Further Numerical Methods

• Further Numerical Methods encompasses techniques utilized to find approximate solutions to complex mathematical problems that are not directly solvable.
• Newton-Raphson Method: A root-finding algorithm which uses an iterative process to approach one root of a real valued function. The sequence converges quadratically, meaning it doubles the number of correct digits in each successive step when it is close to the root.
• The method of Bisection: This technique is applied to locate the roots of a function in an interval where the function crosses the x-axis. The interval is repeatedly halved until it becomes as small as desired.
• Regula Falsi or the False Position Method: This method combines Bisection and Linear Interpolation methodologies to determine roots in an interval. It uses a non-linear function to divide the interval and typically converges much quicker than the Bisection method.
• The Fixed Point Iteration method: This approach finds a point at which a function becomes its own input, also known as the fixed point. It’s based on the principle of successive substitution until the solutions converge to a fixed point.
• The importance of accuracy estimation in methods: The successive approximation methods usually work on the basis of reducing the error within acceptable boundaries. This makes the understanding of the error factor and ability to estimate the accuracy of the final solution extremely vital.
• The concept of Convergence: Understanding when a sequence of operations is going to converge to a fixed point and when it will diverge is crucial. This is largely based on the selection of roots and the expected behaviour (like oscillation, slow convergence) for a given mathematical function.
• Comparing the different methods: Understanding the strengths and weaknesses of different numerical methods is crucial in numerically solving mathematical problems. Variables to consider include the nature of the function, required precision, processing power and speed of computation.