Question

for the linear system of equations [1 2 - 2,1 1 1, 2 2 1][x y z] =[1 3 5] set up the gauss - jacobi and gauss - seidel iteration schemes in matrix form. also check the convergence of the two schemes.

Answer 1

Answer on Question #74014 – Math – Quantitative Methods

Question

For the linear system of equations [1 2 - 2, 1 1 1, 2 2 1][x y z] = [1 3 5] set up the gauss - jacobi and gauss - seidel iteration schemes in matrix form. also check the convergence of the two schemes.

Solution

Consider a system of linear equations $Au = b$ with

$$A = \left[ \begin{array}{ccc} 1 & 2 & -2 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{array} \right], u = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right], b = \left[ \begin{array}{c} 1 \\ 3 \\ 5 \end{array} \right]$$

1. The matrix form of Jacobi iterative method is

$$u^{k+1} = D^{-1}(b - R u^k)$$

where $u(k)$ is the $k$th approximation or iteration of $u$ and $u(k+1)$ is the next or $k+1$ iteration of $u$ and $A = D + R$.

$$A = \left[ \begin{array}{ccc} 1 & 2 & -2 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{array} \right] = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] + \left[ \begin{array}{ccc} 0 & 2 & -2 \\ 1 & 0 & 1 \\ 2 & 2 & 0 \end{array} \right]$$ $$D = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] = D^{-1}, \qquad R = \left[ \begin{array}{ccc} 0 & 2 & -2 \\ 1 & 0 & 1 \\ 2 & 2 & 0 \end{array} \right]$$

The convergence condition is: $\rho(B) < 1$ where $B = D^{-1}R$ and $\rho(B) = \max\{|l_1|, |l_2|, |l_3|\}$, $|1, |2, |3\}$ - eigenvalues of a matrix $B$.

$$l_{1,2,3} = \left[ \begin{array}{c} 0.000005 + 0.000009i \\ 0.000005 - 0.000009i \\ -0.000011 \end{array} \right] \to \max\{|l_1|, |l_2|, |l_3|\} = 0.000011 < 1$$ $$\begin{array}{l} u^{k+1} = D^{-1}(b - R u^k) = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left( \left[ \begin{array}{c} 1 \\ 3 \\ 5 \end{array} \right] - \left[ \begin{array}{ccc} 0 & 2 & -2 \\ 1 & 0 & 1 \\ 2 & 2 & 0 \end{array} \right] \cdot u^k \right) \\ = \left[ \begin{array}{c} 1 \\ 3 \\ 5 \end{array} \right] - \left[ \begin{array}{ccc} 0 & 2 & -2 \\ 1 & 0 & 1 \\ 2 & 2 & 0 \end{array} \right] \cdot \left[ \begin{array}{c} x^k \\ y^k \\ z^k \end{array} \right] = \left[ \begin{array}{c} 1 - 2y^k + 2z^k \\ 3 - x^k - z^k \\ 5 - 2x^k - 2y^k \end{array} \right] \end{array}$$

2. The matrix form of gauss - seidel iterative method is

$$u^{k+1} = L^{-1}(b - M u^k)$$

where $u(k)$ is the kth approximation or iteration of $u$ and $u(k+1)$ is the next or $k+1$ iteration of $u$ and $A = L + M$.

$$A = \left[ \begin{array}{ccc} 1 & 2 & -2 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{array} \right] = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 2 & 2 & 1 \end{array} \right] + \left[ \begin{array}{ccc} 0 & 2 & -2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right]$$ $$L = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 2 & 2 & 1 \end{array} \right], L^{-1} = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -2 & 1 \end{array} \right], M = \left[ \begin{array}{ccc} 0 & 2 & -2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right]$$

The convergence condition is: $\rho(B) < 1$ where $B = L^{-1}M$ and $\rho(B) = \max\{|l_1|, |l_2|, |l_3|\}$, I1, I2, I3 - eigenvalues of a matrix B.

$$l_{1,2,3} = \left[ \begin{array}{c} 0 \\ -2 \\ -2 \end{array} \right] \to \max \bigl\{|l_1|, |l_2|, |l_3|\bigr\} = 2 > 1$$ $$u^{k+1} = L^{-1}(b - M u^k) = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -2 & 1 \end{array} \right] \left( \left[ \begin{array}{c} 1 \\ 3 \\ 5 \end{array} \right] - \left[ \begin{array}{ccc} 0 & 2 & -2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right] \cdot u^k \right)$$ $$= \left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -2 & 1 \end{array} \right] \left[ \begin{array}{c} 1 - 2y^k + 2z^k \\ 3 - z^k \\ 5 \end{array} \right] = \left[ \begin{array}{c} 1 - 2y^k + 2z^k \\ 2 + 2y^k - 3z^k \\ 2z^k - 1 \end{array} \right]$$

**Answer:**

1. jacobi scheme: $\begin{bmatrix} x^{k+1} \\ y^{k+1} \\ z^{k+1} \end{bmatrix} = \begin{bmatrix} 1 - 2y^k + 2z^k \\ 3 - x^k - z^k \\ 5 - 2x^k - 2y^k \end{bmatrix}$, the convergence condition is satisfied

2. gauss - seidel scheme: $\begin{bmatrix} x^{k+1} \\ y^{k+1} \\ z^{k+1} \end{bmatrix} = \begin{bmatrix} 1 - 2y^k + 2z^k \\ 2 + 2y^k - 3z^k \\ 2z^k - 1 \end{bmatrix}$, the convergence condition is not satisfied

**Answer provided by https://www.AssignmentExpert.com**