ZGEBRD(3)

💖 NAME 💖

okay bestie, welcome to the Library Functions Manual page for the source file /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgebrd.f — yes, the NAME section is literally a path, and honestly? iconic. 🫶

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

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

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🌸 Function/Subroutine Documentation 🌸

   subroutine ZGEBRD (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 lwork,
       integer info)
       ZGEBRD

🎀 Purpose 🎀

ZGEBRD 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. no cap, that's the whole glow-up. 💅

🦋 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(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — LWORK is 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 — 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 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 203 of file zgebrd.f.

🌈 Author 🌈

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LAPACK Version 3.12.0 — /home/abui...RC/zgebrd.f(3) — bye bestie, go slay that bidiagonal reduction 💅✨