MATH 174Numerical Analysis I
Section BFirst Semester A.Y. 20142015
Mathematics DivisionInstitute of Mathermatical Sciences and PhysicsUniversity of the Philippines Los Ba˜nos
September 12, 2014
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 1 / 11
Chapter Two.
Interpolation of Functions
Suppose that a mathematical model for the spread of an epidemicproduces that following table showing the number of deaths due to thedisease on
t
days after the outbreak.
t
0 10 25 50 100deaths 0 8 4251 20,677 357What if an expert wants to analyze weekly data?
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 2 / 11
What if we want to get the optimal selling price for a commodity (in termsof gross sales) and the most eﬃcient way to gather data for a model is tosell it at varying prices and observing the number of products sold?Suppose the data gathered is as follows:Price 70 82.5 137.5 187.50Number of Units Sold 1,250 750 550 272.50How will you ﬁnd the optimal price?
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 3 / 11
A common mathematical problem
Given a set of points
(
x
i
,f
(
x
i
))
for
i
= 0
,
1
,
2
,...,n
where the nodes
x
i
’sare distinct values of the independent variable. Then eitherapproximate the value of
f
at some value of
x
not included in the list;ordetermine a function
g
that mimics the behavior of the data in somesense.These problems give rise to a two diﬀerent areas of study:
Interpolation and Approximation
.
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 4 / 11
Interpolation vs Approximation
Interpolationthe function
g
is determined by requiring zero errors at the nodes, i.e.,
f
(
x
i
)
−
g
(
x
i
) = 0
,
∀
i
= 0
,
1
,...,n
Approximationthe function
g
is chosen such that some measure of error is minimized, forexample
n
i
=0
(
f
(
x
i
)
−
g
(
x
i
))
2
< .
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 5 / 11
Consider the function
f
(
x
) =
e
x
.Using the nodes
−
2
,
−
1
,
0
,
1
, the interpolating polynomial is
P
(
x
) =
−
x
3
+ 3
e
x
2
+
x
−
2
x
+
e
3
x
2
+ 3
x
+ 2
x
6
e
2
+
−
3
e
2
x
3
+ 2
x
2
−
x
−
2
+
x
6
e
2
.
While an approximating polynomial for
f
is
Q
(
x
) = 1 +
x
+
x
2
2! +
x
3
3!
.
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 6 / 11
Interpolation, a special case of Approximation
Why focus on interpolation?Common forms of interpolationpolynomialpiecewise polynomialrationaltrigonometricexponential
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 7 / 11
Why polynomials?
they are the ’simplest functions’ practical for computerstheir derivatives and integrals easy to compute and are stillpolynomialsfor any required accuracy, there is a polynomial approximating
f
Theorem
Weierstrass (First) Approximation Theorem.Let
f
∈ C
[
a,b
]
. Then
∀
>
0
there is a polynomial
P
(
x
)
such that

f
(
x
)
−
p
(
x
)

< ,
∀
x
∈
[
a,b
]
.
there is one and only one interpolating polynomial for aset of points
(
x
i
,f
(
x
i
))
for
i
= 0
,
1
,
2
,...,n
.
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 8 / 11
Algebraic Interpolating Polynomial
A polynomial
P
n
(
x
)
≡
P
n
(
x,f,x
1
,x
2
,...,x
n
+1
)
of degree no greaterthat
n
that has the form
P
n
(
x
) =
c
0
+
c
1
x
+
...
+
c
n
x
n
and coincides with the data points
(
x
i
,f
(
x
i
))
for
i
= 1
,
2
,...,n,n
+ 1
is called the algebraic interpolating polynomial of
f
.
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 9 / 11
Unisolvence TheoremTheorem
There is a unique algebraic interpolating polynomial
P
n
(
x,f,x
1
,x
2
,...,x
n
+1
)
.
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 10 / 11
Ngayon pa lang...The accuracy of the interpolating polynomial isguaranteed
ONLY FOR ABSCISSAS between theleast and greatest nodes.
NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 11 / 11