Numerical Methods and Analysis PDF

Computer Engineering Numerical Methods and Analysis Engr. Ranzel Dimaculangan, ECT  01 Numerical Methods and Analysis Numerical Analysis Definition: Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic or analytic...

Computer Engineering Numerical Methods and Analysis Engr. Ranzel Dimaculangan, ECT  01 Numerical Methods and Analysis Numerical Analysis Definition: Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic or analytical computation) for the problems of mathematical analysis. Focus: It is concerned with the accuracy, stability, and efficiency of these algorithms. Numerical analysis seeks to understand the errors involved in approximations and how they propagate through the computations. Purpose: The goal is to provide a theoretical foundation for why and how these methods work, including convergence, error bounds, and robustness. Numerical analysis often deals with the development of new algorithms that are more accurate or efficient. Example Topics: Error analysis, stability of algorithms, convergence of sequences and series, numerical differentiation and integration, and eigenvalue problems. Numerical Methods Definition: Numerical methods are the techniques or algorithms used to obtain numerical solutions to mathematical problems that may not have exact analytical solutions. Focus: It is more applied, focusing on the implementation of specific techniques to solve practical problems. Numerical methods involve the actual procedures or algorithms to approximate the solution of mathematical problems. Purpose: The goal is to develop and apply techniques that can be used in practice to solve problems in science, engineering, and other fields. Numerical methods emphasize the implementation and computational aspects. Example Topics: Methods for solving linear and nonlinear equations (e.g., Newton's method), numerical integration (e.g., Simpson's rule), finite difference methods for differential equations, and Monte Carlo methods. Analytical vs Numerical Solutions Analytical Solutions These are exact, derived using mathematical methods, and are typically preferred when the problem allows it. They provide deep insights but are limited in scope. Numerical Solutions These are approximate, obtained through computational methods, and are necessary for complex or real-world problems where analytical solutions are not feasible. They offer flexibility but at the cost of potential numerical errors and the need for computational resources. Example Solve the quadratic equation x2 - 3x + 2 = 0 using analytical and numerical computation. Using Analytical Computation Solve the quadratic equation x2 - 3x + 2 = 0 using analytical and numerical computation. Method: Using the quadratic formula −𝑏 ± 𝑏 ! − 4𝑎𝑐 𝑥= 2𝑎 Solution: x1 = 2, x2 = 1 Using Numerical Computa