
% This LaTeX was auto-generated from MATLAB code.
% To make changes, update the MATLAB code and republish this document.

\documentclass{article}
\usepackage{graphicx}
\usepackage{color}

\sloppy
\definecolor{lightgray}{gray}{0.5}
\setlength{\parindent}{0pt}

\begin{document}

    
    
\section*{Eigenvalue problems}

\begin{par}
This script explores how to use Matlab to solve eigenvalue problems. Both the naive and 'proper' way are shown.
\end{par} \vspace{1em}

\subsection*{Contents}

\begin{itemize}
\setlength{\itemsep}{-1ex}
   \item The Eig function
   \item Root Solving
   \item Large matrices
\end{itemize}


\subsection*{The Eig function}

\begin{par}
The simplest way to compute eigenvalues and eigenvectors in Matlab is to use the built in "eig" function.
\end{par} \vspace{1em}
\begin{verbatim}
A=rand(10);
tic
EigVal1=eig(A)
toc
\end{verbatim}

        \color{lightgray} \begin{verbatim}
EigVal1 =

   5.0201 + 0.0000i
  -0.0691 + 0.8980i
  -0.0691 - 0.8980i
   0.7199 + 0.0000i
   0.3337 + 0.5856i
   0.3337 - 0.5856i
  -0.6606 + 0.0000i
  -0.3257 + 0.3368i
  -0.3257 - 0.3368i
  -0.0840 + 0.0000i

Elapsed time is 0.010119 seconds.
\end{verbatim} \color{black}
    \begin{par}
This will produce a column vector of the eigenvalues of A. To also get the eigenvectors, return two outputs:
\end{par} \vspace{1em}
\begin{verbatim}
tic
[EigVec2,EigVal2]=eig(A);
toc
\end{verbatim}

        \color{lightgray} \begin{verbatim}Elapsed time is 0.002891 seconds.
\end{verbatim} \color{black}
    \begin{par}
This will return the eigenvectors as the columns of EigVec, and the eigenvalues as the entries of the diagonal matrix EigVal.
\end{par} \vspace{1em}


\subsection*{Root Solving}

\begin{par}
We can also use Matlab to solve an eigenvalue problem in the direct way one would do it on paper. We can calculate the characteristic polynomial of our matrix and then find the roots. Using the built in functions charpoly and roots, this is:
\end{par} \vspace{1em}
\begin{verbatim}
tic
EigVal3=roots(charpoly(A))
toc
\end{verbatim}

        \color{lightgray} \begin{verbatim}
EigVal3 =

   5.0201 + 0.0000i
  -0.0691 + 0.8980i
  -0.0691 - 0.8980i
   0.7199 + 0.0000i
   0.3337 + 0.5856i
   0.3337 - 0.5856i
  -0.6606 + 0.0000i
  -0.3257 + 0.3368i
  -0.3257 - 0.3368i
  -0.0840 + 0.0000i

Elapsed time is 1.887849 seconds.
\end{verbatim} \color{black}
    

\subsection*{Large matrices}

\begin{par}
As you can see in the timings, this method is significantly slower than using the built in function. For large enough matrices, it is not feasible to solve the problem algebraically.
\end{par} \vspace{1em}
\begin{verbatim}
A=rand(1000);
tic
EigVal4=eig(A)
toc
tic
[EigVec5,EigVal5]=eig(A);
toc
\end{verbatim}

