DLAQR3(3)

🎀 DLAQR3(3) 🎀

okay bestie, welcome to the Library Functions Manual page for DLAQR3(3) — it’s literally giving “aggressive early deflation” and we’re gonna get through every single argument together. 💅✨

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlaqr3.f — that’s the file this whole page is named after. No cap. 🫶

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🌸 SYNOPSIS 🌸

💫 Functions/Subroutines 💫

       subroutine DLAQR3 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz,
           z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work,
           lwork)

DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix — aka aggressive early deflation. Yes bestie, that’s the whole vibe. 🦋

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🔥 Function/Subroutine Documentation 🔥

🧚 subroutine DLAQR3 🧚

   subroutine DLAQR3 (logical wantt, logical wantz, integer n, integer ktop,
       integer kbot, integer nw, double precision, dimension( ldh, * ) h,
       integer ldh, integer iloz, integer ihiz, double precision, dimension(
       ldz, * ) z, integer ldz, integer ns, integer nd, double precision,
       dimension( * ) sr, double precision, dimension( * ) si, double
       precision, dimension( ldv, * ) v, integer ldv, integer nh, double
       precision, dimension( ldt, * ) t, integer ldt, integer nv, double
       precision, dimension( ldwv, * ) wv, integer ldwv, double precision,
       dimension( * ) work, integer lwork)

Same energy as above: DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). 💖

🎯 Purpose: 🎯

okay so here’s the whole purpose, bestie: this is aggressive early deflation. DLAQR3 accepts as input an upper Hessenberg matrix H and performs an orthogonal 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 an orthogonal similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. We are all praying for those zeros, ngl 😭✨

🔍 Parameters 🔍

• WANTT (LOGICAL) — if .TRUE., then the Hessenberg matrix H is fully updated so that the quasi-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 orthogonal matrix Z is updated so so that the orthogonal 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 orthogonal 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 (DOUBLE PRECISION 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 an orthogonal similarity transformation, perturbed, and the 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) — it’s an INTEGER, bestie. The row-range details are right below in IHIZ. 💅
• IHIZ (INTEGER) — specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
• Z (DOUBLE PRECISION array, dimension (LDZ,N)) — if WANTZ is .TRUE., then on output, the orthogonal 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.
• SR (DOUBLE PRECISION array, dimension (KBOT)) — real-parts array. The exact storage ranges are described with SI below. 🍓
• SI (DOUBLE PRECISION array, dimension (KBOT)) — on output, the real and imaginary parts of approximate eigenvalues that may be used for shifts are stored in SR(KBOT-ND-NS+1) through SR(KBOT-ND) and SI(KBOT-ND-NS+1) through SI(KBOT), respectively. The real and imaginary parts of converged eigenvalues are stored in SR(KBOT-ND+1) through SR(KBOT) and SI(KBOT-ND+1) through SI(KBOT), respectively.
• V (DOUBLE PRECISION 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 (DOUBLE PRECISION array, dimension (LDT,NW)) — workspace array, described by LDT and NH. 🌈
• 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 (DOUBLE PRECISION array, dimension (LDWV,NW)) — workspace array, described by LDWV and NV. 🎀
• LDWV (INTEGER) — the leading dimension of W just as declared in the calling subroutine. NW <= LDV. (Yes, it says W and LDV — keeping the source exactly, bestie. 💀)
• WORK (DOUBLE PRECISION 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; DLAQR3 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.

Definition at line 272 of file dlaqr3.f.

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📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

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okay bestie, that’s the whole DLAQR3(3) page. The footer vibes are LAPACK Version 3.12.0 with the source path /home/abui...RC/dlaqr3.f(3) — that’s the version/source/date line as given, no extra date string, periodt. We made it through every parameter and honestly? Iconic. 💅✨🎀