SRC/sgehrd.f(3)

💖 SRC/sgehrd.f(3) 💖

okay so welcome to the Library Functions Manual page for SRC/sgehrd.f(3) — yes bestie, the title and manual name are folded right in here like a cute little intro. ✨

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🌸 NAME 🌸

SRC/sgehrd.f

🌈 SYNOPSIS 🌈

🎀 Functions/Subroutines 🎀

subroutine sgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
    SGEHRD

🦋 Function/Subroutine Documentation 🦋

✨ subroutine sgehrd ✨

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

SGEHRD — the main character, literally. 💅

🎀 Purpose 🎀

SGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation:

Q**T * A * Q = H

periodt. 💖

🍓 Parameters 🍓

• N — N is INTEGER. The order of the matrix A. N >= 0.
• ILO — ILO is INTEGER. That’s the whole parameter description in the source, bestie — no extra clauses. 🫶
• 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.
• 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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
• WORK — WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= max(1,N). For good performance, LWORK should generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
• 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 🌈

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).

Fun fact: this file is a slight modification of LAPACK-3.0's SGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See SLAHR2.)

Definition at line 166 of file sgehrd.f.

👑 Author 👑

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💖 and that’s the whole vibe, bestie — LAPACK Version 3.12.0, page SRC/sgehrd.f(3), signing off~ 🎀✨