SRC/cunbdb2.f(3)

🌸 NAME

SRC/cunbdb2.f is giving Library Functions Manual realness today, and this whole page is about the <hotpink>CUNBDB2</hotpink> subroutine. Yes bestie, we are getting into tall-and-skinny matrix bidiagonalization, and it is lowkey iconic. 💅

✨ SYNOPSIS

🫶 Functions/Subroutines

subroutine cunbdb2 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
    taup2, tauq1, work, lwork, info)
    CUNBDB2

🦋 Function/Subroutine Documentation

Here is the full signature, character for character, no cap:

subroutine cunbdb2 (integer m, integer p, integer q, complex,
       dimension(ldx11,*) x11, integer ldx11, complex, dimension(ldx21,*) x21,
       integer ldx21, real, dimension(*) theta, real, dimension(*) phi,
       complex, dimension(*) taup1, complex, dimension(*) taup2, complex,
       dimension(*) tauq1, complex, dimension(*) work, integer lwork, integer
       info)
CUNBDB2

📌 Purpose

CUNBDB2 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. P must be no larger than M-P, Q, or M-Q. Routines CUNBDB1, CUNBDB3, and CUNBDB4 handle cases in which P 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 P-by-P bidiagonal matrices represented implicitly by angles THETA, PHI. Yes, that is literally what the source says, bestie. 💖

💅 Parameters

• M — M is INTEGER. It's the number of rows in X11 plus the number of rows in X21. 🌈
• P — P is INTEGER. The number of rows in X11. Constraint: 0 <= P <= min(M-P,Q,M-Q). 🎀
• Q — Q is INTEGER. The number of columns in X11 and X21. Constraint: 0 <= Q <= M. ✨
• X11 — X11 is COMPLEX 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. Constraint: LDX11 >= P. 💜
• X21 — X21 is COMPLEX 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. Constraint: LDX21 >= M-P. 🫶
• THETA — THETA is REAL array, dimension (Q). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details. 👑
• PHI — PHI is REAL 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 array, dimension (P-1). The scalar factors of the elementary reflectors that define P1. 🌟
• TAUP2 — TAUP2 is COMPLEX array, dimension (Q). The scalar factors of the elementary reflectors that define P2. 🔥
• TAUQ1 — TAUQ1 is COMPLEX array, dimension (Q). The scalar factors of the elementary reflectors that define Q1. 🧚
• WORK — WORK is COMPLEX array, dimension (LWORK). 💻
• LWORK — LWORK is INTEGER. The dimension of the array WORK. Constraint: 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 CUNCSD for details. 🌀

P1, P2, and Q1 are represented as products of elementary reflectors. See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR and CUNGLQ. 💅

📚 References

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

Definition at line 200 of file cunbdb2.f.

🤖 Author

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Okay bestie, that's the whole page — LAPACK Version 3.12.0, source SRC/cunbdb2.f(3). 💖✨