Mathematics : Three question about Cholesky decomposition, Jacobi method, Taylor series, centered finite-difference approximation and 4th-order RK method.

The complete question file was attached below. Please have a look.

The first question is for a symmetric matrix system

(a) perform a Cholesky decomposition by hand and then solve the system;

(b) use the Jacobi method to try it again by starting with an initial estimate of {−1 2 −1}T , and discuss your results.

The second question:

Use a Taylor series expansion to derive a centered finite-difference approximation to the third derivate that is second-order accurate. To do this, you will have to use four different expansions for the points xi-2, xi-1, xi+1 and xi+2. In each case, the expansion will be around the point xi. The interval ∆x will be used in each case of i −1 and i +1, and 2∆x will be used in each case of i − 2 and i + 2 . The four equations must then be combined in a way to eliminate the first and second derivatives. Carry enough terms along in each expansion to evaluate the first term that will be truncated to determine the order of the approximation.

The last question is using 4th order RK method to do calculations.

The complete question file was attached below. Please have a look.

Answers

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