mdh.diva-portal.orgmdh.diva-portal.org/smash/get/diva2:1329121/FULLTEXT01.pdfAbstract This thesis aims to provide a brief exposition of some chosen modes of convergence; namely uniform

School of Education, Culture and CommunicationDivision of Applied Mathematics

BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Introduction to some modes of convergence

Theory and applications

by

Milosz Bolibrzuch

Kandidatarbete i matematik / tillämpad matematik

DIVISION OF APPLIED MATHEMATICSMÄLARDALEN UNIVERSITY

SE-721 23 VÄSTERÅS, SWEDEN

School of Education, Culture and CommunicationDivision of Applied Mathematics

Bachelor thesis in mathematics / applied mathematics

Date:28-09-2017

Project name:Introduction to some modes of convergence: theory and applications

Author:Milosz Bolibrzuch

Supervisor:dr Linus Carlsson

Reviewer:dr Richard Bonner

Examiner:Prof. Anatoliy Malyarenko

Comprising:15 ECTS credits

Abstract

This thesis aims to provide a brief exposition of some chosen modes of convergence; namelyuniform convergence, pointwise convergence and L1 convergence. Theoretical discussion iscomplemented by simple applications to scientific computing. The latter include solving dif-ferential equations with various methods and estimating the convergence, as well as modellingproblematic situations to investigate odd behaviors of usually convergent methods.

1

Acknowledgements

I would like to thank my supervisor, dr Linus Carlsson, who provided assistance in developing,writing and reviewing the thesis at all stages. He has also helped me solve multiple problemsand understand many new concepts, which allowed me to learn a great deal more mathematicsthan I would be able to do on my own in such a short time.

2

Contents

Abstract 1

Acknowledgements 2

List of Figures 5

1 Introduction 61.1 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Modes of convergence: basic theory exposition 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Important definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Pointwise convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Convergence at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 L1 convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Numerical applications with MATLAB 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Uniform convergence in estimating solutions to simple differential equations . 21

3.3.1 Computing convergence of error by comparison with a known solution 213.3.2 Computing convergence of error without access to a known solution . 23

4 Problem areas in numerical computation 264.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Non-smooth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Conclusion 30

Bibliography 31

3

Index 32

A MATLAB code 33A.1 Script 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.2 Script 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.3 Script 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.4 Script 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.5 Script 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.6 Script 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4

List of Figures

2.1 The sequence of isosceles triangles with the properties defined in Example 2,converging pointwise to 0. For clarity of visualisation, sequence started withthe second triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 The sequence of isosceles triangles as defined in the Example 7. It can beeasily observed that they seem to converge to 0 in area, but never converge atpoints 0 or 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Numerical solution to the problem y′ = xy, y(0) = 1 for x between 0 and 1. . 223.2 Convergence of error of the solution to the problem y′ = xy, y(0) = 1. Com-

puted by comparison to the analytic solution, for forward difference methodand central difference method. Plotted in log-log scale, logarithm of base 2. . 23

3.3 Estimated solution to the equation y′ = sin(xy)+ y, y(0) = 1 for x between 0and 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 The empirical convergence of error for central difference method in estimatingthe solution of y′ = sin(xy)+y, y(0) = 1. Plot in log-log scale, logarithm base 2. 25

4.1 The lesser error in trapezoidal method, with the cusp of the function fallingexactly on a grid point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 The larger error in trapezoidal method, with a cusp of the function falling inthe middle of a subinterval. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 The empirical convergence in trapezoidal method for solving the equationfrom the Section 4.2. Plot in log-log scale, logarithm base 2. Second ordermethod slope included for reference. . . . . . . . . . . . . . . . . . . . . . . 29

4.4 The convergence of error in the central difference scheme when solving y′ =23x−1/3, y(0) = 0, as described in the Section 4.3. Plot in log-log scale withbase 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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Chapter 1

Introduction

1.1 Thesis overviewThis thesis consists of three main parts. The first and major part is a theoretical introductionto some modes of convergence, containing definitions, additional descriptions and examples.The examples were constructed with the aim of rendering the concepts simpler to grasp. Themodes included in the thesis are pointwise convergence, uniform convergence, convergence ata point and convergence in L1 norm.In the second part we investigate convergence of some chosen numerical methods. In par-ticular, we employ central difference method and forward difference method to solve simpledifferential equations. Included are definitions and descriptions of some useful concepts, suchas the O(h) notation and log-log scale plotting. A simple solver, which was constructed inMATLAB for this thesis, is used.The last part contains examples of problematic areas in numerical computation, such as non-smooth functions and singularities. It includes introducing the trapezoidal rule along the finitedifference methods and investigating what happens to the convergence of these methods whenthe aforementioned problems occur. Once again, we employ the MATLAB solver to analysethe issues numerically and visualise the results.Additionally, the scripts containing the MATLAB code used in the project are included in theAppendix A: MATLAB code.

1.2 Literature reviewThe idea of convergence is far from a new concept, and it might be extremely difficult ifnot impossible to pinpoint its precise origins. It might, however, be possible to find the firstdistinction between divergent and convergent series, which according to [2] was done in the17th century by James Gregory. The distinction between the modes of convergence introducedin this thesis would come at a later date. The first major influence on the birth of the conceptof uniform convergence was by Cauchy in 1821. He provided an errouneous proof of the factthat a convergent sum of continuous functions is always continuous. This led to discoveries ofuniform convergence by Stokes and Seidel later in the same century. Karl Weierstrass has also

6

discovered this notion and was perhaps the first one to do so, according to [6] (although thefirst use of the term "convergence in a uniform way" was by Gudermann in [5]) . Presently,more refined notions can be found within analysis textbooks, such as [11].Addressing the numerical methods, many of them are in fact ancient in the literal sense. Itwas recently discovered (see [10]) that the trapezoidal rule, one of the methods used in thisthesis, was already known in a similar form to ancient babylonians. The analysis of error andstability of these methods as we know it today, however, was developed mostly in the 20thcentury. For more on this, see e.g. [13].

