Table of Contents
Table of Contents
“Exercises of Numerical Analysis”
INTRODUCTION
THEORETICAL OUTLINE
EXERCISES
“Exercises of Numerical Analysis”
“Exercises of Numerical Analysis”
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
numerical calculation of the roots of a polynomial
numerical solving of matrices, linear and nonlinear systems
numerical computation of the integral and derivatives
finite difference method and numerical solving of ordinary differential equations
finite element method and weak formulation of partial differential equations
Initial theoretical hints are also presented to make the conduct of the exercises understandable.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – THEORETICAL OUTLINE
Calculation of the roots of a polynomial
Resolutions of matrix systems
Interpolation of functions
Definitions
Discretization of the integral
Finite difference method
Finite element method
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II – EXERCISES
Exercise1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
INTRODUCTION
INTRODUCTION
In this workbook, some examples of calculations relating to numerical analysis are carried out.
Furthermore, the main theoretical results of this field of mathematics are presented.
Numerical analysis makes it possible to solve all the operations of mathematical analysis through the use of finite systems, ie electronic calculators.
Therefore, the adaptation of all analytic operations to numerical computation is necessary for an effective solution of mathematical problems.
In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.
What is presented in this exercise book is generally addressed in the courses of numerical calculus and numerical analysis offered at university level.
I
THEORETICAL OUTLINE
THEORETICAL OUTLINE
Calculation of the roots of a polynomial
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A first large category of numerical methods consists in calculating the roots of a polynomial or of a generic transcendental function.
Horner's algorithm allows to evaluate a polynomial performing N additions and N multiplications, instead of the normal N additions and N(N+1)/2 multiplications required .
We have to rewrite any polynomial in another equivalent form:
The value of the polynomial is calculated in this recursive form:
The simplest method for finding the roots of an equation is given by the bisection method or dichotomous method.
This method assumes that if a function assumes values with different signs in a given interval then the root of this function is included in that interval.
This result is known as the zeros theorem and is valid under suitable conditions.
In formulas we have:
The bisection method divides the interval between a and b in half.
If in the middle of the interval the function is null, then that value is the root sought.
In the opposite case, the same algorithm is applied recursively considering only the half of the interval where there is still inversion of the sign.
At the n-th step, the approximation on the real value of the root is given by:
A method that varies the size of the interval is the linear interpolation method.
Instead of always taking half of the given interval, we take an intermediate value given by a weighted average:
In Muller's method a second order polynomial is used as an interpolating function.
An historical method for the numerical calculation of the value of a function in a given point is given by the method of false position in Fibonacci (or regula falsi) which always applies the bisection method, but the interval in which to find the root is calculated according to proportional criteria.
A better method for calculating the roots of a function is given by the so-called method of tangents (or Newton's method) .
By drawing the curve corresponding to the function on a Cartesian plane, this method consists in substituting, in a determined point, the geometric tangent in place of the
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 23.04.2023
ISBN: 978-3-7554-4004-8
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