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# 数学代写|数值分析代写Numerical analysis代考|Gauss-Seidel Method and SOR

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## 数学代写数值分析代写Numerical analysis代考|Gauss-Seidel Method and SOR

Closely related to the Jacobi Method is an iteration called the Gauss-Seidel Method. The only difference between Gauss-Seidel and Jacobi is that in the former, the most recently updated values of the unknowns are used at each step, even if the updating occurs in the current step. Returning to Example 2.19, we see that Gauss-Seidel looks like this:
\begin{aligned} & {\left[\begin{array}{l} u_0 \ v_0 \end{array}\right]=\left[\begin{array}{l} 0 \ 0 \end{array}\right]} \ & {\left[\begin{array}{l} u_1 \ v_1 \end{array}\right]=\left[\begin{array}{l} \frac{5-v_0}{3} \ \frac{5-u_1}{2} \end{array}\right]=\left[\begin{array}{c} \frac{5-0}{3} \ \frac{5-5 / 3}{2} \end{array}\right]=\left[\begin{array}{c} \frac{5}{3} \ \frac{5}{3} \end{array}\right]} \ & {\left[\begin{array}{l} u_2 \ v_2 \end{array}\right]=\left[\begin{array}{l} \frac{5-v_1}{3} \ \frac{5-u_2}{2} \end{array}\right]=\left[\begin{array}{c} \frac{5-5 / 3}{3} \ \frac{5-10 / 9}{2} \end{array}\right]=\left[\begin{array}{c} \frac{10}{9} \ \frac{35}{18} \end{array}\right]} \ & {\left[\begin{array}{l} u_3 \ v_3 \end{array}\right]=\left[\begin{array}{l} \frac{5-v_2}{3} \ \frac{5-u_3}{2} \end{array}\right]=\left[\frac{5-35 / 18}{3}\right]=\left[\begin{array}{c} \frac{55}{54} \ \frac{5-55 / 54}{2} \end{array}\right] .} \end{aligned}
Note the difference between Gauss-Seidel and Jacobi: The definition of $v_1$ uses $u_1$, not $u_0$. We see the approach to the solution $[1,2]$ as with the Jacobi Method, but somewhat more accurately at the same number of steps. Gauss-Seidel often converges faster than Jacobi if the method is convergent. Theorem 2.11 verifies that the GaussSeidel Method, like Jacobi, converges to the solution as long as the coefficient matrix is strictly diagonally dominant.

Gauss-Seidel can be written in matrix form and identified as a fixed-point iteration where we isolate the equation $(L+D+U) x=b$ as
$$(L+D) x_{k+1}=-U x_k+b$$

## 数学代写|数值分析代写Numerical analysis代考|Convergence of iterative methods

In this section we prove that the Jacobi and Gauss-Seidel Methods converge for strictly diagonally dominant matrices. This is the content of Theorems 2.10 and 2.11 .
The Jacobi Method is written as
$$x_{k+1}=-D^{-1}(L+U) x_k+D^{-1} b$$
Theorem A.7 of Appendix A governs convergence of such an iteration. According to this theorem, we need to know that the spectral radius $\rho\left(D^{-1}(L+U)\right)<1$ in order to guarantee convergence of the Jacobi Method. This is exactly what strict diagonal dominance implies, as shown next.

Proof of Theorem 2.10. Let $R=L+U$ denote the nondiagonal part of the matrix. To check $\rho\left(D^{-1} R\right)<1$, let $\lambda$ be an eigenvalue of $D^{-1} R$ with corresponding eigenvector $v$. Choose this $v$ so that $|v|_{\infty}=1$, so that for some $1 \leq m \leq n$, the component $v_m=1$ and all other components are no larger than 1 . (This can be achieved by starting with any eigenvector and dividing by the largest component. Any constant multiple of an eigenvector is again an eigenvector with the same eigenvalue.) The definition of eigenvalue means that $D^{-1} R v=\lambda v$, or $R v=\lambda D v$.

Since $r_{m m}=0$, taking absolute values of the $m$ th component of this vector equation implies
\begin{aligned} & \left|r_{m 1} v_1+r_{m 2} v_2+\cdots+r_{m, m-1} v_{m-1}+r_{m, m+1} v_{m+1}+\cdots+r_{m n} v_n\right| \ & \quad=\left|\lambda d_{m m} v_m\right|=|\lambda|\left|d_{m m}\right| . \end{aligned}
Since all $\left|v_i\right| \leq 1$, the left-hand side is at most $\sum_{j \neq m}\left|r_{m j}\right|$, which, according to the strict diagonal dominance hypothesis, is less than $\left|d_{m m}\right|$. This implies that $|\lambda|\left|d_{m m}\right|<$ $\left|d_{m m}\right|$, which in turn forces $|\lambda|<1$. Since $\lambda$ was an arbitrary eigenvalue, we have shown $\rho\left(D^{-1} R\right)<1$, as desired. Now Theorem A.7 from Appendix A implies that Jacobi converges to a solution of $A x=b$. Finally, since $A x=b$ has a solution for arbitrary $b$, $A$ is a nonsingular matrix.
Putting the Gauss-Seidel Method into the form of (2.43) yields
$$x_{k+1}=-(L+D)^{-1} U x_k+(L+D)^{-1} b$$

## 数学代写数值分析代写Numerical analysis代考|Gauss-Seidel Method and SOR

$$\left[\begin{array}{ll} u_0 & v_0 \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \end{array}\right] \quad\left[u_1 v_1\right]=\left[\frac{5-v_0}{3} \frac{5-u_1}{2}\right]=\left[\frac{5-0}{3} \frac{5-5 / 3}{2}\right]=\left[\frac{5}{3} \frac{5}{3}\right]\left[u_2 v_2\right]=\left[\frac{5-v_1}{3} \frac{5-u_2}{2}\right]=\left[\frac{5-5 / 3}{3} \frac{5-10 / 9}{2}\right]=\left[\frac{10}{9} \frac{35}{18}\right]$$

Gauss-Seidel 可以写成矩阵形式并确定为定点迭代，我们在其中分离方程 $(L+D+U) x=b$ 作为
$$(L+D) x_{k+1}=-U x_k+b$$

## 数学代写|数值分析代写Numerical analysis代考|Convergence of iterative methods

$$x_{k+1}=-D^{-1}(L+U) x_k+D^{-1} b$$

$$\left|r_{m 1} v_1+r_{m 2} v_2+\cdots+r_{m, m-1} v_{m-1}+r_{m, m+1} v_{m+1}+\cdots+r_{m n} v_n\right| \quad=\left|\lambda d_{m m} v_m\right|=|\lambda|\left|d_{m m}\right| .$$

$$x_{k+1}=-(L+D)^{-1} U x_k+(L+D)^{-1} b$$

## MATLAB代写

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