Calculate the roots of many polynomials

Question

Given a matrix Athat represents polynomials in each column. How can one efficiently calculate the roots of each polynomial without cycles?

Answer 1

3 methods are compared here:

• A simple loop through all the lines, using rootsfor each line.
• A fully loop approach based on the YBE idea of ​​using a block-diagonal matrix, using sparseas an intermediate
• A simple loop through all the lines, but this time using the "built-in" code from roots.

Code:

%// The polynomials
m = 15;
n = 8;
N = 1e3;

X = rand(m,n);


%// Simplest approach
tic
for mm = 1:N

    R = zeros(n-1,m);
    for ii = 1:m
        R(:,ii) = roots(X(ii,:));
    end

end
toc


%// Completely loopless approach
tic
for mm = 1:N

    %// Indices for the scaled coefficients
    ii = repmat(1:n-1:m*(n-1), n-1,1);
    jj = 1:m*(n-1);

    %// Indices for the ones
    kk = bsxfun(@plus, repmat(2:n-1, m,1), (n-1)*(0:m-1).');  %'
    ll = kk-1;

    %// The block diagonal matrix
    coefs = -bsxfun(@rdivide, X(:,2:end), X(:,1)).';  %'
    one   = ones(n-2,m);
    C = full(sparse([ii(:); kk(:)], [jj(:); ll(:)],...
        [coefs(:); one(:)]));

    %// The roots
    R = reshape(eig(C), n-1,m);

end
toc


%// Simple loop, roots() "inlined"
tic    
R = zeros(n-1,m);
for mm = 1:N

    for ii = 1:m            
        A = zeros(n-1);
        A(1,:) = -X(ii,2:end)/X(ii,1);
        A(2:n:end) = 1;
        R(:,ii) = eig(A);            
    end

end
toc

Results:

%// m=15, n=8, N=1e3:
Elapsed time is 0.780930 seconds. %// loop using roots()
Elapsed time is 1.959419 seconds. %// Loopless
Elapsed time is 0.326140 seconds. %// loop over inlined roots()

%// m=150, n=18, N=1e2:
Elapsed time is 1.785438 seconds. %// loop using roots()
Elapsed time is 110.1645 seconds. %// Loopless
Elapsed time is 1.326355 seconds. %// loop over inlined roots()

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Answer 2

arrayfun roots, .

n = size(A,2);
t = arrayfun(@(x)roots(A(:,x)), 1:n, 'UniformOutput', 0);

cell2mat, . : r = cell2mat(t),

r = cell2mat(arrayfun(@(x)roots(A(:,x)), 1:n, 'UniformOutput', 0));

Answer 3

Practically, what rootsis to find the eigenvalues ​​of the accompanying matrix.

roots(p) = eig(compan(p))

So, here is my example, which builds a block diagonal matrix from the companion matrices of each polynomial, than it finds the eigenvalues ​​of a block diagonal matrix.

>> p1=[2 3 5 7];
>> roots(p1)

ans =

  -0.0272 + 1.5558i
  -0.0272 - 1.5558i
  -1.4455          

>> eig(compan(p1))

ans =

  -0.0272 + 1.5558i
  -0.0272 - 1.5558i
  -1.4455          

>> p2=[1 2 9 5];
>> roots(p2)

ans =

  -0.6932 + 2.7693i
  -0.6932 - 2.7693i
  -0.6135          

>> p3=[5 1 4 7];
>> roots(p3)

ans =

   0.3690 + 1.1646i
   0.3690 - 1.1646i
  -0.9381          

>> A=blkdiag(compan(p1),compan(p2),compan(p3))

A =

   -1.5000   -2.5000   -3.5000         0         0         0         0         0         0
    1.0000         0         0         0         0         0         0         0         0
         0    1.0000         0         0         0         0         0         0         0
         0         0         0   -2.0000   -9.0000   -5.0000         0         0         0
         0         0         0    1.0000         0         0         0         0         0
         0         0         0         0    1.0000         0         0         0         0
         0         0         0         0         0         0   -0.2000   -0.8000   -1.4000
         0         0         0         0         0         0    1.0000         0         0
         0         0         0         0         0         0         0    1.0000         0

>> eig(A)

ans =

  -0.0272 + 1.5558i
  -0.0272 - 1.5558i
  -1.4455          
  -0.6932 + 2.7693i
  -0.6932 - 2.7693i
  -0.6135          
   0.3690 + 1.1646i
   0.3690 - 1.1646i
  -0.9381