MATLAB: How to calculate the min and max degree/order of a rational polynomial

Symbolic Math Toolbox

I have a algebraic function of 't':

syms t;
p=5/t^2 - 9/t - 9*t + 5*t^2 - 1/t^4 - t^3 + 19;

How do I find the min and max order of 't'?

For example, min = -4, max = 3.

Best Answer

Currently, there is no built-in function to calculate the min and max order of a rational polynomial.

There are two workarounds. If I have the following rational polynomials for example:

p{1} = t^5 + 2*t^4 + 3*t^3 + 4*t^2 + 5*t + 6 + 7/t + 8/t^2 + 9/t^3;
p{2} = t^5 + 2*t^4 + 3*t^3 + 4*t^2;

The workarounds are:

[num,den]=numden(p{i});
orderNum = polynomialDegree(num);
orderDen = polynomialDegree(den);
% workaround 1
if orderNum~=0 && orderDen~=0
    minOrder = -orderDen;
    maxOrder = orderNum - orderDen;       
elseif orderDen==0
    maxOrder = orderNum;
    indexNonZeroCoeff = find(fliplr(coeffs(p{i},'all'))>0);
    minOrder = indexNonZeroCoeff(1)-1;
end
% workaround 2
[n, d] = numden(p{i});
maxo = polynomialDegree(n)-polynomialDegree(d);
[d, n] = numden(subs(p{i}, t, 1/t));
mino = polynomialDegree(n) - polynomialDegree(d);

Please see the attached file 'exampleFindOrder.m' for the full example code.

For more detailed information about the 'polynomialDegree' function:

https://www.mathworks.com/help/symbolic/polynomialdegree.html

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When considering whether a given ODE solver can exactly integrate an ODE of a particular order, the answer has to do with the number of slope evaluations in each step of the algorithm. Euler's method evaluates the slope only once at the beginning of each interval, so it can only exactly integrate constant functions. A 4th order Runge-Kutta solver uses 4 evaluations, so it can exactly integrate cubic functions. A 5th order Runge-Kutta solver actually uses 6 terms, so it can be used for quartic and quintic functions.

Since "ode45" uses both 4th and 5th order methods to obtain error estimates in each step, these methods will have a growing error discrepancy when you go from cubic to quartic functions and beyond, as you have noticed. This is because the 4th order method will not be able to integrate functions of order higher than a cubic very accurately, and it will do progressively worse as the order increases.

in order to obtain a higher accuracy for polynomials of order 4 and above, reduce the "RelTol" and "AbsTol" properties using the "odeset" function.

MATLAB: How to find degree of polynomial in Matlab

In the case you use, where you are working with ratios of symbolic polynomials, then

degree_num = length(coeffs(num,p)) - 1;
degree_den = length(coeffs(den,p)) - 1;

You can also simplify (at least logically) the taking of the limit:

[n, d] = numden(z(p));
coeffn = coeffs(n,p);
coeffd = coeffs(d,p);
lenn = length(coeffn);
lend = length(coeffd);
if lenn > lend + 1
  error('limit would have been infinite')
elseif lenn <= lend
  error('limit would have been zero');
else
  a = coeffn(end);
end

If you are sure that neither of the two exception cases can occur, then

t = coeff(numden(z(p)));
a = t(end);

Best Answer

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