SHSEQR(3)

SHSEQR(3) 💖

okay so welcome to the SHSEQR page from the LAPACK Library Functions Manual — it’s giving real Schur factorization main character energy, no cap, fr fr. ✨

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/shseqr.f

That’s the source-file path this page is about, bestie. 🫶

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine SHSEQR (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)
    SHSEQR

🎀 Function/Subroutine Documentation 🎀

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: 💅

okay so here’s the tea: 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. Iconic. 🦋

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

Slay. 💅

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 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. Vibe check: LDH >= max(1,N).

`WR` 🩷

WR is REAL array, dimension (N).

`WI` 🩷

WI is 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 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 REAL 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 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 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, 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.)

If INFO > 0 and JOB = 'E', 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 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: 🌈

(nothing listed under this heading in part 1)

okay so bestie, before we get into the actual deflation window tea, there's a whole vibe check on the default values that ILAENV supplies. The call looks like this:

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

And like, the man page itself is like “hey, maybe tweak these defaults to get the best performance in your particular computational environment” — no cap, every machine is different. 💅

ISPEC=12 — The SLAHQR vs SLAQR0 crossover point. Default: 75. (Must be at least 11.) slay.
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. it's giving math but make it fashion ✨
ISPEC=14 — Nibble crossover point. (See IPARMQ for details.) Default: 14% of deflation window size. lowkey cute.
ISPEC=15 — Number of simultaneous shifts in a multishift QR iteration. Here's the table, bestie — keep it exactly as is:

                      If IHI-ILO+1 is ...

                      greater than      ...but less    ... the
                      or equal to ...      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 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.

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. periodt.

📚 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.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

💖 that's the whole vibe check from LAPACK Version 3.12.0, file shseqr.f — the date's right there in the footer if you're curious, but like, it's giving ✨ generated by Doxygen ✨. mwah, bestie! 💋💅

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