Numerical Analysis Intro to Scientific Computing

Numerical Methods Numerical Methods: Algorithms that are used to obtain numerical solutions of a mathematical problem. Why do we need them? 1. No analytical solution exists, 2. An analytical solution is difficult to obtain or not practical.

Why use Numerical Methods? To solve problems that cannot be solved exactly

Introduction 1. Introduction to numerical methods for engineering as a general and fundamental tool for all engineering disciplines. We plan to cover (almost) the main topics of numerical analysis. 2. We will use commercial software widely used in science and engineering: MATLAB and Excel. 3. We will illustrate and discuss how numerical methods are used in practice. We will consider examples from Engineering.

5 Modelling in Industry: Automobiles

6 Example of Solving an Engineering Problem

7 Modelling in Industry: Aerospace

8 Modelling in Industry: Electronics

9 Bascule Bridge THG

10 Trunnion Hub Girder Bascule Bridge THG

11 Trunnion-Hub-Girder Assembly Procedure Step1.Trunnion immersed in dry-ice/alcohol Step2.Trunnion warm-up in hub Step3.Trunnion-Hub immersed in dry-ice/alcohol Step4.Trunnion-Hub warm-up into girder

12 Problem After Cooling, the Trunnion Got Stuck in Hub

13 Why did it get stuck? Magnitude of contraction needed in the trunnion was 0.015” or more. Did it contract enough?

Course overview 1.Finding roots of functions of one variable 2.Approximation, errors, and precision. 3.System of linear equations 4.Numerical integration and differentiation.

Introduction Why are Numerical Methods so widely used in Engineering?  Engineers use mathematical modeling (equations and data) to describe and predict the behavior of systems.  Closed-form (analytical) solutions are only possible and complete for simple problems (geometry, properties, etc.).  Computers are widely available, powerful, and (relatively) cheap.  Powerful software packages are available (special or general purpose).

Applications of Numerical Methods in Engineering Communication/power Network simulation Train and traffic networks Computational Fluid Dynamics (CFD): Weather prediction Groundwater & pollutant movement

Electronic Communication by Computer assignments will be submitted as attachments via Text files, Excel & MATLAB documents as attachments. documents will be distributed via the AAST web page.

Useful info Course website: MATLAB instructions: Mathworks, the creator of MATLAB: OCTAVE = free MATLAB clone Available for download at

Scientific computing Design and analysis of algorithms for numerically solving mathematical problems in science and engineering Considers the effect of approximations and performs error analysis modern simulations of engineering applications

Develop mathematical model (usually requires a combination of math skills and some a priori knowledge of the system) Come up with numerical algorithm (numerical analysis skills) Implement the algorithm (software skills) Run, debug, test the software Visualize the results Interpret and validate the results Computational problems: attack strategy Mathematical modeling

Computational problems: well-posedness The problem is well-posed, if (a) solution exists (b) it is unique (c) it depends continuously on problem data Simplification strategies: Infinite finite Nonlinear linear High-order low-order Only approximate solution can be obtained this way!

Sources of numerical errors Before computation modeling approximations empirical measurements, human errors previous computations During computation truncation or discretization Rounding errors Perturbations during computation may be amplified by algorithm Abs_error = approx_value – true_value Rel_error = abs_error/true_value Approx_value = (true_value)x(1+rel_error) Can be controlled through error analysis Cannot be controlled

Representing Real Numbers You are familiar with the decimal system: Decimal System: Base = 10, Digits (0,1,…,9) Standard Representations:

Normalized Floating Point Representation Normalized Floating Point Representation: Scientific Notation: Exactly one non-zero digit appears before decimal point. Advantage: Efficient in representing very small or very large numbers.

Binary System Binary System: Base = 2, Digits {0,1}

IEEE 754 Floating-Point Standard Single Precision (32-bit representation) 1-bit Sign + 8-bit Exponent + 23-bit Fraction Double Precision (64-bit representation) 1-bit Sign + 11-bit Exponent + 52-bit Fraction SExponent 8 Fraction 23 SExponent 11 Fraction 52 (continued)

Machine precision

Calculator Example Suppose you want to compute: * using a calculator with two-digit fractions 3.57 * = True answer:

29 Bridge crane design example Design elements : top plate, bottom plate, web, etc. Bridge girder

Stability Algorithm is stable if result produced is relatively insensitive to perturbations during computation Stability of algorithms is analogous to conditioning of problems For stable algorithm, effect of computational error is no worse than effect of small data error in input

Accuracy Accuracy : closeness of computed solution to true solution of problem Accuracy depends on conditioning of problem as well as stability of algorithm

48.9 Significant Digits - Example

Rounding and Chopping Rounding: Replace the number by the nearest machine number. Chopping: Throw all extra digits.

Can be computed if the true value is known: Error Definitions – True Error

We say that the estimate is correct to n decimal digits if: We say that the estimate is correct to n decimal digits rounded if: Notation

Solution of Nonlinear Equations Some simple equations can be solved analytically: Many other equations have no analytical solution:

Methods for Solving Nonlinear Equations o Bisection Method o Newton-Raphson Method o Secant Method

Solution of Systems of Linear Equations

Cramer’s Rule is Not Practical

Methods for Solving Systems of Linear Equations o Gaussian Elimination o Gaussian Elimination with Scaled Partial Pivoting o Gauss- Jordan

Integration Some functions can be integrated analytically:

Methods for Numerical Integration o Trapezoid Method o Simpson Method o Mid-point method