WUT_Computer_Science/ENUME/references/ENUME_Project1

ENUME_Project1

A Numerical Method project cointaining 4 tasks.

1.

Write a program finding macheps in the MATLAB environment on a lab computer or your computer.

2.

Write a general program solving a system of n linear equations Ax = b using the indicated method Apply the program to solve the system of linear equations for given matrix A and vector b, for increasing numbers of equations n = 10,20,40,80,160,… until the solution time becomes prohibitive (or the method fails), for:

a)

Aij { for i = j -> 13, for i = j-1 or i = j+1 -> 4, other -> 0
Bi = 2.4 + 0.6i,
i,j = 1..n;

b)

Aij = 4/[5(i + j – 1)], Bi = 1/(2 i), i – odd; Bi = 0, i – even, i, j = 1,…,n;

3

Write a general program for solving the system of n linear equations Ax = b using the Gauss-Seidel and Jacobi iterative algorithms. Apply it for the system:
13x1 + 2x2 – 8x3 + x4 =16
x1 + 10x2 + 5x3 – 2x4 = 24
6x1 + 2x2 – 23x3 + 15x4 = 184
x1 + 2x2 – x3 + 13x4 = 82

4

Write a program of the QR method for finding eigenvalues of 5×5 matrices:
a) without shifts;
b) with shifts calculated on the basis of an eigenvalue of the 2×2 right-lower-corner submatrix.