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An Introduction To Computational Methods

An Introduction To Computational Methods Periodic (3D): Use PBCs in all three dimensions It will consist of a slab (continu-ous in say the x and y directions) with a vacuum or some other Aperiodic (2D): Use PBCs only in the plane of the crystal (eg x and y plane) In both cases you need to

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INTRODUCTION TO COMPUTATIONAL MATHEMATICS

Introduction to Computational Mathematics The goal of computational mathematics put simply is to ﬁnd or develop algo- rithms that solve mathematical problems computationally (ie using comput- ers) In particular we desire that any algorithm we develop fulﬁlls four primary properties: • Accuracy

PDF	COMPUTATIONAL MATHEMATICS COMPUTATIONAL MATHEMATICS Models Methods and Analysis with MATLAB

PDF	Numerical Methods for Computational Science and Engineering

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COMPUTATIONAL PROBLEMS AND METHODS

COMPUTATIONAL PROBLEMS AND METHODS RICHARD E QUANDT* Princeton University Contents 1 Introduction 2 Matrix methods 2 1 Methods for solving 4a = c 2 2 Singular value decomposition 2 3 Sparse matrix methods 3 Common functions requiring optimization 3 I Likelihood functions 3 2 Generalized distance functions 3 3

How do you calculate the computational cost of a summation algorithm?
In order to compute the computational cost of this algorithm, we consider two counters: Let M be the number of multiplications or divisions and let A be the number of additions or subtractions. The following summation identities will come in handy in performing our analysis: n−1 1 p2 = n(n 1)(2n 1).
Where can I find computational mathematics education material?
Related computational mathematics education material at the rst-year and second-year undergraduate level can be found at the Shodor Education Foun-dation, whose founder is Robert M. Pano , website and in Zachary's book on programming .
Do undergraduate physics students need computational tools?
There is an increasing need for undergraduate students in physics to have a core set of computational tools. Most problems in physics benefit from numerical methods, and many of them resist analytical solution altogether.
What is computational science?
Computational science is a blend of applications, computations, and mathe-matics. It is a mode of scienti c investigation that supplements the traditional laboratory and theoretical methods for acquiring knowledge. This is done by formulating mathematical models whose solutions are approximated by com-puter simulations.

Lecture 1, Sept 19, 2013: Introduction

Peter Arbenz Computer Science Department, ETH Zurich E-mail: arbenz@inf.ethz.ch www2.math.ethz.ch

Outline of today's lecture

What is numerical methods for CSE Survey of the lecture Organization of the lecture (exercises/examination) References Start of the lecture Scienti c Computing www2.math.ethz.ch

Focus

on algorithms (principles, scope, and limitations), on (e cient, stable) implementations in Matlab, I on numerical experiments (design and interpretation). www2.math.ethz.ch

No emphasis on

theory and proofs (unless essential for understanding of algorithms) hardware-related issues (e.g. parallelization, vectorization, memory access) (These aspects will be covered in the course \\High Performance Computing for Science and Engineering" ered by D-INFK) www2.math.ethz.ch

Goals

Knowledge of the fundamental algorithms in numerical mathematics Knowledge of the essential terms in numerical mathematics and the techniques used for the analysis of numerical algorithms Ability to choose the appropriate numerical method for concrete problems Ability to interpret numerical results Ability to implement numerical algorithms e cientl

Assistants:

Stefan Pauli Daniel Hupp Christian Schuller Robert Carnecky Laura Scarabosio Jonas Sukys Cecilia Pagliantini Alexander Lobbe Sebastian Grandis Sharan Jagathrakashakan arbenz@inf.ethz.ch stefan.pauli@inf.ethz.ch huppd@inf.ethz.ch schuellc@inf.ethz.ch crobi@inf.ethz.ch laura.scarabosio@sam.math.ethz.ch jonas.sukys@sam.math.ethz.ch cecilia.pagliantini

Assignments

The assignment sheets will be uploaded on the course webpage on Monday every week the latest. The exercise should be solved until the following tutorial class. (Hand them in to the assistant or grade yourself.) www2.math.ethz.ch

Examination

Three-hour written examination involving coding problems to be done at the computer on www2.math.ethz.ch

TBA

Dry-run for computer based examination: Does not exist anymore. Try out a computer in the student labs in HG. Pre-exam question session: TBA Examination (cont.) Topics of examination: All topics, that have been addressed in class or in a homework assignment. One exam question will be one of the homework assignment. Lecture slides will be available

Problem solving environment: Matlab

We use Matlab for the exercises. Although most of the algorithm we are dealing with have been implemented in Matlab, it is useful when you program them yourselves. These (little) programs will be building blocks when you will solve more complex problems in your future. Matlab help Matlab commands help/doc Matlab online documentation, e.g., http://w

Types of errors

Errors in the formulation of the problem to be solved. Errors in the mathematical model. Simpli cations. Error in input data. Measurements. Approximation errors Discretization error. Convergence error in iterative methods. Discretization/convergence errors may be assessed by an analysis of the method used. Roundo errors Roundo errors arise everywhe

Algorithm properties

Performance features that may be expected from a good numerical algorithm. www2.math.ethz.ch

Accuracy

Relates to errors. How accurate is the result going to be when numerical algorithm is run with some particular input data. www2.math.ethz.ch

E ciency

How fast can we solve a certain problem? Rate of convergence. Floating point operations ( ops). How much memory space do we need? These issues may a ect each other. Robustness (Numerical) software should run under all circumstances. Should yield correct results to within an acceptable error or should fail gracefully if not successful. www2.math.ethz.ch

Complexity I

Complexity/computational cost of an algorithm :() number of elementary operators Asymptotic complexity ^= \\leading order term" of complexity w.r.t. large problem size parameters The usual choice of problem size parameters in numerical linear algebra is the number of independent real variables needed to describe the input data (vector length, matrix

Complexity II

To a certain extent, the asymptotic complexity allows to predict the dependence of the runtime of a particular implementation of an algorithm on the problem size (for large problems). For instance, an algorithm with asymptotic complexity O(n2) is likely to take 4 as much time when the problem size is doubled. One may argue that the memory accesses

Scaling

Scaling multiplication with diagonal matrices (with non-zero diagonal entries) from left and/or right. It is important to know the di erent e ects of multiplying with a diagonal matrix from left or right: www2.math.ethz.ch

Elementary matrices

Matrices of the form A = I + uvT are called elementary. Again we can apply A to a vector x in a straightforward and a more clever way: Ax = (I + uvT )x or www2.math.ethz.ch

Qualitatively speaking:

The problem is ill-conditioned if a small perturbation in the data may produce a large di erence in the result. The problem is well-conditioned otherwise. The algorithm is stable if its output is the exact result of a slightly perturbed input. www2.math.ethz.ch

An unstable algorithm

Ill-conditioned problem of computing output values y from input values x by y = g(x): when x is slightly perturbed to x, the result y = g(x) is far from y. www2.math.ethz.ch

A stable algorithm

An instance of a stable algorithm for computing y = g(x): the output y is the exact result, y = g(x), for a slightly perturbed input, i.e., x which is close to the input x. Thus, if the algorithm is stable and the problem is well-conditioned, then the computed result y is close to the exact y. www2.math.ethz.ch

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