CGEBRD(3)

💅 CGEBRD(3) — Library Functions Manual 💅

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cgebrd.f — bestie, this is the source file for the CGEBRD routine. That's the name section, and yes it's a whole path; LAPACK said "be specific" and meant it, fr fr.

✨ SYNOPSIS ✨

🌟 Functions/Subroutines 🌟

subroutine CGEBRD (m, n, a, lda, d, e, tauq, taup, work, lwork, info)

CGEBRD is the function/subroutine in question. Main character energy, iykyk. 💅

🦋 Function/Subroutine Documentation 🦋

subroutine CGEBRD (integer m, integer n, complex, dimension( lda, * ) a,
    integer lda, real, dimension( * ) d, real, dimension( * ) e, complex,
    dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension( *
    ) work, integer lwork, integer info)

CGEBRD — the full signature, exactly as listed. No cap. ✨

🎯 Purpose 🎯

CGEBRD reduces a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation: Q**H * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. That's the whole point, and it ate. Periodt. 💅🔥

🌸 Parameters 🌸

• M — INTEGER. The number of rows in the matrix A. M >= 0.
• N — INTEGER. The number of columns in the matrix A. N >= 0.
• A — COMPLEX array, dimension (LDA,N). On entry, the M-by-N general matrix to be reduced. On exit:
  • if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors;
  • if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.
  • See Further Details.
• LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• D — REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
• E — REAL array, dimension (min(M,N)-1). The off-diagonal elements of the bidiagonal matrix B:
  • if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
  • if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
• TAUQ — COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
• TAUP — COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
• WORK — COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — INTEGER. The length of the array WORK. LWORK >= max(1,M,N). For optimum performance LWORK >= (M+N)*NB, where NB is the optimal blocksize.

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 matrices Q and P are represented as products of elementary reflectors:

If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n,

     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by the following examples:

  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    (  v1  v2  v3  v4  v5 )

where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

Definition at line 204 of file cgebrd.f.

🪄 Author 🪄

Generated automatically by Doxygen for LAPACK from the source code.

---

Okay that's the whole page, bestie. Footer tea: LAPACK Version 3.12.0, page file /home/abui...RC/cgebrd.f(3). Go reduce something gorgeous. 💖✨