# 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)))

Did you look on the file exchange? (Clearly not.)
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.
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          1.65068012388579weights =         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 xint((x.^5 -3*x.^2 - 2)*exp(-x^2),[-inf,inf])ans =-(7*pi^(1/2))/2vpa(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 cfun = @(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 xint(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