SRC/zgghrd.f(3)

💅 SRC/zgghrd.f(3) — Library Functions Manual 💅

🌸 NAME 🌸

SRC/zgghrd.f — that's the whole page, bestie. Straight out of the Library Functions Manual, and it's all about the ZGGHRD routine. Let's get into it 💖

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine zgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
    ldz, info)

ZGGHRD

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💖 Function/Subroutine Documentation 💖

🫶 subroutine zgghrd 🫶

subroutine zgghrd (character compq, character compz, integer n, integer ilo,
    integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16,
    dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldq, * ) q,
    integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, integer
    info)

ZGGHRD

✨ Purpose ✨

Okay so ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is a general matrix and B is upper triangular. The generalized eigenvalue problem is

A*x = lambda*B*x

and B is typically made upper triangular by computing its QR factorization and moving the unitary matrix Q to the left side of the equation.

This subroutine simultaneously reduces A to a Hessenberg matrix H:

Q**H*A*Z = H

and transforms B to another upper triangular matrix T:

Q**H*B*Z = T

in order to reduce the problem to its standard form

H*y = lambda*T*y

where y = Z**H*x.

The unitary matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that

Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H

If Q1 is the unitary matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then ZGGHRD reduces the original problem to generalized Hessenberg form.

🎀 Parameters 🎀

• COMPQ — COMPQ is CHARACTER*1:
  • 'N': do not compute Q;
  • 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned;
  • 'V': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned.
• COMPZ — COMPZ is CHARACTER*1:
  • 'N': do not compute Z;
  • 'I': Z is initialized to the unit matrix, and the unitary matrix Z is returned;
  • 'V': Z must contain a unitary matrix Z1 on entry, and the product Z1*Z is returned.
• N — N is INTEGER. The order of the matrices A and B. N >= 0.
• ILO — ILO is INTEGER.
• IHI — IHI is INTEGER.

ILO and IHI mark the rows and columns of A which are to be reduced. 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 ZGGBAL; otherwise they should be set to 1 and N respectively. 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 rest is set to zero.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• B — B is COMPLEX*16 array, dimension (LDB, N). On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**H B Z. The elements below the diagonal are set to zero.
• LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• Q — Q is COMPLEX*16 array, dimension (LDQ, N). On entry, if COMPQ = 'V', the unitary matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the unitary matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'.
• LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
• Z — Z is COMPLEX*16 array, dimension (LDZ, N). On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'.
• LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
• INFO — INFO is INTEGER:
  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.

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👑 Author 👑

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

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🌈 Further Details 🌈

This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in Matrix_Computations_, by Golub and van Loan (Johns Hopkins Press).

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Definition at line 202 of file zgghrd.f.

📝 Author (Doxygen's bit) 📝

Generated automatically by Doxygen for LAPACK from the source code.

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That's the whole page, bestie 💖 Sign-off: LAPACK — Version 3.12.0 — tagged SRC/zgghrd.f(3). That's the footer's source/version slot exactly, and there's no extra date shown on this one. Bye! 🫶✨