ZGEHD2(3)

📦 version 3.12.0, in opensuse-leap 16.0

💜 NAME 💜

ok so bestie, gather round 💖 this page is all about /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgehd2.f — and it lives in the Library Functions Manual, which is where all the good stuff hangs out 🫶

✨ SYNOPSIS ✨

🎀 Functions/Subroutines

subroutine ZGEHD2 (n, ilo, ihi, a, lda, tau, work, info)
    ZGEHD2 reduces a general square matrix to upper Hessenberg form
    using an unblocked algorithm.

🦋 Function/Subroutine Documentation 🦋

so here's the whole glow-up, signature and all 🌸

subroutine ZGEHD2 (integer n, integer ilo, integer ihi, complex*16,
    dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau,
    complex*16, dimension( * ) work, integer info)

ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. no blocking, no shortcuts, just vibes and math 💅

🌟 Purpose 🌟

!>
!> ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
!> by a unitary similarity transformation:  Q**H * A * Q = H .
!>

like literally it takes your matrix and transforms it into H through Q**H * A * Q = H, iconic behavior ngl 👑

🧚 Parameters 🧚

💫 N

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>

the order of the matrix A, and it's gotta be N >= 0 ✨

💫 ILO

!>          ILO is INTEGER
!>

that's it, that's the whole description, she's mysterious 🙈

💫 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 ZGEBAL; otherwise they should be
!>          set to 1 and N respectively. See Further Details.
!>          1 <= ILO <= IHI <= max(1,N).
!>

ok so heads up bestie — A is assumed to already be upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to ZGEBAL; if you don't have that, set them to 1 and N respectively. See Further Details 👀 and yeah, 1 <= ILO <= IHI <= max(1,N) fr fr

💫 A

!>          A is COMPLEX*16 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 unitary matrix Q as a product of elementary
!>          reflectors. See Further Details.
!>

COMPLEX*16 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 get overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal — together with the array TAU — represent the unitary 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).
!>

the leading dimension of the array A, and LDA >= max(1,N) — don't skip this check, she WILL notice 😭

💫 TAU

!>          TAU is COMPLEX*16 array, dimension (N-1)
!>          The scalar factors of the elementary reflectors (see Further
!>          Details).
!>

COMPLEX*16 array, dimension (N-1), holding the scalar factors of the elementary reflectors (see Further Details 🎀)

💫 WORK

!>          WORK is COMPLEX*16 array, dimension (N)
!>

COMPLEX*16 array, dimension (N). that's it, she just works 💅

💫 INFO

!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value.
!>

the receipts 📋 = 0 means successful exit, < 0 means if INFO = -i then the i-th argument had an illegal value. so if you see a negative number, that's your sign bestie 🙈

👑 Author 👑

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

look at that squad 🫶

🔥 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**H
!>
!>  where tau is a complex scalar, and v is a complex 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).
!>

so the tea ☕: Q is a product of (ihi-ilo) elementary reflectors, Q = H(ilo) H(ilo+1) . . . H(ihi-1). each H(i) looks like H(i) = I - tau * v * v**H, where tau is a complex scalar and v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) gets stored on exit in A(i+2:ihi,i), and tau in TAU(i) 🌈

in the picture above, a is an element of the original matrix A, h is a modified element of the upper Hessenberg matrix H, and vi is an element of the vector defining H(i). the before/after is giving transformation arc, honestly obsessed 😭💖

Definition at line 148 of file zgehd2.f. 📍

🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code. 🤖 (yes the robots wrote this one, slay)

and that's the whole page bestie!! 💜 it's LAPACK, Version 3.12.0, straight out of /home/abui...RC/zgehd2.f(3) — go forth and Hessenberg-ify responsibly 🎀✨🦋

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