1. foundations of numerics from advanced mathematics · tu munc¨ hen we also changed the name of...

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1. Foundations of Numerics from Advanced Mathematics

Mathematical Essentials and Notation

Miriam Mehl: 1. Foundations of Numerics from Advanced Mathematics

Mathematical Essentials and Notation, October 22, 2012 1

http://www5.in.tum.de/

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The main purpose of this first chapter (about 4 lectures) is to recall those topics fromyour Advanced Mathematics courses (linear algebra, calculus, stochastics) typical forthe first two years of bachelor’s programs in science and engineering that are ofparticular importance for numerical algorithms and, hence, for the whole CSE master’sprogram.

We do this,• since you can hardly go successfully through a thorough numerical education

without these foundations;

• since we made the experience that the CSE freshmen’s mathematicalbackgrounds are quite heterogeneous (and not always at hand ...);

• since TUM’s CSE program has a methodological (i. e. mathematical andinformatical) point of view that goes beyond the usual and widespread engineeringapproach and way of thinking;

• and since the two numerics courses have been the most serious roadblock forCSE students since the program’s launch (too high failure rates – something wewant to reduce without touching the level of education).

If you are familiar with all this stuff, don’t feel bored – just consider this as a warm-up tothe numerical contents to be discussed later on in this course.




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We also changed the name of the courses from “Numerical Analysis” to “NumericalProgramming”, to indicate that there are mathematical topics to be addressed, but witha clear focus on algorithmics, programming, and applications (instead of proofs etc.).

This introductory part won’t be a complete lecture with all explanations etc. Rather, itwill be a “guided tour” through important topics, mentioning notions and buzzwords thatshould have some meaning for you. If they don’t, you know that you have to close thegaps as soon as possible, with the help of the references provided or by doingadditional exercises etc.

Also use the tutorials to refresh your knowledge!




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1.1. Mathematical Essentials and Notation

Symbols and Notions

Everyone familiar with• the symbol∞;

• the symbols ∃, ∃1, and ∀ (so-called quantifiers);

• the symbolsn∑

i=1

and∏i 6=k

;

• the notions maximum, minimum, infimum, and supremum;

• Kronecker symbol δij ;

• the Landau symbol O(N), O(h2)

• the symbol⇒;

• the meaning of sufficient and necessary;

• the meaning of iff: sufficient and necessary;

• the meaning of associative, commutative, and distributive?




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Visualization∞

(Hubble ultra deep field)




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Visualization ∃

∃ ?




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Exercise min, max, inf, sup

Do min, max, inf, sup of the following sets exists? Determine ifpossible.

A := {−2,−1,0,1,2}

B := {n2;n ∈ N}

C :={1

n ;n ∈ N}

D :={1

n + 3−m;n,m ∈ N}

Solve before reading the solution on the next slide!




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Exercise min, mac, inf, sup – Solution

inf A = min A = −2; sup A = max A = 2.

inf B = min B = 1; no sup no max.

no min; inf C = 0; max C = sup C = 1.

no min; inf D = 0; max D = sup D = 113 .




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Exercise Landau Symbol

Which of the following terms is O(N) for N →∞?

N + 10 · log N + 10,000 ·√

N

N + 10−2 · N2 + 10−4 · N4 + . . .




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Exercise Landau Symbol – Solution

N + 10 · log N + 10,000 ·√

N = O(N)

N + 10−2 · N2 + 10−4 · N4 + . . . 6= O(N)




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Exercise Landau Symbol

Which of the following terms is O(h2) for h→ 0?

10−3 · h2 + 100 · h + 1,000 ·√

h

20 · h2 + 0.1 · h3 + 106 · h4 + 108 · h5




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Exercise Landau Symbol – Solution

10−3 · h2 + 100 · h + 1,000 ·√

h = O(√

h)

20 · h2 + 0.1 · h3 + 106 · h4 + 108 · h5 = O(h2)




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Visualization neccessary, sufficient & iff




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Numbers

• Booleans: true/false; logical operations; relations of logics to set theory (seebelow)

• natural numbers, integers N,Z: factorials; binomial coefficients; Pascal’s triangle

• rational numbers Q: countable/non-countable

• real numbers R:

– field property (allows for arithmetic operations)– order property (allows for comparison)– completeness property (each interval nesting defines exactly one real

number)– supremum/infimum property

• Q is dense in R• different classes of irrational numbers:

√2, e, ...

