Сonvert the coefficients of the Hermite polynomial into a function

I want to make a function from the output of Matlab Hermite function (for example, if we had an output from Hermite function ```[8 0 -12 0]``` it would be ```8x^3 - 12x``` polynomial) and then integrate this function using the **Simpson's 3/8 Rule**.

I have already created a function in Matlab that integrate any function using this rule and also I have created function that returns coefficients of Hermite's polynomial (with the recursion relation) in the vector form.

My questions:

1) If it's possible, in Hermite function I want from this output ```[8 0 -12 0]``` make this output ```8x^3 - 12x```. This output I will able to integrate. How can I do this?

2) Can I combine these two functions and integrate Hermite's polynomial without convention the output of the first function?

Code of Hermite polynomial function, where n is the order of the polynomial:

```

function h = hermite_rec(n)

if( 0==n ), h = 1;

elseif( 1==n ), h = [2 0];

else

h1 = zeros(1,n+1);

h1(1:n) = 2*hermite_rec(n-1);

h2 = zeros(1,n+1);

h2(3:end) = 2*(n-1)*hermite_rec(n-2);

h = h1 - h2;

end

```

Code of Simpson function, that integrate function using the Simpson 3/8 Rule. a is a lower limit of integral, b is a upper limit of integral:

```function IS2 = NewtonIntegral(a,b)

n = 3;

h = (b-a)/(3*n); %3h = (b-a)/n

IS2=0;

for i=1:n

IS2 = IS2+(f(a+(3*i-3)*h) + 3*f(a+(3*i-2)*h) + 3*f(a+(3*i-1)*h) + f(a+(3*i)*h))*3*h/8;

end

end

```

Thank you for any advice!

Answer 1

[Torsten]

https://de.mathworks.com/help/symbolic/poly2sym.html

https://de.mathworks.com/help/symbolic/sym.int.html