laqr4(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🦋 laqr4(3) — Library Functions Manual 🦋

okay so bestie, welcome to laqr4(3) — this is part of the Library Functions Manual, and it's basically the group chat of LAPACK's Hessenberg-eigenvalue subroutines: four of them, all matching, all doing the Schur thing. Pull up a chair, we're going in 💅

✨ NAME ✨

  • laqr4 — laqr4: eig of Hessenberg, step in hseqr

like, that's the whole vibe: eigenvalues of a Hessenberg matrix, one step in hseqr. iconic.

🌈 SYNOPSIS 🌈

💫 Functions

  • claqr4 — complex flavor 💜
subroutine claqr4 (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z,
    ldz, work, lwork, info)

CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

  • dlaqr4 — double precision flavor 🍓
subroutine dlaqr4 (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz,
    ihiz, z, ldz, work, lwork, info)

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

  • slaqr4 — single precision flavor 🌸
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.

  • zlaqr4 — double complex flavor 👑
subroutine zlaqr4 (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z,
    ldz, work, lwork, info)

ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

🎀 Detailed Description 🎀

okay so literally nothing here — this section is empty in the source. she's a placeholder. we respect her privacy 🫶

🔥 Function Documentation 🔥

🧚 subroutine claqr4

subroutine claqr4 (logical wantt, logical wantz, integer n, integer ilo,
    integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex,
    dimension( * ) w, integer iloz, integer ihiz, complex, dimension( ldz, *
    ) z, integer ldz, complex, dimension( * ) work, integer lwork, integer
    info)

CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

💖 Purpose

CLAQR4 implements one level of recursion for CLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by CLAQR0 and, for large enough deflation window size, it may be called by CLAQR3. This subroutine is identical to CLAQR0 except that it calls CLAQR2 instead of CLAQR3.

CLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition

H = Z T Z**H

where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors.

Optionally Z may be postmultiplied into an input unitary 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 unitary matrix Q:

A = Q*H*Q**H = (QZ)*H*(QZ)**H
✨ the way this routine just hands you the whole factorization, no cap.

🌸 Parameters

  • 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 description 🙈 — no extra notes here, bestie, she's self-explanatory apparently)
  • 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 CGEBAL, and then passed to CGEHRD when the matrix output by CGEBAL 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 COMPLEX 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 triangular matrix T from the Schur decomposition (the Schur form).
    • If INFO = 0 and WANT 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).
  • W — W is COMPLEX array, dimension (N). The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in W(ILO:IHI). 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 W(i) = H(i,i).
  • ILOZ — ILOZ is INTEGER. (again, no notes attached — matching energy 🎀)
  • 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 COMPLEX 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 COMPLEX 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 CLAQR4 does a workspace query. In this case, CLAQR4 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.
  • INFO — INFO is INTEGER.
    • = 0 : successful exit 🎉
    • > 0 : if INFO = i, CLAQR4 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 unitary 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 unitary matrix in (*) (regardless of the value of WANTT.)

  • If INFO > 0 and WANTZ is .FALSE., then Z is not accessed.
💀 reminder bestie: when INFO > 0 stuff is only partially converged, and H / Z come back in a very specific state. read the bullet you need, don't skim.

👑 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

🍓 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.

Definition at line 246 of file claqr4.f.

🌟 subroutine dlaqr4

subroutine dlaqr4 (logical wantt, logical wantz, integer n, integer ilo,
    integer ihi, double precision, dimension( ldh, * ) h, integer ldh,
    double precision, dimension( * ) wr, double precision, dimension( * )
    wi, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z,
    integer ldz, double precision, dimension( * ) work, integer lwork,
    integer info)

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

💜 Purpose

DLAQR4 implements one level of recursion for DLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by DLAQR0 and, for large enough deflation window size, it may be called by DLAQR3. This subroutine is identical to DLAQR0 except that it calls DLAQR2 instead of DLAQR3.

