I have this linear algebra question concerning the jacobi method and the frobenius norm that I am having a lot of trouble on, I have an exam soon and I would appreciate any help. NOTE: I have read the entire article of wikipedia on jacobi method and frobnenius norm but I have just started the linear algebra topic and there's only so much I can make out of it.
Consider the following system of equations…
$$
\left\{
\begin{array}{c}
2x + y =5\\
x – 3y =6
\end{array}
\right.
$$
1. Compute the Frobenius norm for the Matrix C under the Jacobi iterative method(would you expect this problem to converge)?
2.Starting at (x, y) =
(0, 0) perform 2 iterations of Jacobi Iterative method for this problem.
I dont understand what the Frobenius norm is exactly, so I started at question 2…
First I converted the system of equations to the respective augmented matrix…
$$
\begin{bmatrix}
2 & 1 & 5 \\
1 & -3 & 6 \\
\end{bmatrix}
$$
From here I can see that…
$$
\left\{
\begin{array}{c}
x^{new} = (-y^{old} + 5)/2\\
y^{new} = (6 – x^{old})/-3
\end{array}
\right.
$$
So I set up the array…
$$
\begin{array}{c|lcr}
Iteration & \text{x^{old}} & \text{y^{old}} & \text{x^{new}} & \text{y^{new}}\\
\hline
1 & 0 & 0 & 2.5 & -2 \\
2 & 2.5 & -2 & 3.5 & -7/6 \\
\end{array}
$$
And that is my answer to number 2, any help with how to do number 1 would be greatly appreciated, thank you for your time.
Best Answer
Hint: The Frobenius norm of an $m \times n$ matrix $A$ is defined as the square root of the sum of the absolute squares of its elements.
Example: Consider matrix
$$A = \left( \begin{array}{ccc} ~~~~1 & -2 & ~~~~~~3 \\ -4 & ~~5 & ~-6 \\ ~~~7 & -8 & ~~~~~~9 \\ \end{array} \right)$$
$\lVert A \rVert _{F} = \sqrt (|1|^2+|-2|^2+|3|^2+|-4|^2+|5|^2+|-6|^2+|7|^2+|-9|^2 + |-8|^2) = \sqrt(285) = 16.8819$
For convergence of Jacobi iteration method you need to find iteration matrix $P = -D^{-1}(L+U)$. Note that the Jacobi iterative scheme will converge if $\lVert P\rVert_{F} $ is strictly less than $1$, where $F$ stands for the Frobenius norm. There may be some other matrix norm (such as the $1$-norm ) that is strictly less than $1$, in which case convergence is still guaranteed. In any case, however, the condition $\lVert P\rVert <1 $ is only a sufficient condition for convergence, not a necessary one.
For the given example
$D = \left( \begin{array}{ccc} ~~1 & ~0 & ~0 \\ ~~0 & ~5 & ~0 \\ ~~0 & ~0 & ~9 \\ \end{array} \right)$ $L = \left( \begin{array}{ccc} ~~~0 & ~~0 & ~0 \\ -4 & ~~0 & ~0 \\ ~~~~7 & -8 & ~0 \\ \end{array} \right)$ $U = \left( \begin{array}{ccc} ~~0 & -2 &~~3 \\ ~~0 & ~~~0 & -6 \\ ~~0 & ~~~0 & ~~~0 \\ \end{array} \right)$
Added: Consider to solve $3\times 3$ size system of linear equation $Ax = b$, where coefficient matrix $A = \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{32} \\ \end{array} \right)$
Assume coefficient matrix $A$ has no zeros on its main diagonal i.e. $a_{11}$, $a_{22}$, $a_{23}$ are non zeros, then $D = \left( \begin{array}{ccc} ~~\frac{1}{a_{11}} & 0 & 0 \\ 0 & \frac{1}{a_{22}} & ~0 \\ ~~0 & 0 & \frac{1}{a_{33}}\\ \end{array} \right)$
$L = \left( \begin{array}{ccc} 0 & 0 & 0 \\ a_{21} & 0 & ~0 \\ a_{31}& a_{32} & 0\\ \end{array} \right)$ $U = \left( \begin{array}{ccc} 0 & a_{12} & a_{13} \\ 0 & 0 & a_{23} \\ 0 & 0 & 0\\ \end{array} \right)$