Exercises 45-54
Exercise 55.
Exercise 56.

Solve the system linear of equations

2x1 + 7x2 + x3 = 19.
4x1 + x2 - x3 = 3
x1 - 3x2 + 12x3 = 31
a) by the Jacobi method
b) by the Gauss-Seidel method (3 iterations).

In part a), compute the iteration matrix G and find out if it has norm smaller that 1 for some of the matrix norms.
Hint: First change the order of equations to obtain a strictly diagonally dominant coefficient matrix.

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Exercise 57.

The coefficient matrix of the linear system

2x + y + z = 4
x + 2y + z = 4
x + y + 2z = 4
is not strictly diagonally dominant. Use
a) the Jacobi method
b) the Gauss-Seidel method
and compute iteration (3). Start from (0) = . Compute the iteration matrices G and study the convergence of the iteration sequences.

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Exercise 58.

Use the Gauss-Seidel method and solve the system

3x1 +x3 = 4
-x1 -x2 +3x3 = 1
x1 +2x2 = 3
Choose (0) = and stop, when || (k) - (k-1)|| < 0,1.

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Exercise 59.

Consider the over determined system

x1 - x3 = 4
x1 - 3x3 = 6
x2 + x3 = - 1
- x2 + x3 = 2
Find the least squares solution. Calculate the 2-norm of the residual vector . Find the pseudo inverse of the coefficient matrix A.

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Exercise 60.

Verify, by straight-forward calculation, that the matrix

A = 1-1 1
0 0 1
0 0 1

satisfies its own characteristic equation. Use the Cayley-Hamilton theorem and compute A5.

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Exercises 61-64
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