ZGGHD3(3)

💅 ZGGHD3(3) — the manual page, but make it make sense 💅

okay so bestie, welcome to the Library Functions Manual page for ZGGHD3(3). This is the LAPACK subroutine page for the file below, and yes, the source path is part of the whole vibe.

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgghd3.f

That's literally the NAME section: the source file. The manual-page queen we're talking about is ZGGHD3(3). 💖

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ZGGHD3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
    ldz, work, lwork, info)

ZGGHD3

🫶 Function/Subroutine Documentation 🫶

subroutine ZGGHD3 (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, complex*16,
    dimension( * ) work, integer lwork, integer info)

ZGGHD3

🎀 Purpose 🎀

okay so here's the tea, no cap: ZGGHD3 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 has the form:

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 ZGGHD3 reduces the original problem to generalized Hessenberg form.

This is a blocked variant of CGGHRD, using matrix-matrix multiplications for parts of the computation to enhance performance. Slay. 🔥

💜 Parameters 💜

• COMPQ — COMPQ is CHARACTER*1. Here's the menu, bestie:
  • = '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. Same energy, different matrix:
  • = '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.
• 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 >= 1. For optimum performance LWORK >= 6*N*NB, where NB is the optimal blocksize. 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: 🦋

This routine reduces A to Hessenberg form and maintains B in triangular form using a blocked variant of Moler and Stewart's original algorithm, as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti (BIT 2008). 💅

🍓 Definition at line 225 of file zgghd3.f. 🍓

Definition at line 225 of file zgghd3.f.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

---

💖 sign-off: LAPACK Version 3.12.0, source /home/abui...RC/zgghd3.f(3). The original footer doesn't list a date, so I'm not inventing one — this is the whole footer vibe, bestie. Mwah. 💖