DHSEQR(3)

DHSEQR(3) 🎀

Library Functions Manual bestie! The manual name is DHSEQR(3), and this page is about to give you the full LAPACK tea. 💅

💖 NAME 💖

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

✨ SYNOPSIS ✨

🌈 Functions/Subroutines 🌈

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

DHSEQR

🧚 Function/Subroutine Documentation 🧚

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 — pick your vibe:

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

COMPZ

COMPZ is CHARACTER*1 — this one 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 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 set 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 is INTEGER

• = 0: successful exit
• < 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 🦋

Nothing extra is listed under this heading in this part. 🫶

Okay so, the default values are supplied by

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

and the docs are literally like: it is suggested that these defaults be adjusted in order to attain best performance in each particular computational environment. Translation: tune these for your machine, bestie ✨

• 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 → 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 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.

💅 AUTHOR 💅

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That’s the whole vibe from LAPACK Version 3.12.0, page sourced at /home/abui...RC/dhseqr.f(3) — the footer date didn’t show up in this slice, but the version/source is right there, bestie 🫶💖🎀