Welcome to the course web page for MATH 361 Numerical Analysis1, here you will find a link to the course syllabus and other useful information, feel free to explore the site. What is this course about? That is a great question!
Scientific computing is a scientific discipline that seeks to develop and implement algorithms that may be used to address mathematical problems that arise in our attempts to study the world around us. Scientific computing is routinely involved in all of the natural sciences, engineering, economics and finance, machine learning, etc. Numerical analysis is the mathematical theory behind the algorithms of scientific computing, see also the Scholarpedia article on numerical analysis. Thus, numerical analysis (sometimes called numerical mathematics) is a branch of mathematics that provides the mathematical foundations for scientific computing.
Scientific computing arises as a result of the complexity of the natural world. Consider as an example the problem of weather forecasting. Weather is the result of entirely physical processes that determine the state of the atmosphere, and weather prediction is concerned with predicting how the atmosphere will evolve over a given period of time. That is, we want to input the current atmospheric conditions and output what the atmospheric conditions will be over some future time period. The atmosphere is a fluid with three velocity component variables u, v, and w as well as a pressure p, density ρ, temperature T, and specific humidity q. Each of these seven variables are multi-variable functions of time t and three spatial variables x, y, and z, and determining what these functions are requires one to solve a large system of nonlinear partial differential equations (PDEs) subject to a set of initial and boundary conditions. It is impractical and perhaps even impossible to find exact solutions to this system of PDEs. Thus, one relies on numerical algorithms to solve this mathematical problem.
This example provides us with another important perspective on numerical analysis. The PDEs just described are a mathematical model for the atmosphere and thus represent a mathematical abstraction of the physical system. It is an abstraction because these PDEs do not incorporate all possible physical detail. For example, they do not include all of the atomic interactions that occur within the atmosphere. The point is that numerical computation may be viewed as a process for taking mathematical abstractions, e.g., mathematical models, and producing concrete numerical results, and it is important to analyze in a careful manner the consequences of such a process.
Let us consider a couple of more example problems for those that remain unsure of the role and value of numerical analysis. Consider the problem of finding a solution to the equation f(x)=0, where y=f(x) is a polynomial function. If f is a quadratic function, the problem has a solution by way of the well-known quadratic formula. At least two issues arise, one is that the solution may be an irrational number which can not be represented exactly on a computer. (If you find this statement confusing do not be concerned as we will discuss this issue in detail in the course.) Another issue is that the algebraic theory of polynomials (known as Galois theory) tells us that in general there is no analog to the quadratic formula for polynomials of arbitratry degree. Thus, if we want to solve f(x)=0 for a general polynomial f, one must result to numerical methods.
Finally, consider the problem of computing a definite integral
∫baf(x) dx,
where f is a continuous function. If f has an anti-derivative F then we may (at least in principle) evaluate the integral by way of the fundamental theorem of calculus. As a specific example, suppose we want to evaluate
∫21e−x22 dx.
Definite integrals of the function f(x)=e−x22 appear routinely in probability and statistics. However, a proof can be given that demonstrates that f(x)=e−x22 does not possess an explicit antiderivative. In light of this, we must resort to a numerical method to solve this problem. Of course, our numerical answer will not be exact. Even if an antiderivative is known, just evaluating function values F(a) or F(b) may return irrational numbers. Thus approximation is inevitable in practical setttings. What are the consequences of approximating an abstract mathematical value by a concrete numerical value? Such questions are only understandable within the framework provided to us by numerical analysis.
Hopefully we have provided some motivation for the study of numerical analysis.
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