        \color{lightgray} \begin{verbatim}
EigVal4 =

   1.0e+02 *

   5.0007 + 0.0000i
   0.0905 + 0.0155i
   0.0905 - 0.0155i
   0.0584 + 0.0708i
   0.0584 - 0.0708i
   0.0903 + 0.0089i
   0.0903 - 0.0089i
   0.0630 + 0.0648i
   0.0630 - 0.0648i
   0.0722 + 0.0541i
   0.0722 - 0.0541i
   0.0878 + 0.0000i
   0.0141 + 0.0900i
   0.0141 - 0.0900i
   0.0357 + 0.0829i
   0.0357 - 0.0829i
   0.0667 + 0.0593i
   0.0667 - 0.0593i
   0.0759 + 0.0456i
   0.0759 - 0.0456i
   0.0740 + 0.0486i
   0.0740 - 0.0486i
   0.0462 + 0.0758i
   0.0462 - 0.0758i
   0.0422 + 0.0781i
   0.0422 - 0.0781i
   0.0516 + 0.0716i
   0.0516 - 0.0716i
   0.0534 + 0.0692i
   0.0534 - 0.0692i
   0.0657 + 0.0576i
   0.0657 - 0.0576i
   0.0682 + 0.0537i
   0.0682 - 0.0537i
   0.0571 + 0.0644i
   0.0571 - 0.0644i
   0.0388 + 0.0784i
   0.0388 - 0.0784i
   0.0841 + 0.0282i
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  -0.0098 + 0.0905i
  -0.0098 - 0.0905i
  -0.0120 + 0.0891i
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  -0.0332 + 0.0839i
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  -0.0161 + 0.0883i
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  -0.0493 + 0.0754i
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  -0.0289 + 0.0850i
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  -0.0499 + 0.0743i
  -0.0499 - 0.0743i
  -0.0918 + 0.0059i
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  -0.0595 + 0.0666i
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  -0.0237 + 0.0863i
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   0.0805 + 0.0000i
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   0.0166 + 0.0626i
   0.0166 - 0.0626i
  -0.0078 + 0.0638i
  -0.0078 - 0.0638i
   0.0636 + 0.0021i
   0.0636 - 0.0021i
  -0.0253 + 0.0583i
  -0.0253 - 0.0583i
   0.0600 + 0.0179i
   0.0600 - 0.0179i
  -0.0321 + 0.0535i
  -0.0321 - 0.0535i
   0.0511 + 0.0338i
   0.0511 - 0.0338i
   0.0552 + 0.0282i
   0.0552 - 0.0282i
  -0.0396 + 0.0464i
  -0.0396 - 0.0464i
  -0.0002 + 0.0632i
  -0.0002 - 0.0632i
   0.0271 + 0.0562i
   0.0271 - 0.0562i
   0.0034 + 0.0629i
   0.0034 - 0.0629i
  -0.0365 + 0.0487i
  -0.0365 - 0.0487i
   0.0305 + 0.0536i
   0.0305 - 0.0536i
  -0.0625 + 0.0011i
  -0.0625 - 0.0011i
   0.0230 + 0.0570i
   0.0230 - 0.0570i
   0.0369 + 0.0457i
   0.0369 - 0.0457i
   0.0151 + 0.0601i
   0.0151 - 0.0601i
  -0.0056 + 0.0614i
  -0.0056 - 0.0614i
   0.0463 + 0.0358i
   0.0463 - 0.0358i
   0.0417 + 0.0402i
   0.0417 - 0.0402i
  -0.0606 + 0.0043i
  -0.0606 - 0.0043i
   0.0163 + 0.0587i
   0.0163 - 0.0587i
  -0.0280 + 0.0514i
  -0.0280 - 0.0514i
  -0.0498 + 0.0300i
  -0.0498 - 0.0300i
  -0.0392 + 0.0419i
  -0.0392 - 0.0419i
  -0.0193 + 0.0570i
  -0.0193 - 0.0570i
  -0.0442 + 0.0362i
  -0.0442 - 0.0362i
   0.0590 + 0.0127i
   0.0590 - 0.0127i
