Does ordinary "eig" calculate eigenvalues faster than "polyeig"?

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Masoud
Masoud 2018-8-3
评论： Masoud Arash 2024-10-29，8:39
I want to compare the computational complexity (computational order) of obtaining eigenvalues (not eigenvectors) of two methods:
  • Ordinary eigenvalue calculation, such as "eig" function.
  • Polynomial approach, which corresponds to the "polyeig" function.
The main point is that MATLAB consumes less time with "eig" function than "polyeig", however the matrices in the "polyeig" method are smaller in size. Also, there is a line in explanation of "polyeig" which says: " The polyeig function uses the qz factorization to find intermediate results in the computation of generalized eigenvalues. polyeig uses the intermediate results to determine if the eigenvalues are well-determined". Furthermore "eig" uses LU-decomposition in its function, where "polyeig" uses qz decomposition.
My questions are:
1) I think MATLAB's "tic-toc" function is not a proper tool for comparison between these two functions, due to the fact that they uses different methods and "polyeig" checks its answers. Is this suggestion right?
2) For a fair comparison, should I write my own program which calculates eigenvalues in both methods (polynomial and ordinary)?
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Christine Tobler
Christine Tobler 2018-8-9
That's correct, all solvers for polynomial eigenvalue problems I am aware of are in some way generalizations of solvers for the simple or generalized eigenvalue problems. They will typically be slower when applied to the same problem.
The solver used in MATLAB calls into EIG directly, with matrices of size n*p, where n is the size of the input matrices, and p is the polynomial degree of the eigenvalue problem.
Since the computational complexity of EIG is O(n^3), the computational complexity of POLYEIG is O(p^3*n^3).

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回答（1 个）

Pavl M.
Pavl M. 2024-10-24，10:45
Ran in:
clc
clear all
close all
%format native-bit
format longg
rand('state',1)
%for ProblemComplexty = 2:1:1000
ProblemComplexity = 4;
%[a1,b1] = eig(ones(4,4))
%[a,b,c] = polyeig([ones(1,4);zeros(1,4);zeros(1,4);zeros(1,4)],[zeros(1,4);ones(1,4);zeros(1,4);zeros(1,4)],[zeros(1,4);zeros(1,4);ones(1,4);zeros(1,4)],[zeros(1,4);zeros(1,4);ones(1,4);zeros(1,4)])
A = randn(ProblemComplexity,ProblemComplexity)
A = 4×4
-0.432564811528221 -1.14647135068146 0.327292361408654 -0.588316543014189 -1.6655843782381 1.190915465643 0.174639142820925 2.1831858181971 0.125332306474831 1.1891642016521 -0.186708577681439 -0.136395883086596 0.287676420358549 -0.0376332765933176 0.725790548293303 0.11393131352081
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
B = randn(ProblemComplexity,ProblemComplexity)
B = 4×4
1.06676821135919 0.29441081639264 -0.691775701702287 -1.44096443190102 0.0592814605236053 -1.3361818579378 0.857996672828263 0.571147623658178 -0.095648405483669 0.714324551818952 1.25400142160253 -0.399885577715363 -0.832349463650022 1.62356206444627 -1.59372957644748 0.689997375464345
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
tstart = cputime;
[V,D,W] = eig(A,B)
V = 4×4
1 -1 -0.564096814047548 -0.577094789193665 0.420551114175413 -0.233324018341873 0.208587294472125 0.569423599904986 0.234364513652587 -0.0245396546049334 -1 0.432158863800365 0.960947796142888 -0.53513947825968 -0.220998889485382 -1
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
D = 4×4
3.93938105751949 0 0 0 0 -2.89854741781082 0 0 0 0 -0.409481554894135 0 0 0 0 0.470532974588248
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
W = 4×4
0.245304083290022 -0.954105799698367 0.300585105170963 0.442313649434121 1 -1 -0.0419455839846326 -0.0115586114369713 0.57350462982635 -0.228567950330134 -0.196945103378633 1 0.770511810816906 -0.474830861513806 1 0.328909476081192
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
first_method_took_time_sec = cputime - tstart
first_method_took_time_sec =
0.0599999999999454
tstart2 = cputime;
[X,e,s] = polyeig(A,-B)
X = 4×4
1 -1 -0.564096814047548 -0.577094789193665 0.420551114175413 -0.233324018341873 0.208587294472125 0.569423599904986 0.234364513652587 -0.0245396546049334 -1 0.432158863800365 0.960947796142888 -0.53513947825968 -0.220998889485382 -1
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
e = 4×1
3.93938105751949 -2.89854741781082 -0.409481554894135 0.470532974588248
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
s = 4×1
4.03813327484369 5.81249808987839 1.36550981888017 2.18896807862899
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
second_method_took_time_sec = cputime - tstart2
second_method_took_time_sec =
0.0300000000002001
speedup = 100*(first_method_took_time_sec - second_method_took_time_sec) / (first_method_took_time_sec)
speedup =
49.999999999621
%end for
%Constructed from needing help code by
%https://independent.academia.edu/PMazniker
%+380990535261
%https://diag.net/u/u6r3ondjie0w0l8138bafm095b
%https://github.com/goodengineer
%https://orcid.org/0000-0001-8184-8166
%https://willwork781147312.wordpress.com/portfolio/cp/
%https://www.youtube.com/channel/UCC__7jMOAHak0MVkUFtmO-w
%https://nanohub.org/members/130066
%https://pangian.com/user/hiretoserve/
%https://substack.com/profile/191772642-paul-m

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