• complex numbers C: imaginary unit i , Re, and Im; conjugate complex

• fundamental theorem of algebra: each polynomial of degree n with complexcoefficients has at least one complex root

• what else can be said of roots of polynomials?




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Exercise Booleans and Logical Operations

1 ∨ 0 = ?

1 ∧ 0 = ?

1 ∧ ¬0 = ?




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Exercise Booleans and Logical Operations – Solution

1 ∨ 0 = 1.

1 ∧ 0 = 0.

1 ∧ ¬0 = 1.




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Exercise Complex Numbers and Polynomials

Solve 2z2 − 8z + 9 = 0.




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Exercise Complex Numbers and Polynomials – Solution

z1,2 = 8±√

64−4·2·94 = 8±

√−8

4 = 2± 1√2i .




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Exercise Complex Numbers

Compute Re z, Im z, and |z|.

z = (3 + i)(1− 4i)

z = 2−i1+4i

z =∑4

n=0 in +∑100

n=96 in




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Exercise Complex Numbers – Solution

z = (3 + i)(1− 4i) = 7− 11i = 7 + 12i ;

Re z = 7, Im z = 12, |z| =√

72 + 112 =√

170.

z = 2−i1+4i =

(2−i)(1−4i)(1+4i)(1−4i) =

−2−9i17 ;

Re z = −217 , Im z = −9

17 , |z| =√

4+81172 = 1

17

√85.

z =∑4

n=0 in +∑100

n=96 in =

1 + i − 1− i + 1 + 1 + i − 1− i + 1 = 2, Re z = 2, Imz = 0, |z| = 2.




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Binomial Coefficients(

nk

)& Pascal’s Triangle

source: http://did.mat.uni-

bayreuth.de/studium/veranstaltungen/wintersemester/19992000/arithmetik und algebra im unterricht/jagusch/index.html.html




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Sets

• notions of sets, subsets, and elements

• set operations: union, intersection, difference, complement

• symbols ∈, ⊂, ⊆

• power set

• Cartesian product of sets

• appearances:

– explicit {1, 2, 3, ...}– implicit {x ∈ R : f (x) = 0}

• already here a bit of topology: open sets, closed sets, bounded sets, compactsets




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Visualization Sets




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Visualization Operations on Sets




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Visualization Topology 1




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Exercise Sets

Are the following sets open, bounded, closed, compact?

A := [0;1[

B :=]0;∞[

C :={1

n ;n ∈ N}

D :={1

n ;n ∈ N}∪ 0




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Exercise Sets

A is only bounded.

B is only open.

C is only bounded.

D is bounded and closed.




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Relations

• definition: relation R between two sets A and B as a subset of A× B: R ⊆ A× B

• notation: aRb or (a, b) ∈ R

• important examples for A = B: <, ≤, >, ≥, ...

• properties of relations:

– transitive– reflexive– symmetric– asymmetric– antisymmetric– connex

• notion of an equivalence relation




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Exercise Relations

Are the following relations transitive, reflexive, symmetric,asymmetric, antisymmetric, connex?

R1 := {(a;b);a ≤ b,a,b ∈ R}

R2 := {(a;b);a = b,a,b ∈ R}

R3 := {(a;b);a|b,a,b ∈ N} (a divides b)




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Exercise Relations – Solution

R1 is transitive, reflexive, asymmetric, antisymmetric, andconnex.

R2 is transitive, reflexive, symmetric, and antisymmetric.

R3 is transitive, reflexive, asymmetric, and antisymmetric.




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Mappings and Functions

• mapping or function (here used in a synonymous way)

f : A→ B : ∀x ∈ A ∃1y ∈ B such that y = f (x);

write x 7→ y

• properties of mappings:

– injective– surjective– bijective

• f−1(x) = ?

• inverse mapping: existence and meaning




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Visualization Injectivity




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Visualization Surjectivity




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Visualization Bijectivity




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Exercise Mappings

Are the following mappings injective, surjective, bijective?Compute the inverse mapping if it exists!

f1 : R→ R, x 7−→ x2

f2 : R→ R+0 , x 7−→ x2

f3 : R−0 → R+0 , x 7−→ x2




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Exercise Mappings – Solution

f1 has none of these properties, the inverse does not exist.

f2 is surjective, but neither injective nor bijective, the inversedoes not exist.

f3 is injective, surjective, and bijective, the inverse is

f3−1 : R+0 → R−0 , x 7−→ −

√x .




