ZLABRD(3)

🦋 NAME 🦋

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

okay so that's literally the whole name section, bestie — this page is brought to you by the LAPACK 3.12.0 source tree and it's just showing you where the file lives. iconic of it, honestly 💅

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✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ZLABRD (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)

ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. that's the whole vibe 🎀

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👑 Function/Subroutine Documentation 👑

subroutine ZLABRD (integer m, integer n, integer nb, 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( ldx, * ) x,
    integer ldx, complex*16, dimension( ldy, * ) y, integer ldy)

ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. no cap, that's the entire job description 💖

🫶 Purpose

ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q**H * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. ✨

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by ZGEBRD — so like, it's the bestie in the background doing the real work while ZGEBRD takes the credit. we see you, ZLABRD 👑

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🌸 Parameters 🌸

• M — M is INTEGER. The number of rows in the matrix A.
• N — N is INTEGER. The number of columns in the matrix A.
• NB — NB is INTEGER. The number of leading rows and columns of A to be reduced. (so like, how much of the front of the matrix we're touching, not the whole thing)
• A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. 💀 (yes bestie, it gets wrecked — read carefully:)
  • If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.
  • If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.
  • See Further Details. (there's a whole diagram down there, go look 👀)
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M). fr fr, respect the bound or it's over for you 🚨
• D — D is DOUBLE PRECISION array, dimension (NB). The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i). 💖
• E — E is DOUBLE PRECISION array, dimension (NB). The off-diagonal elements of the first NB rows and columns of the reduced matrix.
• TAUQ — TAUQ is COMPLEX*16 array, dimension (NB). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
• TAUP — TAUP is COMPLEX*16 array, dimension (NB). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
• X — X is COMPLEX*16 array, dimension (LDX,NB). The m-by-nb matrix X required to update the unreduced part of A.
• LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,M).
• Y — Y is COMPLEX*16 array, dimension (LDY,NB). The n-by-nb matrix Y required to update the unreduced part of A.
• LDY — LDY is INTEGER. The leading dimension of the array Y. LDY >= max(1,N).

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🧚 Author 🧚

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

(the whole squad, we stan a collaborative effort 🫶)

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⭐ Further Details ⭐

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

Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

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. 🎀

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 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).

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

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U**H which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y**H - X*U**H. 🔥

The contents of A on exit are illustrated by the following examples with nb = 2:

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

  (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
  (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
  (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
  (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
  (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
  (  v1  v2  a   a   a  )

where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). ✨ (look at the little diagram, it's literally so cute and it tells you exactly what survived the reduction 🦋)

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Definition at line 210 of file zlabrd.f.

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💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code. (yes bestie, a robot wrote this part, not a human — doxygen said "I got it" 🙈)

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✨ and that's the page! ✨ this absolute icon of a doc came to you from LAPACK, Version 3.12.0, dated at the bottom of the original as part of the /home/abui...RC/zlabrd.f(3) footer — mwah, go reduce something bidiagonal, bestie 💅🎀🫶