In this chapter we study the notion of analytic functions and their properties. It will be shown that a complex function is differentiable if and only if there is an important compatibility relationship between its real and imaginary parts.

A dynamical system is a mechanical, electrical, chemical, or biological system that evolves in time.Dynamical systems theory provides one of the most powerful and pervasive applications of matrix methods in science and engineering.These qualitative and quantitative tools and methods allow for the determination and characterization of the number and types of solutions, including their stability, of complex, often nonlinear, systems.These methods include phase-plane analysis, bifurcation diagrams, stability theory, Poincare diagrams illustrated using linear and nonlinear physical examples, including the Duffing equation and the Saltzman-Lorenz model.

The algebraic eigenproblem is the mathematical answer to the physical questions:What are the principal stresses in a solid or fluid and on what planes do they act?What are the natural frequencies of a system?Is the system stable to small disturbances?What is the best basis with respect to which to solve a system of linear algebraic equations with a real symmetric coefficient matrix?What is the best basis with respect to which to solve a system of linear ordinary differential equations?What is the best basis with respect to which to represent an experimental or numerical data set?

A central goal of scientists and engineers is obtaining solutions of the differential equations that govern their physical systems.This can be done numerically for large and/or complex systems using finite-difference methods, finite-element methods, or spectral methods.This chapter gives an introduction and the formal basis for these methods, with particular emphasis on finite-difference methods.Second-order partial differential equations are classified as elliptic, parabolic, or hyperbolic, and the numerical methods developed for such equations must be faithful to their mathematical properties.

Optimization and root finding are closely aligned techniques for determining the extremums and zeros, respectively, of a function.Newton's method is the workhorse of both types of algorithms for nonlinear functions, and the conjugate-gradient and GMRES methods are also covered.Optimization of linear, quadratic, and nonlinear functions are addressed with and without constraints, which may be equality or inequality.In the linear programming case, emphasis is placed on the simplex method.

Reduced-order modeling is an active area of research by which simplified models of experimental or numerical data can be generated that are faithful to the behavior of the unerlying system.These methods are based on Galerkin projection, which is motivated by variational methods, or some other method of weighted residuals and allow for the projection of any governing differential equation onto an appropriate set of basis vectors or functions.These basis vectors or functions can be obtained using proper-orthogonal decomposition (POD) or one of its extensions or alternatives.Galerkin projection and POD are applied to continuous and discrete data sets.