SRC/zunbdb1.f(3)

💖 NAME 💖

SRC/zunbdb1.f — bestie, this is the Library Functions Manual page for zunbdb1, a LAPACK baddie that does simultaneous bidiagonalization. No cap, it's giving linear algebra main character energy. 💅

✨ SYNOPSIS ✨

🌸 Functions/Subroutines 🌸

subroutine zunbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
    taup2, tauq1, work, lwork, info)
    ZUNBDB1

🌸 Function/Subroutine Documentation 🌸

subroutine zunbdb1 (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(*) work, integer lwork, integer info)
    ZUNBDB1

💫 Purpose 💫

ZUNBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonormal columns:

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

X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P, M-P, or M-Q. Routines ZUNBDB2, ZUNBDB3, and ZUNBDB4 handle cases in which 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 Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI.

🎀 Parameters 🎀

• M — INTEGER. The number of rows X11 plus the number of rows in X21.
• P — INTEGER. The number of rows in X11. 0 <= P <= M.
• Q — INTEGER. The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).
• X11 — 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 — INTEGER. The leading dimension of X11. LDX11 >= P.
• X21 — 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 — INTEGER. The leading dimension of X21. LDX21 >= M-P.
• THETA — DOUBLE PRECISION array, dimension (Q). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
• PHI — DOUBLE PRECISION array, dimension (Q-1). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
• TAUP1 — COMPLEX*16 array, dimension (P). The scalar factors of the elementary reflectors that define P1.
• TAUP2 — COMPLEX*16 array, dimension (M-P). The scalar factors of the elementary reflectors that define P2.
• TAUQ1 — COMPLEX*16 array, dimension (Q). The scalar factors of the elementary reflectors that define Q1.
• WORK — COMPLEX*16 array, dimension (LWORK).
• LWORK — 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 — 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 201 of file zunbdb1.f.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

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💖 cute sign-off: LAPACK Version 3.12.0, from SRC/zunbdb1.f(3). Bye bestie, slay your linear algebra! 💅✨