ZGEHRD(3)

💅 ZGEHRD(3) — Library Functions Manual ✨

Okay so bestie, welcome to the zgehrd page in the Library Functions Manual. This is the LAPACK file at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgehrd.f, and it is literally here to reduce a complex general matrix. 🦋

🌸 NAME 🌸

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

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

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

ZGEHRD

💖 Function/Subroutine Documentation 💖

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

ZGEHRD

🌈 Purpose 🌈

Bestie, ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation:

Q**H * A * Q = H .

The original really does say “an unitary” — we are not fixing it, we are respecting the icon. 💅

🦋 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.
• 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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
• WORK — WORK is COMPLEX*16 array, dimension (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**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).

This file is a slight modification of LAPACK-3.0's ZGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See ZLAHR2.)

Definition at line 166 of file zgehrd.f.

⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code.

---

💖 Cute sign-off: LAPACK, Version 3.12.0, source /home/abui...RC/zgehrd.f(3). Date? Not on this footer, bestie — just pure version energy. Bye and slay! 💅✨