SRC/zlaqr3.f(3)

🌸 NAME 🌸

okay so this page is from the Library Functions Manual, and the main character is SRC/zlaqr3.f. That's the whole NAME section, bestie — short, sweet, iconic.

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine zlaqr3 (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)
    ZLAQR3 performs the unitary similarity transformation of a
    Hessenberg matrix to detect and deflate fully converged eigenvalues
    from a trailing principal submatrix (aggressive early deflation).

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💖 Function/Subroutine Documentation 💖

subroutine zlaqr3 (logical wantt, logical wantz, integer n, integer ktop,
    integer kbot, integer nw, complex*16, dimension( ldh, * ) h, integer
    ldh, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z,
    integer ldz, integer ns, integer nd, complex*16, dimension( * ) sh,
    complex*16, dimension( ldv, * ) v, integer ldv, integer nh, complex*16,
    dimension( ldt, * ) t, integer ldt, integer nv, complex*16, dimension(
    ldwv, * ) wv, integer ldwv, complex*16, dimension( * ) work, integer
    lwork)

ZLAQR3 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 🎯

okay so this is the aggressive early deflation moment. ZLAQR3 accepts as input an upper Hessenberg matrix H and performs a unitary similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output, H has been overwritten by a new Hessenberg matrix that is a perturbation of a unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. Fingers crossed, bestie. 🫶

🔍 Parameters 🔍

• WANTT — 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 — LOGICAL. If .TRUE., then the unitary matrix Z is updated so that the unitary Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced.
• N — INTEGER. The order of the matrix H and (if WANTZ is .TRUE.) the order of the unitary matrix Z.
• KTOP — 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 — 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 — INTEGER. Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
• H — COMPLEX*16 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 returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries.
• LDH — INTEGER. Leading dimension of H just as declared in the calling subroutine. N <= LDH.
• ILOZ — INTEGER. That's the whole note here, bestie.
• IHIZ — INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
• Z — COMPLEX*16 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 — INTEGER. The leading dimension of Z just as declared in the calling subroutine. 1 <= LDZ.
• NS — INTEGER. The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine.
• ND — INTEGER. The number of converged eigenvalues uncovered by this subroutine.
• SH — COMPLEX*16 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).
• V — COMPLEX*16 array, dimension (LDV,NW). An NW-by-NW work array.
• LDV — INTEGER. The leading dimension of V just as declared in the calling subroutine. NW <= LDV.
• NH — INTEGER. The number of columns of T. NH >= NW.
• T — COMPLEX*16 array, dimension (LDT,NW).
• LDT — INTEGER. The leading dimension of T just as declared in the calling subroutine. NW <= LDT.
• NV — INTEGER. The number of rows of work array WV available for workspace. NV >= NW.
• WV — COMPLEX*16 array, dimension (LDWV,NW).
• LDWV — INTEGER. The leading dimension of W just as declared in the calling subroutine. NW <= LDV.
• WORK — COMPLEX*16 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 — 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; ZLAQR3 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.

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👑 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

Definition at line 264 of file zlaqr3.f.

📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole tea, bestie 💅 This little icon is from LAPACK Version 3.12.0, and the source file is SRC/zlaqr3.f(3). slay. ✨🦋