György Bicsák, Dávid Sziroczák, Aaron Latty

Numerical methods


Rectangular Rule

One of the easiest method is the rectangular rule, or constant function value, or zero order rule. In this case, the integral value (abfxdx) is approximated by creating a rectangle in between a and b endpoints, which gives the width of the rectangler: b-a. The only question is that how we can determine the height of the rectangle? Actually, the rectangular rule gives 3 possibilities immediately: we can take the left-hand endpoint of the interval, the right-hand endpoint or the function value in the midpoint of the interval. That’s why they are called as left-hand rule, right-hand rule and midpoint rule. Although, the method itself is really simple, the accuracy is not the best. As Figure 21 illustrates, if the [a,b] interval is wide and additionally, the f(x) function over it has significant changes (high curvature), then all of the rectangular rules give high error. So, if we need precise integrate values, the rectangular rule is not the best solution, it can give a rough estimation for the order of the function, but the method used globally over the interval is not the best choice: the solution should not be accepted.

Numerical methods

Tartalomjegyzék


Kiadó: Akadémiai Kiadó

Online megjelenés éve: 2019

ISBN: 978 963 454 283 4

This book on Numerical Methods is part of the Transportation and Vehicle Engineering Faculty’s curriculum at the Budapest University of Technology and Economics, created under the Stipendium Hungaricum Scholarship Programme. The book demonstrates the aim of developing and using numerical methods. Instead of relying on more complicated and expensive analytical solution methods, simpler arithmetical equations are used to approximate the accurate solution with the required level of accuracy. After discussing the various sources and propagation of solution errors, the book discusses the solution of single variable equations and system of equations. Following chapters describe the two main curve fitting methods; approximation and interpolation. These chapters are followed by the numerical differentiation and integration methods, then finally the numerical solutions for ordinary and partial differentiation equations are outlined. MATLAB is one of the most popular choice today for numerical computing, with its user friendly, yet powerful computing capability. It is practically an industry standard for prototyping and new developments, and there is great value for students to be introduced to solving problems with MATLAB. Many chapters in the notes show examples implemented in MATLAB, with actual working code available to use and practice.

Hivatkozás: https://mersz.hu/bicsak-numerical-methods//

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