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Exercise Mappings


g1 : [0;1]→ R, x 7−→ 3x + 2

g2 : [0;1]→ [2;5], x 7−→ 3x + 2




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g1 is injective, but neither surjective nor bijective, the inversedoes not exist.

g2 is injective, surjective, and bijective, the inverse is

g2−1 : [2;5]→ [0;1], x 7−→ x−23 .




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Exercise Mappings


h :]0;∞[→ R, x 7−→ 2 ln x3 − 1




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h :]0;∞[→ R, x 7−→ 2 ln x3 − 1 is injective, surjective, and

bijective, the inverse is

h−1 : R→]0;∞[, x 7−→ 3 exp( x+1

2

).




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Building Blocks ...

... of a math course / book / presentation:

• definition: new notions etc. are defined and, thus, introduced

• theorem / proposition:

– a central statement, typically consisting of conditions and a conclusion (“if thisand that holds, then the following is valid ...”)

– the more restrictions are made, the more can be concluded (but also the lessgeneral the statements are)

• lemma: similar to a theorem w.r.t. its structure, but usually only an auxiliarystatement of minor importance by itself (that marks just a step on the way to atheorem, e.g.)

• corollary: a statement that follows immediately from a previous theorem etc.

• proof: a precise argumentation showing clearly that a theorem, lemma, orcorollary is correct

Note that all this is typically formulated as general and generic as possible – a factwhich is frequently misinterpreted as “not concrete” or “without practical relevance”.




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Building Blocks Visualization




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A Short Remark on Proofs

Why proofs – or how much of proofs?

• Proofs are the essence of mathematical argumentation – they make the latterrigorous.

• Proofs are a permanent source of misunderstandings and problems:

– math professors often do not want to do anything without proofs – even incourses for non-mathematicians

– non-math students often think that only the results or statements are relevant,but not the proofs (which they suppose to be something for hardcoremathematicians only)

– note that both points of view are problematic

• hence: proofs for non-mathematicians (such as CSE students)?

– yes, if the way of proving a statement helps to understand it– no, if just for itself (i.e. just to prove it)




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Standard Proof Techniques

• forward: A⇒ B ⇒ C ⇒ D

• by contradiction (“what if”): ¬D ⇒ ...⇒ ¬A

• by counterexample: to refute the assertion that all students are smart, just findone stupid and the job is done

• complete search: to prove that all students are smart, check them all

• mathematical / complete induction: show the statement for n = 1, and show theconclusion from n to n + 1 (does it work for the smart student example?)




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Exercise Proofs

Proof that all students are smart.

Contradiction? Counter Example? Complete Search?Induction?




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Exercise Proofs – Solution

Proof that all students are smart.

Complete Search!




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Exercise Proofs

Refute that all students are smart.





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Refute that all students are smart.

Counter Example!




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Exercise Proofs

Proof that∑N

q=0(1

2

)q< 2 for all N.

Contradiction? Counter Example? Induction?




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Proof that∑N

q=0(1

2

)q< 2 for all N.

Induction!




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Exercise Proofs

Proof that students passing the exam in NumericalProgramming are smart.





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Proof that students passing the exam in NumericalProgramming are smart.

Contradiction!




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Exercise Proofs

Show by mathematical induction that every natural numbern ≥ 1 can be represented as a product of prime numbers.




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Show by mathematical induction that every natural numbern ≥ 2 can be represented as a product of prime numbers.

Start n = 2 : trivial

Induction Step: Consider any natural number n. n is either aprime number (which is trivial to write as a product of primenumbers) or can be written as a product n1 · n2 with n1,n2 < n.By induction asspunption, n1 and n2 can be written as aproduct of prime numbers. �




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Exercise Proofs

Show by mathematical induction the Bernoulli inequality

∀n ∈ N : (1 + x)n ≥ 1 + nx

if x ∈ [−1;∞[⊂ R.




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Show by mathematical induction the Bernoulli inequality

∀n ∈ N : (1 + x)n ≥ 1 + nx if x ∈ [−1;∞[⊂ R.

Start n = 1: trivial

Induction Step:

(1 + x)n = (1 + x)(1 + x)n−1ind. assump.

≥x ≥ −1

(1 + x)(1 + (n − 1)x) =

1 + x + (n − 1)x + (n − 1)x2 ≥ 1 + nx . �




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