DLAQR4 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.

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
🌸 note the glow-up from claqr4: this one's real, so it's Z**T not Z**H, and T is quasi-triangular. same energy, different dtype.

⭐ Parameters

  • WANTT — WANTT is LOGICAL.
    • = .TRUE. : the full Schur form T is required;
    • = .FALSE. : only eigenvalues are required.
  • WANTZ — (...and that's exactly where this part of the page cuts off — the rest of WANTZ and every parameter after it is waiting for you in part 2 💖)
  • WANTZ — WANTZ is LOGICAL. Bestie, this is the “do I want the Schur vectors?” switch.
    • = .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. Math girlies stay winning.
  • ILO — ILO is INTEGER. No extra lore here, it just exists.
  • IHI — IHI is INTEGER. Okay so here’s the tea: it’s assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N, and if ILO > 1, then H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to DGEBAL, and then passed to DGEHRD when the matrix output by DGEBAL 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. Precision, periodt.
  • H — H is DOUBLE PRECISION 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. It’s giving “I contain multitudes and also maybe zeros.”
  • LDH — LDH is INTEGER. The leading dimension of the array H. LDH >= max(1,N). Respect the leading dimension or else.
  • WR — WR is DOUBLE PRECISION array, dimension (IHI). Real parts go here, bestie.
  • WI — WI is DOUBLE PRECISION 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). Complex conjugate couples stay adjacent, that’s the rule.
  • ILOZ — ILOZ is INTEGER. Short and sweet.
  • 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. Bounds check slay.
  • Z — Z is DOUBLE PRECISION 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.) The Schur vectors are literally having their main character moment.
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. If WANTZ is .TRUE. then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. Don’t skip this or the vibes will be off.
  • WORK — WORK is DOUBLE PRECISION array, dimension LWORK. On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK. Workspace query era.
  • 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 DLAQR4 does a workspace query. In this case, DLAQR4 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. No cap, the query mode is iconic.
  • INFO — INFO is INTEGER.
    • = 0: successful exit;
    • > 0: if INFO = i, DLAQR4 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.) Rare, but not never, so pay attention.

If INFO > 0 and WANT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigen-values 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 an 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. Leave Z alone, it said no.

👑 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

📚 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.

Definition at line 261 of file dlaqr4.f.

🦋 subroutine slaqr4 🦋

   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. Another one, and yes she’s also booked and busy.

✨ Purpose ✨

SLAQR4 implements one level of recursion for SLAQR0. It is 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.

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.

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. Factorization but make it fashion.

🎀 Parameters 🎀

  • WANTT — WANTT is LOGICAL
    • = .TRUE. : the full Schur form T is required;
    • = .FALSE.: only eigenvalues are required. Choose your fighter.
  • 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. Same energy as before, but now in single precision land.
  • ILO — ILO is INTEGER. Still just vibing.
  • 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. Notice the S in SGEBAL and SGEHRD — we’re in REAL mode, bestie.
  • 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. REAL array, real consequences.
  • LDH — LDH is INTEGER. The leading dimension of the array H. LDH >= max(1,N). Leading dimension stays undefeated.
  • WR — REAL array, dimension (IHI). Short and sweet, no notes. 💖
  • WI — REAL array, dimension (IHI). Okay, this one is carrying the whole imaginary agenda: 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). Iconic behavior, honestly. Periodt. ✨
  • ILOZ — INTEGER. That's the whole entry. Iykyk.
  • IHIZ — INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. Read the bounds twice, bestie.
  • Z — 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.) No cap, that parenthetical matters.
  • LDZ — INTEGER. The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. Lowercase "if" is canon, don't come for me.
  • WORK — REAL array, dimension LWORK. On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK. Workspace query girlies rise.
  • LWORK — 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 if you're just querying, bestie, nothing gets touched.

  • INFO — INTEGER. The full breakdown is literally:
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 eigen-
      values 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 (*) (regard-
      less of the value of WANTT.)

