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This study investigates the characteristics of random polynomials up to degree 100 and random matrices, focusing on the average number of real zeros and real eigenvalues. By creating 1,000 random polynomials and analyzing them using the Kac Formula, we examine variations in coefficient selection from standard normal and uniform distributions. We simulate random matrices to evaluate the number of real eigenvalues, comparing the results with theoretical expectations. Join us in exploring the fascinating relationships between coefficients, real roots, and eigenvalues in our mathematical simulations.
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Random Polynomials & Random Matrices Anne Calder & Alan Cordero
Goals • Create 1000 random polynomials up to degree 100 • Calculate the number of real zeros on average for each degree • Check our results with the Kac Formula • Change the way our coefficients are chosen and check the average number of real zeros • Create random matrices and check the number of real eigenvalues
Review • Kac Formula for finding the expected amount of real zeros of a random polynomial • Coefficients are chosen from a standard normal distribution
Random Polynomial Example • Create a random polynomial of degree 5 >> x=randn([1 6]) -1.2025 0.4054 0.1872 1.1233 -1.3993 0.2030 • What polynomial would look like -1.2025x5+0.4054x4+0.1872x3+1.1233x2-1.3993x+0.2030=0
Roots of the polynomial • >> roots(x) ans = -0.7096 + 0.8911i -0.7096 - 0.8911i 0.7938 + 0.3759i 0.7938 - 0.3759i 0.1687
Simulating 1000 random polynomials N=1000; % 1000 simulations x=zeros(1000, 5); % creates a empty 1000 x 5 matrix for k=1:N x(k,:) = roots(randn([1 6])); % calculates the roots end after each simulation size(x(imag(x)==0)) /1000 % displays the average number of real zeros
Kac Plot Comparison N=100; En = zeros(1,N); for i=1:N En(i) = 4/pi*quad(@(x)sqrt(1./(1-x.^2).^2 - (i+1)^2*x.^(2*i)./(1-x.^(2*i+2)).^2),0,1); end plot(1:N,En)
Different Coefficients • rand – a random number from the standard uniform distribution on the open interval (0,1) • rand -.5 – a random number from the uniform distribution on the open interval (-.5,.5)
How many eigenvalues of a random matrix are real? • The expected number of real eigenvalues with independent standard normal entries is:
Eigenvalue CodeSimulating 1000 Random Matrices n=1000; Y=zeros(1,n); %creates an empty 1:n empty matrix X=[1:n]; fori=1:n A=randn(i); %creates a matrix size n with random entries e=eig(A); %eigenvalues of a s=size(e(imag(e)==0)); %amount of real eigenvalues p=s(1) Y(i)=p; plot(X,Y,'b.') title('Amount of real eigenvalues') xlabel('n - Size of square matrix') ylabel('Amount of real eigenvalues') end
The data doesn’t change dramatically with the change of how the coefficients are chosen.
