SRC/zunbdb4.f(3)

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version 3.12.1 · 4 versions
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💖 SRC/zunbdb4.f 💖

hey bestie, this one’s straight from the Library Functions Manual and it’s all about SRC/zunbdb4.f — she’s giving very niche linear-algebra main character energy, and yes, we’re getting into it 💅✨

✨ NAME ✨

SRC/zunbdb4.f

🦋 SYNOPSIS 🦋

🧚 Functions/Subroutines 🧚

subroutine zunbdb4 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
    taup2, tauq1, phantom, work, lwork, info)
    ZUNBDB4

🌈 Function/Subroutine Documentation 🌈

subroutine zunbdb4 (integer m, integer p, integer q, complex*16,
    dimension(ldx11,*) x11, integer ldx11, complex*16, dimension(ldx21,*)
    x21, integer ldx21, double precision, dimension(*) theta, double
    precision, dimension(*) phi, complex*16, dimension(*) taup1, complex*16,
    dimension(*) taup2, complex*16, dimension(*) tauq1, complex*16,
    dimension(*) phantom, complex*16, dimension(*) work, integer lwork,
    integer info)
    ZUNBDB4

💫 Purpose 💫

okay so ZUNBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonormal columns. the glow-up is literally this:

                           [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                           [  0  ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P, M-P, or Q. Routines ZUNBDB1, ZUNBDB2, and ZUNBDB3 handle cases in which M-Q is not the minimum dimension.

The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors.

B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented implicitly by angles THETA, PHI.

real talk: those dimensions and constraints are not decorative, bestie — they’re the whole vibe check of whether this routine makes sense.

🌸 Parameters 🌸

  • M — M is INTEGER. she’s the number of rows in X11 plus the number of rows in X21.
  • P — P is INTEGER. the number of rows in X11. 0 <= P <= M.
  • Q — Q is INTEGER. the number of columns in X11 and X21. 0 <= Q <= M and M-Q <= min(P,M-P,Q).
  • X11 — X11 is COMPLEX*16 array, dimension (LDX11,Q). on entry, the top block of the matrix X to be reduced. on exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1.
  • LDX11 — LDX11 is INTEGER. the leading dimension of X11. LDX11 >= P.
  • X21 — X21 is COMPLEX*16 array, dimension (LDX21,Q). on entry, the bottom block of the matrix X to be reduced. on exit, the columns of tril(X21) specify reflectors for P2.
  • LDX21 — LDX21 is INTEGER. the leading dimension of X21. LDX21 >= M-P.
  • THETA — THETA is DOUBLE PRECISION array, dimension (Q). the entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. see Further Details.
  • PHI — PHI is DOUBLE PRECISION array, dimension (Q-1). the entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. see Further Details.
  • TAUP1 — TAUP1 is COMPLEX*16 array, dimension (M-Q). the scalar factors of the elementary reflectors that define P1.
  • TAUP2 — TAUP2 is COMPLEX*16 array, dimension (M-Q). the scalar factors of the elementary reflectors that define P2.
  • TAUQ1 — TAUQ1 is COMPLEX*16 array, dimension (Q). the scalar factors of the elementary reflectors that define Q1.
  • PHANTOM — PHANTOM is COMPLEX*16 array, dimension (M). the routine computes an M-by-1 column vector Y that is orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and Y(P+1:M), respectively.
  • WORK — WORK is COMPLEX*16 array, dimension (LWORK).
  • LWORK — LWORK is INTEGER. the dimension of the array WORK. LWORK >= M-Q.

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.

  • INFO — INFO is INTEGER.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.

👑 Author 👑

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

🔍 Further Details 🔍

The upper-bidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry in each bidiagonal band is a product of a sine or cosine of a THETA with a sine or cosine of a PHI. See [1] or ZUNCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors. See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR and ZUNGLQ.

📚 References 📚

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

Definition at line 210 of file zunbdb4.f.

🌈 Author 🌈

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mwah 💋 — cute little sign-off from SRC/zunbdb4.f(3), LAPACK Version 3.12.0; the footer doesn’t give us a date, so I’m not inventing one, bestie. 🫶✨

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