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Interpolation 5-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 5-Numerical Analysis-Lecture Slides and more Slides Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Chapter 5Chapter 5InterpolationInterpolation
r^
r^
r
i^
i^
i
y^
y^
y
^
^
1
1 1
,
, (^
1),..., k^
k^
k
i^
i^
i
y^
y^
y
i^ n
n^
k ^
^
^
^
(1 2)^
n^
n^
n
i^
i^
i
y^
y^
y
^
^
^
^
Thus
(^
)^
( )
x^
x^ h^
x
y^
y^
y^
f^ x^
h^
f^ x
^
^
^
^
^
2 x^
x^ h^
x
y^
y^
y
^
^ ( )
(
)
x^
x^
x^ h y^
y^
y^
f^ x^
f^ x^
h
^
^
^
^
^
(^ / 2)^
(^ / 2)^
2
2
x^ x^
h^
x^ h
h^
h
y^ y
y^
f^ x^
f^ x
^
^
^
^
^
^
^
^
^
^
^
^
Similarly
The inverse operator
-1 E
is defined as
1 ( )
(^
)
E^
f^ x
f^ x
h
^
^
Similarly,
( )
(^
)
n E^
f^ x
f^ x
nh
^
^
Average Operator,
(^ / 2)^
(^ / 2) 1 ( )^
2
2
2
1 2
x^ h^
h^ x h
h
f^ x^
f^ x^
f^ x
y^
y
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Important Results
1 E
1
1 1
E E^
E ^
^
^
1/ 2^
E^
E
^ 1/ 2^
1/ 2
1 (^
) E 2
E
^
^
log hD^
E
Newton’sNewton’sForwardForwardDifferenceDifferenceInterpolationInterpolationFormulaFormula
( )
(^
).
p E f^
x^
f^ x
ph ^
0
0
0
(^
)^
(^ )^
(^
)^ (
)
p^
p
f^ x^
ph^
E^ f^
x^
f^ x
^
^
^ 2
3
0
(^ 1)
(^ 1)(
(^ )
2!^
p p^
p p^
p
p^
f^ x
^
^
^
^ ^
^
^ ^
^
^
docsity.com
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 ^
^
^
^
^
^
^
^
^
^ ^
^
This is known as Newton’sforward difference formula forinterpolation, which gives thevalue of
f^ ( x^0
+^ ph
) in terms of
f^ ( x^0
and its leading differences.
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ExerciseFind a cubic polynomial in
x
which takes on the values-3, 3, 11, 27, 57 and 107,when
x^ = 0, 1, 2, 3, 4 and 5 respectively.
SolutionHere, the observations aregiven at equal intervals of unitwidth.To determine the requiredpolynomial, we first constructthe difference table
Since the 4
th^ and higher
order differences are zero, therequired Newton’sinterpolation formula
2
0
0
0
0
3 0
(^ 1)
(^
)^ (
)^
(^ )^
(^ ) 2
(^ 1)(
(^
) 6
p p
f^ x^
ph^
f^ x^
p f^ x
f^ x
p p^
p^
f^ x
^
^
^ ^
^
^
^
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(^01)
(^ )
(^6) ( )^
2 (^ )
6 x^ x
x
p^
x
h f^ x f^
^ x f x
^
^
^
^
^
Here,