   0.0516 + 0.0271i
   0.0516 - 0.0271i
   0.0577 + 0.0097i
   0.0577 - 0.0097i
   0.0463 + 0.0328i
   0.0463 - 0.0328i
   0.0527 + 0.0208i
   0.0527 - 0.0208i
  -0.0223 + 0.0550i
  -0.0223 - 0.0550i
   0.0496 + 0.0261i
   0.0496 - 0.0261i
  -0.0577 + 0.0142i
  -0.0577 - 0.0142i
   0.0100 + 0.0583i
   0.0100 - 0.0583i
  -0.0585 + 0.0088i
  -0.0585 - 0.0088i
  -0.0111 + 0.0574i
  -0.0111 - 0.0574i
  -0.0526 + 0.0264i
  -0.0526 - 0.0264i
  -0.0567 + 0.0000i
  -0.0546 + 0.0204i
  -0.0546 - 0.0204i
  -0.0526 + 0.0253i
  -0.0526 - 0.0253i
   0.0008 + 0.0586i
   0.0008 - 0.0586i
  -0.0140 + 0.0561i
  -0.0140 - 0.0561i
   0.0301 + 0.0482i
   0.0301 - 0.0482i
  -0.0554 + 0.0167i
  -0.0554 - 0.0167i
  -0.0242 + 0.0509i
  -0.0242 - 0.0509i
  -0.0305 + 0.0466i
  -0.0305 - 0.0466i
   0.0268 + 0.0489i
   0.0268 - 0.0489i
   0.0190 + 0.0535i
   0.0190 - 0.0535i
  -0.0469 + 0.0308i
  -0.0469 - 0.0308i
  -0.0365 + 0.0408i
  -0.0365 - 0.0408i
   0.0144 + 0.0546i
   0.0144 - 0.0546i
   0.0122 + 0.0553i
   0.0122 - 0.0553i
   0.0571 + 0.0000i
   0.0386 + 0.0383i
   0.0386 - 0.0383i
  -0.0508 + 0.0207i
  -0.0508 - 0.0207i
   0.0313 + 0.0447i
   0.0313 - 0.0447i
   0.0463 + 0.0267i
   0.0463 - 0.0267i
  -0.0154 + 0.0535i
  -0.0154 - 0.0535i
  -0.0050 + 0.0555i
  -0.0050 - 0.0555i
   0.0416 + 0.0331i
   0.0416 - 0.0331i
   0.0071 + 0.0552i
   0.0071 - 0.0552i
  -0.0547 + 0.0115i
  -0.0547 - 0.0115i
  -0.0409 + 0.0322i
  -0.0409 - 0.0322i
  -0.0359 + 0.0379i
  -0.0359 - 0.0379i
   0.0520 + 0.0169i
   0.0520 - 0.0169i
   0.0340 + 0.0393i
   0.0340 - 0.0393i
   0.0017 + 0.0549i
   0.0017 - 0.0549i
   0.0550 + 0.0036i
   0.0550 - 0.0036i
   0.0537 + 0.0100i
   0.0537 - 0.0100i
   0.0324 + 0.0395i
   0.0324 - 0.0395i
   0.0491 + 0.0170i
   0.0491 - 0.0170i
   0.0033 + 0.0534i
   0.0033 - 0.0534i
  -0.0182 + 0.0500i
  -0.0182 - 0.0500i
   0.0239 + 0.0468i
   0.0239 - 0.0468i
  -0.0543 + 0.0079i
  -0.0543 - 0.0079i
   0.0522 + 0.0028i
   0.0522 - 0.0028i
  -0.0210 + 0.0483i
  -0.0210 - 0.0483i
   0.0168 + 0.0489i
   0.0168 - 0.0489i
  -0.0537 + 0.0038i
  -0.0537 - 0.0038i
   0.0493 + 0.0000i
   0.0280 + 0.0434i
   0.0280 - 0.0434i
  -0.0075 + 0.0517i
  -0.0075 - 0.0517i
   0.0359 + 0.0339i
   0.0359 - 0.0339i
  -0.0352 + 0.0360i
  -0.0352 - 0.0360i
  -0.0436 + 0.0260i
  -0.0436 - 0.0260i
  -0.0299 + 0.0397i
  -0.0299 - 0.0397i
  -0.0488 + 0.0155i
  -0.0488 - 0.0155i
  -0.0272 + 0.0412i
  -0.0272 - 0.0412i
  -0.0450 + 0.0223i
  -0.0450 - 0.0223i
  -0.0486 + 0.0143i
  -0.0486 - 0.0143i
   0.0266 + 0.0421i
   0.0266 - 0.0421i
  -0.0101 + 0.0487i
  -0.0101 - 0.0487i
  -0.0016 + 0.0500i
  -0.0016 - 0.0500i
  -0.0378 + 0.0301i
  -0.0378 - 0.0301i
   0.0469 + 0.0129i
   0.0469 - 0.0129i
   0.0487 + 0.0068i
   0.0487 - 0.0068i
   0.0105 + 0.0475i
   0.0105 - 0.0475i
  -0.0488 + 0.0000i
