Computational Physics (Making programs for interpolation and differentiation)

[jhosamelly]

Homework Statement

Chapter 4
1. Write a program that implements the first order (linear) interpolation
2. Write a program that implemets n-point Lagrange interpolation. Trean n as an imput parameter.
3. Apply the program to study the quality of the Lagrange interpolation to functions
f(x)=sin(x2), f(x)=esin(x), and f(x)=0.2/[(x-3.2)2+0.04]
initially calculated in 10 unifirm points in the interval [0.0, 5.0]. Compare the results with the cubic spline interpolation.
4. Use third and seventh order polynomial interpolation to interpolate Runge's function:

at the n=11 points xi=-1.0, -0.8, ..., 0.8, 1.0. Compare the results with the cubic spline interpolation
Study how the number of data points for interpolation affects the quality of interpolation of 5. Runge's function in the example above.Chapter 5

1. Write a program to calculate sin(x) and cos(x) and determine the forward differentiation.
2. Do the same but use central difference.
3. Plot all derivatives, and compare with the analytical derivative.
4. The half-life t1/2 of 60Co is 5.271 years. Write a program which calculates the activity as a function of time and amount of material. Design your program in such a way, that you could also input different radioactive materials.
5. Write a program which will calculate the first and secon derivative for any function you give.

Homework Equations

Here is the link for the problem.
its chapter 4 and 5.
http://ww2.odu.edu/~agodunov/book/solutions.html

The Attempt at a Solution

well, This has not been discussed to us yet. but our professor is out of the country so he asked us to do these programs which I have no idea how to do. Can someone please help me? We're using fortran.

 

[mark726]

Hello,

Thank you for posting your question. I am a scientist and I would be happy to help you with your programming assignment.

For the first problem in Chapter 4, you will need to write a program that implements the first order (linear) interpolation. This means that you will need to create a function that takes in two points (x1,y1) and (x2,y2) and calculates the value of the function at any point in between those two points using the formula for linear interpolation: f(x) = y1 + (x-x1)(y2-y1)/(x2-x1).

For the second problem, you will need to write a program that implements n-point Lagrange interpolation. This means that you will need to create a function that takes in n points and calculates the value of the function at any point using the Lagrange interpolation formula. This formula involves a sum of terms, each of which includes one of the given points and its corresponding function value.

For the third problem, you will need to apply the program you created in the second problem to study the quality of Lagrange interpolation for the given functions. This means that you will need to calculate the values of the functions at the 10 uniform points in the given interval and then use your program to calculate the interpolated values at any point in between those points. You can then compare the results with the cubic spline interpolation, which is a different method of interpolating data.

For the fourth problem, you will need to use third and seventh order polynomial interpolation to interpolate Runge's function at the given points. This means that you will need to create two different functions, one for each order of polynomial, and use them to calculate the values of the function at any point in between the given points. You can then compare the results with the cubic spline interpolation again.

For the fifth problem in Chapter 5, you will need to write a program that calculates the sine and cosine functions, and then use these functions to determine the forward differentiation and central difference of a given function. You can then plot these derivatives and compare them with the analytical derivative.

For the last problem, you will need to write a program that calculates the activity of a radioactive material as a function of time and amount of material. This will involve using the half-life equation: A(t) = A0 * exp(-λt), where A0 is the initial activity, λ is the decay constant, and t is time. You will