SRC/chseqr.f(3)

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🌸 SRC/chseqr.f(3) — Library Functions Manual 🌸

okay so welcome to SRC/chseqr.f(3), bestie 💖 this one's from the Library Functions Manual and it's literally the whole chseqr situation, broken down for you. no cap, we're starting from the top.

✨ NAME ✨

SRC/chseqr.f — that's the file, that's the vibe 🎀

🌈 SYNOPSIS 🌈

💅 Functions/Subroutines

so here's the cast, main character energy only:

  • subroutine chseqr (job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info) 🦋

this is the subroutine, the one we're all here for. look at all those arguments — job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info — iconic lineup fr fr 👑

  • CHSEQR ⭐

yeah. that's the name. it's giving uppercase, it's giving confidence, it's giving CHSEQR 💜

more parts incoming, don't go anywhere 🫶✨

💻 Function/Subroutine Documentation 💻

   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

🧩 Parameters 🧩

JOB

JOB is CHARACTER*1, bestie — here are the choices:

  • = '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 unitary matrix Q on entry, and the product Q*Z is returned.

N

N is INTEGER. It’s 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 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.) 👀

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. It’s 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. It’s 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. It’s 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 🔍

Default values supplied by

ILAENV(ISPEC,'CHSEQR',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 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 — the default is NS = 2(+)
  • greater than or equal to 30 but less than 60 — the default is NS = 4(+)
  • greater than or equal to 60 but less than 150 — the default is NS = 10(+)
  • greater than or equal to 150 but less than 590 — the default is NS = **
  • greater than or equal to 590 but less than 3000 — the default is NS = 64
  • greater than or equal to 3000 but less than 6000 — the default is NS = 128
  • 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 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.

🎀 AUTHOR 🎀

ok so nobody actually hand-wrote this page, bestie — it was generated automatically by Doxygen for LAPACK from the source code. so the "author" is a robot that read the Fortran and typed all this up. iconic of it, honestly. no human to @ here, just a doc generator doing its thing 💅

and that's the whole page!! 🫶 tysm for reading all the way through, you absolute legend. this little doc situation lives in LAPACK, Version 3.12.0, and it's comin' at you straight from SRC/chseqr.f(3) — go forth and compute, bestie 💖✨

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