  -0.0353 + 0.0324i
  -0.0353 - 0.0324i
  -0.0181 + 0.0438i
  -0.0181 - 0.0438i
  -0.0226 + 0.0417i
  -0.0226 - 0.0417i
  -0.0391 + 0.0270i
  -0.0391 - 0.0270i
   0.0465 + 0.0000i
   0.0441 + 0.0177i
   0.0441 - 0.0177i
   0.0389 + 0.0254i
   0.0389 - 0.0254i
   0.0377 + 0.0265i
   0.0377 - 0.0265i
   0.0420 + 0.0201i
   0.0420 - 0.0201i
   0.0160 + 0.0439i
   0.0160 - 0.0439i
  -0.0271 + 0.0351i
  -0.0271 - 0.0351i
   0.0305 + 0.0322i
   0.0305 - 0.0322i
  -0.0457 + 0.0105i
  -0.0457 - 0.0105i
  -0.0299 + 0.0321i
  -0.0299 - 0.0321i
  -0.0006 + 0.0465i
  -0.0006 - 0.0465i
  -0.0067 + 0.0448i
  -0.0067 - 0.0448i
  -0.0091 + 0.0439i
  -0.0091 - 0.0439i
   0.0199 + 0.0399i
   0.0199 - 0.0399i
   0.0443 + 0.0063i
   0.0443 - 0.0063i
   0.0399 + 0.0191i
   0.0399 - 0.0191i
   0.0281 + 0.0336i
   0.0281 - 0.0336i
  -0.0236 + 0.0373i
  -0.0236 - 0.0373i
   0.0022 + 0.0439i
   0.0022 - 0.0439i
   0.0076 + 0.0428i
   0.0076 - 0.0428i
   0.0028 + 0.0435i
   0.0028 - 0.0435i
   0.0337 + 0.0260i
   0.0337 - 0.0260i
  -0.0451 + 0.0030i
  -0.0451 - 0.0030i
  -0.0424 + 0.0137i
  -0.0424 - 0.0137i
  -0.0406 + 0.0167i
  -0.0406 - 0.0167i
  -0.0432 + 0.0073i
  -0.0432 - 0.0073i
  -0.0426 + 0.0118i
  -0.0426 - 0.0118i
  -0.0365 + 0.0235i
  -0.0365 - 0.0235i
   0.0390 + 0.0178i
   0.0390 - 0.0178i
   0.0244 + 0.0344i
   0.0244 - 0.0344i
   0.0422 + 0.0000i
   0.0418 + 0.0092i
   0.0418 - 0.0092i
  -0.0162 + 0.0380i
  -0.0162 - 0.0380i
  -0.0367 + 0.0204i
  -0.0367 - 0.0204i
   0.0209 + 0.0344i
   0.0209 - 0.0344i
  -0.0289 + 0.0291i
  -0.0289 - 0.0291i
  -0.0200 + 0.0347i
  -0.0200 - 0.0347i
  -0.0305 + 0.0265i
  -0.0305 - 0.0265i
  -0.0404 + 0.0000i
  -0.0111 + 0.0383i
  -0.0111 - 0.0383i
   0.0120 + 0.0381i
   0.0120 - 0.0381i
   0.0003 + 0.0406i
   0.0003 - 0.0406i
   0.0052 + 0.0401i
   0.0052 - 0.0401i
  -0.0263 + 0.0260i
  -0.0263 - 0.0260i
   0.0404 + 0.0000i
  -0.0189 + 0.0315i
  -0.0189 - 0.0315i
   0.0144 + 0.0352i
   0.0144 - 0.0352i
   0.0331 + 0.0193i
   0.0331 - 0.0193i
   0.0307 + 0.0213i
   0.0307 - 0.0213i
   0.0258 + 0.0271i
   0.0258 - 0.0271i
   0.0389 + 0.0068i
   0.0389 - 0.0068i
   0.0084 + 0.0383i
   0.0084 - 0.0383i
   0.0274 + 0.0248i
   0.0274 - 0.0248i
  -0.0238 + 0.0271i
  -0.0238 - 0.0271i
  -0.0076 + 0.0373i
  -0.0076 - 0.0373i
  -0.0372 + 0.0069i
  -0.0372 - 0.0069i
  -0.0110 + 0.0361i
  -0.0110 - 0.0361i
  -0.0028 + 0.0377i
  -0.0028 - 0.0377i
   0.0386 + 0.0026i
   0.0386 - 0.0026i
   0.0364 + 0.0097i
   0.0364 - 0.0097i
  -0.0348 + 0.0146i
  -0.0348 - 0.0146i
  -0.0353 + 0.0100i
  -0.0353 - 0.0100i
  -0.0154 + 0.0320i
  -0.0154 - 0.0320i
  -0.0267 + 0.0222i
  -0.0267 - 0.0222i
  -0.0202 + 0.0271i
  -0.0202 - 0.0271i
  -0.0339 + 0.0074i
  -0.0339 - 0.0074i
  -0.0074 + 0.0333i
  -0.0074 - 0.0333i
  -0.0337 + 0.0013i
