SRC/dlahr2.f(3)

💖 dlahr2, but make it main-character 💖

okay so bestie, this is the SRC/dlahr2.f(3) page from the LAPACK Library Functions Manual, and it's all about dlahr2 — a tiny auxiliary routine with serious behind-the-scenes energy. She reduces columns, builds reflectors, and hands back the auxiliary matrices. 💅

🌸 NAME 🌸

SRC/dlahr2.f — the source page for the dlahr2 subroutine, aka the little auxiliary routine that helps DGEHRD do its thing. ✨

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

• subroutine dlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy) — DLAHR2 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.

subroutine dlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)

🧚 Function/Subroutine Documentation 🧚

💫 subroutine dlahr2 💫

subroutine dlahr2 (integer n, integer k, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( nb ) tau, double precision, dimension( ldt, nb ) t, integer ldt, double precision, dimension( ldy, nb ) y, integer ldy)

DLAHR2 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. 🫶

🎯 Purpose 🎯

DLAHR2 reduces the first NB columns of a real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q**T * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.

This is an auxiliary routine called by DGEHRD.

🎀 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.
• NB — NB is INTEGER. The number of columns to be reduced.
• A — A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB). The scalar factors of the elementary reflectors. See Further Details.
• T — T is DOUBLE PRECISION 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 DOUBLE PRECISION 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**T

where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).

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 DLAHRD 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 DLAHRD routine. (This subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)

🚨 bestie, this one is not backward compatible with LAPACK-3.0's DLAHRD — the A(1:K,2:NB) entries are different, and that's real. 💀

📚 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 dlahr2.f.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

---

💖 that's the whole page, bestie! signed with LAPACK Version 3.12.0, source SRC/dlahr2.f(3). stay slay, and may your subdiagonals be zero. ✨