1.3 NotationIn this section we briefly introduce the notation used in the thesis.

d(a,b) a distance between a and b∃ a there exists an a∀a for all aa ∈ A a is an element of set Asupa∈A f (a) supremum of f on the set AA⊂ B set A is a subset of set BA∪B a union of A and Bsupp( f ) support of a function fO( f ) asymptotic notationR set of real numbersQ set of rational numbersN set of natural numbersCk class of functions for which the first k derivates exist and are continuous(a,b) an open interval from a to b[a,b] a closed interval from a to b|a| absolute value of a‖u‖ a norm of uuh an approximation of u obtained with a step size of h(R, | · |) the real line with a distance defined as the absolute value

7

Chapter 2

Modes of convergence: basic theoryexposition

2.1 IntroductionThis chapter will cover a few selected modes of convergence, specifically convergence at apoint, pointwise convergence, uniform convergence and L1 convergence. We will introducethe necessary theory to discuss these concepts in terms of their properties, differences betweenthem, potential problems within the way they are defined and subsequently their utility.

Before we move forward with the aforementioned topics, it is important to draft a fewinitial notes. In particular, it will prove particularily useful to outline a few important definitonswhich provide necessary context for the discussion about convergence.

In all of the examples given in this part we let the function range be in (R, |.|).

2.2 Important definitions

Firstly, we need to introduce some concept of a space. As we will repeatedly find out duringthis exposition, the convergence of a sequence is very much dependent on a type of spacewithin which it is being considered. Topological space is the most general concept that appearsin many areas of modern mathematics. They are key to discussing convergence, continuityand other ideas. In particular, we are interested in metric spaces, which are a special case oftopological spaces. A metric space is, in essence, a topological space equipped with a metric(also known as a distance function).

Definition 1. Let M be any non-empty set. Let d be a function defined on M×M→ R. If dsatisfies the conditions

1. d(x,y)≥ 0 non-negativity2. d(x,y) = 0⇔ x = y indiscernibility3. d(x,y) = d(y,x) symmetry4. d(x,y)+d(y,z)≥ d(x,z) triangle inequality

8

then d is a metric and the pair (M,d) is called a metric space.

Now that we have defined a metric space, we can follow it up with another essential defin-ition that will help us discuss convergence-related concepts: a Cauchy sequence. Cauchysequences are useful when considering convergence in various spaces; for example, we candefine property of a space by stating that Cauchy sequences converge or do not converge onthat space.

Definition 2. A sequence{

x j}∞

j=1 in a metric space M = (M,d) is called a Cauchy sequenceif for every ε > 0 there exists an integer N = N(ε) such that

d(xm,xn)< ε for every m,n > N.

We also want to be able to use the idea of a support of a function. In essence, supportis a subset of the function’s domain where the function does not vanish. In particular, we areinterested in the closed support. For that, we need to first define the closure of a set.

Definition 3. For an open set A, the closure of A is the intersection of all closed sets containingA.

Definition 4. Let X be a topological space and f be a function defined on that space. Thenthe support of f is the closure of the subset of X where f is non-zero, i.e.

supp( f ) = {x ∈ X : f (x) 6= 0}.

Finally, we need to introduce supremum and infimum. These concepts are similar tomaximum and minimum which the reader should be familiar with, but there is a key difference.Firstly, we need to introduce lower and upper bounds.

Definition 5. Let set A be a subset of real numbers. Then A is bounded from above if thereexists a real number M, called the upper bound of A, such that x < M for every x ∈ A. In thesame fashion, A is bounded from below if there exists a real number m, called the lower boundof A, such that x > m for every x ∈ A. The set A is called bounded if it is bounded from belowand bounded from above.

Definition 6. The supremum of a set A is its least upper bound. The infimum of a set A is itsgreatest lower bound. If the set is not bounded above, we say that supA = ∞. If the set is notbouned below, we say that infA =−∞.

In contrast to maximum, supremum does not need to be an element of that given set. Thismeans that when maximum exists, supremum also must exist, but not necessarily the otherway around. The same is of course true for the relation between minimum and infimum.For a deeper introduction to metric spaces, we refer the reader to any standard textbook in thefield, e.g. [8].

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2.3 Pointwise convergencePointwise convergence is one of the first theoretical notions of convergence that was welldefined. This is at least partly due to the fact that the way we define pointwise convergenceseems on the first glance like the most obvious and natural way to define convergence for func-tions. However, as we will clearly demonstrate later in this text (see Examples 1, 2), thinkingabout pointwise convergence for functions can often lead to somewhat confusing results. Thisis why today we would often conrast it with the idea of uniform convergence, a stronger no-tion, which we will explore in Section 2.4. To begin discussing pointwise convergence, weshall first introduce a formal definition.

Definition 7. Let fn be a sequence of functions defined on a metric space X with its range ona metric space (Y,d). Then fn converges pointwise to a limit function f if and only if

∀ x ∈ X , ∀ ε > 0, ∃M > 0

such that d( fn(x)− f (x))< ε when n > M.

This definition of convergence will quite often prove sufficient. However, we should ob-serve that according to this formulation it is only necessary to show that the function sequenceconverges for every fixed point x. This directly brings about one rather awkward result, whichwe shall demonstrate with an example.

Example 1. Let us consider a function sequence fn(x) = xn defined on a closed set X = [0,1]on the real line. It is readily seen that fn converges pointwise according to the aforementioneddefinition, to f (x) = 0 for all x ∈ X except for x = 1, for which it converges pointwise tof (x) = 1. This is a case of a sequence of continuous functions converging to a discontinuouslimit - an interesting and rather unintuitive result.

There is yet another staple situation in which pointwise convergence exhibits odd behavior.It is directly connected with the construction of an object called a Dirac delta measure, oftenreferred to as Dirac delta function, or simply a delta function, despite formally not being afunction. It is what we call a generalized function or a measure. Measure theory, however, isout of the scope of this paper, and thus we will try to present this concept without delving toodeep. The reader interested in learning more can be referred to [4].The main properties of the delta function are as follows: it has the value of zero everywhereon the real line except for the origin, where it spikes into infinity, and the integral of this objectover the real line is one. This is why it cannot be precisely defined as a function; a functionwith value of zero everywhere except at one point must consequently have the integral equalto zero. Furthermore, it is not possible to define a function as having infinite value at onespecified point. One way to define the delta function is through integration; such definition willalso allow us to pinpoint the aforementioned odd behavior concerning pointwise convergence.Let us now consider the following scenario.

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Example 2. Let f be any continuous function defined on the real line and Tn be any functionsequence defined on the same metric space, with the properties

1. Tn(x)≥ 0, Tn is continuous.

2. limn→∞

n⋂i=1

supp(Ti) = {0} .

3.∫

∞

−∞

Tn = 1.

