SRC/zgehd2.f(3)

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🎀 SRC/zgehd2.f 🎀

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✨ NAME ✨

SRC/zgehd2.f

🦋 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 💖

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.

🌟 Purpose 🌟

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

🍓 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 ZGEBAL; otherwise they should be set to 1 and N respectively. See Further Details.

1 <= ILO <= IHI <= max(1,N).

🧸 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.

🧸 LDA

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

🧸 TAU

TAU is COMPLEX*16 array, dimension (N-1)

The scalar factors of the elementary reflectors (see Further Details).

🧸 WORK

WORK is COMPLEX*16 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 🔥

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).

Definition at line 148 of file zgehd2.f.

🖨️ Author 🖨️

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