ROOTS OF EQUATIONS.

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Presentation transcript:

ROOTS OF EQUATIONS

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

The Bisection Method

The Bisection Method

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.

EXAMPLE

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.

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.

Derivation and Error Analysis of the Newton-Raphson Method

The Secant Method