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Interpolation 8-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 8-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 ^
^
^
^
^
^
^
^
^
^ ^
^
NEWTON’SBACKWARDDIFFERENCE INTERPOLATION
FORMULA
The formula is,
2
3 (^
)^ (^
)^
(^ )
(^ 1)^
(^ ) 2! ( 1)(^
2)^
(^ ) 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
The Lagrange Formulafor Interpolation
1 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
^
^
^ ^
^
^
Also
( )^
( )
( )^
(^ )
(^ ) ( ) (^
)^ k^ (^ )
k
k
k^ k
k
k^
k P^ x
P^ x
L^ x
P
x^
x x x^ x
x ^
^
^
^
Finally, the Lagrange’sinterpolation polynomial ofdegree n can be written as
0 0
( )^
( )^
(^ )
(^
)^ (^
( )^ (^
)^
n
k
k^
k^
k
n^
n k^
k^
k^ k
k^
k
x
y x^
f^ x^
f^ x
x^ x^
x
L^ x f
x^
L^ x y
^
^
^
^
^
^
Let us assume that thefunction
y^ =^
f^ ( x ) is known for
several values of
x, (x
, y) , for ii
i=0,1,..n
. The divided differences oforders 0, 1, 2, …,
n^ are now
defined recursively as:
0
0
0
[^
]^
(^
)
y^ x
y x
y
^
The zero-th orderdivided difference
Second order divideddifference
1 2
0 1
0 1
2
2 0 [^ ,^
]^ [
,^
]
[^ ,^
,^ ]^
y^ x^
x^
y x^
x
y x^
x^ x^
x x ^
Generally
(^10) 2
0 1
0 1
1 [^ ,^ , ,^ ] 1 [^ ,^ , ,^ ]
[^ ,^ , ,^ n ]
n
n
n
y x^ x^
x
y x^ x^
x^
y x^ x^
x
x^ x^
^
^
^
^