hseqr(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🫶 hseqr(3) 🫶

okay bestie, welcome to hseqr(3) from the Library Functions Manual — this is the hseqr: Hessenberg eig, QR iteration page, and it is lowkey a whole eigenvalue factory. ✨ The vibe is Hessenberg eig + QR iteration, no cap.

💖 NAME 💖

hseqr - hseqr: Hessenberg eig, QR iteration

that’s the name, that’s the assignment, periodt. 💅

✨ SYNOPSIS ✨

🎀 Functions 🎀

subroutine chseqr (job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work,
    lwork, info)
    CHSEQR
subroutine dhseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz,
    work, lwork, info)
    DHSEQR
subroutine shseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz,
    work, lwork, info)
    SHSEQR
subroutine zhseqr (job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work,
    lwork, info)
    ZHSEQR

four flavors, same main-character energy. 🦋

🌈 Detailed Description 🌈

this is the detailed-description zone, and honestly it’s mostly the runway before the function docs below. no extra lore in this part, bestie.

🧚 Function Documentation 🧚

🌟 chseqr 🌟

subroutine chseqr (character job, character compz, integer n, integer ilo,
    integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex,
    dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex,
    dimension( * ) work, integer lwork, integer info)

CHSEQR

💅 Purpose 💅

okay so CHSEQR 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)*T*(QZ)**H

slay. that’s the whole purpose, fr fr. 💖

🎀 Parameters 🎀

  • JOB — JOB is CHARACTER*1. this is the “what do you want from me” switch:
    • = 'E': compute eigenvalues only;
    • = 'S': compute eigenvalues and the Schur form T.
  • COMPZ — COMPZ is CHARACTER*1. this controls the Schur-vector situation:
    • = 'N': no Schur vectors are computed;
    • = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned;
    • = 'V': Z must contain a unitary matrix Q on entry, and the product Q*Z is returned.
  • N — N is INTEGER. the order of the matrix H. N >= 0.
  • ILO — ILO is INTEGER. that’s it, that’s the description in this part. 🫶
  • IHI — IHI is INTEGER. the tea is: it is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to CGEBAL, and then passed to ZGEHRD 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 JOB = 'S', H contains the upper triangular matrix T from the Schur decomposition (the Schur form).
    • if INFO = 0 and JOB = 'E', the contents of H are unspecified on exit.
    • the output value of H when INFO > 0 is given under the description of INFO below.
    • heads up: unlike earlier versions of CHSEQR, this subroutine may explicitly 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. if JOB = 'S', 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). 💜
  • Z — Z is COMPLEX array, dimension (LDZ,N).
    • if COMPZ = 'N', Z is not referenced.
    • if COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, Z contains the unitary matrix Z of the Schur vectors of H.
    • if COMPZ = 'V', on entry Z must contain an N-by-N matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, if INFO = 0, Z contains Q*Z.
    • normally Q is the unitary matrix generated by CUNGHR after the call to CGEHRD which formed the Hessenberg matrix H.
    • 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 COMPZ = 'I' or COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1.
  • WORK — WORK is COMPLEX array, dimension (LWORK). on exit, if INFO = 0, 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 and delivers very good and sometimes optimal performance.
    • however, LWORK as large as 11*N may be required for optimal performance.
    • a workspace query is recommended to determine the optimal workspace size.
    • if LWORK = -1, then CHSEQR does a workspace query. In this case, CHSEQR 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, the i-th argument had an illegal value
    • > 0: if INFO = i, CHSEQR failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of W contain those eigenvalues which have been successfully computed. (Failures are rare.) 😭
    • if INFO > 0 and JOB = 'E', 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 JOB = 'S', 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 COMPZ = 'V', then on exit:
(final value of Z)  =  (initial value of Z)*U

where U is the unitary matrix in (*) (regardless of the value of JOB).

  • if INFO > 0 and COMPZ = 'I', then on exit:
(final value of Z)  = U

where U is the unitary matrix in (*) (regardless of the value of JOB).

