SRC/claqr3.f(3)

📦
version 3.12.1 · 7 versions
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🌸 NAME 🌸

Okay so this page is SRC/claqr3.f(3), straight out of the Library Functions Manual — and the NAME section is literally just SRC/claqr3.f. No extra lore, bestie. 🎀

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

       subroutine claqr3 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz,
           z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork)

CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Slay, honestly. 💅

🧚 Function/Subroutine Documentation 🧚

subroutine claqr3 (logical wantt, logical wantz, integer n, integer ktop,
     integer kbot, integer nw, complex, dimension( ldh, * ) h, integer ldh,
     integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz,
     integer ns, integer nd, complex, dimension( * ) sh, complex, dimension(
     ldv, * ) v, integer ldv, integer nh, complex, dimension( ldt, * ) t,
     integer ldt, integer nv, complex, dimension( ldwv, * ) wv, integer ldwv,
     complex, dimension( * ) work, integer lwork)

CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). 💖

🌟 Purpose 🌟

Aggressive early deflation, periodt:

CLAQR3 accepts as input an upper Hessenberg matrix H and performs an unitary similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been over-written by a new Hessenberg matrix that is a perturbation of an unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. That's the goal, bestie — zero subdiagonal entries where possible. 🙏

💜 Parameters 💜

  • WANTT — WANTT is LOGICAL. If .TRUE., then the Hessenberg matrix H is fully updated so that the triangular Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then only enough of H is updated to preserve the eigenvalues. 🫶
  • WANTZ — WANTZ is LOGICAL. If .TRUE., then the unitary matrix Z is updated so so that the unitary Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced. Yes, the “so so” is in the original vibe. 😭
  • N — N is INTEGER. The order of the matrix H and (if WANTZ is .TRUE.) the order of the unitary matrix Z. 🔢
  • KTOP — KTOP is INTEGER. It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix. 🧩
  • KBOT — KBOT is INTEGER. It is assumed without a check that either KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix. 🫡
  • NW — NW is INTEGER. Deflation window size. 1 <= NW <= (KBOT-KTOP+1). 🪟
  • H — H is COMPLEX array, dimension (LDH,N). On input the initial N-by-N section of H stores the Hessenberg matrix undergoing aggressive early deflation. On output H has been transformed by a unitary similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries. ✨
  • LDH — LDH is INTEGER. Leading dimension of H just as declared in the calling subroutine. N <= LDH. 📏
  • ILOZ — ILOZ is INTEGER. That's it. That's the description. 💀
  • IHIZ — IHIZ is INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N. 🎯
  • Z — Z is COMPLEX array, dimension (LDZ,N). IF WANTZ is .TRUE., then on output, the unitary similarity transformation mentioned above has been accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ is .FALSE., then Z is unreferenced. 🦋
  • LDZ — LDZ is INTEGER. The leading dimension of Z just as declared in the calling subroutine. 1 <= LDZ. 📐
  • NS — NS is INTEGER. The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine. Yep, it says SR and SI right there. 🙈
  • ND — ND is INTEGER. The number of converged eigenvalues uncovered by this subroutine. ✅
  • SH — SH is COMPLEX array, dimension (KBOT). On output, approximate eigenvalues that may be used for shifts are stored in SH(KBOT-ND-NS+1) through SR(KBOT-ND). Converged eigenvalues are stored in SH(KBOT-ND+1) through SH(KBOT). (And yes, the source text says SR(KBOT-ND) in that first range.) 🌈
  • V — V is COMPLEX array, dimension (LDV,NW). An NW-by-NW work array. 🛠️
  • LDV — LDV is INTEGER. The leading dimension of V just as declared in the calling subroutine. NW <= LDV. 📏
  • NH — NH is INTEGER. The number of columns of T. NH >= NW. 🗂️
  • T — T is COMPLEX array, dimension (LDT,NW). That's the whole description, bestie. 🎀
  • LDT — LDT is INTEGER. The leading dimension of T just as declared in the calling subroutine. NW <= LDT. 📐
  • NV — NV is INTEGER. The number of rows of work array WV available for workspace. NV >= NW. 🧵
  • WV — WV is COMPLEX array, dimension (LDWV,NW). Again: that's the whole description. 💅
  • LDWV — LDWV is INTEGER. The leading dimension of W just as declared in the calling subroutine. NW <= LDV. (Yes, it says W and LDV here — preserving the original exactly.) 🫠
  • WORK — WORK is COMPLEX array, dimension (LWORK). On exit, WORK(1) is set to an estimate of the optimal value of LWORK for the given values of N, NW, KTOP and KBOT. 🔮
  • LWORK — LWORK is INTEGER. The dimension of the work array WORK. LWORK = 2*NW suffices, but greater efficiency may result from larger values of LWORK. 🚀
If LWORK = -1, then a workspace query is assumed; CLAQR3 only estimates the optimal workspace size for the given values of N, NW, KTOP and KBOT. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed. ⚠️

👑 Author 👑

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

🌈 Contributors: 🌈

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA. Absolute icons. 💖

Definition at line 263 of file claqr3.f.

🎀 Author 🎀

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Byeee bestie 💖 — SRC/claqr3.f(3), LAPACK Version 3.12.0, signing off with maximum sparkle. 🎀✨

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