CGEHRD(3)

CGEHRD(3) — 📚 Library Functions Manual ✨

NAME

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SYNOPSIS

Functions/Subroutines

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

CGEHRD — that’s the main character 💫.

Function/Subroutine Documentation

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

CGEHRD again, because the docs said so 🫶.

Purpose

okay so the whole point: CGEHRD reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q**H * A * Q = H. That’s the vibe. ✨

Parameters

• N — N is INTEGER. The order of 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 CGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. Constraints: 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
• A — A is COMPLEX 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 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 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 CGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See CLAHR2.)

Definition at line 166 of file cgehrd.f.

Author — generated automatically edition

Generated automatically by Doxygen for LAPACK from the source code.

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