SRC/zgghd3.f(3)

📦
version 3.12.1 · 4 versions
, in arch rolling

💜 NAME 💜

SRC/zgghd3.f — bestie, welcome to the group chat of linear algebra. This is from the Library Functions Manual, and it's the Fortran source page for a little icon called ZGGHD3. ✨

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine zgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
    ldz, work, lwork, info)

ZGGHD3 — that's the whole vibe. 🎀

🌸 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 picture this: 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 form of 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. Iconic behavior, honestly. ✨

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. Slay. 💖

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. Lowkey the main character of this whole routine. 🔥

✨ 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. (That's it, that's the whole description, no notes. 🙈)
  • 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 (MAX(1,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). No cap, that's the whole lore. 📚

Definition at line 225 of file zgghd3.f.

💫 Author 💫

Generated automatically by Doxygen for LAPACK from the source code. 🤖

ok that's the whole page bestie, you ate that 🫶

And for the cute sign-off: this little masterpiece is brought to you by LAPACK, Version 3.12.0, source SRC/zgghd3.f(3) — go forth and reduce some Hessenbergs, ily 💖🌸

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