hiteshsai/numerical-approximation-of-integrals

This program approximates numerical one dimensional integrals and find the area under the curve

Numerical-Approximation-Of-Integrals

This program approximates numerical one dimensional integrals and find the area under the curve

Task and approach

We were assigned the task to compute 1D integrals. We used many ways for integration

1. Riemann sum(left, right and centre)
2. Trapezoidal rule
3. Monte-carlo integration(1D and 2D)
4. Simpson rule
5. Boole’s rule

We plotted the Riemann left, right and centre rectangles, trapezoids and we also plotted random points for monte-carlo on the given curve to make it clear what is going on.

References

We took reference from T.M.Apostol and the following websites:

• http://www.intmath.com/integration/6-simpsons-rule.php
• http://en.wikipedia.org/wiki/Monte_Carlo_integration
• http://en.wikipedia.org/wiki/Boole's_rule
• http://matplotlib.org/mpl_toolkits/mplot3d/tutorial.html

Working of each function:

1. Riemann sum: We define riemann_left, riemann_right and riemann_centre which takes 4 arguments: f (function), xmin, xmax and n (number of rectangles). The entire region is divided into n rectangles and the area of the rectangles are found and then summed up to find the value of the integral.

2. Monte-Carlo integration: We define the function monte_carlo which takes 4 arguments: f (function), xmin, xmax and n (number of random uniform points). In the function we define a rectangle_area= (xmax-xmin)(ymax-ymin). The function then takes n random points in this rectangle and checks how many of them are above and below the curve. If there are k points below the curve then the function computes the integral as area of the rectangle probability of getting a point below the curve.

3. Boole’s rule: We define the function boole which takes 4 arguments: f (function), a, b, k. It approximates the integral by using the value of f at 5 equally spaced points.

4. Trapezoidal rule: We define the function trapez which takes 4 arguments :( function), a, b and n (number of trapezoids). It approximates the region under the graph as a trapezoid and calculating its area.

5. Simpson Rule: We define the function simpson which takes 4 arguments: f (function), a, b, n (number of intervals). The function approximates the integral using a polynomial of degree 2.

6. Monte-Carlo 2D integration: We define the function monte_carlo_cuboid which takes 6 arguments: f (function), xmin, xmax,ymin,ymax and n (number of random uniform points). In the function we define a cuboid_volume= (xmax-xmin)(ymax-ymin)(zmax-zmin). The function then takes n random points in this cuboid and checks how many of them are above and below the curve. If there are k points below the curve then the function computes the integral as volume of the cuboid* probability of getting a point below the curve.

Plotting makes use of inbuilt function pylab.

1. Plotting Riemann rectangles: We defined the functions plot_riemann_left, plot_riemann_right and plot_riemann_centre which take 5 arguments: f (function), xmin, xmax, name (name of the function) and n (no. of rectangles). The functions plot the Riemann left, right and centre rectangles on the curve of the function.

2. Plotting Trapezoidal: We define the function plot_trapezoid which takes 5 arguments: f (function), xmax, xmin, name (name of the function) and n (no. of trapezoids). The function plots the trapezoids under the curve which is used for the approximation of the integral.

3. Plotting Monte-Carlo: We define the function plot_monte_carlo which takes 5 arguments: f (function), xmax, xmin, name (name of the function) and n (no. of random points). The function plots the random points in the rectangle that we have defined around the function.

4.* Plotting Monte-Carlo 2_D:* We define the the function plot_monte_carlo_cuboid .We managed to make Monte carlo 2D graph by looking into website for plotting 3D curves.

Difficulties faced:

We faced difficulty in Monte carlo 2D and the plotting of its curve was even more difficult but after doing a bit research in websites relating to plotting of 3D curves we finally succeded.