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M 277 (60 h)
Discrete Mathematics & Logic
Bibliography
Discrete Mathematics and its applications, Kenneth
H. Rosen
Numerical Analysis, Richard L. Burden, J. Douglas
Faires, Albert C. Reynolds. Prindle, Weber & Schmidt
Boston, Massachusetts.
Applied Numerical Analysis, Curtis F. Gerald, Patrick
O. Wheatley. Addison-wesley publishing company.
Contents
• Algorithms: Introduction, Complexity, Recursive
definitions, Recursive algorithms, program
correctness
• Solve for the roots of a nonlinear equation.
• Solve large systems of linear equations.
• Interpolate to find intermediate values within a table
of data.
• Integrate any function even when known only as a
table of values.
• Solve ordinary differential equations when given
initial values for the variables.
• Cubic Spline interpolation
Introduction
• Numerical analysis is a way to higher mathematics
problems on a computer, a technique widely used by
scientists and engineers to solve their problems.
• A major advantage for numerical analysis is that a
numerical answer can be obtained even when a
problem has no “analytical” solution.
• Actually, evaluating an analytical result to get the
numerical answer (approximation) for a specific
application is subject to same errors.
Using a computer to do numerical
analysis
• Numerical analysis is so important that
extensive commercial software packages are
available (MatLab, MATH / LIBRARY)).
• Many such symbolic algebra programs are
available (Maple, Mathematica,..)
Four general steps
1)
2)
3)
4)
State the problem clearly, including any simplifying
assumptions.
Develop a mathematical statement of the problem in
a form that can be solved for a numerical answer.
Solve the equation(s) that result from step 2.
Interpret the numerical result to arrive at a decision.
This will require experience and an understanding
of the situation…
Introduction
The rules of logic give precise meaning to mathematical
statements. these rules are used to distinguish between
valid and invalid mathematical arguments.
Algorithms
• An algorithm is a definite procedure for solving a
problem using a finite number of steps.
• Example: Describe an algorithm for finding the
largest element in a finite sequence of integers.
Procedure max(a1,a2,…an: integers)
max = a1
For i = 2 to n
if max < ai then max = ai
(max is the largest element)
• Properties of algorithm:
 Input: an algorithm has input values from a specified
set.
 Output: from each set of input values an algorithm
produces out values from a specified set. The output
values comprise the solution to the problem.
 Definiteness: The steps of an algorithm must be
defined precisely.
 Finiteness: An algorithm should produce the desired
output after a finite number of steps for any input in
the set.
 Effectiveness: It must be possible to perform each
step of an algorithm exactly and in a finite amount of
time.
 Generality: The procedure should be applicable for
all problems of the desired form, not just for a
particular set of input values.
• Searching algorithms: Locate an element x in a list of
distinct elements a1,a2,…an, or determine that it is not in
the list.
• First algorithm: Linear search or sequential search.
Procedure linear search(x: integer, a1,a2,…an: distinct integers)
i=1
while ( i ≤ n and x ≠ ai )
i = i +1
if (i ≤ n) then location = i
else location = 0
(location is the subscript of term that equals x, or is 0 if x is not
found)
• Second algorithm: Binary search algorithm.
• This algorithm can be used when the list has terms occurring in
order of increasing size (smallest to largest).
Procedure binary search algorithm(x: integer, a1,a2,…an:distinct
integers)
i=1
j=n
while ( i < j )
begin
m = (i + j)/2
if x > am then i = m +1
else j = m
end
if x = ai then location = i
else location = 0
(location is the subscript of term that equal to x or zero if x is not
found)
• The following example demonstrates how a binary search
works.
• To search for 19 in the list
1 2 3 5 6 7 8 10 12 13 15 16 18 19 20 22
First split this list, which has 16 terms, into two smaller lists
with eight terms each, namely
1 2 3 5 6 7 8 10
12 13 15 16 18 19 20 22
Then, compare 19 and the largest term in the first list. Since
10 < 19, the search for 19 can be restricted to the list
containing the ninth through the sixteenth terms of the
original list.
12 13 15 16
18 19 20 22
18 19
20 22
Since 19 is not greater than the largest term of the first of
these two lists, which is also 19, the search is restricted
to the first list,…
• Devise an algorithm that finds the sum of all the
integers in a list.
• Solution
procedure sum(a1,a2,…,an: integers)
sum = a1
for i = 2 to n
sum = sum + ai
(sum has desired value)
• Describe an algorithm that interchanges the values
of the variables x and y.
• Solution
procedure interchange (x, y: real numbers)
z=x
x=y
y=z
(The minimum number of assignments needed is threes)
• Describe an algorithm that will count the number of
ones in a bit string by examining each bit of the string
to determine whether it is a one bit.
• Procedure ones(a: bit string, a = a1a2…an)
ones = 0
for i = 1 to n
begin
if ai = 1 then
ones = ones +1
end
{ones is the number of ones in the bit strings a}
• Describe an algorithm that, given the binary
expansions of the integers a and b, determine whether
a > b, a < b, or a = b.
• Procedure compare(a, b: positive integers and
a = (anan-1…a1a0)2, b = (bnbn-1…b1b0)2 )
k=n
while ak = bk and k > 0
k = k -1
if ak= bk then print “ a equals b”
if ak > bk then print “ a is greater than b”
if ak < bk then print “ a is less than b”
Complexity of Algorithms
• Time complexity: The time required to solve a
problem (number of operation).
• Space complexity: The computer memory required
to solve a problem.
Speed Computer
Algorithmes
Calculs
Instructions
Operations
Ips
Mips
Instructions per
second
Méga (Million)
Instructions per
second
Ops
operations per second
Flops
(floating operations per
second)
Mops
Mflops
Gops
Gflops
Mega and Giga operations
per second
Example
year
Name
Gflops
1976

