SRC/slaqr2.f(3)

💖 NAME 💖

SRC/slaqr2.f — the Library Functions Manual page for this Fortran source file, bestie. That’s the whole NAME section, no cap. 💅

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🎀 SYNOPSIS 🎀

Functions/Subroutines

subroutine slaqr2 (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)

SLAQR2 performs the orthogonal 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 slaqr2 (logical wantt, logical wantz, integer n, integer ktop,
    integer kbot, integer nw, real, dimension( ldh, * ) h, integer ldh,
    integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz,
    integer ns, integer nd, real, dimension( * ) sr, real, dimension( * )
    si, real, dimension( ldv, * ) v, integer ldv, integer nh, real,
    dimension( ldt, * ) t, integer ldt, integer nv, real, dimension( ldwv, *
    ) wv, integer ldwv, real, dimension( * ) work, integer lwork)

SLAQR2 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 SLAQR2 is identical to SLAQR3 except that it avoids recursion by calling SLAHQR instead of SLAQR4. That’s the tea. ☕

Aggressive early deflation, bestie: this subroutine 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. Fingers crossed, etc. 🫶

👑 Parameters 👑

WANTT

WANTT is 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

WANTZ is LOGICAL — If .TRUE., then the orthogonal matrix Z is updated — the source says “so so that”, and honestly same — so that the orthogonal Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced.

N

N is INTEGER — The order of the matrix H and (if WANTZ is .TRUE.) the order of the orthogonal 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 REAL 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 — direct quote from the source, grammar and all — “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, bestie. The source gives no extra description, just the type. 🫡

IHIZ

IHIZ is INTEGER — Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. (yes, two dots in the original, iconic). 1 <= ILOZ <= IHIZ <= N.

Z

Z is REAL 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

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.

ND

ND is INTEGER — The number of converged eigenvalues uncovered by this subroutine.

SR

SR is REAL array, dimension (KBOT) — That’s the whole description in the source; the output locations live under SI because the original puts them there.

SI

SI is REAL 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-ND), 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

V is REAL 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 REAL array, dimension (LDT,NW) — Yep, that’s the whole line. Work array energy, no extra lore.

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 REAL array, dimension (LDWV,NW) — That’s the line, bestie.

LDWV

LDWV is INTEGER — The leading dimension of W just as declared in the calling subroutine. NW <= LDV. (Yes, the original says LDV here, not LDWV; I’m just reporting the page exactly. 💀)

WORK

WORK is REAL 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; SLAQR2 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 📌

Definition at line 275 of file slaqr2.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 page 💖 cute sign-off time: the footer says LAPACK Version 3.12.0 and SRC/slaqr2.f(3). There’s no date printed on the page, so I’m not inventing one — slay responsibly. 🎀✨