ZLAHR2(3)

📦 version 3.12.0, in opensuse-leap 16.0

✨ ZLAHR2(3) ✨ — the LAPACK glow-up you didn't know you needed

okay bestie, buckle up 💅 this page lives in the Library Functions Manual for LAPACK, and it's literally the doc for zlahr2.f — yes, the whole path is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlahr2.f. That's the file. That's the icon. 🎀

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlahr2.f

like, that's it, that's the name, she's a whole path and she's proud of it 💖

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine ZLAHR2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)

ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. no cap, that's the whole gig 🔥

🌈 Function/Subroutine Documentation 🌈

subroutine ZLAHR2 (integer n, integer k, integer nb, complex*16, dimension(
    lda, * ) a, integer lda, complex*16, dimension( nb ) tau, complex*16,
    dimension( ldt, nb ) t, integer ldt, complex*16, dimension( ldy, nb ) y,
    integer ldy)

ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. (yes, same sentence, she said it twice and honestly? she ate 💅)

💜 Purpose 💜

ZLAHR2 reduces the first NB columns of a complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by a unitary similarity transformation Q**H * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T. 🫶

This is an auxiliary routine called by ZGEHRD. iykyk ✨

🎀 Parameters 🎀

  • N — N is INTEGER, the order of the matrix A. 👑
  • K — K is INTEGER, the offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. and bestie, note this: K < N. ⚠️
  • NB — NB is INTEGER, the number of columns to be reduced. 🍓
  • A — A is COMPLEX*16 array, dimension (LDA,N-K+1). On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 💖
  • LDA — LDA is INTEGER, the leading dimension of the array A. LDA >= max(1,N). 🚨
  • TAU — TAU is COMPLEX*16 array, dimension (NB), the scalar factors of the elementary reflectors. See Further Details. ✨
  • T — T is COMPLEX*16 array, dimension (LDT,NB), the upper triangular matrix T. 💜
  • LDT — LDT is INTEGER, the leading dimension of the array T. LDT >= NB. 🚨
  • Y — Y is COMPLEX*16 array, dimension (LDY,NB), the n-by-nb matrix Y. 🦋
  • LDY — LDY is INTEGER, the leading dimension of the array Y. LDY >= N. 🚨

👑 Author 👑

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

the whole squad, fr fr 🫶

🔥 Further Details 🔥

The matrix Q is represented as a product of nb elementary reflectors:

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i). 💫

The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form:

A := (I - V*T*V**H) * (A - Y*V**H).

The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:

   ( a   a   a   a   a )
   ( a   a   a   a   a )
   ( a   a   a   a   a )
   ( h   h   a   a   a )
   ( v1  h   a   a   a )
   ( v1  v2  a   a   a )
   ( v1  v2  a   a   a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). 🌸

⚠️ okay so listen — this subroutine is a slight modification of LAPACK-3.0's ZLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3.0's ZLAHRD routine. (This subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.) like, she changed her whole vibe and did NOT leave a forwarding address 💀

🧚 References 🧚

Gregorio Quintana-Orti and Robert van de Geijn, "Improving theperformance of reduction to Hessenberg form," ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

(yes, "theperformance" is exactly how it's spelled in the source — we don't fix, we respect the icon 🙈)

Definition at line 180 of file zlahr2.f. ⭐

💖 Author (the other one) 💖

Generated automatically by Doxygen for LAPACK from the source code. beep boop, no human hands touched this one, she was born in a build script 🎀

and that's the whole page bestie!! 🌈✨ this was LAPACK, Version 3.12.0, signed off from /home/abui...RC/zlahr2.f(3) — mwah, go reduce some Hessenberg form, you've got this 💅👑🦋

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