CODEWITHSUNDEEP
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jbl
jbl

Reputation: 135

Finding zeros, minima, and maxima of a polynomial in Matlab

How can I find the zeros, maxima, and minima of the polynomial x^3 - 10x + 20? So far, I have done the following

fhandle1 = @(x) x.^3 - 10.*x + 20; 
figure(1);
hold on;
fplot(fhandle1,[0, 10]); 
[xMin, yMin] = fminbnd(fhandle1, 2.5, 4);
plot(xMin, yMin, 'or')
[xzero,yzero] = fzero(fhandle1,5);
plot(xzero,yzero,'*g')

How can I find the maxima of the polynomial?

Upvotes: 1

Views: 572

Answers (1)

Wolfie
Wolfie

Reputation: 30047

fminbnd will return the max/min value of a function, which isn't necessarily the same as the maxima/minima.

The most complete way to approach this for any given polynomial leans on two facts

  1. Extrema of a function f(x) occur when f'(x) = 0
  2. You can classify the extrema as maxima/minima by evaluating f''(x)

You can use roots to find all roots of a polynomial from its coefficients, and you can use polyder to find the derivative coefficients for a polynomial.

Output

plot

Code

coeffs = [1, 0, -10, 20]; % 1*x^3 + 0*x^2 - 10*x + 20
% Find zeros
r = realRoots( coeffs );
% Get the coeffs of the derivative
dcoeffs = polyder( coeffs );
% Minima/maxima are at roots of derivative
dr = realRoots( dcoeffs );
% Get coeffs of next derivative
ddcoeffs = polyder( dcoeffs );
% Minima: ddcoeffs > 0, Maxima: ddcoeffs < 0, Inflection pt: ddcoeffs = 0 
minima = dr( polyval( ddcoeffs, dr ) > 0 );
maxima = dr( polyval( ddcoeffs, dr ) < 0 );
inflection = dr( polyval( ddcoeffs, dr ) == 0 );

% Plot (ranges would need to be based on extrema for a generic polynomial)
figure(1); clf; hold on; grid on;
x = -10:0.1:4; 
plot( x, polyval( coeffs, x ), 'LineWidth', 1 );                       % function line
plot( r, polyval( coeffs, r ), 'ok', 'LineWidth', 2 );                 % roots
plot( minima, polyval( coeffs, minima ), 'vk', 'LineWidth', 2 );       % minima
plot( maxima, polyval( coeffs, maxima ), '^k', 'LineWidth', 2 );       % maxima
plot( inflection, polyval( coeffs, inflection ), 'xk', 'LineWidth', 2 ); % inflection
ylim( [-20, 50] );

% Helper function to return the real roots from polynomial coeffs
function r = realRoots( c )
    r = roots( c );        % all roots
    r = r( imag(r) == 0 ); % real roots
end

Upvotes: 3

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