First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 1. INTRODUCTION 9 Exercise 1.1-1: x 0 =0. Find the second order Taylor polynomial for f(x) =e x sin x about a) Compute P 2 (0.4) to approximate f(0.4). Use the remainder term R 2 (0.4) to find an upper bound for the error |P 2 (0.4)−f(0.4)|. Compare the upper bound with the actual error. b) Compute ∫ 1 P 0 2(x)dx to approximate ∫ 1 f(x)dx. Find an upper bound for the error 0 using ∫ 1 R 0 2(x)dx, and compare it to the actual error. 1.2 Julia basics The first step is to download the Julia programming language from the Julia webpage https://julialang.org. In this book I used Julia 1.1.0. There are different environments and editors to run Julia. Here I will use the Jupyter environment https://jupyter.org. Instructions on how to install software do not age well: I ran the command using IJulia; jupyterlab() on the Julia terminal to install the Jupyter environment, but this could change in the future versions of the software. There are several tutorials and other resources on Julia at https://julialang.org where one can find up-to-date information on installing Julia and Jupyter. The Jupyter environment uses the so-called Jupyter notebook where one can write and edit a Julia code, run the code, and export the work into various file formats including Latex and pdf. Most of our interaction with Julia will be through the Jupyter notebooks. One exception is when we need to install a package. A Julia package provides additional functionality to the core programs, and there are numerous packages from visualization to parallel computing and machine learning. To install a package, open Julia by clicking on the software icon, which will open the Julia terminal, called Julia REPL (Read-Evaluate- Print-Loop). In the Julia REPL, press ] to enter the package mode. (To get back to the Julia REPL press backspace, delete, or shift C.) Once in the package mode, typing "add PackageName" will install the package. To learn more about packages, type Pkg in the Julia 1.1 documentation https://docs.julialang.org/en/v1/index.html. After installing Julia and Jupyter, open a Jupyter notebook. Here is a screenshot of my notebook:
CHAPTER 1. INTRODUCTION 10 Let’s start with some basic computations. In [1]: 2+3 Out[1]: 5 In [2]: sin(pi/4) Out[2]: 0.7071067811865475 One way to learn about a function is to search for it online in the Julia documentation https://docs.julialang.org/. For example, the syntax for the logarithm function is log(b, x) where b is the base. In [3]: log(2,4) Out[3]: 2.0 Arrays Here is the basic syntax to create an array: In [4]: x=[10,20,30]
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Chapter 3 Interpolation In this cha
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CHAPTER 3. INTERPOLATION 92 basis f
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CHAPTER 3. INTERPOLATION 100 With t
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CHAPTER 3. INTERPOLATION 120 Exerci
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CHAPTER 3. INTERPOLATION 140 Recrea
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CHAPTER 4. NUMERICAL QUADRATURE AND
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Chapter 5 Approximation Theory 5.1
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Index Absolute error, 38 Beasley-Sp
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INDEX 220 Chebyshev, 203 Julia code
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First Semester in Numerical Analysis with Julia, 2020a

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