Introduction to Numerical Analysis

• Mathematics is the study of topics such as quantity (numbers), structure, space, and change.
• Numerical analysis provides algorithms that use numeric approximation for the problems of mathematical analysis.
• Operation research is a discipline that deals with the application of advanced analytical methods to help make better decisions

Numerical Analysis

• Numerical analysis is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics.
• Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life sciences, social sciences, medicine, business, and even the arts.
• Numerical analysis focuses on developing methods for approximating solutions to mathematical problems that are often too complex to solve analytically.
• Algorithms in numerical analysis must be stable (small errors do not grow uncontrollably during computation) and convergent (approximations approach the true solution as the number of steps increases).
• Numerical analysis also involves quantifying the error in the approximation.

Key Areas in Numerical Analysis

• Root finding: Algorithms for finding roots of equations.
• Solving systems of equations: Numerical methods for solving linear and nonlinear systems of equations.
• Interpolation and regression: Constructing functions that pass through given data points or fit a set of data.
• Integration: Numerical methods for approximating definite integrals.
• Differential equations: Numerical methods for solving ordinary and partial differential equations.
• Optimization: Techniques for finding the maximum or minimum of a function.

Operation Research

• Operations research (OR) is a discipline that deals with the application of advanced analytical methods to help make better decisions.
• Operations Research is also known as management science.
• Operations research employs techniques from mathematical sciences to arrive at optimal or near-optimal solutions to complex decision-making problems.
• Operations research often concerns itself with determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some objective function.
• Operations research often overlaps with other disciplines, such as industrial engineering and logistics.

Key Areas in Operation Research

• Linear programming: Optimizing linear objective functions subject to linear constraints.
• Integer programming: Linear programming problems with integer constraints.
• Network optimization: Models and algorithms for optimizing flows and paths in networks.
• Queuing theory: Mathematical models for analyzing waiting lines and service systems.
• Simulation: Creating computer models to simulate complex systems and evaluate different scenarios.
• Decision analysis: Frameworks for making decisions under uncertainty.
• Inventory management: Techniques for optimizing inventory levels.

Mathematical Modeling

• Mathematical modeling is the art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application.
• Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering, mechanical engineering).
• Model validation is an essential step in the modeling process, where the model is tested against experimental data or real-world observations.

Optimization

• Optimization is the process of finding the best solution to a problem, usually in terms of maximizing rewards or minimizing costs.
• Unconstrained optimization deals with finding the maximum or minimum of a function without any restrictions on the variables.
• Constrained optimization involves finding the maximum or minimum of a function subject to a set of constraints.
• Linear Programming (LP) is used for optimization problems where the objective function and the constraints are linear.
• Nonlinear Programming (NLP) deals with optimization problems where the objective function or the constraints are nonlinear.

Simulation

• Simulation is the imitation of the operation of a real-world process or system over time.
• Simulation is used in many contexts, such as simulation of technology for performance optimization, safety engineering, testing, training, education.
• Discrete-event simulation (DES) models the operation of a system as a sequence of discrete events in time.
• Monte Carlo simulation is a computational technique that uses random sampling to obtain numerical results.

Linear Algebra

• Linear algebra is a branch of mathematics concerning linear equations such as: a1x1 + ... + anxn = b, linear functions such as: (x1, ..., xn) ↦ a1x1 + ... + anxn and their representations in vector spaces and through matrices.
• Linear algebra is central to almost all areas of mathematics.
• Linear algebra is fundamental in modern presentations of geometry.
• Common uses of linear algebra are in computer graphics, physics, economics, and engineering.

Calculus

• Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
• Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
• Differential calculus focuses on the rate of change.
• Integral calculus focuses on the accumulation of quantities and the areas under and between curves.