX21,Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX21 — INTEGER. The leading dimension of X11. LDX21 >= MAX(1,M-P). (Yeah it says X11 — that's what the page says, not me, don't come for me 🙈)
  • X22 — DOUBLE PRECISION array, dimension (LDX22,M-Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX22 — INTEGER. The leading dimension of X11. LDX22 >= MAX(1,M-P). (Same deal as above, bestie — verbatim.)
  • THETA — DOUBLE PRECISION array, dimension (R), in which R = MIN(P,M-P,Q,M-Q). The angles themselves:
    • C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
    • S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
  • U1 — DOUBLE PRECISION array, dimension (LDU1,P). If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
  • LDU1 — INTEGER. The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= MAX(1,P).
  • U2 — DOUBLE PRECISION array, dimension (LDU2,M-P). If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal matrix U2.
  • LDU2 — INTEGER. The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= MAX(1,M-P).
  • V1T — DOUBLE PRECISION array, dimension (LDV1T,Q). If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal matrix V1**T.
  • LDV1T — INTEGER. The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= MAX(1,Q).
  • V2T — DOUBLE PRECISION array, dimension (LDV2T,M-Q). If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal matrix V2**T.
  • LDV2T — INTEGER. The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= MAX(1,M-Q).
  • WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). The scratch space, and she's kind of a diary:
    • 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 — 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. (So like, -1 is not a crime, it's a question ✨)
  • IWORK — INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q)).
  • INFO — INTEGER. The receipts:
    • = 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.

📚 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.

🌈 recursive subroutine sorcsd — the REAL (single precision) version 🌈

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 ✨

Girl, it's the exact same story, just in REAL. SORCSD computes the CS decomposition of an M-by-M partitioned orthogonal matrix X:

                                [  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).

Same decomposition, same drama, half the bytes 💅

🎀 Parameters 🎀

  • JOBU1 — CHARACTER. Decides if U1 gets computed:
    • = 'Y': U1 is computed;
    • otherwise: U1 is not computed.
  • JOBU2 — CHARACTER. Decides if U2 gets computed:
    • = 'Y': U2 is computed;
    • otherwise: U2 is not computed.
  • JOBV1T — CHARACTER. Decides if V1T gets computed:
    • = 'Y': V1T is computed;
    • otherwise: V1T is not computed.
  • JOBV2T — CHARACTER. Decides if V2T gets computed:
    • = 'Y': V2T is computed;
    • otherwise: V2T is not computed.
  • TRANS — CHARACTER. Storage order, once again:
    • = '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 — CHARACTER. Picking which block goes nonpositive:
    • = 'O': The lower-left block is made nonpositive (the convention);
    • otherwise: The upper-right block is made nonpositive (the convention).
  • 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 — REAL array, dimension (LDX11,Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX11 — INTEGER. The leading dimension of X11. LDX11 >= MAX(1,P).
  • X12 — REAL array, dimension (LDX12,M-Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX12 — INTEGER. The leading dimension of X12. LDX12 >= MAX(1,P).

…and we are back in the SORCSD parameter roll-call, bestie, no time to waste 💅

  • LDX12 — INTEGER. The leading dimension of X12, and the requirement is LDX12 >= MAX(1,P). That's the law, fr fr.
  • X21 — REAL array, dimension (LDX21,Q). On entry, part of the orthogonal matrix whose CSD is desired. She's a supporting character but she matters 💖
  • LDX21 — INTEGER. The leading dimension of X11. LDX21 >= MAX(1,M-P). (Yes, the docs really say X11 here — not my typo, I'm just the messenger 💀)
  • X22 — REAL array, dimension (LDX22,M-Q). On entry, part of the orthogonal matrix whose CSD is desired.
  • LDX22 — INTEGER. The leading dimension of X11. LDX22 >= MAX(1,M-P). (Again with the X11 — the source said it, so we say it 🫶)
  • THETA — REAL array, dimension (R), in which R = MIN(P,M-P,Q,M-Q). And the vibes are:
C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