  -0.0337 - 0.0013i
   0.0250 + 0.0243i
   0.0250 - 0.0243i
   0.0039 + 0.0343i
   0.0039 - 0.0343i
   0.0210 + 0.0271i
   0.0210 - 0.0271i
   0.0316 + 0.0146i
   0.0316 - 0.0146i
  -0.0279 + 0.0154i
  -0.0279 - 0.0154i
   0.0113 + 0.0318i
   0.0113 - 0.0318i
   0.0335 + 0.0086i
   0.0335 - 0.0086i
   0.0184 + 0.0274i
   0.0184 - 0.0274i
   0.0332 + 0.0000i
   0.0245 + 0.0197i
   0.0245 - 0.0197i
   0.0301 + 0.0119i
   0.0301 - 0.0119i
  -0.0001 + 0.0320i
  -0.0001 - 0.0320i
   0.0326 + 0.0054i
   0.0326 - 0.0054i
   0.0141 + 0.0283i
   0.0141 - 0.0283i
  -0.0304 + 0.0103i
  -0.0304 - 0.0103i
  -0.0140 + 0.0279i
  -0.0140 - 0.0279i
  -0.0071 + 0.0293i
  -0.0071 - 0.0293i
   0.0239 + 0.0166i
   0.0239 - 0.0166i
   0.0200 + 0.0199i
   0.0200 - 0.0199i
  -0.0269 + 0.0143i
  -0.0269 - 0.0143i
   0.0043 + 0.0300i
   0.0043 - 0.0300i
  -0.0290 + 0.0000i
  -0.0248 + 0.0142i
  -0.0248 - 0.0142i
   0.0058 + 0.0285i
   0.0058 - 0.0285i
  -0.0201 + 0.0185i
  -0.0201 - 0.0185i
  -0.0139 + 0.0229i
  -0.0139 - 0.0229i
   0.0113 + 0.0241i
   0.0113 - 0.0241i
   0.0267 + 0.0054i
   0.0267 - 0.0054i
   0.0278 + 0.0000i
   0.0165 + 0.0194i
   0.0165 - 0.0194i
   0.0069 + 0.0243i
   0.0069 - 0.0243i
  -0.0259 + 0.0074i
  -0.0259 - 0.0074i
  -0.0081 + 0.0234i
  -0.0081 - 0.0234i
  -0.0258 + 0.0042i
  -0.0258 - 0.0042i
  -0.0139 + 0.0197i
  -0.0139 - 0.0197i
  -0.0002 + 0.0255i
  -0.0002 - 0.0255i
   0.0177 + 0.0138i
   0.0177 - 0.0138i
   0.0241 + 0.0032i
   0.0241 - 0.0032i
  -0.0234 + 0.0086i
  -0.0234 - 0.0086i
   0.0105 + 0.0195i
   0.0105 - 0.0195i
   0.0159 + 0.0142i
   0.0159 - 0.0142i
  -0.0030 + 0.0234i
  -0.0030 - 0.0234i
   0.0030 + 0.0216i
   0.0030 - 0.0216i
   0.0216 + 0.0000i
   0.0214 + 0.0000i
   0.0106 + 0.0154i
   0.0106 - 0.0154i
   0.0052 + 0.0193i
   0.0052 - 0.0193i
  -0.0237 + 0.0000i
  -0.0152 + 0.0135i
  -0.0152 - 0.0135i
  -0.0094 + 0.0172i
  -0.0094 - 0.0172i
  -0.0178 + 0.0083i
  -0.0178 - 0.0083i
  -0.0186 + 0.0000i
  -0.0030 + 0.0173i
  -0.0030 - 0.0173i
  -0.0138 + 0.0122i
  -0.0138 - 0.0122i
  -0.0181 + 0.0036i
  -0.0181 - 0.0036i
  -0.0102 + 0.0132i
  -0.0102 - 0.0132i
  -0.0076 + 0.0127i
  -0.0076 - 0.0127i
   0.0155 + 0.0049i
   0.0155 - 0.0049i
   0.0135 + 0.0082i
   0.0135 - 0.0082i
   0.0167 + 0.0011i
   0.0167 - 0.0011i
   0.0107 + 0.0097i
   0.0107 - 0.0097i
   0.0054 + 0.0133i
   0.0054 - 0.0133i
  -0.0049 + 0.0121i
  -0.0049 - 0.0121i
   0.0002 + 0.0135i
   0.0002 - 0.0135i
  -0.0093 + 0.0076i
  -0.0093 - 0.0076i
  -0.0101 + 0.0050i
  -0.0101 - 0.0050i
  -0.0009 + 0.0108i
  -0.0009 - 0.0108i
  -0.0107 + 0.0005i
  -0.0107 - 0.0005i
   0.0088 + 0.0009i
   0.0088 - 0.0009i
  -0.0018 + 0.0061i
  -0.0018 - 0.0061i
   0.0063 + 0.0066i
   0.0063 - 0.0066i
   0.0029 + 0.0048i
   0.0029 - 0.0048i
  -0.0001 + 0.0000i
  -0.0026 + 0.0000i

