Numerical Analysis $-$Homework $2-$ Final topics

Jacobi and Gauss$-$Seidel Iteration Methods

Solve the following system of linear equations using the Jacobi and Gauss$-$Seidel methods. Perform five iterations for each method and compare the results. Use an initial guess of $x_1=0,x_2=0,x_3=0.$

$4x_1-x_2+x_3=7$

$-2x_1+5x_2-2x_3=-8$

$x_1-2x_2+4x_3=6$

Trapezoidal Rule

Use the trapezoidal rule to approximate the integral:

${\displaystyle \underset{0}{\overset{2}{}}}{e}^{-x^2}dx$

Divide the interval into $n=4$ subintervals.

$3.$ Simpson's $\frac{1}{3}$ Rule

Using Simpson's $\frac{1}{3}$ Rule, approximate the integral:

${\displaystyle \underset{0}{\overset{1}{}}}\frac{1}{1+x^2}dx$

Divide the interval into $n=6$ subintervals.

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