Figure 2.1: The sequence of isosceles triangles with the properties defined in Example 2, con-verging pointwise to 0. For clarity of visualisation, sequence started with the second triangle.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5X axis

0

2

4

6

8

10

12

14

16

Y ax

is

T1

T3

T7

A much studied example of such functions are the so called bump functions. Bump func-tion is a function that is both smooth as well as compactly supported - meaning that its supportis closed and bounded. Another, easy to imagine example of such function sequence could bea sequence of isosceles triangles on the real line with height 2n and base length of 1/n, startingat the origin (see Figure 2.1.). If we think about limiting behavior of sequences formulatedthis way, it is readily seen that we are able to construct the Dirac delta function by going tothe limit. Furthermore, it will hold that

limn→∞

∫∞

−∞

f (x)Tn(x)dx = f (0).

This can be quickly demonstrated with a proof.

Proof. ∫∞

−∞

f (x)Tn(x)dx =∫

supp(Tn)f (x)Tn(x)dx

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by first mean value theorem for definite integrals (see e.g. [1]), the above expression is equalto

f (an)∫

supp(Tn)Tn(x)dx = f (an), for some an ∈ supp(Tn)

Since supp(Tn)→ 0 as n→ ∞, we have that limn→∞ f (an) = f (0), since f is continuous.

The relation we have just proven is one way to formulate the definition of a delta functionand contain its most basic properties. Now, concerning pointwise convergence, we are able toshow that Tn(x)→ 0 pointwise.

Proof. To prove that Tn(x)→ 0, we will first divide the problem into two cases. The first caseis trivial.

1. If a point x0 ∈ R\ (0,1), then Tn(x0) = 0. Thus,

limn→∞

Tn(x0) = 0.

The second part is a bit more tricky, as we’re looking at the non-trivial subset of the domain.

2. If a point x0 ∈ (0,1), then x0 > 0 and thus 1x0∈ R.

Hence, there must exist a natural number N such that 1N < 1

x0. Therefore, supp(Tn) ⊂

(0, 1

N

)for all n > N. Finally, we obtain the expression below.

limn→∞

Tn(x0) = limn→∞n>N

Tn(x0) = limn→∞

0 = 0.

This result brings about a particular dissonance. We can immediately observe one oddconsequence, as we have

limn→∞

∫∞

−∞

Tn(x)dx = 1 6= 0 =∫

∞

−∞

limn→∞

Tn(x)dx.

As we can see, we are not able to freely switch the order of operations in this case, which isoften a desired result.

Remark. In Example 2, the condition on support of Tn is not necessary to construct a Diracdelta. We introduce it for clarity of exposition.

The above examples are some of the reasons why it is not always optimal to rely only onpointwise convergence. This is why we often refer to another, stronger mode, called uniformconvergence.

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2.4 Uniform convergenceAs previously mentioned, the pointwise convergence is often contrasted with a stronger mode,called uniform convergence. Uniform convergence directly implies pointwise convergence,but not the other way around. In fact, we will demonstrate how in the previous, problematicexamples of pointwise convergence, the function sequences in question do not attain uniformconvergence (see e.g. [4]).Firstly, however, we should introduce a proper definition for uniform convergence.

Definition 8. Let fn be a sequence of real-valued functions defined on R. Then fn convergesuniformly on R to a limit function f if and only if

∀ ε > 0, ∃M > 0, such that

supx∈R| fn(x)− f (x)|< ε when n > M.

For a more general definition of uniform convergence, which we shall not need in thisthesis, see e.g. [11]. Now let us take a closer look at the introduced definition. One key differ-entiating factor between uniform and pointwise convergence is that the former does not allowconsidering fixed x points. Instead, we have to determine whether the sequence converges as awhole, or as the name suggests, in a uniform fashion on it’s entire domain. We will soon findout that this more rigorous approach yields better results when it comes to problem areas ofpointwise convergence; in particular, continuity and interchanging of limit processes.Firstly, however, we ought to introduce some examples.

Example 3. Let us consider a function sequence fn(x) = 1x+ 1

non the interval (0,∞). It is

readily seen that this function sequence converges pointwise to f (x) = 1x .

Proof. First, we fix a point x ∈ (0,∞) and let ε > 0. Then we have

| fn(x)− f (x)| =

∣∣∣∣∣ 1x+ 1

n

− 1x

∣∣∣∣∣ =∣∣∣∣∣x− (x+ 1

n)

x(x+ 1n)

∣∣∣∣∣=

1n· 1

x(x+ 1n)

<1n· 1

x2 < ε, if n >1

εx2 .

Thus, by Definition 4, fn converges pointwise to 1x .

However, this function sequence does not converge uniformly on the real line. One wayto demonstrate that is by choosing a subsequence and disproving the uniform convergence bydefinition for that subsequence.

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Proof. Given ε > 0, we will show that the series does not converge uniformly to f (x) = 1x .

We will choose a subsequence of fn by only considering xn =1n over (0,1). This gives us the

following expression.

supx∈(0,1)

| fn(x)− f (x)| ≥∣∣∣∣ fn

(1n

)− f

(1n

)∣∣∣∣=

∣∣∣∣∣ 11n +

1n

− 11n

∣∣∣∣∣ =∣∣∣∣∣12

n

− 11n

∣∣∣∣∣ = ∣∣∣n2−n∣∣∣

=n2> ε, if n > 2ε.

Thus, by Definition 5, fn does not converge uniformly to 1x .

Remark. It might be interesting to point out that, have we looked at another interval, namely(1,∞), this function sequence would actually converge uniformly, since

| fn(x)− f (x)|< 1n· 1

x2 <1n·1 < ε, if n >

1ε.

With that out of the way, let us consider Examples 1 and 2 in light of our new definition ofconvergence.

Example 4. In the first example we looked at fn(x) = xn on the interval [0,1] and we wereable to show that it converges pointwise to a discontinuous limit despite being a sequence ofcontinuous functions. In this part, we will demonstrate that fn does not converge uniformlyon that same interval.

Proof. To begin, we pick an ε = 1/2. Then we look at

supx∈[0,1]

| fn(x)− f (x)|

Since for x = 1 this distance is always fixed and equal to 0, we can exclude that point and writeas below.

supx∈[0,1)

| fn(x)− f (x)| = supx∈[0,1)

|xn−0|>

((34

) 1n)n

=34>

12.

Thus, by Definition 5, fn does not converge uniformly to a discontinuous limit of

f (x) =

{0 if 0≤ x < 11 if x = 1.

In fact, uniform convergence guarantees that this dissonance is always avoided.

