Quick Revision

Root-Finding Check

• Root exists in $[a,b]$ if $f(x)$ is continuous on $[a,b]$ and $f(a)f(b)<0$.
• Linear eqn: highest power of variable is $1$.
• Nonlinear eqn: power $>1$ or curve-type relation.

Graphical Method

• Plot $y=f(x)$.
• Root = point where graph cuts $x$-axis.
• For $$x^3-2x-5=0$$ sign changes: $$f(2)=-1,\quad f(3)=16$$ and $$f(2.09)\approx -0.0507,\quad f(2.10)\approx 0.061$$
• Approx root: $$x\approx 2.09$$

Bisection Method

Conditions:

• Continuous function
• Opposite signs at endpoints

Formula:

$$c=\frac{a+b}{2}$$

Rule:

• If $f(a)f(c)<0$, new interval $[a,c]$
• Otherwise new interval $[c,b]$

Stopping:

$$\frac{b-a}{2}<\mathrm{TOL}$$

Exam result:

$$x^3-7=0 \Rightarrow x\approx 1.910$$

Fixed Point Iteration

Write

$$x=g(x)$$

Iteration:

$$x_{n+1}=g(x_n)$$

Good exam form:

$$x=\sqrt[3]{6-x}$$

For

$$x^3+x-6=0$$

with $x_0=1.5$:

$$x\approx 1.6346$$

Secant Method

Formula:

$$x_{n+1}=x_n-\frac{f(x_n)(x_n-x_{n-1})}{f(x_n)-f(x_{n-1})}$$

No derivative needed.

For

$$x^3-4x-9=0$$

use

$$x_0=2,\quad x_1=3$$

After 5 iterations:

$$x\approx 2.706528$$

Newton-Raphson Method

Formula:

$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$

Stopping:

$$|x_{n+1}-x_n|<\varepsilon \quad \text{or} \quad |f(x_{n+1})|<\varepsilon$$

For

$$f(x)=x^3-2x^2+3$$ $$f'(x)=3x^2-4x$$

Iteration form:

$$x_{n+1}=x_n-\frac{x_n^3-2x_n^2+3}{3x_n^2-4x_n}$$

With $x_0=-1.5$:

$$x\approx -1$$

Gaussian Elimination

Steps:

1. Write augmented matrix.
2. Make zeros below pivots.
3. Convert to upper triangular form.
4. Use back substitution.

Exam result for

$$\begin{aligned} 3x+y-z&=4\\ 2x-4y+5z&=-3\\ x+2y+3z&=10 \end{aligned}$$ $$x=0.84,\quad y=2.72,\quad z=1.24$$

Partial Pivoting

Use when pivot is zero or very small.

Why:

• avoids division by tiny number
• reduces round-off error
• improves stability

Exam trigger:

• first pivot $0.2$ too small, so swap rows first

Exam result:

$$x\approx -0.8491,\quad y\approx 4.6415,\quad z\approx -5.7547$$

Gauss-Seidel Method

Use when:

• system is diagonally dominant
• iterative approx acceptable

Rearrange each equation:

$$x=\frac{11-y+z}{10},\qquad y=\frac{12-2x-z}{10},\qquad z=\frac{14+2x-3y}{10}$$

Diagonal dominance check:

$$|10|>2,\quad |10|>3,\quad |10|>5$$

Start with

$$x_0=y_0=z_0=0$$

After 4 iterations:

$$x\approx 1.15493,\quad y\approx 0.83090,\quad z\approx 1.38172$$

Exam Hit List

• Sign-change test
• Bisection interval update
• Fixed-point form choice
• Secant formula
• Newton-Raphson derivative + iteration
• Augmented matrix
• Upper triangular form
• Partial pivoting row swap
• Diagonal dominance test
• Gauss-Seidel iteration table

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