Numerical solution of equations

Numerical Methods for Solving Equations

Numerical Methods: Techniques used for finding approximate solutions to complex mathematical problems that cannot be solved exactly.
Approximate Solution: A numerical value that is close to, but not exactly equal to, the true solution of a mathematical problem.

Iteration: A procedure for approximating solutions in which an initial estimate is improved through an iterative process.
Predictor-Corrector Method: An iterative method where the result at each stage is used as the starting point for the next stage.

Change of Sign Method: Effective when a continuous function crosses the x-axis, indicating that a root lies between two x-values.
Bisection Method: A specific change of sign method, which involves halving the interval until it becomes sufficiently small.
Error Bounds: Interval around the approximate solution where the exact solution is known to lie.

Newton-Raphson Method: Efficient iterative method for finding roots, especially when the initial approximation is close to the root.
Derivative: In this method, the tangent at a point is used to provide the next approximation, requiring the calculation of the derivative of the function.

Rearranging Equations: Effective method when equations can be rearranged into the form x = g(x).
Fixed Point Iteration: Involves repeatedly applying the function g(x) until the value of x stabilises.
Convergence: The property of iteration methods where the sequence of approximations gets closer and closer to the actual solution.

Choosing a Method: The choice of method depends on what information is known about the function and the nature of the root.
Sensitivity to Starting Values: Some methods, like Newton-Raphson and iteration, may give different solutions with different starting values.
Accuracy and Precision: Determine how close the approximate solution is to the true solution and how consistently it can be reproduced.
Rate of Convergence: How quickly the method arrives at the approximation; fast convergence is usually desirable.

Real-world Applications: Numerical methods are crucial in areas like physics, engineering, finance, and computer science where exact solutions are sometimes unattainable.