March 2014
Intermediate to advanced
412 pages
11h 16m
English
Content preview from Numerical Methods and Optimization
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118 Numerical Methods and Optimization: An Introduction
5.2.2 The method of undetermined coefficients
Alternatively, we could use the power form for the interpolating polyno-
mial,
p(x)=a
0
+ a
1
x + a
2
x
2
+ ···+ a
n
x
n
,
and use the fact that p(x
i
)=y
i
,i=0,...,n, to find its coefficients by solving
the following linear system for a
0
,...,a
n
:
a
0
+ a
1
x
0
+ a
2
x
2
0
+ ···+ a
n
x
n
0
= y
0
a
0
+ a
1
x
1
+ a
2
x
2
1
+ ···+ a
n
x
n
1
= y
1
.
.
.
a
0
+ a
1
x
n
+ a
2
x
2
n
+ ···+ a
n
x
n
n
= y
n
.
The same system in the matrix form is given by Va= y,where
V =
⎡
⎢
⎢
⎢
⎣
1 x
0
x
2
0
··· x
n
0
1 x
1
x
2
1
··· x
n
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1 x
n
x
2
n
··· x
n
n
⎤
⎥
⎥
⎥
⎦
,a=
⎡
⎢
⎢
⎢
⎣
a
0
a
1
.
.
.
a
n
⎤
⎥
⎥
⎥
⎦
,y=
⎡
⎢
⎢
⎢
⎣
y
0
y
1
.
.
.
y
n
⎤
⎥
⎥
⎥
⎦
.
The matrix V is a Vandermonde matrix and is known to ha
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