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Example 5. In the next example we are looking at the more intriguing case of a functionsequence that seemingly vanishes, yet its integral never goes to zero. That prevents us frombeing able to freely interchange the order of limit operations by only asserting pointwise con-vergence (see Example 2). We proved that it converges pointwise to 0. Here, we want to showthat the sequence in question does not converge uniformly.

Proof. Recall that uniform convergence directly implies pointwise convergence. We knowthat Tn(x) converges pointwise to 0. Thus, if Tn converges uniformly, it has to converge uni-formly to 0. Following this, if Tn converges uniformly, we have for any ε > 0

|Tn(x)−0|< ε ∀ x.

Observe that if we choose a sequence xn =1n , we have Tn(xn) = 2n. Then

|Tn(xn)−0|= |2n| ≥ 2 ∀ n ∈ {1,2,3, ...} .

Now, by choosing ε = 1 we have |Tn(xn)−0| ≥ 2 ≮ ε and thus we have disproved uniformconvergence for Tn.

In general terms, uniform convergence guarantees a lot of intuitive results; continuousfunctions converging to continuous limits, ability to interchange the order of limit operationsand so on. This is why whenever it is possible to prove uniform convergence, it will alwaysbe more beneficial than only proving pointwise convergence. Obviously, proving uniformconvergence is not always possible; in some situations we can only prove weaker modes ofconvergence, and that has merits of its own. We will end this section with a well knowntheorem. For its proof, see e.g. [11].

Theorem 1. Uniform convergence theorem. If a sequence fn of continuous functions definedon a metric space M converges uniformly to a function f defined on M, then f is continuous.

2.5 Convergence at a pointThe weakest of the modes of convergence chosen to be discussed in this thesis is convergenceat a point. It is also arguably the most basic. The formal definition follows.

Definition 9. Let fn be a sequence of functions defined on a metric space X with a range on ametric space (Y,d). Then fn converges at a point x0 to a limit a if and only if

∀ ε > 0, ∃M > 0

such that d( fn(x0)−a)< ε when n > M.

In fact, we can quickly observe that considering convergence of a function sequence at apoint is the same as considering the convergence of a sequence of numbers. For a fixed pointx0, a function sequence fn(x0) is precisely that - a sequence of numbers, which we can moreconveniently (and appropriately) denote as simply {yn}.It is readily seen that this mode of convergence does not imply any previously introducedmodes; however, both uniform and pointwise convergence by definition imply convergence atany point.

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2.6 L1 convergenceThe L1 space is a particular case of a mathematical notion called Lp spaces, or Lebesguespaces. We will only concern ourselves with the L1 space consisting of real-valued functionswhose domain are all real numbers. Considering such objects in detail is out of the scope ofthis paper. We shall only use it to look at the objects

‖ f‖Lp =

(∫R| f |p

)1/p

.

In particular, with p = 1, the expression becomes

‖ f‖L1 =∫R| f | .

Remark. The objects above are not norms when considering the Riemann integral, but wekeep the standard notation, ||.||.

We will consider convergence in L1 to be convergence under these conditions. It is yetanother interesting concept to look at when attempting to establish a basic understanding ofthe relations between the modes of convergence. One sensible way to look at this type ofconvergence, prior to attaining mathematically rigorous knowledge on the topic, is to thinkof objects converging in terms of area. As we will shortly demonstrate, this does not exactlytranslate to previously described modes. Indeed, it is possible to find examples for pointwiseand even uniformly convergent function sequences which do not converge in L1, as well asexamples of L1 convergent objects that do not converge even pointwise. Furthermore, wewill demonstrate an example of an L1 convergent function sequence which does not attainconvergence at any single fixed point in the domain. Firstly, we will need a working definitionof what we mean by L1 convergence.

Definition 10. Let fn be a sequence of real valued functions defined on R. Then fn convergesin L1 on R to a limit function f if and only if

∀ ε > 0, ∃M > 0

such that∫R| fn(x)− f (x)|< ε when n > M.

Equipped with this definition, let us look for examples that show the relations betweenL1 convergence and the other modes. First, let us try to construct two examples of functionsequences which will converge in L1, but not uniform or pointwise.

Example 6. We start with a function f (x) such that

f (x) =

{0 if x ∈ R\Q1 if x ∈Q.

16

In other words, f disappears for all irrational numbers, but is equal to 1 for all rational num-bers. We know that irrational numbers are uncountably infinite, while rational numbers are acountable set (see e.g. [14]). This means that we can interpret taking an integral of f (x) overthe real line as taking an integral of a g(x) = 0 with an infinite (but countable) number of holesthat take on the value of 1. These points are just that; points, and thus can not affect the valueof the integral, which remains 0.Now, we want to construct a sequence of functions using f . We will shift it back and forth byπ , letting

f2n(x) = f (x)

f2n+1(x) = f (x−π)

Observe that∫

∞

−∞fn(x) = 0 for all n ∈ N, thus obviously this function sequence converges

to 0 in L1. Notice, however, that for all of the points that start off as rational numbers, aswell as those irrational numbers that can be expressed as p

q + π , the values of fn will keeposcillating between 0 and 1 forever. Hence we know that this function sequence can notconverge pointwise.

Example 7. Recall the setting of the Example 2. In this example we will be using a similarsetting, however this time we want our function sequence, which we will call Tn, to convergeto 0 in terms of area, while not converging uniformly or pointwise at the same time. Again,it will be easy to imagine the sequence of isosceles triangles. We want all of our trianglesto have fixed height of 1, and their base length to go to 0 as n→ ∞ and the support of Tn togo to {0}∪ {1}. To complete the example, we let T1 be centered at x0 = 1, T2 centered at0, T3 centered at 1 again and so on in perpetuity (see Figure 2.2.). It is readily seen that thissequence converges in L1 to 0. However, we can observe that it does not converge uniformly orpointwise for x0 = 1 nor x0 = 0. At these points, the values of the functions in the sequence willforever oscillate between 0 and 1. It might be worth pointing out that the sequence convergesuniformly on all closed subsets of the domain which do not contain 0 or 1, as for all the otherpoints sooner or later we arrive at 0 and stay at that value.

Example 8. Finally, we want to briefly go over the opposite situation; a uniformly convergentsequence which does not converge in L1. This is a bit tricky, since we are stepping a bit outof the scope of this thesis. There exist functions that are not in L1, but we are not able to gointo much detail on them due to this being an undergraduate level thesis. A reader interestedin gaining deeper understanding can be referred to [3, p. 20], where one such example is givenand elaborated on. For this thesis we only need to acknowledge that functions like that existand can be uniformly convergent. For instance, we may take some f not in L1 and use it tocreate a simple sequence fn = f +1/n. Obviously, it converges uniformly to f , but there canbe no consideration of L1 convergence due to our assumption of f not being in L1 at all.

