SLAQR4(3)

🌸 SLAQR4(3) — slaqr4.f, aka the LAPACK subroutine that's literally the main character of the multi-shift QR algorithm 💅

okay so bestie, you know how the file lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/slaqr4.f? that's the whole NAME of this page, no cap — it's a Fortran source file in LAPACK, and this is its manual page (the man name is slaqr4.f(3), keep that in your back pocket 🫶).

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✨ SYNOPSIS ✨

Functions/Subroutines 🦋

subroutine SLAQR4 (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz,
    ihiz, z, ldz, work, lwork, info)

SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. that's the whole vibe 🔥

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👑 Function/Subroutine Documentation

the full signature, iconically long 💖

subroutine SLAQR4 (logical wantt, logical wantz, integer n, integer ilo,
    integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension(
    * ) wr, real, dimension( * ) wi, integer iloz, integer ihiz, real,
    dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer
    lwork, integer info)

SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. (yes i said it twice, it's that important ✨)

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💜 Purpose

okay so here's what's actually going on, listen up:

• SLAQR4 implements one level of recursion for SLAQR0. it's a complete implementation of the small bulge multi-shift QR algorithm. it may be called by SLAQR0 and, for large enough deflation window size, it may be called by SLAQR3. this subroutine is identical to SLAQR0 except that it calls SLAQR2 instead of SLAQR3. tiny detail, huge consequences, iykyk 🎀
• SLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. iconic behavior, honestly 👑
• optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. slay 🌈

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🌸 Parameters

the cast of characters, one by one 🧚

• WANTT — WANTT is LOGICAL
  • = .TRUE. : the full Schur form T is required;
  • = .FALSE.: only eigenvalues are required.
• WANTZ — WANTZ is LOGICAL
  • = .TRUE. : the matrix of Schur vectors Z is required;
  • = .FALSE.: Schur vectors are not required.
• N — N is INTEGER
  • The order of the matrix H. N >= 0.
• ILO — ILO is INTEGER (that's it, that's the whole description, short king 💀)
• IHI — IHI is INTEGER
  • It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO > 1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to SGEBAL, and then passed to SGEHRD when the matrix output by SGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N > 0, then 1 <= ILO <= IHI <= N. If N = 0, then ILO = 1 and IHI = 0. 🫶
• H — H is REAL array, dimension (LDH,N)
  • On entry, the upper Hessenberg matrix H.
  • On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.)
  • This subroutine may explicitly set H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
• LDH — LDH is INTEGER
  • The leading dimension of the array H. LDH >= max(1,N).
• WR — WR is REAL array, dimension (IHI) (again, short and mysterious 💅)
• WI — WI is REAL array, dimension (IHI)
  • The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i). 🍓
• ILOZ — ILOZ is INTEGER (mystery girl 💀)
• 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.
• Z — Z is REAL array, dimension (LDZ,IHI)
  • If WANTZ is .FALSE., then Z is not referenced.
  • If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO > 0 is given under the description of INFO below.)
• LDZ — LDZ is INTEGER
  • The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
• WORK — WORK is REAL array, dimension LWORK
  • On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK.
• LWORK — LWORK is INTEGER
  • The dimension of the array WORK. LWORK >= max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. 👑
  • If LWORK = -1, then SLAQR4 does a workspace query. In this case, SLAQR4 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed. (so efficient, so demure ✨)
• INFO — INFO is INTEGER
  • (the source literally has !> \verbatim sitting in here too, keeping it for the historical record, don't @ me 🙈)
  • INFO is INTEGER
    • = 0: successful exit
    • > 0: if INFO = i, SLAQR4 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) 💀
  • If INFO > 0 and WANT 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 a 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(ILO:IHI,ILOZ:IHIZ)
 =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

• where U is the orthogonal matrix in (*) (regardless of the value of WANTT.)
• If INFO > 0 and WANTZ is .FALSE., then Z is not accessed.

🎀 heads up bestie: notice that one of those says WANT, not WANTT. that's exactly how the source has it. i'm not fixing it, i'm just reporting the tea. ☕

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🦋 Author

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

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🧚 Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA. legends, honestly 🫶

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⭐ References:

• K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.
• K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

like, they wrote papers about this. academic royalty fr fr 👑

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🔥 Definition at line 263 of file slaqr4.f.

(and the whole page itself was generated automatically by Doxygen for LAPACK from the source code — so if the prose feels a little feral, blame the generator, not me 💅✨)

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and that's the page!! LAPACK, Version 3.12.0, dated /home/abui...RC/slaqr4.f(3) — mwah, go compute some eigenvalues bestie, you've got this 🎀💖🌟