The angles literally become the C and S matrices. Iconic behaviour 🌸

  • U1 — REAL array, dimension (LDU1,P). If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
  • LDU1 — INTEGER. The leading dimension of U1. If JOBU1 = 'Y', then LDU1 >= MAX(1,P).
  • U2 — REAL array, dimension (LDU2,M-P). If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal matrix U2.
  • LDU2 — INTEGER. The leading dimension of U2. If JOBU2 = 'Y', then LDU2 >= MAX(1,M-P).
  • V1T — REAL array, dimension (LDV1T,Q). If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal matrix V1**T. (They wrote "matrix orthogonal matrix", and honestly? same energy 😭)
  • LDV1T — INTEGER. The leading dimension of V1T. If JOBV1T = 'Y', then LDV1T >= MAX(1,Q).
  • V2T — REAL array, dimension (LDV2T,M-Q). If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal matrix V2**T.
  • LDV2T — INTEGER. The leading dimension of V2T. If JOBV2T = 'Y', then LDV2T >= MAX(1,M-Q).
  • WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. And here's the spicy part: 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. So yeah, if things go sideways, the leftovers are literally sitting in WORK 👀
  • 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. Sneaky little query mode, we love her ✨

  • IWORK — INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q)). That's the whole description. Short queen. 👑
  • INFO — 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.

That > 0 case is the "she didn't converge" moment — go reread the WORK entry, it's all up there 🔥

🔖 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 295 of file sorcsd.f.

🦋 ZUNCSD

Okay so now we go complex, bestie — literally. Here's the signature, copied exactly, no notes:

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)

Yeah. That's the whole cast list. Deep breath 🫶

✨ Purpose

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 ]

Look at that factorization. She's stunning. The way those identity blocks and C and S blocks just slot into place? Main character energy, no cap 💖

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).

So R is the little one — the minimum of the four block dimensions. She's small but she runs the whole show 🎀

💅 Parameters

  • JOBU1 — CHARACTER.
= 'Y':      U1 is computed;
otherwise:  U1 is not computed.
  • JOBU2 — CHARACTER.
= 'Y':      U2 is computed;
otherwise:  U2 is not computed.
  • JOBV1T — CHARACTER.
= 'Y':      V1T is computed;
otherwise:  V1T is not computed.
  • JOBV2T — CHARACTER.
= 'Y':      V2T is computed;
otherwise:  V2T is not computed.

Four job flags, all the same deal: 'Y' or literally anything else. Pick your faves 😌

  • TRANS — 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.

Note it's all five of them that flip together — X, U1, U2, V1T, and V2T. No mixing and matching, bestie 🌈

  • SIGNS — CHARACTER.
= 'O':      The lower-left block is made nonpositive (the
             convention);
otherwise:  The upper-right block is made nonpositive (the
             convention).

Both branches are described as "(the convention)", which is deeply funny to me and I refuse to edit it 😭

  • M — INTEGER. The number of rows and columns in X. Square, obviously, she's built like that.
  • 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, 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, 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, part of the unitary matrix whose CSD is desired.
  • LDX21 — INTEGER. The leading dimension of X11. LDX21 >= MAX(1,M-P). (The "X11" is in the original — I'm not out here renaming things 💀)
  • X22 — COMPLEX*16 array, dimension (LDX22,M-Q). On entry, part of the unitary matrix whose CSD is desired.
  • LDX22 — INTEGER. The leading dimension of X11. LDX22 >= MAX(1,M-P). (Same deal. X11 again. The docs said it, so it's canon.)
  • 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)) ).

Same angles-to-matrices pipeline as before, but now the angles are DOUBLE PRECISION because the matrix is complex. Balance 💐

  • 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', then 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', then LDU2 >= MAX(1,M-P).
  • V1T — COMPLEX*16 array, dimension (LDV1T,Q). If JOBV1T = 'Y', V1T contains the Q-by-Q matrix unitary matrix V1**H. (Yes: "matrix unitary matrix". We keep it. We respect it. 🫡)
  • LDV1T — INTEGER. The leading dimension of V1T. If JOBV1T = 'Y', then 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', then 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. Same trick as her real-valued sister — ask nicely, get the size back, no XERBLA shouting 🌟

  • RWORK —

RWORK 🌸

RWORK is a 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 💅

LRWORK is INTEGER — it's 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 🎀

IWORK is an INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q)).

INFO 🔥

INFO is INTEGER:

  • INFO = 0 — successful exit.
  • INFO < 0 — if INFO = -i, the i-th argument had an illegal value.
  • INFO > 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 at line 314 of file zuncsd.f.

Author (Doxygen edition) 🫶

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