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Interpolation 6-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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Chapter 5Chapter 5InterpolationInterpolation
r^
r^
r
i^
i^
i
y^
y^
y
ļ^
ļ^
ļ½ ļ^
ļ ļ 1
1 ,^1
k^ k^ , (^ 1),...,
k i^
i^
i
y^
y^
y
i^ n^
n^
k ļ^
ļ ļ
ļ^
ļ½ ļ^
ļ ļ ļ½^
ļ^1
1 (1 2)^
(1 2)
n^
n^
n
i^
i^
i
y^
y^
y
ļ¤^
ļ¤
ļ¤ ļ^
ļ ļ«^
ļ
ļ½^
ļ
Thus
(^
)^ ( )
x^ x
h^
x y^ y
y
f^
x^ h^
f^ x
ļ« ļ^ ļ½
ļ
ļ½^
ļ«^ ļ
2 x^
x^ h^
x
y^
y^
y
ļ^
ļ½ ļ^
ļ ļ^ ( )^
(^ )
x^ x^
x^ h y^ y
y^
f^ x^
f^ x^ h ļ ļ^ ļ½
ļ^
ļ½^
ļ^
ļ
(^ / 2)^
(^ / 2)^
x^ x^ h
x^
h
h^
h
y^ y^
y^
f^ x^
f^ x
ļ¤^
ļ«^
ļ
ļ¦^ ļ¶
ļ¦^
ļ½^
ļ^
ļ½^ ļ«
ļ^
ļ§^ ļ·
ļ§^
ļØ^ ļø
ļØ^
Similarly
The inverse operator
-1 E
is defined as
1 ( )
(^
)
E^ f
x^
f^ x^
h
ļ^
ļ½^
ļ
Similarly,
( )^
(^
)
n E f^
x^
f^ x
nh
ļ^
ļ½^
ļ
Average Operator,
(^ / 2)^
(^ / 2) 1 ( )^
2
2
2
1 x^2
h^
h^ x h
h
f^ x^
f^ x^
f^ x
y^
y
ļ«^
ļ ļ©^
ļ¹
ļ¦^
ļ¶^ ļ¦
ļ¶
ļ½^
ļ«^ ļ«
ļ ļ§^
ļ·^ ļ§
ļ·
ļŖ^
ļŗ
ļØ^
ļø^ ļØ
ļø
ļ«^
ļ»
ļ©^
ļ¹
ļ½^
ļ« ļ«^
ļ»
Important Results
(^1) E ļ ļ½^
ļ^
1
1 1
Eļ ļ (^) E E ļ ļ½^ ļ^
ļ½
1/ 2^
1/ 2 E^
E ļ¤^
ļ ļ½^
ļ^ 1/ 2^
1/ 2 1 (^
) E^2
E ļ^
ļ ļ½^
ļ«
log hD^
E ļ½
NewtonāsNewtonāsForwardForwardDifferenceDifferenceInterpolationInterpolationFormulaFormula
2
0
0
0 3 0
0 (^ 1) 2! (^ 1)(
- 3! (^ 1)^
(^
1)^
Error ! x
n p^ p y^ y^
p^ y^
y
p^ p^
p^
y p^ p^
p^ n^
y n
ļ ļ½^ ļ«^
ļ^ ļ«^
ļ ļ^ ļļ«
ļ^ ļ« ļ^
ļ^ ļ« ļ«^
ļ ļ^ ļ« ļ
This is also known as Newton-Gregory forward differenceinterpolation formula.Here p=(x-x
)/h. 0
An alternate expression is
NEWTONāSBACKWARDDIFFERENCE INTERPOLATION
FORMULA
Alternatively, this formula can alsobe written as
2 (^ 1) 2!^3 (^ 1)(^
- 3! (^ 1)(^
2)^ (^
1)^
Error ! x^ n^
n^
n n
n n p p y^ y^
p^ y^
y
p p^
p^
y p p^
p^
p^ n^
y ļ« n ļ½^ ļ«^ ļ^ ļ«^
ļ ļ«^ ļ«ļ«
ļ^ ļ« ļ«^ ļ«^
ļ«^ ļ ļ«^
ļ ļ^ ļ« ļ
x^ x^ ļ^ np ļ½ h
Here
LAGRANGEāSINTERPOLATIONFORMULA
If the values of theindependent variable arenot given at equidistantintervals, then we have thebasic formula associatedwith the name of Lagrangewhich will be derived now.
Let^ y
=^ f^ ( x
) be a function
which takes the values, y, y^0
,ā¦yn^
corresponding to x^ , x^0
, ā¦x^ n
. Since there are ( n^ + 1) values of
y
corresponding to (
n^ + 1)
values of
x , we can represent the function
f^ ( x ) by a polynomial of degree
n****.