17

Figure 2.2: The sequence of isosceles triangles as defined in the Example 7. It can be easilyobserved that they seem to converge to 0 in area, but never converge at points 0 or 1.

-0.5 0 0.5 1 1.5 20

0.5

1T1

-0.5 0 0.5 1 1.5 20

0.5

1T2

-0.5 0 0.5 1 1.5 20

0.5

1T3

-0.5 0 0.5 1 1.5 20

0.5

1T4

-0.5 0 0.5 1 1.5 20

0.5

1T5

18

Chapter 3

Numerical applications with MATLAB

3.1 IntroductionAcknowledging that this is an undergraduate level thesis and we are not able to delve verydeep into the theoretical aspects of convergence, it might be of a good idea to make up forthat with some applications to scientific computing. Practical implementation is often a veryuseful tool for better understanding and exercising freshly obtained theoretical knowledge,especially when the knowledge in question is quite abstract in some ways.This is why for this thesis we will include a chapter dedicated to providing numerical andcomputational examples where some of the previously introduced theoretical concepts canbe used, in an attempt form a link between the theory and application and make both moreaccessible to the reader. In the following chapters we restrict ourselves to real-valued functionsdefined on subsets of the real line.

3.2 Important conceptsAs detailed above, this thesis tackles the topic of modes of convergence from two related,yet not identical perspectives, attempting to provide the reader with a solid outlook on thematter. Having covered the underlying mathematics necessary to understand these concepts,we will now be going through a series of numerical problems. In many of them we will beusing appropriate iterations of a simple solver script (see the Appendix A), which we haveconstructed in MATLAB for this thesis. This solver uses primarily finite difference methodsfor solving differential equations, both central and forward. The relationships used for thesemethods are as follows.

Central: f ′(x) =f (x+h)− f (x−h)

2h+O(h2).

Forward: f ′(x) =f (x+h)− f (x)

h+O(h).

They are based on Taylor series. The derivation from Taylor series for central and forwardmethods are given in[9]. This solver will be used to approximate solutions to chosen differen-tial equations with initial values as well as computing definite integrals. It contains the code

19

to compute and graphs empirical errors to help consider the convergence. Particular versionsof the script and underlying algorithms will be explained in more detail in the sections corres-ponding to the problems for which those versions were developed.The last section will focus on demonstrating some of the potential problems that one canencounter when trying to assess convergence of a numerical method. This part will also incor-porate MATLAB scripts to help solidify and visualise the issues. Before we go any further,however, let us stop to consider some basic notions which we will require in order to expandinto numerics.Convergence within the numerical context usually involves analysing an error of some sort. Itwould be of value to introduce a common error notation at this point, commonly referred toas big O notation (O(h)), also known as asymptotic notation. We have already used it above,as a part of the description for the central difference method. The big O notation describeslimiting behavior of a function given arguments tending towards infinity or a particular value.In our case we will mostly be talking about arguments tending to zero, as the main purpose ofthis notation, within our context, is to express a bound on the difference between a functionand its approximation. This allows us to formulate the error in a simple form by omitting theterms that become much less significant for small values of h. Let us attempt to define thisnotation a bit more precisely (for more on this, see e.g. [7]).

Definition 11. Let f and g be functions defined on an open interval containing 0. Then

f (h) = O(g(h)) for sufficiently small h

if and only if there exist positive numbers ε and N such that

| f (h)| ≤ N|g(h)| when 0 < |h|< ε.

Here are some basic rules for O(h) calculations that follow from the definition.

O(hp)±O(hp) = O(hp)

O(hp)±O(hq) = O(hmin(p,q))

O(hp) ·O(hq) = O(hp+q)

C ·O(hp) = O(hp).

We should now briefly state what we mean by the order of convergence. For the error ex-pressed as follows

|uh−u| ≤Chp,

the order of convergence is p, given some constant C and sufficiently small h.Another notion worth introducing before delving deeper into numerical problems are the log-log plots. The idea behind using these plots, which employ logarithmic scale on both axis, isto make presenting and estimating convergence rate for error simpler. In general, log-log plotsare used to present the relationships of type y = axk, called monomials, as straight lines on thegraph. Taking logarithm of both sides of a monomial, we get

logy = k logx+ loga.

20

In particular, a common representation of error would be (see e.g. [12])

|uh−u|=Chp +O(hp+1) =Chp(

1+O(hp+1)

Chp

)log |uh−u|= logChp + log(1+O(hp+1−p))

log |uh−u|= log |C|+ p logh+O(h).

This last relation makes it significantly easier to read the convergence of a given method offof the plot, which is why we will employ this technique for most graphs to come in this thesis.When h is small the graph ends up looking like a straight line with its slope equal to theconvergence rate p.

3.3 Uniform convergence in estimating solutions to simpledifferential equations

In this section, our goal is to present an instance where uniform convergence would be themost desired mode. We will try to accomplish that with approximating solutions to differentialequations. In such type of problems it is obviously important that all the points converge tothe right values, or else the solution would be simply incorrect.

3.3.1 Computing convergence of error by comparison with a known solu-tion

Firstly we shall investigate an example in which we are able to easily obtain the error es-timates by comparing with the real values. To demonstrate this in practice we shall use theaforementioned solver and the example of the following initial value problem.

y′ = xy, y(0) = 1.

This iteration of the solver utilizes the following algorithm:

1 Outer while loop through step sizes until n reaches maximum n2

3 Decrease step size to 1/2.^n4 Declare initial values5

6 Inner for loop, stepping the interval7

8 Use central difference method to estimate the value of y9 at the current step and store

10 Use forward difference method to estimate the value of y11 at the current step and store12 Compute the real value of y at the current step and store13

21

14 End inner loop15

16 Compute error as a maximum difference between approximations17 of y through central difference method and the real values18 at the current step size19

20 Compute error as a maximum difference between approximations21 of y through forward difference method and the real values22 at the current step size23

24 End outer loop25

26 Plot the errors in a log-log form

We use the step size of 1/2n for n going from 1 to some chosen number (for plotting wechose maximum n to be 17). Maximum is being used a substitute of supremum from the uni-form convergence definition, since we are in the MATLAB environment, which is based onmatrices. Finally, we adjust the results to the conventional log-log form before plotting (usinglog2). Included are the plot of an estimated solution and the plot of uniform convergence forboth methods used by solver in this case, central difference method and forward differencemethod. For the script in MATLAB, see the Appendix A, Script 1. The first plot, when com-

Figure 3.1: Numerical solution to the problem y′ = xy, y(0) = 1 for x between 0 and 1.

