ZGEBD2(3)

💅 ZGEBD2(3) — Library Functions Manual 💅

hey bestie, welcome to the ZGEBD2(3) page from the Library Functions Manual — the header path vibe is /home/abui...RC/zgebd2.f(3). No cap, we're getting into some LAPACK bidiagonal magic. ✨

💖 NAME 💖

The NAME is literally this path, bestie:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgebd2.f

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine ZGEBD2 (m, n, a, lda, d, e, tauq, taup, work, info)

ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. iconic.

🌸 Function/Subroutine Documentation 🌸

subroutine ZGEBD2 (integer m, integer n, complex*16, dimension( lda, * ) a,
    integer lda, double precision, dimension( * ) d, double precision,
    dimension( * ) e, complex*16, dimension( * ) tauq, complex*16,
    dimension( * ) taup, complex*16, dimension( * ) work, integer info)

ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. periodt.

🎀 Purpose 🎀

okay so the Purpose is: ZGEBD2 reduces a complex general m by n matrix A to upper or lower real 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. slay.

✨ Parameters ✨

• M — M is INTEGER. The number of rows in the matrix A. M >= 0.
• N — N is INTEGER. The number of columns in the matrix A. N >= 0.
• A — A is COMPLEX*16 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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• D — D is DOUBLE PRECISION array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
• E — E is DOUBLE PRECISION 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 — TAUQ is COMPLEX*16 array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
• TAUP — TAUP is COMPLEX*16 array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
• WORK — WORK is COMPLEX*16 array, dimension (max(M,N)).
• 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: 🔥

okay bestie, the Further Details are literally this:

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, 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). obsessed with this diagram, ngl.

Definition at line 188 of file zgebd2.f.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

---

okay that's the whole page, bestie! This one is LAPACK Version 3.12.0 and the footer path is /home/abui...RC/zgebd2.f(3). 💖✨