1. ROOTS OF EQUATIONS

2. ROOTS OF EQUATIONS Bracketing Methods Open Methods
The Bisection Method
The False-Position Method
Open Methods
Simple Fixed-Point Iteration
The Secant Method

3. The Bisection Method

4. The Bisection Method

5. The False-Position Method
A graphical depiction of the method of false position. Similar triangles used to derive the formula for the method are shaded.

7. EXAMPLE

8. Open Methods
In (a), which is the bisection method, the root is constrained within the interval prescribed by xl and xu. In contrast, for the open method depicted in (b) and (c), a formula is used to project from xi to xi+1 in an iterative fashion. Thus, the method can either (b) diverge or (c) converge rapidly, depending on the value of the initial guess.

9. The Newton-Raphson Method
Graphical depiction of the Newton-Raphson method. A tangent to the function of xi [that is, f’(xi)] is extrapolated down to the x axis to provide an estimate of the root at xi+1.

10. Derivation and Error Analysis of the Newton-Raphson Method

12. The Secant Method