Elapsed time is 0.973892 seconds.
Elapsed time is 1.285288 seconds.
\end{verbatim} \color{black}
    \begin{par}
If the matrix you are considering is very large, you may not want all of the eigenvalues, in this case you can use:
\end{par} \vspace{1em}
\begin{verbatim}
tic
EigVal6=eigs(A);
toc
\end{verbatim}

        \color{lightgray} \begin{verbatim}Elapsed time is 2.473976 seconds.
\end{verbatim} \color{black}
    \begin{par}
This returns the largest (in magnitude) six eigenvalues of the matrix. The function works the same as the previous one to calculate eigenvectors.
\end{par} \vspace{1em}
\begin{verbatim}
tic
[EigVec7,EigVal7]=eigs(A)
toc
\end{verbatim}

        \color{lightgray} \begin{verbatim}Warning: Only 5 of the 6 requested eigenvalues converged. 

EigVec7 =

  Columns 1 through 4

   0.0322 + 0.0000i  -0.0378 - 0.0003i  -0.0378 + 0.0003i  -0.0053 - 0.0113i
   0.0313 + 0.0000i  -0.0057 + 0.0388i  -0.0057 - 0.0388i  -0.0095 - 0.0098i
   0.0306 + 0.0000i   0.0223 - 0.0130i   0.0223 + 0.0130i  -0.0057 - 0.0062i
   0.0328 + 0.0000i  -0.0173 - 0.0136i  -0.0173 + 0.0136i   0.0101 + 0.0007i
   0.0328 + 0.0000i   0.0167 + 0.0010i   0.0167 - 0.0010i   0.0026 + 0.0143i
   0.0318 + 0.0000i  -0.0206 + 0.0513i  -0.0206 - 0.0513i   0.0065 - 0.0170i
   0.0317 + 0.0000i  -0.0068 + 0.0004i  -0.0068 - 0.0004i  -0.0103 - 0.0403i
   0.0333 + 0.0000i   0.0069 - 0.0074i   0.0069 + 0.0074i   0.0233 - 0.0232i
   0.0325 + 0.0000i  -0.0245 - 0.0474i  -0.0245 + 0.0474i  -0.0027 + 0.0016i
   0.0304 + 0.0000i  -0.0063 + 0.0471i  -0.0063 - 0.0471i   0.0013 - 0.0101i
   0.0317 + 0.0000i  -0.0122 + 0.0114i  -0.0122 - 0.0114i   0.0103 + 0.0259i
   0.0320 + 0.0000i  -0.0246 + 0.0060i  -0.0246 - 0.0060i   0.0476 + 0.0197i
   0.0312 + 0.0000i  -0.0495 - 0.0093i  -0.0495 + 0.0093i   0.0150 + 0.0241i
   0.0317 + 0.0000i   0.0234 + 0.0027i   0.0234 - 0.0027i   0.0038 - 0.0187i
   0.0316 + 0.0000i  -0.0064 + 0.0173i  -0.0064 - 0.0173i  -0.0170 + 0.0361i
   0.0310 + 0.0000i   0.0194 - 0.0260i   0.0194 + 0.0260i   0.0078 - 0.0017i
   0.0314 + 0.0000i  -0.0029 + 0.0420i  -0.0029 - 0.0420i  -0.0369 + 0.0080i
   0.0308 + 0.0000i   0.0025 - 0.0019i   0.0025 + 0.0019i  -0.0001 + 0.0306i
   0.0313 + 0.0000i   0.0199 - 0.0158i   0.0199 + 0.0158i   0.0069 - 0.0640i
   0.0312 + 0.0000i  -0.0390 + 0.0092i  -0.0390 - 0.0092i  -0.0231 - 0.0539i
   0.0314 + 0.0000i   0.0151 - 0.0314i   0.0151 + 0.0314i  -0.0207 + 0.0015i
   0.0312 + 0.0000i   0.0001 - 0.0107i   0.0001 + 0.0107i   0.0024 - 0.0057i
   0.0313 + 0.0000i   0.0129 - 0.0090i   0.0129 + 0.0090i  -0.0065 + 0.0455i
   0.0324 + 0.0000i   0.0151 - 0.0166i   0.0151 + 0.0166i  -0.0229 - 0.0265i
   0.0323 + 0.0000i  -0.0278 - 0.0149i  -0.0278 + 0.0149i   0.0005 - 0.0327i
   0.0327 + 0.0000i  -0.0013 + 0.0236i  -0.0013 - 0.0236i  -0.0267 + 0.0859i
   0.0315 + 0.0000i   0.0077 + 0.0194i   0.0077 - 0.0194i   0.0227 + 0.0149i
   0.0321 + 0.0000i   0.0057 - 0.0174i   0.0057 + 0.0174i   0.0016 + 0.0420i
   0.0317 + 0.0000i   0.0175 - 0.0116i   0.0175 + 0.0116i  -0.0035 - 0.0353i
   0.0312 + 0.0000i  -0.0271 + 0.0063i  -0.0271 - 0.0063i   0.0081 + 0.0196i
   0.0316 + 0.0000i  -0.0328 - 0.0073i  -0.0328 + 0.0073i  -0.0066 - 0.0195i
   0.0298 + 0.0000i  -0.0175 - 0.0594i  -0.0175 + 0.0594i  -0.0264 - 0.0147i
   0.0329 + 0.0000i  -0.0459 + 0.0257i  -0.0459 - 0.0257i   0.0074 + 0.0282i
   0.0323 + 0.0000i  -0.0302 - 0.0003i  -0.0302 + 0.0003i  -0.0335 + 0.0055i
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  Columns 5 through 6

