ข้อสอบเก่ามิดเทอม Numerical methods 2566/1

1. เขียนโค้ด Graphical Methods ประสิทธิภาพสูง หาค่า $127/13$

2. Find result of a program (Bisection $x_m = \dfrac{x_l + x_r}{3}$ $[0,10]$ $f(x)=1-2x$)

    #include <iostream>
    #include <math.h>
   
    using namespace std;
   
    double fx(double x) {
      return 1 - (2 * x);
    }
   
    double err(double xold, double xnew) {
      return fabs((xnew - xnew) / xnew) * 100;
    }
   
    int main() {
      double xl = 0.0, xr = 100.0, xm, ea, e = 0.000001, fxr, fxm;
      int iter = 0;
   
      do {
        iter += 1;
        xm = (xl + xr) / 3;
        fxr = fx(xr);
        fxm = fx(xm);
   
        if(fxr * fxm > 0) {
          ea = err(xr, xm);
          xr = xm;
        } else if(fxr * fxm < 0) {
          ea = err(xl, xm);
          xl = xm;
        }
   
        cout << iter << " " << xm << " " << ea << endl;
      } while(ea < e)
   
      return 0;
    }

   1. What result gonna be after iter 3?
   2. Will this code converge?

3. One-point iteration methods หา iteration ที่ 5

   $$\begin{align*} & Given \\ & f(x)=e^{-x/4}(2-x)-1 \\ & x_1 = 1.00 \end{align*}$$

   $x$	$value$
   $x_1$	$1.00$
   $x_2$	$0.7159746$
   $x_3$	$0.8039869$

4. Guass elimination

   $$\begin{bmatrix} a_{11} & a_{12} & a_{13} & | & b_{1} \\ 0 & a'_{22} & a'_{23} & | & b'_{2} \\ 0 & 0 & a''_{33} & | & b''_{3} \\ \end{bmatrix}$$
   1. Find $x_1$ as variable equation
   2. Back Substiution find $b'_2$ given some equation

5. Linear algebra equation

   $${ \begin{align*} \begin{equation} -9x_1 + 7x_2 - 19x_3 = 4000 \space\space\space\space\space \end{equation} \\ \begin{equation} 14x_1 - 13bx_2 - 6x_3 = 1200 \space\space\space\space\space \end{equation} \\ \begin{equation} 16x_1 + x_2 - 4x_3 = 2350 \space\space\space\space\space \end{equation} \\ \end{align*} }$$
   1. Cramer's rules
   2. Matrix inversion
   3. LU Decomposition