Modified 2nd order newton rapson method

I want to solve the following equation:

integration{v^4.exp{a+b.v+c.v^2}dv} from (0,vmax) = 1/N * sum (1,N) v^2

where a,b,c are unknowns to be found

Can anybody explain me how to solve this equation using Modified 2nd order newton rapson method

Answer 1

[Ronit]

Hello Gaurav,

To solve the given equation using a modified 2nd order Newton-Raphson method in MATLAB, you can refer to the following steps:

1. Use numerical methods to evaluate the integral on the left-hand side. MATLAB provides the integral function for this purpose, which is suitable for handling complex integrands.
2. Initialize the parameters ‘a’, ‘b’, and ‘c’. These initializations can be arbitrary but should be based on the context of the problem.
3. The modified 2nd order version involves using both the first and second derivatives (gradient and Hessian) to update the parameters.
4. Gradient (First Derivative): Compute the gradient of the integral with respect to each parameter. This involves differentiating the integrand with respect to ‘a’, ‘b’, and ‘c’.
5. Hessian (Second Derivative): Compute the Hessian matrix, which contains the second derivatives of the integral with respect to the parameters.
6. Update the parameters iteratively using the Newton-Raphson update rule:

delta = -hessian \ gradient;
a = a + delta(1);
b = b + delta(2);
c = c + delta(3);

Now, continue iterating until the change in parameters is below a certain tolerance level or a maximum number of iterations is reached. This ensures that the solution converges to the desired accuracy.

Please refer to the documentation of integral which can be used during implementation: https://www.mathworks.com/help/matlab/ref/integral.html

I hope it helps you query!