      If INFO > 0 and WANTZ is .FALSE., then Z is not
      accessed.

👑 Author

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

Shoutout to the institutions.

💖 Contributors

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

📚 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.

Definition at line 263 of file slaqr4.f.

subroutine zlaqr4 (logical wantt, logical wantz, integer n, integer ilo,
    integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16,
    dimension( * ) w, integer iloz, integer ihiz, complex*16, dimension(
    ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork,
    integer info)

ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. Main character subroutine energy.

🎯 Purpose

ZLAQR4 implements one level of recursion for ZLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by ZLAQR0 and, for large enough deflation window size, it may be called by ZLAQR3. This subroutine is identical to ZLAQR0 except that it calls ZLAQR2 instead of ZLAQR3.

ZLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors.

Optionally Z may be postmultiplied into an input unitary 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 unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.

🧾 Parameters

  • WANTT — LOGICAL. = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required. Choose your fighter.
  • WANTZ — LOGICAL. = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required. Same vibe, different output.
  • N — INTEGER. The order of the matrix H. N >= 0. Zero is allowed, bestie.
  • ILO — INTEGER. That's it. The whole description. No extra lore.
  • IHI — 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 ZGEBAL, and then passed to ZGEHRD when the matrix output by ZGEBAL 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. This is the boundary tea, don't skip it.
  • H — COMPLEX*16 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 triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and WANT 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. Yes, it can just zero those out. That's the note.

  • LDH — INTEGER. The leading dimension of the array H. LDH >= max(1,N). Respect the leading dimension.
  • W — COMPLEX*16 array, dimension (N). The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in W(ILO:IHI). 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 W(i) = H(i,i). Eigenvalue storage, slay.
  • ILOZ — INTEGER. Again, that's the whole entry. Iykyk.
  • IHIZ — INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. Bounds check, bestie.
  • Z — COMPLEX*16 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.) Same parenthetical, same importance.
  • LDZ — INTEGER. The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. Lowercase "if" again, we keep it exactly.
  • WORK — COMPLEX*16 array, dimension LWORK. On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK. Workspace query, but make it complex.
  • LWORK
  • LWORK — INTEGER, bestie, and it's the dimension of the array WORK. Here's the tea: LWORK >= max(1,N) is sufficient — that's the bare minimum, it'll work — but honestly? LWORK typically needs to be as large as 6*N for optimal performance. So if you want it to actually slay and not chug, size it up. A workspace query to figure out the optimal workspace size is recommended (do it, seriously, it's free real estate).

And if LWORK = -1, then ZLAQR4 does a workspace query. In that case, ZLAQR4 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. That estimate gets returned in WORK(1). No error message related to LWORK is issued by XERBLA. And neither H nor Z are accessed — like, at all. Just a little peek at your parameters, no touching the actual matrices 💅

  • INFO — INTEGER, the vibe report when it's all done:
    • = 0: successful exit ✨
    • > 0: if INFO = i, ZLAQR4 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. fr fr, extremely rare.)

And now the deets for the INFO > 0 situations, because they each hit different:

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 unitary 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 unitary matrix in (*) (regardless of the value of WANTT — yeah, even then, iconic of it).

If INFO > 0 and WANTZ is .FALSE., then Z is not accessed. Not even looked at. Respectful distance 🫶

🌸 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

🎀 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.
Little note: those two references are literally split across parts I and II of the same paper, so if you're reading one, bestie, read both. iykyk 📚

Definition at line 245 of file zlaqr4.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code. So like, a robot wrote this bit and we're just vibing with it 🤖💖

That's the whole page, babe! And for the cute sign-off: this gorgeous thing is stamped <b>LAPACK, Version 3.12.0</b>, sitting pretty on the laqr4(3) footer line — go forth, size your LWORK, and slay 💅✨🦋

📦 where this page comes from