SRC/zlahr2.f(3)

💅 NAME 💅

okay bestie, this is SRC/zlahr2.f from the LAPACK Library Functions Manual. That's the whole name situation — now let's get into the serve. 🫶

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✨ SYNOPSIS ✨

🌈 Functions/Subroutines 🌈

• subroutine zlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy) — ZLAHR2 reduces however many first columns you specified of a general rectangular matrix A so that everything below the specified subdiagonal is zero, and it returns the auxiliary matrices you need to apply that transformation to the unreduced part of A. 💖

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🌸 Function/Subroutine Documentation 🌸

💜 subroutine zlahr2 💜

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. No cap, that's the whole assignment. ✨

🎯 Purpose 🎯

okay so picture this: ZLAHR2 takes the first NB columns of a complex general n-BY-(n-k+1) matrix A and reduces them so that the elements below the k-th subdiagonal are zero. The reduction is done 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 it also returns the matrix Y = A * V * T. This is an auxiliary routine called by ZGEHRD. iconic behavior, honestly. 👑

🧸 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. K < N. Keep that in mind, bestie. 🌸
• 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.

🧚 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). 🧚‍♀️

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.

⚠️ heads up, bestie: this subroutine is not backward compatible with LAPACK-3.0's ZLAHRD. That's a real caveat, not a vibe. 💀

📚 References 📚

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

Definition at line 180 of file zlahr2.f. 📍

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🫶 Author 🫶

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byeee 💋 — footer check: LAPACK Version 3.12.0, page SRC/zlahr2.f(3). That's the source/version vibe, periodt. 🫶