1. Foundations of Numerics from Advanced Mathematics. Mathematical Essentials and Notation

Transcription

1 1. Foundations of Numerics from Advanced Mathematics Mathematical Essentials and Notation Mathematical Essentials and Notation, October 22,

2 The main purpose of this first chapter (about 4 lectures) is to recall those topics from your Advanced Mathematics courses (linear algebra, calculus, stochastics) typical for the first two years of bachelor s programs in science and engineering that are of particular importance for numerical algorithms and, hence, for the whole CSE master s program. 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 mathematical backgrounds are quite heterogeneous (and not always at hand...); since TUM s CSE program has a methodological (i. e. mathematical and informatical) point of view that goes beyond the usual and widespread engineering approach and way of thinking; and since the two numerics courses have been the most serious roadblock for CSE students since the program s launch (too high failure rates something we want 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 to the numerical contents to be discussed later on in this course. Mathematical Essentials and Notation, October 22,

3 We also changed the name of the courses from Numerical Analysis to Numerical Programming, to indicate that there are mathematical topics to be addressed, but with a 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, it will be a guided tour through important topics, mentioning notions and buzzwords that should have some meaning for you. If they don t, you know that you have to close the gaps as soon as possible, with the help of the references provided or by doing additional exercises etc. Also use the tutorials to refresh your knowledge! Mathematical Essentials and Notation, October 22,

4 1.1. Mathematical Essentials and Notation Symbols and Notions Everyone familiar with the symbol ; the symbols, 1, and (so-called quantifiers); n the symbols and i=1 i k; the notions maximum, minimum, infimum, and supremum; Kronecker symbol δ ij ; the Landau symbol O(N), O(h 2 ) the symbol ; the meaning of sufficient and necessary; the meaning of iff: sufficient and necessary; the meaning of associative, commutative, and distributive? Mathematical Essentials and Notation, October 22,

5 Visualization (Hubble ultra deep field) Mathematical Essentials and Notation, October 22,

6 Visualization? Mathematical Essentials and Notation, October 22,

7 Exercise min, max, inf, sup Do min, max, inf, sup of the following sets exists? Determine if possible. A := { 2, 1, 0, 1, 2} B := {n 2 ; n N} C := { 1 n ; n N} D := { 1 n + 3 m ; n, m N } Solve before reading the solution on the next slide! Mathematical Essentials and Notation, October 22,

8 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 = Mathematical Essentials and Notation, October 22,

9 Exercise Landau Symbol Which of the following terms is O(N) for N? N + 10 log N + 10, 000 N N N N Mathematical Essentials and Notation, October 22,

10 Exercise Landau Symbol Solution N + 10 log N + 10, 000 N = O(N) N N N O(N) Mathematical Essentials and Notation, October 22,

11 Exercise Landau Symbol Which of the following terms is O(h 2 ) for h 0? 10 3 h h + 1, 000 h 20 h h h h 5 Mathematical Essentials and Notation, October 22,

12 Exercise Landau Symbol Solution 10 3 h h + 1, 000 h = O( h) 20 h h h h 5 = O(h 2 ) Mathematical Essentials and Notation, October 22,

13 Visualization neccessary, sufficient & iff Mathematical Essentials and Notation, October 22,

14 Numbers Booleans: true/false; logical operations; relations of logics to set theory (see below) 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 complex coefficients has at least one complex root what else can be said of roots of polynomials? Mathematical Essentials and Notation, October 22,

15 Exercise Booleans and Logical Operations 1 0 =? 1 0 =? 1 0 =? Mathematical Essentials and Notation, October 22,

16 Exercise Booleans and Logical Operations Solution 1 0 = = = 1. Mathematical Essentials and Notation, October 22,

17 Exercise Complex Numbers and Polynomials Solve 2z 2 8z + 9 = 0. Mathematical Essentials and Notation, October 22,

18 Exercise Complex Numbers and Polynomials Solution z 1,2 = 8± = 8± 8 4 = 2 ± 1 2 i. Mathematical Essentials and Notation, October 22,

