MATLAB: Is there a matlab code for gaussion integration!!!

integrationsimulation

 syms x;
 syms E;
  syms r;
  syms D;
  syms c;
syms m;
 g(x)=exp((-2./h).*(sqrt(2*m*(E-c))).*(sqrt(x*(x+r))-r*log((sqrt(r)+sqrt(x+r))-D)))

Best Answer

Did you look on the file exchange? (Clearly not.)
https://www.mathworks.com/matlabcentral/fileexchange/8679-gaussquad
https://www.mathworks.com/matlabcentral/fileexchange/1130-quadg
They both look decent, though guassquad is purely a gauss-legendre code, gaussg a more general code for standard weight functions, though it is a guass-kronrod scheme.
In fact, even my sympoly toolbox provides gaussian quadrature rules, thus the weights and nodes for any standard weight function class. So look at the guassquadrule function.
https://www.mathworks.com/matlabcentral/fileexchange/9577-symbolic-polynomial-manipulation
So using guassquadrule from sympoly, the nodes and weights for a 4 point Gauss-Hermite rule would be:
[nodes,weights] = gaussquadrule(4,'hermite')
nodes =
         -1.65068012388579         -0.52464762327529          0.52464762327529          1.65068012388579
weights =
         0.081312835447245         0.804914090005513         0.804914090005513        0.0813128354472448
To compute the integral of (x^5 - 3*x^2 - 2)*exp(-x.^2), from -inf to inf, I might do:
fun = @(x) (x.^5 -3*x.^2 - 2);
dot(weights,fun(nodes))
ans =
          -6.2035884781693
Syms confirms that it was correct:
syms x
int((x.^5 -3*x.^2 - 2)*exp(-x^2),[-inf,inf])
ans =
-(7*pi^(1/2))/2
vpa(ans)
ans =
-6.203588478169306095543586191694
However, will you find a tool that performs gaussian integration for a function that contains symbolic parameters? No, you won't find anything for a good reason. Gaussian integration is a numerical integration procedure, not really designed to do symbolic integration.
Can you use gaussian quadrature for a symbolic function? Well, yes.
So, here a Gauss-Laguerre integral of (a*x*2 + b*x + c)*exp(-x), over the domain [0,inf].
syms a b c
fun = @(x) a*x.^2 + b*x + c;
[nodes,weights] = gaussquadrule(5,'laguerre');
vpa(dot(weights,fun(nodes)),13)
ans =
2.0*a + 1.0*b + 1.0*c
What Gaussian quadrature rule you expect to apply to this function:
g(x)=exp((-2./h).*(sqrt(2*m*(E-c))).*(sqrt(x*(x+r))-r*log((sqrt(r)+sqrt(x+r))-D)))
I have no idea, since I don't see any domain provided, nor do I see any standard weight function in there that can be extracted. So I have a funny feeling that you may be confused as to the purpose and utility of classical Gaussian quadratures.
If you look at Gaussian quadrature rules, they presume a weight function from among several standard forms, AND a domain of integration. Gauss-Legendre assumes a unit weight function, so is applicable to integration of a general function, over the interval [-1,1]. For example, suppose you wanted to compute the integral of cos(exp(x)), over the interval [0,1].
syms x
int(cos(exp(x)),[-1,1])
ans =
cosint(exp(1)) - cosint(exp(-1))
vpa(ans)
ans =
0.67038594208938451613294763492665
Luckily, the symbolic TB is smart enough to do the work, because I'm way too sleepy to think.
Gauss-Legendre presumes an interval of [-1,1], although the nodes and weights can be transformed for other intervals.
fun = @(x) cos(exp(x));
[nodes,weights] = gaussquadrule(15,'legendre');
dot(weights,fun(nodes))
ans =
         0.670385942089469
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