Iterative method of Engineering Mathematics |Fixed Point Method| Method of Successive Approximations

Iterative method of Engineering Mathematics |Fixed Point Method| Method of Successive Approximations

The fixed point method is a numerical technique used to find the roots of an equation. The core idea is to transform the equation into the form x = g(x), where a fixed point is a value x such that g(x) = x.

How it works:
Rearrange the equation: Convert the given equation into the form x = g(x).
Choose an initial guess: Select a starting value, x0, as an approximation to the root.
Iterate: Apply the formula x_(n+1) = g(x_n) repeatedly to generate a sequence of approximations.
Convergence: If the sequence converges to a value x, then x is a fixed point and an approximate root of the original equation.
Key points:
Convergence: Not all functions g(x) lead to convergent iterations. The method's convergence depends on the properties of g(x).
Rate of convergence: The speed at which the iterations approach the root can vary. Some functions converge faster than others.
Applications: The fixed point method is used in various fields, including engineering, physics, and economics, to solve equations numerically.

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