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   0.0109 - 0.0233i  -0.0486 + 0.0000i
   0.0192 + 0.0145i  -0.0330 + 0.0000i
   0.0089 + 0.0265i   0.0393 + 0.0000i
   0.0045 + 0.0134i  -0.0202 + 0.0000i
  -0.0123 + 0.0000i  -0.0061 + 0.0000i
   0.0050 + 0.0154i   0.0487 + 0.0000i
  -0.0148 - 0.0131i  -0.0097 + 0.0000i
  -0.0030 - 0.0295i   0.0136 + 0.0000i
   0.0074 + 0.0507i   0.0791 + 0.0000i
  -0.0010 + 0.0091i   0.0109 + 0.0000i
   0.0147 - 0.0485i  -0.0590 + 0.0000i
   0.0505 - 0.0203i   0.0024 + 0.0000i
  -0.0213 - 0.0269i  -0.0526 + 0.0000i
   0.0151 + 0.0015i   0.0201 + 0.0000i
   0.0310 - 0.0347i  -0.0066 + 0.0000i
   0.0108 + 0.0320i  -0.0046 + 0.0000i
  -0.0165 + 0.0009i   0.0302 + 0.0000i
  -0.0023 - 0.0425i   0.0434 + 0.0000i
  -0.0247 + 0.0006i  -0.0296 + 0.0000i
  -0.0086 + 0.0115i  -0.0093 + 0.0000i
  -0.0265 + 0.0049i   0.0155 + 0.0000i
  -0.0288 - 0.0145i   0.0013 + 0.0000i
   0.0117 + 0.0595i   0.0193 + 0.0000i
  -0.0017 + 0.0073i  -0.0073 + 0.0000i
   0.0311 + 0.0124i   0.0259 + 0.0000i
  -0.0215 - 0.0333i  -0.0637 + 0.0000i
   0.0293 + 0.0428i   0.0131 + 0.0000i
  -0.0101 + 0.0352i  -0.0009 + 0.0000i
  -0.0246 + 0.0394i  -0.0646 + 0.0000i
   0.0181 - 0.0153i  -0.0025 + 0.0000i
  -0.0038 - 0.0049i  -0.0288 + 0.0000i
   0.0305 - 0.0069i   0.0533 + 0.0000i
  -0.0002 + 0.0378i   0.0090 + 0.0000i
  -0.0273 - 0.0169i  -0.0265 + 0.0000i
  -0.0180 + 0.0240i  -0.0251 + 0.0000i
  -0.0009 - 0.0312i  -0.0107 + 0.0000i
  -0.0002 + 0.0028i   0.0102 + 0.0000i
  -0.0224 + 0.0554i  -0.0122 + 0.0000i
  -0.0217 + 0.0080i  -0.0464 + 0.0000i
   0.0230 + 0.0294i  -0.0184 + 0.0000i
  -0.0105 + 0.0117i  -0.0512 + 0.0000i
   0.0177 - 0.0301i   0.0152 + 0.0000i
   0.0008 - 0.0189i   0.0562 + 0.0000i
   0.0207 - 0.0111i   0.0532 + 0.0000i
   0.0068 - 0.0304i   0.0271 + 0.0000i
   0.0077 - 0.0034i   0.0363 + 0.0000i
   0.0258 + 0.0085i  -0.0458 + 0.0000i
  -0.0139 - 0.0379i  -0.0348 + 0.0000i
   0.0005 - 0.0421i   0.0230 + 0.0000i
   0.0098 - 0.0046i  -0.0091 + 0.0000i
   0.0328 + 0.0100i   0.0147 + 0.0000i
  -0.0011 + 0.0411i  -0.0383 + 0.0000i
   0.0013 - 0.0175i   0.0439 + 0.0000i
   0.0019 + 0.0278i  -0.0383 + 0.0000i
  -0.0039 + 0.0033i  -0.0249 + 0.0000i
  -0.0096 - 0.0184i   0.0550 + 0.0000i
   0.0223 - 0.0099i  -0.0294 + 0.0000i
  -0.0106 + 0.0073i   0.0175 + 0.0000i
  -0.0089 - 0.0286i  -0.0576 + 0.0000i
   0.0184 + 0.0243i  -0.0656 + 0.0000i
  -0.0133 + 0.0045i  -0.0175 + 0.0000i
   0.0150 + 0.0030i   0.0333 + 0.0000i
   0.0188 + 0.0709i  -0.0343 + 0.0000i
   0.0266 - 0.0526i  -0.0145 + 0.0000i
  -0.0022 - 0.0079i   0.0369 + 0.0000i