  • if INFO > 0 and COMPZ = 'N', then Z is not 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

🦋 Further Details 🦋

(this heading is here, but there’s literally nothing listed under it in this part.)

okay so these defaults are supplied by ILAENV(ISPEC,'CHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK) — literally that exact call, bestie. And the suggestion is to adjust these defaults so each particular computational environment can hit its best performance. ✨

  • ISPEC=12 — the CLAHQR vs CLAQR0 crossover point. Default: 75. (Must be at least 11.) 🚨
  • ISPEC=13 — recommended deflation window size. This depends on ILO, IHI and NS. NS is the number of simultaneous shifts returned by ILAENV(ISPEC=15). (See ISPEC=15 below.) The default for (IHI-ILO+1) <= 500 is NS. The default for (IHI-ILO+1) > 500 is 3*NS/2.
  • ISPEC=14 — nibble crossover point. (See IPARMQ for details.) Default: 14% of deflation window size.
  • ISPEC=15 — number of simultaneous shifts in a multishift QR iteration.

If IHI-ILO+1 is ...

  • greater than or equal to 1 but less than 30 → NS = 2(+)
  • greater than or equal to 30 but less than 60 → NS = 4(+)
  • greater than or equal to 60 but less than 150 → NS = 10(+)
  • greater than or equal to 150 but less than 590 → NS = **
  • greater than or equal to 590 but less than 3000 → NS = 64
  • greater than or equal to 3000 but less than 6000 → NS = 128
  • greater than or equal to 6000 but less than infinity → NS = 256
  • (+) By default some or all matrices of this order are passed to the implicit double shift routine CLAHQR and this parameter is ignored. See ISPEC=12 above and comments in IPARMQ for details.
  • The (**) asterisks indicate an ad-hoc function of N increasing from 10 to 64.
  • ISPEC=16 — select structured matrix multiply. If the number of simultaneous shifts (specified by ISPEC=15) is less than 14, then the default for ISPEC=16 is 0. Otherwise the default for ISPEC=16 is 2.

📚 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 297 of file chseqr.f.

💜 subroutine dhseqr 💜

   subroutine dhseqr (character job, character compz, integer n, integer ilo,
       integer ihi, double precision, dimension( ldh, * ) h, integer ldh,
       double precision, dimension( * ) wr, double precision, dimension( * )
       wi, double precision, dimension( ldz, * ) z, integer ldz, double
       precision, dimension( * ) work, integer lwork, integer info)

DHSEQR

🎯 Purpose 🎯

DHSEQR 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.

🧾 Parameters 🧾

JOB — JOB is CHARACTER*1.

  • = 'E': compute eigenvalues only;
  • = 'S': compute eigenvalues and the Schur form T.

COMPZ — COMPZ is CHARACTER*1.

  • = 'N': no Schur vectors are computed;
  • = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned;
  • = 'V': Z must contain an orthogonal matrix Q on entry, and the product Q*Z is returned.

N — N is INTEGER. The order of the matrix H. N >= 0.

ILO — ILO is INTEGER.

IHI — IHI is INTEGER.

It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to DGEBAL, and then passed to ZGEHRD 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.

H — H is DOUBLE PRECISION array, dimension (LDH,N).

On entry, the upper Hessenberg matrix H.

On exit, if INFO = 0 and JOB = 'S', 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 JOB = 'E', the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.)

Unlike earlier versions of DHSEQR, this subroutine may explicitly 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 DOUBLE PRECISION array, dimension (N).

WI — WI is DOUBLE PRECISION array, dimension (N).

The real and imaginary parts, respectively, of the computed eigenvalues. 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 JOB = 'S', 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).

Z — Z is DOUBLE PRECISION array, dimension (LDZ,N).

If COMPZ = 'N', Z is not referenced.

If COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, Z contains the orthogonal matrix Z of the Schur vectors of H. If COMPZ = 'V', on entry Z must contain an N-by-N matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, if INFO = 0, Z contains Q*Z.

Normally Q is the orthogonal matrix generated by DORGHR after the call to DGEHRD which formed the Hessenberg matrix H. (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 COMPZ = 'I' or COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1.

WORK — WORK is DOUBLE PRECISION array, dimension (LWORK). On exit, if INFO = 0, 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 and delivers very good and sometimes optimal performance. However, LWORK as large as 11*N may be required for optimal performance. A workspace query is recommended to determine the optimal workspace size.