0 0.2 0.4 0.6 0.8 1X axis

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Y ax

is

Solution estimate

pared with the real solution, looks virtually undistinguishable. Using the MATLAB functionpolyfit we find out that the slope of the central difference method’s convergence is 1.9972,while for the forward difference method it is 1.0126. These results are in line with the analyticcalculations; see the derivations in [9] and their result summed up in the Section 3.2. This

22

Figure 3.2: Convergence of error of the solution to the problem y′ = xy, y(0) = 1. Computedby comparison to the analytic solution, for forward difference method and central differencemethod. Plotted in log-log scale, logarithm of base 2.

-18 -16 -14 -12 -10 -8 -6 -4 -2Step size (in powers of 2)

-35

-30

-25

-20

-15

-10

-5

0

Erro

r (in

pow

ers

of 2

)

CentralForward

result tells us that we were indeed able to attain uniform convergence in this case.

3.3.2 Computing convergence of error without access to a known solu-tion

The above example was of a problem to which the solution can be analytically found, whichmakes assessment of the error substantially easier. However, in most cases, the real value isnot available for comparison. There are two main approaches to consider in such situations:1. Compute a single reference value with a very small step size h and then proceed as if thatreference value was a solution. This is a very straightforward method, but not too practicalwhen computing the value for a very small step size takes a long time.2. The second approach is to look at the differences between the consecutive estimates (fordecreasing time steps). In this section, we will be utilizing the latter approach as more sensiblein our context.To present an example, we will use the following problem:

y′ = sin(xy)+ y, y(0) = 1.

This equation does not have an analytic solution that we know of (although, to be precise, wehave not disproved the existence of a potential solution), so we will use the solver to obtain theapproximation and assess convergence. In this iteration, we follow nearly the same procedure

23

as in the first example, but including more loops to compare the initially saved values (see theAppendix A, Script 2). MATLAB polyfit indicates that the slope of the line is 1.9973, whichindicates second order of convergence. This is precisely what we would expect when usingthis method. Included are the relevant plots (see Figures 3.3 and 3.4).

Figure 3.3: Estimated solution to the equation y′ = sin(xy)+ y, y(0) = 1 for x between 0 and1.

0 0.14 0.28 0.42 0.56 0.7 0.84X axis

0.5

1

1.5

2

2.5

3

3.5

4

Y ax

is

Estimatedsolution

24

Figure 3.4: The empirical convergence of error for central difference method in estimating thesolution of y′ = sin(xy)+ y, y(0) = 1. Plot in log-log scale, logarithm base 2.

-16 -14 -12 -10 -8 -6 -4 -2Step size (in powers of 2)

-35

-30

-25

-20

-15

-10

-5

0

Erro

r (in

pow

ers

of 2

)

Convergence oferror estimate

25

Chapter 4

Problem areas in numerical computation

4.1 IntroductionIn this section we will discuss and present examples of potential major issues that could arisewhile using the aforementioned methods. The convergence that we are hoping to obtain canbe weakened or impossible to read from the computations in some particular circumstances.We will focus on two potential problem classes and present examples using MATLAB.

4.2 Non-smooth functionFor a large portion of numerical schemes, it is required that the function in the problem issmooth enough. Different methods might require different degrees of smoothness (differenti-ability). In this instance we will investigate an issue with the trapezoidal rule, caused by thefunction having a cusp on the interval which we would like to integrate on.

To do this, we will first introduce the basic premise of the trapezoidal rule. This methodallows for an approximation of an area under the curve on a given interval. In principle, thedefinite integral is approximated by a trapezoid as∫ b

af (x)dx≈ (b−a)

f (a)+ f (b)2

.

When using the rule, the interval is usually divided into M subintervals of length h = (b−a)/M. In this case the approximation through what is now the composite trapezoidal rule canbe expressed as

T ( f ,h) =h2( f (a)+ f (b))+h

M−1

∑k=1

f (xk).

To express the error term ET ( f ,h), we can state∫ b

af (x)dx = T ( f ,h)+ET ( f ,h).

26

In this case, provided that the function f ∈ C2[a,b], there exists a value c with a < c < b sothat the error term has the form

ET ( f ,h) =−(b−a) f ′′(c)h2

12.

The proof of this result is given in [9]. This shows that the expected order of convergence ofthis particular method is p = 2.

However, as we plan to demonstrate, that might not hold for functions with non-smoothparts. In particular, let us consider the following object, inspired by a very similar examplegiven in [12]. ∫ 1

0(x−a)2/3 dx, 0 < a < 1.

The integrand in this case is a function with a cusp at the point x = a. This can cause issues

Figure 4.1: The lesser error in trapezoidal method, with the cusp of the function falling exactlyon a grid point.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X axis

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Y ax

is

Grid points

ERRORAREA

with estimating the convergence of error. Although the error is bounded by O(h2), dependingon the grid settings the error caused by this problem can be more or less significant. If theinterval is divided into subintervals in such a way that the point a falls precisely or very close tothe start of one of the subintervals, then the error will be really mild (see Figure 4.1). However,if the point a would fall in the middle of a subinterval, the error for that subinterval would besignificantly higher than for all other subintervals, nearing the bound (see Figure 4.2). Thismeans that attempting to compute the convergence by comparing consecutive approximationswill not work properly in this case, as demonstrated with the figure below.

We will use MATLAB to visualise this problem. We will need to fix the parameter a, leta= 1/

√2, which along with step size of h= 1/2n will allow for the situations described above

to take place. We use a simple algorithm written down in MATLAB language (see Appendix

27

Figure 4.2: The larger error in trapezoidal method, with a cusp of the function falling in themiddle of a subinterval.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X axis

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Y ax

is

ERRORAREA

Grid points

A, Script 4), looping through the decreasing step sizes and measuring the convergence bydefining error as a difference between consecutive results. We obtain the following log-logplot, with slopes of methods of order 1 and 2 added for comparison. Since the magnitude ofthe order oscillates between approximations, we are unable to get a usual straight line fromwhich we could read the slope (see Figure 4.3).

4.3 SingularityIt is not uncommon to encounter a mathematical setting in which there are some singularities.Values of functions escaping into infinity at some points can obviously cause major havocwithin a numerical algorithm. In this subsection, we shall consider the following initial valueproblem:

y′ =23

x−1/3, y(0) = 0.