CRAY 1
0.166
1981

CYBER 205 de CDC
0.2
1985

CRAY 2
1-2
2001

G4 d'Apple
5.5
Complixity
P ( x)  a0  a1 x  a2 x 2  ...  an x n
Complixity
n(n  1)



C p ( x )  O nadditions  1  2  3  ...  n 
multiplications    O(n 2 )
2



Horner
P ( x)  a0  xa1  xa2  x...  an ...
nadditions et nmultiplications  O(n)
Matrix Multiplication
• Procedure matrix multiplication (Anxn, Bnxn: matrices)
for i = 1 to n
begin
for j = 1 to n
begin
cij = 0
for q = 1 to n
cij  cij  aiq .bqj
end
end {C is the product of A and b}
Complexity:
O(n3)
A and B are nxn matrices
Read: O(n 2 )
n
trace: tr( A)   a O(n)
ii
i 1
Addition : A  BC such that c  a b O(n 2 )
ij ij ij
n
Multiplica tion: A*BC such that c   a *b O(n3)
ij
ik kj
k 1
a 00 a
01
Determinant (Cramer):
 a a a a
a
a
00 11
01 11
10
11
Temps du calcul
• Nk est le nombre total d’opérations a
effectuer pour calculer un déterminant
d’ordre K: Nk = K.Nk-1 + 2K-1
avec N1 = 0 ; N2 = 3 ;
• Si K = 50
Nk = 1064 opérations
• G4 d'Apple (2001) 5.5 Gflops
• 1064 op ≈ 1054 S ≈ 1049 Jours ≈ 1046 ans
Recursive Definitions
• Sometimes it is difficult to define an object explicitly.
However, it may be easy to define this object in terms of
itself. This process is called recursive.
• We can use recursion to define sequences, functions,
and sets.
• The sequence of powers of 2 are given by an = 2n for n= 0,
1, 2, 3,… However, this sequence can be defined by the
first term of the sequence, namely, a0 = 1, and a rule
finding a term of the sequence from the previous one,
namely, an+1 =2an for n = 0, 1, 2,…
Recursively Defined functions
•
We can define a function the set of
nonnegative integers as its domain by
1. Specifying the value of the function at zero,
2. Giving a rule for finding its value at an integer
from its values at smaller integers.
•
Two conditions
P(0) is true
n  N P(n) P(n 1)
Then n  N is true
Recursively Defined functions
•
1.
2.
•
We can define a function the set of nonnegative integers as its
domain by
Specifying the value of the function at zero,
Giving a rule for finding its value at an integer from its values at
smaller integers.
Ex. Suppose that f is defined recursively by:
f(0) = 3
f(n+1) = 2.f(n) + 3
Find f(1), f(2), f(3) and f(4)
Solution:
f(1) = 2f(0) + 3 = 2.3 + 3 = 9
f(2) = 2f(1) + 3 = 2.9 + 3 = 21
f(3) = 2f(2) + 3 = 2.21 + 3 = 45
f(4) = 2f(3) + 3 = 2.45 + 3 = 93
• Ex.: Give an inductive definition of the factorial
