DGEHRD(3)

💅 DGEHRD(3) 💅

okay bestie, welcome to the Library Functions Manual corner of the group chat 🫶 this one is all about DGEHRD, and yes it’s giving ✨ linear algebra main character energy ✨.

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✨ NAME ✨

The NAME is literally this file path, no cap:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dgehrd.f

🌈 SYNOPSIS 🌈

Functions/Subroutines:

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

and the function/subroutine is DGEHRD itself 💖

📚 FUNCTION/SUBROUTINE DOCUMENTATION 📚

subroutine DGEHRD (integer n, integer ilo, integer ihi, double precision,
    dimension( lda, * ) a, integer lda, double precision, dimension( * )
    tau, double precision, dimension( * ) work, integer lwork, integer info)

DGEHRD 💅

🎯 PURPOSE: 🎯

okay so the purpose is: DGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation:

Q**T * A * Q = H .

periodt. That’s the whole glow-up. ✨

🧸 PARAMETERS 🧸

• N — N is INTEGER. It’s the order of the matrix A. N >= 0. 💖
• ILO — ILO is INTEGER. That’s the whole description here, bestie; no extra lore. 🫶
• 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 DGEBAL; 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 DOUBLE PRECISION array, dimension (LDA,N). On entry, it’s 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. It’s the leading dimension of the array A. LDA >= max(1,N). 🎀
• TAU — TAU is DOUBLE PRECISION array, dimension (N-1). These are 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 DOUBLE PRECISION array, dimension (LWORK). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 💜
• LWORK — LWORK is INTEGER. It’s 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 👑

Credit where it’s due, bestie:

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

🔍 FURTHER DETAILS: 🔍

okay so here’s the tea ☕ 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). 🌈

This file is a slight modification of LAPACK-3.0's DGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See DLAHR2.) 🔥

📍 DEFINITION AT LINE 📍

Definition at line 166 of file dgehrd.f. 💅

🪄 AUTHOR 🪄

Generated automatically by Doxygen for LAPACK from the source code. 🤖✨

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okay bestie that’s the whole page 💖 the footer’s source is LAPACK, the version is 3.12.0, and the path is /home/abui...RC/dgehrd.f(3) — no date printed, just vibes — mwah 💅🎀