Numerical Analysis Course Pdf
Arleen Jerdee
Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant engineering majors and some majors of mathematics.
The structure of the course is very similar in many of the institutions whose syllabi I've looked at: one begins with finite-precision arithmetic, then fixed-point methods for root-finding (usually 1-D problems),interpolation by polynomials, quadrature, numerical differentiation, some standard ODE methods, and perhaps some finite difference methods for PDE. Any rationale for this particular sequence of topics is obscured in the course.
The truly deep and interesting aspects - approximation theory, error analysis, computational complexity - are either not discussed, or not dwelt on. Instead, the typical introductory course is a collection of algorithms for problems which seem contrived. This is a pity. The stronger mathematics student comes away believing numerical analysis is boring and shallow, and the engineer comes away thinking mathematics has nothing to offer a real problem.
The question: Are there examples (links to course outlines or course webpages preferred) of introductory numerical analysis courses which avoid the above-described tedium, and which have a history of attracting strong mathematics students?
John Hubbard tends to take sort of the opposite track, in that he likes to bring a more serious numerical analysis perspective into the 1st and 2nd courses on calculus and differential equations, rather than assuming the students come out of a standard service-stream calculus, differential equations, linear algebra sequence of courses. Usually this includes a discussion of various ways of representing numbers on computers, like floating-point numbers, round-off errors, perhaps even topics like interval arithmatic.
For example, once the idea of ODEs are set up he likes to talk about "fences". I don't know if this is standard terminology anywhere or just his, but it's basically like a Lyapanov function but for time-dependent ODEs. So it gives you regions that trap orbits, but the region may move with time. He gets students used to thinking in this way gradually, by cooking up fences in the 1-dimensional time-dependent ODE case first. Then he moves on to things like the Gronwall inequality, applying it for things like the Euler approximations to ODE solutions to observe error growth rates. He also proves Kantorovich's theorem, which he uses for the implicit and inverse function theorems. He has quite a lot of success getting 1st and 2nd year physics and engineering students thinking about these things. But it's known as the "challenging" calculus stream at Cornell, and less ambitious students have other options. I don't know what their numbers are now, but when I was a TA for the course I think he was getting around 80 students per year in the course.
Motivational example, heat transfer between two Points. We discretized the Problem and derived a way to solve it (1-dimensional FDM). From this, we then moved on to multiple dimensions and time dependency (FTCS, etc..), introducing error estimates along the way.
At this point, we were able to take short detours and look at different other fields of Numerics briefly, for example, Finite Volume Method. We also took a look at things we had left out, like Newton Method (alot of which were introduced in other lectures).
We finalized the course with the Finite Element Method (as this is a core research field at our University), starting with Ritz-Galerkin and ending at a-posteriori error estimates. (Althoug this would need some basic knowledge in Functional Analysis)
Further, I have to point out (as briefly mentioned in point 5), that alot of things were already introduced in some other lectures. Mainly our physics lectures required some basic Numerics, so this was not a complete introduction to Numerics.
When I took a course on numerical analysis a couple of years ago I very much liked the book "An introduction to numerical analysis" by Suli and Mayers, it is very clear and concise. In particular it contains a lot of rigorous error estimates.
I'm teaching within the MPhil for Scientific Computing at the University of Cambridge. has become very popular, but places are limited. So we can be very selective with our students. In March 2016 there will be a book published by CRC Press called "A Concise Introduction to Numerical Analysis" based on the Numerical Analysis lectures of the course.
I was taking a mandatory grad level numerical methods course last year - but my research is in fact "engineering education". So I spent some time researching (first of all what the heck is the purpose of numerical methods, because obviously I missed that in my undergrad intro course to numerical methods) and then searching interesting ways that numerical methods courses could be taught.
Firstly - I will emphasize the extreme importance regularly reminding students what the main point of numerical methods is. Sometimes they will get lost in the math, and forget about the whole point of the course. Regularly remind your students the point of numerical methods vs. analytical methods. Otherwise the knowledge will go at the wayside if they get to know how to jump through quiz "hoops" but really have no context of what the heck the purpose of the course is in their big picture. Check out this info: crosscuttingconcepts site, click articles and "introduction to numerical methods".
-Design a car, and numerical configure how it operates, gear ratios etc as the course progresses (using introductory topics), also the mind map is amazing. Great paper and course designed by Coller and Scott 2009 -niu.edu/assessment/committees/CAN/PresentationsPapersArticles/coller-scott-2009-computers-and-education.pdf
-Motivational elements/examples. How Disney uses numerical methods (maybe higher level) to model life situations... search disney animations, click technology to see how they used numerical methods in the movie frozen.
-All courses like "numerical methods" have a culture and traditional structure (referred to as signature pedagogies). In this book "Exploring Signature Pedagogies: Approaches to Teaching ... " - and you find numerical methods in the computer science section (pg 250) You can see a few recommendations for new ways to adapt the education of topics like this (like "an expectation for interactivity and application to their world").Look at using programming to let them in real time engage in the course material with a wow factor. Good possible platforms may be WebGL, or consult with comp sci visualization faculty. Search "chrome experiments" so see those amazing ways of using numerical methods.
Numerical analysis is a big subject... Stephen Boyd's Convex Optimization (available for download on his web page, or in two pound form from CUP) is a very lucid book, covering both applications and theory.
The goal of the course is to get 2nd year students comfortable with writing mathematical software in a high-level computer language. This semester we are using Python but the specific language is the choice of the instructor.
But along the way we teach them about various elements from numerical analysis and their limitations. We largely do not teach any theory in this course. The course is about learning by example. So students see first-hand the issues that come from round-off error. They see first-hand arbitrary precision floats and integers, and how they can help (and hinder) an investigation.
Programming: Proficiency in programming languages like Python, MATLAB, or R is essential for implementing numerical algorithms and solving complex problems. You should also be comfortable with data structures, loops, conditionals, and functions.
Statistics: Understanding statistical concepts will help you analyze and interpret numerical data accurately. Concepts like probability theory, hypothesis testing, and regression analysis will be useful for various numerical analysis techniques.
Numerical Methods: Familiarity with various numerical algorithms, such as root-finding methods, interpolation, numerical integration, and solving linear systems, is essential. You should also know how to apply these methods effectively to solve real-world problems.
Optimization: Knowledge of optimization techniques, including linear and nonlinear programming, gradient descent, and constrained optimization, will enable you to optimize functions and find optimal solutions in numerical analysis scenarios.
Data Visualization: Being able to visually represent numerical data through graphs, charts, and plots is essential for gaining insights and effectively communicating results. Familiarize yourself with data visualization tools and techniques, such as matplotlib or ggplot.
Quantitative Analyst: Banks, investment firms, and hedge funds often hire individuals proficient in Numerical Analysis to develop quantitative models and strategies for financial forecasting and risk management.
Statistician: Statisticians utilize Numerical Analysis techniques to design experiments, collect and analyze data, and make predictions and recommendations in areas such as healthcare, marketing, social sciences, and more.
Research Scientist: Government agencies, research institutions, and private companies employ research scientists with Numerical Analysis skills to conduct mathematical and statistical research for various fields, like physics, biology, engineering, and more.
Numerical analysis is a branch of mathematics that focuses on developing algorithms and methods to solve mathematical problems using numerical approximations. It involves studying the accuracy, stability, and efficiency of numerical techniques for solving problems that may be too complex or time-consuming to solve analytically. Numerical analysis plays a crucial role in various fields such as engineering, physics, computer science, and finance, where accurate and efficient numerical solutions are essential.
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