Interpolation and Lagrange Polynomials - (3.1)
1. Polynomial Interpolation:
Problem:
Given
n
±
1 pairs of data points
x
i
,
y
i
,
i
²
0,1,...,
n
, we want to find a polynomial
P
k
x
of
lowest
possible degree for which
P
k
x
i
²
y
i
,
i
²
0,1,...,
n
.
The polynomial
P
k
x
is said to
interpolate
the data
x
i
,
y
i
,
i
²
0,1,...,
n
and is called an
interpolating
polynomial.
Graphically,
P
k
x
is an
approximation
to
f
x
and
satisfies the conditions
:
P
k
x
i
²
f
x
i
,
i
²
0,1,...,
n
.
-10
-5
0
5
10
-3
-2
-1
1
2
3
x
—
y
²
f
x
,
-.-.-
y
²
P
k
x
Obviously, for this example
P
k
x
is not a good approximation to
f
x
though
P
k
x
satisfies the conditions:
P
k
x
i
²
y
i
for
i
²
0,1,2.
Note that the differences between a
k
th degree
Taylor polynomial
and a
k
th degree
interpolating polynomial
are:
a.
P
Taylor
x
²
f
x
at only
x
²
x
0
and
P
interpolating
x
²
f
x
, at
x
²
x
0
,
x
1
, ...,
x
n
.
b.
P
Taylor
requires knowledge of
f
U
,
f
UU
, ... but
P
interpolating
requires
f
x
0
,
f
x
1
, ...,
f
x
n
.
2. Lagrange Interpolating Polynomials:
a. Lagrange Polynomials:
For
k
²
0,1,...,
n
, define
L
n
,
k
x
²
±
i
²
0,
i
p
k
n
x
"
x
i
x
k
"
x
i
²
x
"
x
0
...
x
"
x
k
"
1
x
"
x
k
±
1
...
x
"
x
n
x
k
"
x
0
...
x
k
"
x
k
"
1
x
k
"
x
k
±
1
...
x
k
"
x
n
.
L
n
,
k
x
are called
Lagrange polynomials
. For example, let
x
i
²
i
,
i
²
0,1,2,3.
L
3,1
x
²
x
x
"
2
x
"
3
1
"
0
1
"
2
1
"
3
²
1
2
x
x
"
2
x
"
3
Observe that Lagrange Polynomials have the following
properties:
i.
L
n
,
k
x
k
²
1
L
n
,
k
x
k
²
x
k
"
x
0
...
x
k
"
x
k
"
1
x
k
"
x
k
±
1
...
x
k
"
x
n
x
k
"
x
0
...
x
k
"
x
k
"
1
x
k
"
x
k
±
1
...
x
k
"
x
n
²
1
ii.
L
n
,
k
x
i
²
0 for
i
p
k
.
L
n
,
k
x
i
²
x
i
"
x
0
...
x
i
"
x
i
...
x
i
"
x
k
"
1
x
i
"
x
k
±
1
...
x
i
"
x
n
x
k
"
x
0
...
x
k
"
x
i
...
x
k
"
x
k
"
1
x
k
"
x
k
±
1
...
x
k
"
x
n
²
0
1