laqr2(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

✨ NAME ✨

laqr2 — laqr2: step in hseqr. Bestie, this is the laqr2(3) page, and the manual name is folded in right here because we are not copying that boring header line. 💅

🛠️ SYNOPSIS 🛠️

💫 Functions

💖 claqr2

subroutine claqr2 (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)

CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Periodt.

🦋 dlaqr2

subroutine dlaqr2 (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)

DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

🌈 slaqr2

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).

👑 zlaqr2

subroutine zlaqr2 (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)

ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

🌸 Detailed Description 🌸

This section is literally just a heading in the original — no extra prose, no side quest. The real tea is in Function Documentation. 🍓

🫶 Function Documentation 🫶

🦋 claqr2

subroutine claqr2 (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)

CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). No cap, that's the whole vibe.

💡 Purpose

CLAQR2 is identical to CLAQR3 except that it avoids recursion by calling CLAHQR instead of CLAQR4. Slay.

Aggressive early deflation:

This subroutine accepts as input an upper Hessenberg matrix H and performs an 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 an unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. We are literally rooting for those zeros.

🎀 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. Iconic behavior.
  • 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 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 all the original gives us here — no extra description, no cap. 🫶
  • IHIZ — INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
  • Z — COMPLEX 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 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). Yes, the original says SR there — we keep it exactly, no editing. 💀
  • V — COMPLEX 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 array, dimension (LDT,NW). No extra description in the original.
  • 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 array, dimension (LDWV,NW). No extra description in the original.
  • LDWV — INTEGER. The leading dimension of W just as declared in the calling subroutine. NW <= LDV. Yes, it says W and LDV — keeping it exactly. 🙈
  • WORK — COMPLEX 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; CLAQR2 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 266 of file claqr2.f.

🌈 dlaqr2

subroutine dlaqr2 (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)

DLAQR2 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

DLAQR2 is identical to DLAQR3 except that it avoids recursion by calling DLAHQR instead of DLAQR4. Same energy, different letter.

Aggressive early deflation:

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.

🍓 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 is LOGICAL. If .TRUE., then the orthogonal matrix Z is updated so that the orthogonal Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced. 🦋
  • N is INTEGER. The order of the matrix H and (if WANTZ is .TRUE.) the order of the orthogonal matrix Z. 💅
  • 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 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 is INTEGER. Deflation window size. 1 <= NW <= (KBOT-KTOP+1). 🎀
  • H is 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 is INTEGER. Leading dimension of H just as declared in the calling subroutine. N <= LDH.
  • ILOZ is INTEGER. 💜
  • IHIZ is INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
  • Z is 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 is INTEGER. The leading dimension of Z just as declared in the calling subroutine. 1 <= LDZ.
  • 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 is INTEGER. The number of converged eigenvalues uncovered by this subroutine. 🌟
  • SR is DOUBLE PRECISION array, dimension (KBOT).
  • SI is 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-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 is DOUBLE PRECISION array, dimension (LDV,NW). An NW-by-NW work array.
  • LDV is INTEGER. The leading dimension of V just as declared in the calling subroutine. NW <= LDV.
  • NH is INTEGER. The number of columns of T. NH >= NW.
  • T is DOUBLE PRECISION array, dimension (LDT,NW).
  • LDT is INTEGER. The leading dimension of T just as declared in the calling subroutine. NW <= LDT.
  • NV is INTEGER. The number of rows of work array WV available for workspace. NV >= NW.
  • WV is DOUBLE PRECISION array, dimension (LDWV,NW).
  • LDWV is INTEGER. The leading dimension of W just as declared in the calling subroutine. NW <= LDV.
  • WORK is 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 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; DLAQR2 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 275 of file dlaqr2.f.

🧚 subroutine slaqr2

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

SLAQR2 is identical to SLAQR3 except that it avoids recursion by calling SLAHQR instead of SLAQR4.

Aggressive early deflation:

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 over-written 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. 🦋

🌸 Parameters

  • 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 is LOGICAL. If .TRUE., then the orthogonal matrix Z is updated so that the orthogonal Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced.
  • N is INTEGER. The order of the matrix H and (if WANTZ is .TRUE.) the order of the orthogonal matrix Z.
  • 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 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 is INTEGER. Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
  • 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 the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries. 🔥
  • LDH is INTEGER. Leading dimension of H just as declared in the calling subroutine. N <= LDH.
  • ILOZ is INTEGER.
  • IHIZ is INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
  • 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 is INTEGER. The leading dimension of Z just as declared in the calling subroutine. 1 <= LDZ.
  • NS
  • NS — NS is INTEGER. It's the number of unconverged (ie approximate) eigenvalues returned in SR and SI that the calling subroutine may use as shifts. 💅
  • ND — ND is INTEGER. The number of converged eigenvalues uncovered by this subroutine.
  • SR — SR is REAL array, dimension (KBOT).
  • 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).
  • 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).
  • LDWV — LDWV is INTEGER. The leading dimension of W just as declared in the calling subroutine. NW <= LDV.
  • 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 at line 275 of file slaqr2.f.

🦋 subroutine zlaqr2

subroutine zlaqr2 (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)

ZLAQR2 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:

ZLAQR2 is identical to ZLAQR3 except that it avoids recursion by calling ZLAHQR instead of ZLAQR4.

Aggressive early deflation, bestie:

ZLAQR2 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 over-written 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. ✨

🧮 Parameters

  • WANTT — WANTT is 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 — WANTZ is 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 — N is INTEGER. The order of the matrix H and (if WANTZ is .TRUE.) the order of the unitary 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 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 — LDH is INTEGER. Leading dimension of H just as declared in the calling subroutine. N <= LDH.
  • ILOZ — ILOZ is INTEGER.
  • IHIZ — IHIZ is INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
  • Z — Z is 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 — 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.
  • SH — SH is 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 — V is COMPLEX*16 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 COMPLEX*16 array, dimension (LDT,NW).
  • 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 COMPLEX*16 array, dimension (LDWV,NW).
  • LDWV — LDWV is INTEGER. The leading dimension of W just as declared in the calling subroutine. NW <= LDV.
  • WORK — WORK is 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 — 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; ZLAQR2 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 267 of file zlaqr2.f.

👑 Author

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