SRC/zlahqr.f(3)

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

🌸 NAME

SRC/zlahqr.f — that's the file, bestie. This page lives in the Library Functions Manual section, and we're about to get into the QR algorithm of it all. ✨

🧚 SYNOPSIS

🫶 Functions/Subroutines

  • subroutine zlahqr (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info) — ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. It's giving linear algebra main character energy, fr fr. 🔥

📚 Function/Subroutine Documentation

🫶 subroutine zlahqr

   subroutine zlahqr (logical wantt, logical wantz, integer n, integer ilo,
       integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16,
       dimension( * ) w, integer iloz, integer ihiz, complex*16, dimension(
       ldz, * ) z, integer ldz, integer info)

ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. Slay. 💅

🎯 Purpose

ZLAHQR is an auxiliary routine called by CHSEQR to update the eigenvalues and Schur decomposition already computed by CHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. It's the cleanup crew for that specific block, lowkey iconic. 🫡

🧾 Parameters

  • WANTT — WANTT is LOGICAL. = .TRUE.: the full Schur form T is required; = .FALSE.: only eigenvalues are required. Pick your fighter. ⚔️
  • WANTZ — WANTZ is LOGICAL. = .TRUE.: the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required. No extras, no stress. 💅
  • N — N is INTEGER. The order of the matrix H. N >= 0. Simple as that, bestie. ✨
  • ILO — ILO is INTEGER. That's the whole note for this one. 🎀
  • IHI — IHI is INTEGER. It is assumed that H is already upper triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). ZLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE.. 1 <= ILO <= max(1,IHI); IHI <= N. Respect the bounds, periodt. 🛑✨
  • H — H is COMPLEX*16 array, dimension (LDH,N). On entry, the upper Hessenberg matrix H. On exit, if INFO is zero and if WANTT is .TRUE., then H is upper triangular in rows and columns ILO:IHI. If INFO is zero and if WANTT is .FALSE., then the contents of H are unspecified on exit. The output state of H in case INF is positive is below under the description of INFO. (Yes, the page says INF right there — I'm not fixing it, I'm just reporting it. 💀)
  • LDH — LDH is INTEGER. The leading dimension of the array H. LDH >= max(1,N). Do not skip the max, bestie. 🙈
  • W — W is COMPLEX*16 array, dimension (N). The computed eigenvalues ILO to IHI are stored in the corresponding elements of W. If WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i). Cute little diagonal moment. 🍓
  • ILOZ — ILOZ is INTEGER. That's it, that's the note. 🎀
  • IHIZ — IHIZ is INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. Boundaries are cute when they're respected. 🌸
  • Z — Z is COMPLEX*16 array, dimension (LDZ,N). If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by CHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced. It said what it said. 💅
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= max(1,N). Same energy as LDH, different letter. ✨
  • INFO — INFO is INTEGER.
    • = 0: successful exit. We love a clean finish. 👑
    • > 0: if INFO = i, ZLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30 iterations per eigenvalue; elements i+1:ihi of W contain those eigenvalues which have been successfully computed.
    • If INFO > 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H.
    • If INFO > 0 and WANTT is .TRUE., then on exit (initial value of H)*U = U*(final value of H) where U is an orthogonal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI.
    • If INFO > 0 and WANTZ is .TRUE., then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regardless of the value of WANTT.)

📝 Author

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

🧑‍🔬 Contributors:

  • 02-96 Based on modifications by David Day, Sandia National Laboratory, USA
  • 12-04 Further modifications by Ralph Byers, University of Kansas, USA
  • This is a modified version of ZLAHQR from LAPACK version 3.0. It is (1) more robust against overflow and underflow and (2) adopts the more conservative Ahues & Tisseur stopping criterion (LAWN 122, 1997).

📍 Definition

Definition at line 193 of file zlahqr.f.

🤖 Author

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