SRC/sgehd2.f(3)

🌸 NAME

SRC/sgehd2.f — from the Library Functions Manual, bestie, and this is the LAPACK page for the sgehd2 subroutine. 💖

✨ SYNOPSIS

🎀 Functions/Subroutines

• subroutine sgehd2 (n, ilo, ihi, a, lda, tau, work, info) — SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. slay.

💖 Function/Subroutine Documentation

subroutine sgehd2 (integer n, integer ilo, integer ihi, real, dimension(
       lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * )
       work, integer info)

SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. Periodt.

🎯 Purpose

okay so the whole point: SGEHD2 reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q**T * A * Q = H . iconic.

🧸 Parameters

• N — N is INTEGER. The order of the matrix A. N >= 0.
• ILO — ILO is INTEGER.
• IHI — IHI is INTEGER. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to SGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= max(1,N).
• A — A is REAL array, dimension (LDA,N). On entry, the n by n general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• TAU — TAU is REAL array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details).
• WORK — WORK is REAL array, dimension (N).
• INFO — INFO is INTEGER
  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.

👑 Author

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

🦋 Further Details

okay bestie, here's the lore: the matrix Q is represented as a product of (ihi-ilo) elementary reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         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).

Definition at line 148 of file sgehd2.f.

🌈 Author

Generated automatically by Doxygen for LAPACK from the source code.

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This page is from LAPACK Version 3.12.0, source SRC/sgehd2.f(3) — that's the footer receipt, bestie (no date on this one), and yes she ate. 💅✨