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Über dieses Buch

This is the first numerical analysis text to use Sage for the implementation of algorithms and can be used in a one-semester course for undergraduates in mathematics, math education, computer science/information technology, engineering, and physical sciences. The primary aim of this text is to simplify understanding of the theories and ideas from a numerical analysis/numerical methods course via a modern programming language like Sage. Aside from the presentation of fundamental theoretical notions of numerical analysis throughout the text, each chapter concludes with several exercises that are oriented to real-world application. Answers may be verified using Sage.

The presented code, written in core components of Sage, are backward compatible, i.e., easily applicable to other software systems such as Mathematica®. Sage is open source software and uses Python-like syntax. Previous Python programming experience is not a requirement for the reader, though familiarity with any programming language is a plus. Moreover, the code can be written using any web browser and is therefore useful with Laptops, Tablets, iPhones, Smartphones, etc. All Sage code that is presented in the text is openly available on SpringerLink.com.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Fundamentals

Most existing Numerical Analysis textbooks, are centered around some programming language (such as C, C + +, etc.) or some Mathematical programs (e.g., Mathematica

, Maple

, Matlab

, etc.) see [3, 6, 7]. What we are proposing here is a Numerical Analysis book that uses Sage Math. Throughout this book we will refer to it shortly as Sage.

George A. Anastassiou, Razvan A. Mezei

Chapter 2. Solving Nonlinear Equations

In this chapter we’ll deal with finding (or rather approximating) the solution of an equation

f

(

x

) = 0. This is a very common problem in Applied Mathematics, but finding the exact solution is not always an easy task.

George A. Anastassiou, Razvan A. Mezei

Chapter 3. Polynomial Interpolation

The process of finding a polynomial that passes through a given set of data points is called

polynomial interpolation

. This is the topic of the current chapter.

George A. Anastassiou, Razvan A. Mezei

Chapter 4. Numerical Differentiation

In this chapter we discuss different numerical approximation techniques for estimating the derivatives of functions.

George A. Anastassiou, Razvan A. Mezei

Chapter 5. Numerical Integration

Definite integrals have many applications in several fields, including Physics, Mathematics, Probability and Statistics, Computer Science, Economics, and others. In this chapter we will see some numerical methods that can be used to approximate definite integrals.

Quadrature

, a Mathematical term used for the problems of calculating areas, will be used in the names of some of these methods.

George A. Anastassiou, Razvan A. Mezei

Chapter 6. Spline Interpolation

Given the set of

n

+ 1 data points,

x

0

x

1

x

2

x

n

y

0

y

1

y

0

y

n

,

with

x

0

<

x

1

<

<

x

n

,

$$\displaystyle{ \begin{array}{l|l|l|l|l} x_{0} & x_{1} & x_{2} & \ldots & x_{n} \\ \hline y_{0} & y_{1} & y_{0} & \ldots & y_{n} \end{array},\text{ with }x_{0} < x_{1} < \cdots < x_{n}, }$$

we have seen how we can obtain a polynomial function of degree (at most)

n

, 

p

(

x

)

=

a

n

x

n

+

a

n

1

x

n

1

+

+

a

1

x

+

a

0

$$p(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0}$$

that interpolates these points. That is:

p

(

x

0

)

=

y

0

,

p

(

x

1

)

=

y

1

,

,

p

(

x

n

)

=

y

n

.

$$\displaystyle{ p(x_{0}) = y_{0},p(x_{1}) = y_{1},\ldots,p(x_{n}) = y_{n}. }$$

This, however, has a major drawback: the polynomial can have a very high degree (up to

n

) and hence, the interpolating function can oscillate too much. The oscillation may be quite wild even when all the

y

-values of the data set given are essentially constant.

George A. Anastassiou, Razvan A. Mezei

Chapter 7. Numerical Methods for Differential Equations

An equation that involves an unknown function,

y

and one or more of its derivatives is called a

differential equation

. In this chapter, we will consider the case when

y

 = 

y

(

x

) is a function of

x

.

George A. Anastassiou, Razvan A. Mezei

Erratum

George A. Anastassiou, Razvan A. Mezei

Backmatter

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