19 Exercise Complex Numbers Compute Re z, Im z, and z. z = (3 + i)(1 4i) z = 2 i 1+4i z = 4 n=0 in n=96 in Mathematical Essentials and Notation, October 22,

20 Exercise Complex Numbers Solution z = (3 + i)(1 4i) = 7 11i = i; Re z = 7, Im z = 12, z = = 170. z = 2 i 1+4i = (2 i)(1 4i) (1+4i)(1 4i) = 2 9i 17 ; Re z = , Im z = 17, z = 4+81 = z = 4 n=0 in n=96 in = 1 + i 1 i i 1 i + 1 = 2, Re z = 2, Im z = 0, z = 2. Mathematical Essentials and Notation, October 22,

21 ( n Binomial Coefficients k ) & Pascal s Triangle source: und algebra im unterricht/jagusch/index.html.html Mathematical Essentials and Notation, October 22,

22 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, compact sets Mathematical Essentials and Notation, October 22,

23 Visualization Sets Mathematical Essentials and Notation, October 22,

24 Visualization Operations on Sets Mathematical Essentials and Notation, October 22,

25 Visualization Topology 1 Mathematical Essentials and Notation, October 22,

26 Visualization Topology 2 Mathematical Essentials and Notation, October 22,

27 Visualization Topology 3 Mathematical Essentials and Notation, October 22,

28 Visualization Topology 4 Mathematical Essentials and Notation, October 22,

29 Visualization Topology 5 Mathematical Essentials and Notation, October 22,

30 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 Mathematical Essentials and Notation, October 22,

31 Exercise Sets A is only bounded. B is only open. C is only bounded. D is bounded and closed. Mathematical Essentials and Notation, October 22,

32 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 Mathematical Essentials and Notation, October 22,

33 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) Mathematical Essentials and Notation, October 22,

34 Exercise Relations Solution R1 is transitive, reflexive, asymmetric, antisymmetric, and connex. R2 is transitive, reflexive, symmetric, and antisymmetric. R3 is transitive, reflexive, asymmetric, and antisymmetric. Mathematical Essentials and Notation, October 22,

35 Mappings and Functions mapping or function (here used in a synonymous way) write x y properties of mappings: injective surjective bijective f 1 (x) =? f : A B : x A 1 y B such that y = f (x); inverse mapping: existence and meaning Mathematical Essentials and Notation, October 22,

36 Visualization Injectivity Mathematical Essentials and Notation, October 22,

37 Visualization Surjectivity Mathematical Essentials and Notation, October 22,

38 Visualization Bijectivity Mathematical Essentials and Notation, October 22,

39 Exercise Mappings Are the following mappings injective, surjective, bijective? Compute the inverse mapping if it exists! f 1 : R R, x x 2 f 2 : R R + 0, x x 2 f 3 : R 0 R+ 0, x x 2 Mathematical Essentials and Notation, October 22,

40 Exercise Mappings Solution f 1 has none of these properties, the inverse does not exist. f 2 is surjective, but neither injective nor bijective, the inverse does not exist. f 3 is injective, surjective, and bijective, the inverse is f 3 1 : R + 0 R 0, x x. Mathematical Essentials and Notation, October 22,

41 Exercise Mappings Are the following mappings injective, surjective, bijective? Compute the inverse mapping if it exists! g1 : [0; 1] R, x 3x + 2 g2 : [0; 1] [2; 5], x 3x + 2 Mathematical Essentials and Notation, October 22,

42 Exercise Mappings Solution g1 is injective, but neither surjective nor bijective, the inverse does not exist. g2 is injective, surjective, and bijective, the inverse is g2 1 : [2; 5] [0; 1], x x 2 3. Mathematical Essentials and Notation, October 22,