If LWORK = -1, then DHSEQR does a workspace query. In this case, DHSEQR 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 — okay bestie, INFO is an INTEGER, and it’s literally the receipts for how the call went:
    • = 0: successful exit — slay, we love a clean run 💅
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, DHSEQR 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 JOB = 'E', 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 JOB = 'S', 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 quasi-triangular in rows and columns INFO+1 through IHI.
      • If INFO > 0 and COMPZ = 'V', then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regardless of the value of JOB.)
      • If INFO > 0 and COMPZ = 'I', then on exit (final value of Z) = U where U is the orthogonal matrix in (*) (regardless of the value of JOB.)
      • If INFO > 0 and COMPZ = 'N', then Z is not 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

Further Details 🔍

Default values supplied by ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). It is suggested that these defaults be adjusted in order to attain best performance in each particular computational environment.

  • ISPEC=12: The DLAHQR vs DLAQR0 crossover point. Default: 75. (Must be at least 11.)
  • ISPEC=13: Recommended deflation window size. This depends on ILO, IHI and NS. NS is the number of simultaneous shifts returned by ILAENV(ISPEC=15). (See ISPEC=15 below.) The default for (IHI-ILO+1) <= 500 is NS. The default for (IHI-ILO+1) > 500 is 3*NS/2.
  • ISPEC=14: Nibble crossover point. (See IPARMQ for details.) Default: 14% of deflation window size.
  • ISPEC=15: Number of simultaneous shifts in a multishift QR iteration.
    • If IHI-ILO+1 is greater than or equal to 1, but less than 30 → the default is NS = 2(+)
    • If IHI-ILO+1 is greater than or equal to 30, but less than 60 → the default is NS = 4(+)
    • If IHI-ILO+1 is greater than or equal to 60, but less than 150 → the default is NS = 10(+)
    • If IHI-ILO+1 is greater than or equal to 150, but less than 590 → the default is NS = **
    • If IHI-ILO+1 is greater than or equal to 590, but less than 3000 → the default is NS = 64
    • If IHI-ILO+1 is greater than or equal to 3000, but less than 6000 → the default is NS = 128
    • If IHI-ILO+1 is greater than or equal to 6000, but less than infinity → the default is NS = 256
    • (+) By default some or all matrices of this order are passed to the implicit double shift routine DLAHQR and this parameter is ignored. See ISPEC=12 above and comments in IPARMQ for details.
    • (**) The asterisks (**) indicate an ad-hoc function of N increasing from 10 to 64.
  • ISPEC=16: Select structured matrix multiply. If the number of simultaneous shifts (specified by ISPEC=15) is less than 14, then the default for ISPEC=16 is 0. Otherwise the default for ISPEC=16 is 2.

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 314 of file dhseqr.f.

subroutine shseqr — SHSEQR 🌟

   subroutine shseqr (character job, character compz, integer n, integer ilo,
       integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension(
       * ) wr, real, dimension( * ) wi, real, dimension( ldz, * ) z, integer
       ldz, real, dimension( * ) work, integer lwork, integer info)
       SHSEQR

Purpose 🎯

  • SHSEQR 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.

Parameters 💅

  • JOB — JOB is CHARACTER*1
    • = 'E': compute eigenvalues only;
    • = 'S': compute eigenvalues and the Schur form T.
  • COMPZ — COMPZ is CHARACTER*1
    • = 'N': no Schur vectors are computed;
    • = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned;
    • = 'V': Z must contain an orthogonal matrix Q on entry, and the product Q*Z is returned.
  • N — N is INTEGER. The order of the matrix H. N >= 0.
  • ILO — ILO is INTEGER
  • IHI — IHI is INTEGER
    • It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to SGEBAL, and then passed to ZGEHRD 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 a REAL array, dimension (LDH,N)
    • On entry, the upper Hessenberg matrix H.
    • On exit, if INFO = 0 and JOB = 'S', 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 JOB = 'E', the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.)
    • Unlike earlier versions of SHSEQR, this subroutine may explicitly 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 a REAL array, dimension (N)
  • WI — WI is a REAL array, dimension (N)
    • The real and imaginary parts, respectively, of the computed eigenvalues. 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 JOB = 'S', 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).
  • Z — Z is a REAL array, dimension (LDZ,N)
    • If COMPZ = 'N', Z is not referenced.
    • If COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, Z contains the orthogonal matrix Z of the Schur vectors of H.
    • If COMPZ = 'V', on entry Z must contain an N-by-N matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, if INFO = 0, Z contains Q*Z.
    • Normally Q is the orthogonal matrix generated by SORGHR after the call to SGEHRD which formed the Hessenberg matrix H. (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 COMPZ = 'I' or COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1.
  • WORK — WORK is a REAL array, dimension (LWORK). On exit, if INFO = 0, WORK(1) returns an estimate of the optimal value for LWORK.
  • LWORK — and this chunk cuts off right here, bestie — LWORK’s full description is coming in the next part 🫶
  • LWORK — INTEGER. bestie this one's the workspace dimension 🫶 so listen up:
    • LWORK >= max(1,N) is sufficient and honestly delivers very good, sometimes straight-up optimal performance ✨
    • BUT LWORK as big as 11*N might be required for optimal performance, so a workspace query is recommended to figure out the optimal workspace size 🔍
    • if LWORK = -1, then SHSEQR does a workspace query. In this case SHSEQR checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate gets returned in WORK(1). No error message related to LWORK is issued by XERBLA, and neither H nor Z are accessed 🙈 (iconic behaviour honestly)

