Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 1. Foundations of Numerics from Advanced Mathematics 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 1 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics The main purpose of this first chapter (about 4 weeks) 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. 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 2 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 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! 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 3 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 1.1. Mathematical Essentials and Notation Symbols and Notions Everyone familiar with • the symbol ∞ ; • the symbols ∃ , ∃ 1 , and ∀ (so-called quantifiers ); n X Y • 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 ? 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 4 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 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? 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 5 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 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 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 6 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 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 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 7 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics Mappings and Functions • mapping or function (here used in a synonymous way) f : A → B : ∀ x ∈ A ∃ 1 y ∈ B such that y = f ( x ); write x �→ y • properties of mappings: – injective – surjective – bijective • f − 1 ( x ) = ? • inverse mapping: existence and meaning 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 8 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 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”. 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 9 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 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) 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 10 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 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?) 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 11 of 50
Mathematical Essentials and Notation Linear Algebra Calculus Stochastics and Statistics 1.2. Linear Algebra Mathematical Structures • a mathematical structure consists of one or several sets and one or several operations defined on the set(s) • special elements: – neutral element (of an operation) – inverse element (of some element x ) • a group : a structure to add and subtract • a field : a structure to add, subtract, multiply, and divide • a vector space : a set with additional properties, allows for addition and multiplication with scalars • note: sometimes, the association with classical (geometric) vectors is helpful, sometimes it is more harmful 1. Foundations of Numerics from Advanced Mathematics Numerical Programming I (for CSE), Hans-Joachim Bungartz page 12 of 50
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