43 Exercise Mappings Are the following mappings injective, surjective, bijective? Compute the inverse mapping if it exists! h :]0; [ R, x 2 ln x 3 1 Mathematical Essentials and Notation, October 22,

44 Exercise Mappings Solution h :]0; [ R, x 2 ln x 3 1 is injective, surjective, and bijective, the inverse is h 1 : R ]0; [, x 3 exp ( ) x+1 2. Mathematical Essentials and Notation, October 22,

45 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 this and that holds, then the following is valid... ) the more restrictions are made, the more can be concluded (but also the less general the statements are) lemma: similar to a theorem w.r.t. its structure, but usually only an auxiliary statement of minor importance by itself (that marks just a step on the way to a theorem, e.g.) corollary: a statement that follows immediately from a previous theorem etc. proof: a precise argumentation showing clearly that a theorem, lemma, or corollary is correct Note that all this is typically formulated as general and generic as possible a fact which is frequently misinterpreted as not concrete or without practical relevance. Mathematical Essentials and Notation, October 22,

46 Building Blocks Visualization Mathematical Essentials and Notation, October 22,

47 A Short Remark on Proofs Why proofs or how much of proofs? Proofs are the essence of mathematical argumentation they make the latter rigorous. Proofs are a permanent source of misunderstandings and problems: math professors often do not want to do anything without proofs even in courses 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 hardcore mathematicians 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) Mathematical Essentials and Notation, October 22,

48 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 find one 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 the conclusion from n to n + 1 (does it work for the smart student example?) Mathematical Essentials and Notation, October 22,

49 Exercise Proofs Proof that all students are smart. Contradiction? Counter Example? Complete Search? Induction? Mathematical Essentials and Notation, October 22,

50 Exercise Proofs Solution Proof that all students are smart. Complete Search! Mathematical Essentials and Notation, October 22,

51 Exercise Proofs Refute that all students are smart. Contradiction? Counter Example? Complete Search? Induction? Mathematical Essentials and Notation, October 22,

52 Exercise Proofs Solution Refute that all students are smart. Counter Example! Mathematical Essentials and Notation, October 22,

53 Exercise Proofs Proof that N q=0 ( 1 2) q < 2 for all N. Contradiction? Counter Example? Induction? Mathematical Essentials and Notation, October 22,

54 Exercise Proofs Solution Proof that N q=0 ( 1 2) q < 2 for all N. Induction! Mathematical Essentials and Notation, October 22,

55 Exercise Proofs Proof that students passing the exam in Numerical Programming are smart. Contradiction? Counter Example? Complete Search? Induction? Mathematical Essentials and Notation, October 22,

56 Exercise Proofs Solution Proof that students passing the exam in Numerical Programming are smart. Contradiction! Mathematical Essentials and Notation, October 22,

57 Exercise Proofs Show by mathematical induction that every natural number n 1 can be represented as a product of prime numbers. Mathematical Essentials and Notation, October 22,

58 Exercise Proofs Solution Show by mathematical induction that every natural number n 2 can be represented as a product of prime numbers. Start n = 2 : trivial Induction Step: Consider any natural number n. n is either a prime number (which is trivial to write as a product of prime numbers) or can be written as a product n 1 n 2 with n 1, n 2 < n. By induction asspunption, n 1 and n 2 can be written as a product of prime numbers. Mathematical Essentials and Notation, October 22,

59 Exercise Proofs Show by mathematical induction the Bernoulli inequality n N : (1 + x) n 1 + nx if x [ 1; [ R. Mathematical Essentials and Notation, October 22,

60 Exercise Proofs Solution Show by mathematical induction the Bernoulli inequality n N : (1 + x) n 1 + nx if x [ 1; [ R. Start n = 1: trivial Induction Step: ind. assump. (1 + x) n = (1 + x)(1 + x) n 1 x x + (n 1)x + (n 1)x nx. (1 + x)(1 + (n 1)x) = Mathematical Essentials and Notation, October 22,