| 2x1 | + | 7x2 | + | x3 | = | 19 | . | |||||
| 4x1 | + | x2 | - | x3 | = | 3| x1 | - | 3x2 | + | 12x3 | = | 31 | |
Solve the system linear of equations
|
| 2x1 | + | 7x2 | + | x3 | = | 19 | . | |||||
| 4x1 | + | x2 | - | x3 | = | 3| x1 | - | 3x2 | + | 12x3 | = | 31 | |
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.
The coefficient matrix of the linear system
| 2x + y + z = 4| x + 2y + z = 4 | x + y + 2z = 4 | |
(3). Start from
(0) = 
.
Compute the iteration matrices G and study the convergence of the iteration sequences.
Use the Gauss-Seidel method and solve the system
| 3x1 +x3 = 4| -x1 -x2 +3x3 = 1 | x1 +2x2 = 3 | |
(0) = and stop, when
|| (k) - (k-1)||
< 0,1.
Consider the over determined system
| x1 | - | x3 | = | 4| x1 | - | 3x3 | = | 6 | x2 | + | x3 | = | - 1 | - x2 | + | x3 | = | 2 | |
.
Find the pseudo inverse 
of the
coefficient matrix A.
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.