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Interpolation 7-Numerical Analysis-Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization
This course contains solution of non linear equations and linear system of equations, approximation of eigen values, interpolation and polynomial approximation, numerical differentiation, integration, numerical solution of ordinary differential equations. This lecture includes: Interpolation, Finite, Difference, Operators, Newton, Forward, Interpolation, Backward, Lagrange, Cubic, Spline
Typology: Slides
2011/2012
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Download Interpolation 7-Numerical Analysis-Lecture Slides and more Slides Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Chapter 5Chapter 5InterpolationInterpolation
Newton’sNewton’sForwardForwardDifferenceDifferenceInterpolationInterpolationFormulaFormula
0
0
0 2
3
0
0 0
(^
)^ (^
)^
(^ )
(^ 1)^
(^ 1)(
(^ )^
(^ )
2!^
(^ 1)^
(^
1)^
(^ )^ Error !
n
f^ x^
ph^ f
x^
p^ f^ x
p p^
p p^
p f^ x^
f^ x
p p^
p^ n^
f^ x n ^
^
^
^
^
^
^
^
^
^ ^
^
The Newton’s forward differenceformula for interpolation, whichgives the value of
f^ ( x^0
+^ ph
) in
terms of
f^ ( x^0
) and its leading
differences.
NEWTON’SBACKWARDDIFFERENCE INTERPOLATION
FORMULA
The formula is,
2
3
(^
)^ (
)^
(^ )
(^ 1)
3! ( 1)(^
2)^ (
1)^
(^ )^
Error
n^
n^
n n
n
n n
f^ x^
ph^
f^ x^
p^ f^ x
p p^
f^ x
p p^
p^
f^ x
p p^
p^
p^ n^
f^ x
n
^
^
^
^
^
^
^
^
^
^
^
LAGRANGE’SINTERPOLATIONFORMULA
Newton’s interpolationformulae can be used onlywhen the values of theindependent variable
x^ are
equally spaced. Also thedifferences of
y^ must
ultimately become small.
Here the polynomial is of theform
( )^
n^
n
n
f^ x^
A x^
A x^
A
^
^
^
or in the form
0
1
2
1
0
2 2
0
1 0
1
1
( )^
(^
)(^
)^ (^
)
(^
)(^
)^ (^
)
(^
)(^
)^ (^
)
(^
)(^
)^ (^
)
n n n
n^
n
y^ f^
x^ a^
x^ x^
x^ x^
x^ x
a^ x^
x^ x^
x^
x^ x
a^ x^
x^ x^
x^
x^ x
a^ x^
x^ x^
x^
x^ x^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Here, the coefficients
a arek^
so chosen as to satisfy thisequation by the (
n^ + 1) pairs
( x ,^ y i^
). Thus we geti (^) 0 0 0
0 1
0 2
0
(^ )^
(^
)(^
)^ (^
)n
y^ f x^
a^ x^
x^ x^
x^
x^ x
^
^
^
^
0
0
0
1 0
2
0
(^
)(^
)^ (
)n
y
a^
x^
x^ x
x
x^
x
^
^
^
Therefore,
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Substituting the values of a ,^ a^0
, …, 1
a n^
we get 1 2
0 2 0
1
0 1 0
2 0
1 0 1
2 1
(^ )(^
)^ (^
)^ (^
)(^ )^
(^ )
( )^ (^
)(^ )^
(^ )^
(^ )(^
)^ (^
)
n^
n
n^
n
x^ x^ x^
x^ x^
x^
x^ x^ x^
x^ x^
x
y^ f^ x^
y^
y
x^ x^ x^
x^ x^
x^ x
x^ x^
x^ x^
x
^ ^
^
^ ^
^ ^
^
^
^
^ ^
^
^
^
0 1
1
1
0 1
1
1
(^ )(
)^
(^ )(
)^
(^ )
(^ )(
)^
(^
)(^
)^ (^
) i^
i^
n i
i^
i^
i^ i^
i^ i^
i^ n
x^ x^ x
x^
x^ x^
x^ x^
x^ x^
y
x^ x^
x^ x^
x^ x^
x^ x^
x^ x ^
^
^
^
^
^
^
^
^
^
^
^
^
0
1
2
1
0
1
2
1
(^
)(^
)(^
)^ (^
(^
)(^
)(^
)^ (^
n n )
n^
n^
n^
n^ n
x^ x^
x^ x^
x^ x^
x^ x^
y
x^ x^
x^ x^
x^ x^
x x
^
^
^
^
^
^
The Lagrange’s formula forinterpolation
This formula can be usedwhether the values
x ,^0
x , …,^2
x n
are equally spaced or not.Alternatively, this can also bewritten in compact form as
0 0
1 1 ( )^
( )^
( )^
( )^
( ) i^ i^
n^ n
y^ f^ x
L^ x y
L^ x y
L^ x y
L^ x y
^
^
^
^
^
n ( )k^0
k L^ k x y ^ ^
( ) 0 (^ ) n k^
k L^ k x f^ x ^ ^
Further, if we introduce thenotation
0
1
0 ( )^
(^
)^ (^
)(^
)^ (^
)
n
i^
n
x^ x^ i
x^
x^ x^
x^ x^
x^ x
^
^
^ ^
^
That is
is a product of
( n^ + 1) factors. Clearly, itsderivative
contains a sum
of ( n
+ 1) terms in each of which one of the factors ofwill^
( )x ^ ( )x ( )x be absent.
We also define,^ ( )
(^
)
k^
i i^ k P^ x
x^
x ^
which is same as
except
that
the
factor
( x
- x )k^
is
absent. Then
( )x
0
1
( )^
( )^
( )^
( )n
x^
P^ x^
P x^
P^ x
^
^
But, when
x^ =
x , all terms in thek
above sum vanishes except
P^ (xk )k docsity.com