INFO 💀

INFO is INTEGER, and it's the vibe check on how everything went:

  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value — so like, check your inputs bestie 😭
  • > 0: if INFO = i, SHSEQR 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. genuinely rare, don't panic 💅)

when INFO > 0, here's the full tea depending on your flags:

  • If INFO > 0 and JOB = 'E', 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 JOB = 'S', 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 quasi-triangular in rows and columns INFO+1 through IHI.

  • If INFO > 0 and COMPZ = 'V', then on exit:
(final value of Z)  =  (initial value of Z)*U

where U is the orthogonal matrix in (*) (regardless of the value of JOB).

  • If INFO > 0 and COMPZ = 'I', then on exit:
     (final value of Z)  = U

where U is the orthogonal matrix in (*) (regardless of the value of JOB).

  • If INFO > 0 and COMPZ = 'N', then Z is not 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 — absolute icons, we stan 🫶

✨ Further Details: ✨

Default values supplied by:

ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).

It is suggested that these defaults be adjusted in order to attain best performance in each particular computational environment. (iykyk — tune it for your machine, bestie 🔧)

  • ISPEC=12: The SLAHQR vs SLAQR0 crossover point. Default: 75. (Must be at least 11.) ⭐
  • ISPEC=13: Recommended deflation window size. This depends on ILO, IHI and NS. NS is the number of simultaneous shifts returned by ILAENV(ISPEC=15). (See ISPEC=15 below.) The default for (IHI-ILO+1) <= 500 is NS. The default for (IHI-ILO+1) > 500 is 3*NS/2. 🪟
  • ISPEC=14: Nibble crossover point. (See IPARMQ for details.) Default: 14% of deflation window size. 🍪
  • ISPEC=15: Number of simultaneous shifts in a multishift QR iteration. If IHI-ILO+1 is… 👀
    • greater than or equal to 1, but less than 30 → NS = 2(+)
    • greater than or equal to 30, but less than 60 → NS = 4(+)
    • greater than or equal to 60, but less than 150 → NS = 10(+)
    • greater than or equal to 150, but less than 590 → NS = **
    • greater than or equal to 590, but less than 3000 → NS = 64
    • greater than or equal to 3000, but less than 6000 → NS = 128
    • greater than or equal to 6000, up to infinity → NS = 256
    • (+) By default some or all matrices of this order are passed to the implicit double shift routine SLAHQR and this parameter is ignored. See ISPEC=12 above and comments in IPARMQ for details.
    • (**) The asterisks (**) indicate an ad-hoc function of N increasing from 10 to 64. no cap, it's just built different 💀
  • ISPEC=16: Select structured matrix multiply. If the number of simultaneous shifts (specified by ISPEC=15) is less than 14, then the default for ISPEC=16 is 0. Otherwise the default for ISPEC=16 is 2. 🧮

📚 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 314 of file shseqr.f.

🦋 subroutine zhseqr 🦋

okay so this is the complex*16 sibling and she is GIVING main character energy 👑

subroutine zhseqr (character job, character compz, integer n, integer ilo,
    integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16,
    dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz,
    complex*16, dimension( * ) work, integer lwork, integer info)

ZHSEQR

Purpose: 💖

ZHSEQR 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. slay ✨

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)*T*(QZ)**H

literally the whole glow-up pipeline 💅

Parameters 🌈

JOB

JOB is CHARACTER*1:

  • = 'E': compute eigenvalues only
  • = 'S': compute eigenvalues and the Schur form T

COMPZ

COMPZ is CHARACTER*1:

  • = 'N': no Schur vectors are computed
  • = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned
  • = 'V': Z must contain a unitary matrix Q on entry, and the product Q*Z is returned

N

N is INTEGER. The order of the matrix H. N >= 0. 🍓

ILO

ILO is INTEGER — short and sweet, bestie.