This problem has a fairly simple analytic solution, y = x2/3. We will use the previously intro-duced MATLAB solver, based on the central difference method, to demonstrate the problemsin practice. For the script, see the Appendix A, Script 3. It is readily seen that computing closeto zero in this case will yield misleading results, and thus destroy the convergence. An attemptto solve this problem with our solver returns the plot of convergence of error as shown below.As we can observe from the log-log plot, the central difference method is much less useful inthis case, yielding the order of convergence of merely 1.0370 according to the approximationby MATLAB polyfit function, instead of the expected 2 (see Figure 4.4).

28

Figure 4.3: The empirical convergence in trapezoidal method for solving the equation fromthe Section 4.2. Plot in log-log scale, logarithm base 2. Second order method slope includedfor reference.

-16 -14 -12 -10 -8 -6 -4 -2-35

-30

-25

-20

-15

-10

-5

0

5

Error estimateSecond order slope

Figure 4.4: The convergence of error in the central difference scheme when solving y′ =23x−1/3, y(0) = 0, as described in the Section 4.3. Plot in log-log scale with base 2.

-16 -14 -12 -10 -8 -6 -4 -2Step size (in powers of 2)

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Erro

r (in

pow

ers

of 2

)

Convergence oferror estimate

29

Conclusion

In conclusion, we have presented theoretical background for considering different modes ofconvergence. A big limiting factor was the undergraduate level of mathematics; knowledgeof measure theory would have allowed this paper to become more precise as well as moreextensive. We have covered some interesting numerical examples, but there are undoubtedlymany more interesting properties to investigate. There are at least two possible directions forexpanding on this topic; one would include delving into the theory of computation to analyseless trivial algorithms and their convergence, and the other would be to apply measure the-ory and present a more coherent explanation of the mathematical theory, perhaps incorporateMonte Carlo methods and stochastic convergence.

30

Bibliography

[1] Robert A Adams and Christopher Essex. Calculus: a complete course. Vol. 7. PearsonCanada 7th ed. Toronto, 2010.

[2] Walter William Rouse Ball. A short account of the history of mathematics. CourierCorporation, 1908.

[3] Gerald B Folland. Real analysis: modern techniques and their applications. John Wiley& Sons, 2013.

[4] D. H Griffel. Applied functional analysis. eng. New York: Ellis Horwood, 1981. ISBN:0-13-043324-1.

[5] Christoph Gudermann. “Theorie der Modular-Functionen und der Modular-Integrale.”In: Journal für die reine und angewandte Mathematik 18 (1838), pp. 1–54.

[6] Godfrey Harold Hardy. “Sir George Stokes and the concept of uniform convergence”.In: Proc. Cambridge Philos. Soc. Vol. 19. 1918, pp. 148–156.

[7] Donald E Knuth. “The Art of Computer Programming. Volume 1: Fundamental Al-gorithms. Volume 2: Seminumerical Algorithms”. In: Bull. Amer. Math. Soc (1997).

[8] Erwin Kreyszig. Introductory functional analysis with applications. Vol. 1. Wiley NewYork, 1989.

[9] John H Mathews, Kurtis D Fink et al. Numerical methods using MATLAB. Vol. 4. Pear-son London, UK: 2004.

[10] Mathieu Ossendrijver. “Ancient Babylonian astronomers calculated Jupiter’s positionfrom the area under a time-velocity graph”. In: Science 351.6272 (2016), pp. 482–484.

[11] Walter Rudin et al. Principles of mathematical analysis. Vol. 3. McGraw-hill New York,1964.

[12] Olof Runborg. Verifying Numerical Convergence Rates. 2012. URL: http://www.csc.kth.se/utbildning/kth/kurser/DN2255/ndiff13/ConvRate.pdf.

[13] Vidar Thomée. “From finite differences to finite elements: A short history of numericalanalysis of partial differential equations”. In: Journal of Computational and AppliedMathematics 128.1 (2001), pp. 1–54.

[14] Howard J Wilcox and David L Myers. An introduction to Lebesgue integration andFourier series. Courier Corporation, 2012.

31

Index

big O notation, 14bump function, 6

Cauchy sequence, 5central difference method, 13compactly supported, 6convergence at a point, 10countable set, 11

Dirac delta function, 6distance, 4

forward difference method, 13

indiscernibility, 5infimum, 5integral convergence, 11irrational number, 11isosceles triangle, 6

L1 convergence, 11Lp space, 11log-log plot, 14lower bound, 5

maximum, 5mean value theorem for definite integrals, 7metric space, 4metric, 4minimum, 5monomial, 14

non-negativity, 5

pointwise convergence, 5

rational number, 11

support, 5supremum, 5symmetry, 5

Taylor series, 13topological space, 4trapezoidal rule, 19triangle inequality, 5

uncountable set, 11uniform convergence, 8upper bound, 5

32

Appendix A

MATLAB code

The code for MATLAB scripts constructed to use in this thesis can all be found in this ap-pendix.

A.1 Script 1The first script is used to solve a differential equation in Chapter 3.3 and plot the estimatedconvergence.

1 %% Solver for a differential equation with known solution,2 % central/forward. This version of the solver focuses on3 % estimating the convergence of a4 % problem with a known solution, specifically y' = xy, y(0)=1. It5 % accomplishes the following:6 % 1. Plots the convergence in the conventional, easy to read,7 % logarithmic scale, for both the forward difference and central8 % difference. Analytic calculations indicate that the first method9 % should have the order of convergence one, and the second

10 % order of convergence two and the numerical results confirm it.11 % 2. Checks the empirical order of convergence with polyfit MATLAB12 % function.13 %%14 close all15 clear all16 n = 2;17 max_m = 17;18 m = 1;19 error = zeros(1,max_m);20 error1 = zeros(1,max_m);21

22 while m <= max_m23 n = 2.^m;24 h = 1/n;25 H(m) = h;26 y = zeros(1,n+2);

33

27 x = zeros(1,n+2);28 real = zeros(1,n+2);29 y(1) = 1;30 y(2) = 1;31 x(1) = 1;32 x(2) = y(2);33 real(1) = 1;34 real(2) = y(2);35 for i = 1:n-236 x(i+2) = x(i+1) + h*(i+1)*h*x(i+1);37 y(i+2) = y(i) + 2*y(i+1)*h*h*(i+1);38 real(i+2) = exp((((i+2)*h).^2)/2);39 end40 error1(m) = max(abs(x-real));41 error(m) = max(abs(y-real));42 m = m+1;43 end44 %% Prepare log-log plot in conventional form45 Hflip = fliplr(H);46 errorflip = fliplr(error);47 errorflip1 = fliplr(error1);48 plot(log2(Hflip), log2(errorflip))49 hold on50 plot(log2(Hflip), log2(errorflip1))51 legend('Central' , 'Forward');52 %% Check empirical rates of convergence53 Q = polyfit(log2(Hflip(1:12)), log2(errorflip(1:12)),1)54 R = polyfit(log2(Hflip(1:12)), log2(errorflip1(1:12)),1)55 %% For plotting the solution of the equation, comment out the following:56 % plot(y(1:n))57 % plot(save(1:n))58 % hold on59 % plot(real(1:n))

A.2 Script 2The second script is used to estimate the solution as well as convergence of a problem de-scribed in the second part of the Chapter 3.3.

