NCM_2015
Numerical Computing with Matlab
Senior Seminar in Scientific Computing

http://people.sc.fsu.edu/~jburkardt/classes/ncm_2015/ncm_2015.html

NCM_2015 is the home page for the class ISC4932, "Numerical Computing with Matlab", Senior Seminar in Scientific Computing, an undergraduate seminar class offered by the Department of Scientific Computing at Florida State University, Fall Session 2015.

This is a 1-credit seminar class, intended to introduce the students to ideas, algorithms, and programs common to Scientific Computing. The topic this semester will be numerical linear algebra.

The course is based on the book "Numerical Computing with Matlab" by Cleve Moler.

All the chapters of the book are available online for download from the MathWorks website: http://www.mathworks.com/moler/chapters.html

The class meets on ?days, from ?to?, in room ?.

Weekly class presentations will be chosen from the following topic list:

• Introduction to Matlab:
  To familiarize ourselves with Matlab in a scientific computing setting, we will consider several simple problems, involving the Golden Ratio, Fibonacci Numbers, the Fractal Fern, Magic Squares, Cryptography, the 3*n+1 Sequence, and Floating Point Arithmetic. We will learn how to explore the mathematical relationships involved, to carry out iterations, and to use graphics to help us to see the patterns hidden in the numbers we have computed.
• Linear Equations:
  Linear equations provide the simplest way of describing relationships between various quantities, and solving such equations is a requirement in almost every scientific computation. We will look at the technique of Gauss elimination first; then try a new way of seeing it as a factorization of the matrix. We'll look at how Matlab can use special techniques if the matrix is triangular or symmetric. Then we will ask where errors come from - is it because the system is large? can a small set of equations be difficult to solve? Is there any way to measure whether a system of equations will be hard to solve?
• Interpolation:
  Interpolation lets us assume that between a hot noon and a cold evening there is probably a cool evening. Numerically, we are always given a small amount of measurements, but then asked to estimate the situation at places where no measurements were made. We will look at reliable ways to making such estimates, and the choices we have for filling in the gaps, including polynomial interpolation, piecewise linear, piecewise cubic Hermite, and spline methods.
• Zeros and Roots:
  A formula takes your input and gives you an output as answer. A formula might tell you that if you load p pounds of gunpowder into the cannon, the cannonball will land x feet away. Suppose you want the cannonball to land exactly on a target that is y feet away, what do you do? The method for determining the input value that will result in a desired output value is known as root finding. If enough information is known about the formula, an answer can quickly be found using a few steps of an iteration, such as the bisection method, the secant method, or Newton's method. The ideas used for zero finding can be extended to optimizatino, in which we seek the input that results in the maximum or minimum value of a function.
• Least Squares:
  Surveyors were among the first people to encounter mathematical problems with "too much" data. As a simplified example, we may have measurements that essentially tell us that x+2y=10, x-y=2, 2x+y=11. No pair of values (x,y) makes all three equations true. The surveyors assumed that all the equations were only approximately true. Given that no answer will be perfect, is there a "least bad" answer? The method of least squares shows how to pick an approximate answer using the criterion that the sum of the squares of the errors in each equation is the least possible total. Along the way, we will see how the QR method is used to factor the linear system, and how rank deficiency must be dealt with. In many cases, the we are not dealing with a system of linear equations, but with nonlinear functions. Generalizations of the least squares method have been developed to handle them. Seeking an approximate solution to a set of equations arises frequently, and the least squares technique is the appropriate tool to use.
• Quadrature:
  Calculus shows us how to compute the integral of many functions, but in computation, the integrals we encounter are usually not treatable using antiderivatives, and must be estimated instead. A simple idea is to sample the integrand at a few points, compute the interpolating polynomial and integrate that. Another idea is to estimate the integral a second time, using more points. If the two estimates differ greatly, this indicates that we may need to make a third estimate with even more points, continuing to refine our estimate until it settles down.
• Ordinary Differential Equations:
  Many physical systems can be described by specifying an initial condition, and the rules for how the system changes from one instant of time to the next. Matlab includes several functions that can report a sequence of system states over time, by starting at the initial state and taking many small steps. While most physical systems are well-behaved, there are examples such as the Lorenz equations, which show that the computed solution can be very sensitive to small errors. Another problem arises when the physical system includes possibilities for changes at a wide range of time scales, as in descriptions of chemical reactions.
• Fourier Analysis:
  The sine and cosine functions are periodic, and are suitable for describing electronic signals, weather, music, and other phenomena that have an underlying cyclic nature. Fourier analysis is a way of breaking down a signal into contributions to a sequence of harmonic frequencies. We will look at an example involving historical records of sunspots.
• Random Numbers:
  The generation of random numbers on the computer has always been a serious study. Random numbers might seem at first to be meaningless; but what is important is that they can be used as a technique for sampling. Using random numbers allows us to rapidly estimate the behavior of a system by, essentially, averaging its results for many random trials. Often, the normal distribution is preferred, since in many physical systems there is a naturally preferred result, along with a small tendency to variation. In some computations, rapid, repeatable, parallel computation of millions of random numbers is vital, and we discuss how Matlab's Twister algorithm can be used for this purpose.
• Eigenvalues and Singular Values:
  The eigenvalue equation A*x=lambda*x seeks a direction x such that the linear transformation A simply lengthens x, without changing its direction. Eigenvalue analysis is a standard way of thinking about many advanced problems in linear algebra. The related singular value decomposition, A=U*S*V', is a more modern way of analyzing matrices. It can be applied even if the matrix is rectangular, and the factorization does not encounter the many special cases that can arise in eigenanalysis. We will try to understand the meaning of these special numbers, the eigenvalues and singular values, and the special directions, the eigenvectors and singular vectors.
• Partial Differential Equations:
  Partial differential equations model a physical system by using partial derivatives. The classic example is the Poisson equation, which can be used to describe the diffusion of a drop of ink in water, or the spread of heat in a sheet of metal. The finite difference method allows us to approximate the second derivatives in the Poisson equation by difference quotients of data evaluated on a grid. This, in turn, results in a large, sparse system of linear equations. The properties of this linear system turn out to be interesting to analyze. We can also model the wave equation, in which case we will wind up looking at the eigenvalue problem. The analysis of the "L-shaped region" is the origin of the Matlab logo.