IHI

IHI is INTEGER.

It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. 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. 👑

H

H is COMPLEX*16 array, dimension (LDH,N).

  • On entry, the upper Hessenberg matrix H.
  • On exit, if INFO = 0 and JOB = 'S', H contains the upper triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and JOB = 'E', the contents of H are unspecified on exit 😭 (so don't go reading it like it means something). (The output value of H when INFO > 0 is given under the description of INFO below.)
Unlike earlier versions of ZHSEQR, this subroutine may explicitly 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*16 array, dimension (N). The computed eigenvalues. If JOB = 'S', 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). 🎀

Z

Z is COMPLEX*16 array, dimension (LDZ,N).

  • If COMPZ = 'N', Z is not referenced.
  • If COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, Z contains the unitary matrix Z of the Schur vectors of H.
  • If COMPZ = 'V', on entry Z must contain an N-by-N matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, if INFO = 0, Z contains Q*Z.

Normally Q is the unitary matrix generated by ZUNGHR after the call to ZGEHRD which formed the Hessenberg matrix H. (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 COMPZ = 'I' or COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1.

WORK

WORK is COMPLEX*16 array, dimension (LWORK). On exit, if INFO = 0, WORK(1) returns an estimate of the optimal value for LWORK. ⭐

LWORK

…okay the LWORK tea is coming right up 👀

  • LWORK (INTEGER) — okay so this is the dimension of the array WORK. LWORK >= max(1,N) is sufficient and delivers very good and sometimes optimal performance. However, LWORK as large as 11*N may be required for optimal performance. A workspace query is recommended to determine the optimal workspace size.

If LWORK = -1, then ZHSEQR does a workspace query. In this case, ZHSEQR 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 (INTEGER) — bestie, this is the exit-status tea ☕:

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, ZHSEQR failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of W contain those eigenvalues which have been successfully computed. (Failures are rare.)

If INFO > 0 and JOB = 'E', 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 JOB = 'S', 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 COMPZ = 'V', then on exit

(final value of Z)  =  (initial value of Z)*U

where U is the unitary matrix in (*) (regardless of the value of JOB.)

If INFO > 0 and COMPZ = 'I', then on exit

(final value of Z)  = U

where U is the unitary matrix in (*) (regardless of the value of JOB.)

If INFO > 0 and COMPZ = 'N', then Z is not 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

🔍 FURTHER DETAILS: 🔍

Default values supplied by

ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK)

It is suggested that these defaults be adjusted in order to attain best performance in each particular computational environment.

  • ISPEC=12: The ZLAHQR vs ZLAQR0 crossover point. Default: 75. (Must be at least 11.)
  • ISPEC=13: Recommended deflation window size. This depends on ILO, IHI and NS. NS is the number of simultaneous shifts returned by ILAENV(ISPEC=15). (See ISPEC=15 below.) The default for (IHI-ILO+1) <= 500 is NS. The default for (IHI-ILO+1) > 500 is 3*NS/2.
  • ISPEC=14: Nibble crossover point. (See IPARMQ for details.) Default: 14% of deflation window size.
  • ISPEC=15: Number of simultaneous shifts in a multishift QR iteration.

If IHI-ILO+1 is ...

  • greater than or equal to 1, but less than 30 — default is NS = 2(+)
  • greater than or equal to 30, but less than 60 — default is NS = 4(+)
  • greater than or equal to 60, but less than 150 — default is NS = 10(+)
  • greater than or equal to 150, but less than 590 — default is NS = **
  • greater than or equal to 590, but less than 3000 — default is NS = 64
  • greater than or equal to 3000, but less than 6000 — default is NS = 128
  • greater than or equal to 6000, but less than infinity — default is NS = 256

(+) By default some or all matrices of this order are passed to the implicit double shift routine ZLAHQR and this parameter is ignored. See ISPEC=12 above and comments in IPARMQ for details.

(**) The asterisks (**) indicate an ad-hoc function of N increasing from 10 to 64.

  • ISPEC=16: Select structured matrix multiply. If the number of simultaneous shifts (specified by ISPEC=15) is less than 14, then the default for ISPEC=16 is 0. Otherwise the default for ISPEC=16 is 2.

📚 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 297 of file zhseqr.f.

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