SRC/slahr2.f(3)

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💖 SRC/slahr2.f(3) — Library Functions Manual 💖

🌸 NAME 🌸

SRC/slahr2.f

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

subroutine slahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)

SLAHR2 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.

💅 Function/Subroutine Documentation 💅

🧚 subroutine slahr2 🧚

subroutine slahr2 (integer n, integer k, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( nb ) tau, real, dimension( ldt, nb ) t, integer ldt, real, dimension( ldy, nb ) y, integer ldy)

Okay bestie, here’s the tea: SLAHR2 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: 🎀

So the whole vibe is this: SLAHR2 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 SGEHRD. 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. K < N.
  • NB — NB is INTEGER. The number of columns to be reduced.
  • A — A is REAL 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 REAL array, dimension (NB). The scalar factors of the elementary reflectors. See Further Details.
  • T — T is REAL 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 REAL 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 SLAHRD 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 SLAHRD routine.

⚠️ Warning, bestie: this subroutine is not backward compatible with LAPACK-3.0's SLAHRD. No cap, that’s literally in the original.

⭐ 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 slahr2.f.

💜 Author 💜

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