MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 6
Problem 1 (Steepest descent iteration). When plotted, the co
Composition of Functions
Definition: Let A, B, C be sets and let
f : A B and g : B C be functions. The
Composition of g with f is the function
g f : A C , a 7 g(f (a) .
g f is pronounced g after f .
C
A
f
B
g
g(f (a)
a
f (a)
g
f
Lemma: Let A
B C. Then
a)
Recursive Definitions
Example (Fibonacci Numbers): Define
F0 def
=0
F1 def
= 1 and
Fn+1 def
= Fn1 + Fn
for n = 1, 2, 3, . . . .
It is easy to see by Induction, alternate
form, that Fn is welldefined for n = 0, 1, . . .
Recursive Definitions
Example (Fibon
Partial Orders on Finite S
Partial Orders on Finite Sets
Fix henceforth a nonvoid finite partially
ordered set (A, ). We pronounce a b as
a precedes b.
Definitions: An element m A is minimal
(for the order ) if there is no other element a A preceding it:
Lecture 3
Review: Principle 0, Principle 1, and
Theorem: A set of size n has
n def
n!
=
k!(n k)!
k
subsets of size k. Arrangements, Binomial
Theorem, and the Multinomial Formula:
(a1 + a2 + + ar )n
X
n
k1 k2
=
a1 a2 akr r .
k1 k2 kr
Pki0
i ki=n
Problem
Review
Given a 6= b in our universe Z, there are
two generators g and g of the ideal
(a, b) def
= cfw_xa + yb : x, y Z
of all linear combinations of a, b. Both
of them are common divisors of a, b with
the property that any other common divisor of a, b div
Elementary Logic
Logic is about Statements, also called
Propositions. We shall use the following
understanding of what a statement is:
Informal Definition: A statement or
proposition is a pair (d, m) consisting of
a declaratory sentence d together with a
Elementary Logic
Review Propositions, Logical Connectives:
Conjunction, Disjunction, Implication, Equiv
alence. Truth Tables. Tautologies, Logical Equivalence, and Contradictions. The
Laws of Logic.
The Laws of Logic
P P
Law of double negation
(P Q) P Q d
Elementary Logic
Review Propositions, Logical Connectives:
Conjunction, Disjunction, Implication, Equiv
alence. Truth Tables. Tautologies, Logical Equivalence, and Contradictions. The
Laws of Logic.
Quantifiers
Conider the declarative sentence
x > 0
(O)
S
Set Theory
Set theory is about the membership relation ; x S means x is a member of the
set S. This is another informal definition.
The statement S is a set consists of this
sentence plus a verification scheme that
permits us to decide whether any widget
The Division Algorithm
Recall that the universe is Z.
Definition: For a, b Z we say a divides b
and write a|b if there exists a q Z such
that
b = qa .
In this case we also say that b is a <integer>
multiple of a or that a is a divisor of b.
Some Facts: Fo
Review
Lemma: For A, B, C S
P[A B C] = P[A] + P[B] + P[C]
P[A B] P[A C] P[B C]
+ P[A B C] .
Example: You toss n times a coin whose
Head comes up with probability p.
a) Describe a suitable state space.
2
Example: You toss n times a coin whose
Head comes u
Review
Theorem (Division Algorithm): If a, b Z
with a > 0 then there exist unique integers
q, r such that
b = qa + r
and
0r<a.
Definition: A nonvoid subset I of Z is an
Ideal if for all a, b, z Z
a) a, b I = a+b I and
b) z Z a I = za I.
Examples:
The idea
Probability
Probability is the mathematical model to
analyze experiments physical, chemical, sociological, etc. that dont have a predictable
outcome, for example measuring fluctuating voltages or polling voters.
Examples: You toss a coin; H or T? It
is on
Relations
Recall this
Definition: The cartesian Product of two
sets A, B is the set of all ordered pairs:
A B def
= cfw_(a, b) : a A , b B .
Example: If |A| = m and |B| = n then the
size of A B is
Relations
Recall this
Definition: The cartesian Product of
MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 4
Problem 1 (Norms on Rn ). To be a norm, a functional has to
MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 3
Problem 1 (Gaussian elimination). The computation is long an
MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 2
Problem 1 (Taylor series). When one plots the original funct
Depicting Relations
Review
Exercise: If is a partial order on a set A
then so is its transpose T . It is called
the Reverse Order.
Example: Let A def
= cfw_2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
and define by
a b a|b .
This is a partial order on A.
Depicting
Laws of Set Theory
Let A, B U
A=A
Law of double complement
AB =AB
de Morgans law
AB =AB
de Morgans law
AB =BA
Commutative Law
AB =BA
Commutative Law
A (B C) = (A B) C Associativity
A (B C) = (A B) C Associativity
A A = A, A A = A Idempotent Laws
A = A, A
The Integers - Induction
The Universe shall be the set Z of integers. It contains the natural numbers
N = cfw_0, 1, 2, . . .. We assume that we know
all about addition and multiplication of
integers, and about their order. For instance, for all a, b, c Z
Lecture 2
Review: Principle 0, Principle 1, and
Theorem: A set of size n has
n def
n!
=
k
k!(n k)!
subsets of size k if k n, none if k > n.
Pascals Triangle.
Example: To a conference on Middle East
Peace the Americans (A) send five delegates, the Lebane
Theorem: Let f : A B be a function.
Then A1, A2 A and B1, B2 B
B1 B2
A1 A2
f 1(B1 B2)
f 1(B1 B2)
f (A1 A2)
f (A1 A2)
= f 1(B1) f 1(B2) ;
= f (A1) f (A2) ;
= f 1(B1) f 1(B2) ;
= f 1(B1) f 1(B2) ;
= f (A1) f (A2) . And
= f (A1) f (A2) iff f is 11 .
Theorem:
Relations Revisited
We investigate here relations R : A A
from a set A to itself, Relations on A.
Of particular interest are those that have
one or more of the following properties:
Definition: Let R A A be a relation
on A.
a) R is reflexive if aRa a A.
b
Special Functions
Definition: Let A, B, C U. A function
:AB C
is called a Binary Operation. Of particular interest is the case A = B = C. A
Binary Operation on A is a function
: A A A , (a, a0) 7 a a0 .
Examples: (Z, +), (Z, ), (Z, ), (Q, +), (Q, ),
(Q,
MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 5
Problem 1 (Norms on innite dimensional spaces). To show that