   0.0083 + 0.0241i  -0.0268 + 0.0000i
  -0.0215 + 0.0521i  -0.0183 + 0.0000i
   0.0070 - 0.0153i  -0.0423 + 0.0000i
  -0.0077 - 0.0257i   0.0211 + 0.0000i
  -0.0000 + 0.0443i   0.0544 + 0.0000i
  -0.0288 - 0.0010i   0.0053 + 0.0000i
   0.0140 - 0.0087i  -0.0075 + 0.0000i
  -0.0182 - 0.0131i  -0.0004 + 0.0000i
  -0.0253 + 0.0311i   0.0156 + 0.0000i
   0.0111 - 0.0135i   0.0495 + 0.0000i
   0.0226 - 0.0222i   0.0030 + 0.0000i
  -0.0019 - 0.0097i   0.0327 + 0.0000i
  -0.0099 + 0.0307i  -0.0365 + 0.0000i
  -0.0300 - 0.0191i   0.0269 + 0.0000i
   0.0223 - 0.0259i  -0.0375 + 0.0000i
   0.0042 - 0.0234i   0.0025 + 0.0000i
  -0.0022 - 0.0033i  -0.0153 + 0.0000i
  -0.0063 - 0.0038i  -0.0302 + 0.0000i
   0.0035 + 0.0177i  -0.0141 + 0.0000i
   0.0030 + 0.0078i  -0.0143 + 0.0000i
   0.0262 + 0.0083i  -0.0847 + 0.0000i
  -0.0305 + 0.0364i  -0.0741 + 0.0000i
   0.0047 + 0.0262i   0.0096 + 0.0000i
  -0.0164 - 0.0377i   0.0496 + 0.0000i
   0.0037 + 0.0448i  -0.0300 + 0.0000i
  -0.0313 + 0.0463i   0.0361 + 0.0000i
  -0.0073 - 0.0195i  -0.0395 + 0.0000i
   0.0122 - 0.0124i  -0.0311 + 0.0000i
  -0.0047 + 0.0354i  -0.0143 + 0.0000i
   0.0057 + 0.0056i   0.0393 + 0.0000i
  -0.0040 - 0.0012i   0.0236 + 0.0000i
   0.0114 + 0.0045i  -0.0200 + 0.0000i
   0.0328 + 0.0189i   0.0327 + 0.0000i
   0.0286 + 0.0569i   0.0517 + 0.0000i
  -0.0126 - 0.0045i  -0.0066 + 0.0000i
   0.0166 + 0.0090i   0.0624 + 0.0000i
  -0.0057 + 0.0267i  -0.0161 + 0.0000i
   0.0237 - 0.0431i   0.0110 + 0.0000i
  -0.0032 - 0.0054i  -0.0213 + 0.0000i
  -0.0052 - 0.0056i  -0.0027 + 0.0000i
  -0.0022 + 0.0241i  -0.0139 + 0.0000i
   0.0213 + 0.0269i  -0.0380 + 0.0000i
   0.0031 + 0.0014i   0.0308 + 0.0000i


EigVal7 =

   1.0e+02 *

  Columns 1 through 4

   5.0007 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0905 + 0.0155i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0905 - 0.0155i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i  -0.0918 + 0.0059i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i

  Columns 5 through 6

   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i
  -0.0918 - 0.0059i   0.0000 + 0.0000i
   0.0000 + 0.0000i      NaN +    NaNi

Elapsed time is 1.839961 seconds.
\end{verbatim} \color{black}
    \begin{par}
If you specify a number, k, after the matrix, the function will return the largest k eigenvalues.
\end{par} \vspace{1em}
\begin{verbatim}
tic
EigVal8=eigs(A,1)
toc
\end{verbatim}

        \color{lightgray} \begin{verbatim}
EigVal8 =

  500.0662

Elapsed time is 0.058583 seconds.
\end{verbatim} \color{black}
    


\end{document}
    