function F(n) = n! With F(0) = 1
F(n+1) = (n+1).F(n)
F(5) = 5.F(4) = 5.4.F(3) = 5.4.3.F(2) = 5.4.3.2.F(1)
= 5.4.3.2.1.F(0) = 5.4.3.2.1.1 = 120
• Ex.: Give a recursive definition of an where a is a real
number and n is a nonnegative integer.
The recursive definition contains two parts. First a0 is
specify a0 = 1. Then the rule for finding an+1 = a.an for
n = 1, 2, 3,…is given.
Recursive Algorithm
• An algorithm is called recursive if it solves a problem by
reducing it to an instance of the same problem with
smaller input.
• Give a recursive algorithm for computing an where a is a
real number and n is a nonnegative integer.
procedure power ( a : real number, n : nonnegative integer)
if n  0 then power (a, n)  1
else power (a, n)  a * power (a, n - 1)
A Recurcive Procedure for Factorials.
• Procedure factorial(n: positive integer)
if n = 1 then
factorial = 1
else
factorial(n) = n* factorial(n-1)
An Iterative Procedure for Factorials.
• Procedure iterative factorial(n: positive integer)
x=1
for i = 1 to n
x = i* x
{x is n!}
Fibonacci Numbers
• The Fibonacci numbers, f0, f1, f2,…, are defined by
the equations f0 = 0, f1 = 1 and
fn = fn-1 + fn-2 for n = 2, 3, 4, …
f2 =
f3 =
f4 =
f5 =
f6 =
f1
f2
f3
f4
f5
+
+
+
+
+
f0
f1
f2
f3
f4
=1+0=1
=1+1=2
=2+1=3
=3+2=5
= 5 +3 = 8
A Recurcive Algorithm for Fibonacci Numbers
• Procedure Fibonacci(n: nonnegative integer)
if n = 0 then Fibonacci(0) = 0
else if n = 1 Fibonacci(1) = 1
else Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)
An Iterative Algorithm for computing Fibonacci Numbers
• Procedure iterative fibonacci(n: nonnegative integer)
if n = 0 then y = 0
else
begin
x=0
y=1
for i = 1 to n-1
begin
z=x+y
x=y
y=z
end
end
{ y is the nth Fibonacci number}
A Recurcive Procedure for
S = 1 + 2 + …+(n-1) + n
• Procedure Sum(n: positive integer)
if n = 1 then
Sum = 1
else
Sum(n) = n + Sum(n-1)
An Iterative Algorithm for computing Sum = 1 + 2 + …+(n-1) + n
• Procedure Sum(n: nonnegative integer)
Sum = 0
for i = 1 to n
Sum = Sum + i
end
{ output : Sum }
A Recurcive Procedure for
S = 1 + 3 + 5…+ (n-2) + n
n: odd number
• Procedure Sum(n: odd integer number)
if n = 1 then
Sum = 1
else
Sum(n) = n + Sum(n-2)
Commonly used Terminology for the Complexity of
algorithm
Complexity
Terminology
O(1)
O(log n)
O(n)
O(n log n)
O(nb)
O(n!)
Constant complexity
logarithmic complexity
linear complexity
n log(n) complexity
Polynomial complexity
Factorial complexity