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University of Science and Technology of ORAN
Faculty of Mathematics and Computer Science
Departement of Mathematics
Module : English scienti c
Project title :
Translate the first chapter in English of
scientific computing
Realized by :
•
Berached
Hbib
•
Sifedinne
????
•
Douaifia
Redouane
1
Contents
1 General Introduction :
3
2 Discrete approximation (method of least squares) :
3
1.1 Problem formulation : . . . . . . . . . . . . . . . . . . . . . .
1.1.1 De ni on : . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Linear case : . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Nonlinear case : . . . . . . . . . . . . . . . . . . . . . . . . . .
2
3
3
4
4
1
General Introduction :
T he purpose of the approximation is to replace a given f unction (f ) by another
appropriate f unction (φ) there are two cases :
Contunous approximation :
f ∈ C([a ; b])
Discrete approximation :
f is known only on a set of pairs of values .
1.1
Problem formulation :
In both cases the principale is to look f or a f unction φ(x, c) = φ(x, c0 , c1 , . . . , cp )
such as
E(c)=kf (x) − φ(x, c)k be as small as possible .
1.1.1 De ni on :
T he approximation is said to be linear if it has the f ollowing f orm :
φ(x, c) = c0 φ0 (x) + c1 φ1 (x) + · · · + cp φp (x)
otherwise the approximation φ(x, c) is called : nonlinear .
Examples :
φ(x, c1 , c2 ) = c1 + c2 e2x
φ(x, c1 , c2 ) = sin(c1 x) + ln(c2 )
2
(linear)
(nonlinear)
Discrete approximation (method of least squares) :
Lets
(xi , P
yi = f (xi ))
i = 0, n
Let
φ(x, c) = pi=0 ci φi (x)
φi (x) are given f unctions .
it is clear that φ(x, c) is a linear approximation .
3
2.1
Linear case :
Goal :
F ind ci , i = 0, p such as :
E(c) = kf (x) − φ(x, c)k2
n
X
=
(f (xk − φ(xk , c)))2
k=0
n
X
=
(f (xk −
p
X
ci φi (xk )))2
i=0
k=0
be as small as possible .
E(c) as small as possible ⇒
(F)
∂E(c)
∂c0
=0
∂E(c)
∂c1
=0
j = 0, p
=0
(F) ⇔ AC = B ⇒ C = A−1 B
2.2
=0
a system of (p+1) equations and (p+1) unknowns .
..
.
∂E(c)
∂cp
∂E(c)
∂cj
(A inversible ⇔ |A| =
6 0)
Nonlinear case :
Lets
(xi , yi = f (xi ))
, i = 0, n
(n + 1) given points .
φ(x, c) = φ(x, c0 , c1 , . . . , cp ) nonlinear approximation .
Goal :
F indPc0 , c1 , . . . , cp such as :
E(c) = ni=0 (f (xi − φ(xi , c)))2 be as small as possible .
In this case we assume that : c = c∗ + α .
c∗ : appreciation parameter .
4
α : unknown uncertainty .
∗
φ(x, c) = φ(x, c ) +
p
X
∂φ(xi , c∗ )
∂ci
k=0
p
= φ(x, c∗ ) +
X ∂φ(xi , c∗ )
∂ck
k=0
(ck − c∗k )
(αk )
we put βi = f (xi ) − φ(xi , c)
i = 0, n
Pp ∂φ(xi ,c∗ )
∗
βi = φ(xi , c ) + k=0 ∂ck (αk )
i = 0, n
If we put :
Y =
f (x0 ) − φ(x0 , c∗ )
..
.
β0
β = ...
βn
∗
f (xn ) − φ(xn , c )
A=
∂φ(x0 ,c∗ )
∂c0
...
..
.
...
∂φ(xn ,c∗ )
∂c0
...
α0
α = ...
αn
∂φ(x0 ,c∗ )
∂cp
..
.
∂φ(xn ,c∗ )
∂cp
we nd that β = Y − Aα
or E(c) = β t β
theref ore
E(c) = (Y − Aα)t (Y − Aα)
Goal :
F ind the value of α .
∂E(c)
=0
∂α
⇒ 2At Aα − 2At Y = 0
⇒ At Aα = At Y
E(c) be minimal ⇒
if |At A| =
6 0
we f ind
α = (At A)−1 At Y
5
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