SRC/zhseqr.f(3)

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💅 SRC/zhseqr.f(3) — Library Functions Manual (yes bestie, we're reading the man page)

okay so first things first: this whole thing lives at SRC/zhseqr.f(3) in the <purple>Library Functions Manual</purple>. That's the vibe. That's the address. Write it down 🎀

✨ NAME ✨

SRC/zhseqr.f — that's literally it, that's the name, she's iconic and she needs no further introduction 💅

🌸 SYNOPSIS 🌸

🦋 Functions/Subroutines

so under the <mint>Functions/Subroutines</mint> umbrella we have exactly one (1) subroutine, and she's the main character of this whole page:

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

and her name, in all caps, the way she's listed here:

ZHSEQR

thirteen arguments, no cap — job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, and info. That's the whole squad 🫶 keep scrolling because there is so much more coming, we're literally just getting started 💖

💖 Function/Subroutine Documentation 💖

okay bestie, here’s the actual subroutine tea 🫖:

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. 💅

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

= '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

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 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. (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 💅

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 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 💅

INFO is INTEGER

= 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 ...   but less than ...   default is

1                              30                  NS = 2(+)
30                             60                  NS = 4(+)
60                             150                 NS = 10(+)
150                            590                 NS = **
590                            3000                NS = 64
3000                           6000                NS = 128
6000                           infinity            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.

Author 🎀

So this whole page was generated automatically by Doxygen for LAPACK from the source code — like, nobody hand-wrote it, a robot did it while reading the actual Fortran, and honestly? slay, we love an automated king. 💅✨

No cap, that's the entire author credit: Doxygen did it, LAPACK is the vibes it's serving, and the source code is the main character. 🫶

okay bestie, that's the whole SRC/zhseqr.f(3) page — you made it to the end, I'm so proud of you 😭💖 go forth and compute those eigenvalues, you absolute icon. 👑

LAPACK — Version 3.12.0 🌸 SRC/zhseqr.f(3) — fr fr, that's the source file and the version, and that's a wrap. 🎀🦋✨

📦 where this page comes from

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