1 %% This is a solver for differential equations.2 % It uses the central difference method to estimate the solution and3 % convergence of an equation with no known solution.4 % This version tries to estimate the y' = sin(xy)+y, y(0)=1.5 %%6 close all7 clear all8 n = 2;9 max_m = 17;

10 m = 1;

34

11 error1 = zeros(1,n.^(max_m-1));12 error2 = zeros(1,max_m-1);13 save = zeros(n.^(max_m),max_m);14

15

16 while m <= max_m17 n = 2.^m;18 h = 1/n;19 H(m) = h;20 y = zeros(1,n+2);21

22 y(1) = 1;23 % We use the derivative at the first point as the second point, in24 % this case it's 1.25 y(2) = 1+h;% the derivative at x=0 is 1 so y(2) is approx y(1)+h*126 save(1,m) = y(1);27 save(2,m) = y(2);28

29 for i = 1:n30 % is x(i+1)=h*(i+1)31 % when i=1 we calculate x=h32 x_i_plus_1=h*i;33 y(i+2) = y(i) + 2*h*(sin(y(i+1)*x_i_plus_1)+y(i+1));34 save(i+2,m) = y(i+2);35 end36 m = m+1;37 end38

39 m = 1;40 n = 2;41

42 while m <= (max_m - 1)43 n = 2.^m;44 for i = 1:n45 % error1(i) = abs(save(n+1,m)-save((2*n)+1,m+1)); %pointwise46 error1(i)=abs(save(i+1,m)-save((2*i)+1,m+1)); %uniform47 end48 error2(m) = max(error1);49 m = m+1;50 end51 %% Plotting convergence in the conventional log-log form52 Hflip = fliplr(H(1:max_m-1));53 errorflip = fliplr(error2);54 plot(log2(Hflip), log2(errorflip))55 %% Alternatively, plot the solution estimate.56 % plot(y)

35

A.3 Script 3This script is for the Chapter 4.3, where we compute the convergence of error in a setting witha singularity to demonstrate error propagation.

1 %% Error propagation2 % This version attempts to show the problems with having a non-smooth3 % function within the problem. We use (x^2)^(-1/3) as f(x) and solve the4 % DE: y' = f(x), y(0) = 1. We are using the central difference method.5 %%6 close all7 clear all8 n = 2;9 max_m = 17;

10 m = 1;11 save = zeros(n.^(max_m),max_m);12

13 while m <= max_m14 n = 2.^m;15 h = 1/n;16 H(m) = h;17 y = zeros(1,n+2);18 y(1) = 1;19 y(2) = 1+h;20 save(1,m) = y(1);21 save(2,m) = y(2);22

23 for i = 1:n24 x_i_plus_1=h*i;25 y(i+2) = y(i) + 2*h*abs((x_i_plus_1)-(1/sqrt(2)));26 save(i+2,m) = y(i+2);27

28 end29 m = m+1;30 end31

32 m = 1;33 n = 1;34

35 while m <= max_m - 136 n = 2.^m;37 for i = 1:n38 % error1(i) = abs(save(n+1,m)-save((2*n)+1,m+1)); %pointwise39 error1(i)=abs(save(i+1,m)-save((2*i)+1,m+1)); %uniform40 end41 error2(m) = max(error1);42 m = m+1;43 end44

45 Hflip = fliplr(H(1:max_m-1));46 errorflip = fliplr(error2);

36

47 plot(log2(Hflip), log2(errorflip))

A.4 Script 4This script attempts to present a solution for the problem in Chapter 4.2.

1 %% Plotting convergence of error in trapezoidal method.2 % Goal: show that convergence is NOT of order 2 (expected order) when3 % computing for a function that has a cusp and making the grid points4 % on-and off the cusp (big error-small error oscillation).5 % Using built-in trapz function.6 %%7 m = 1;8 max_m = 17;9 error = zeros(1,m);

10 while m <= max_m11 n = 2.^m;12 h = 1/n;13 H(m) = h;14 X = 0:h:1;15 Y = ((X- 1/sqrt(2)).^2).^(1/3);16 value(m) = trapz(X,Y);17 if m >= 218 error(m) = abs(value(m)-value(m-1));19 end20 m = m+1;21 end22

23 %% Plotting the convergence of error in conventional log-log form24 %%25 Hflip = fliplr(H(1:max_m-1));26 errorflip = fliplr(error(2:max_m));27 plot(log2(Hflip), log2(errorflip))

A.5 Script 5This script generates the sequence of triangles used in Chapter 2.3.

1 %% Generating a sequence of isosceles triangles converging pointwise.2 %%3 n = 2;4 max_n = 8;5 while n <= max_n6 X = [0 1/(2*n) 1/n];7 Y = [0 2*n 0];8 plot(X,Y);

37

9 hold on10 n = n+1;11 end

A.6 Script 6This script generates a sequence of triangles used in Chapter 2.6.

1 %% Generate a sequence of isosceles triangles converging in L^1.2 clear all3 n = 1;4 max_n = 5;5 figure6 axis equal7 while n <= max_n8 if mod(n,2) == 09 X = [0-1/n 0 0+1/n];

10 Y = [0 1 0];11 ax(n) = subplot(max_n,1,n);12 plot(X,Y);13 title(['T_' num2str(n)'])14 hold on15 else16 X = [1-1/n 1 1+1/n];17 Y = [0 1 0];18 ax(n) = subplot(max_n,1,n);19 plot(X,Y);20 title(['T_' num2str(n)'])21 hold on22 end23 n = n+1;24 end25

26 linkaxes([ax(1:n-1)],'xy');

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