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Interpolation 9-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 9-Numerical Analysis-Lecture Slides and more Slides Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Chapter 5Chapter 5InterpolationInterpolation
DIVIDEDDIFFERENCES
0
0 0 [^ ]^
(^ ) y^ x^
y x^
y ^
The zero-th orderdivided difference
Second order divideddifference
1 2 0
1 0 1 2 [^ ,^ ]^ [^ ,^ ]^2 y [ , , ] x^ x^ y x^
x y x^ x^ x^
x x
Generally
(^1 ) 0 1
[^ ,^ ,^ ,^ ] 1 0 1 1
[^ ,^ ,^ ,^ ]^
y x^ x^ x^ n n [^ ,^ ,^ ,^ ] nn
y x^ x^ x^
y x x^ x x x
^
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Newton’s divided differenceinterpolation formula^0
0 0 1 0 1 0 1 2 0 1
1 0 1
( )^ (^
) [^ ,^ ]
(^ )(^ ) [
,^ ,^ ]
(^ )(^ )^
(^ ) [^ ,^
,...,^ ] n n
y^ f^ x^ y^
x^ x^ y x^ x x x x x y x^ x^ x
x^ x^ x^ x^
x^ x^ y x^ x^
x
^ ^ ^
^ ^ ^ ^ ^
Newton’s divideddifferences can also beexpressed in terms offorward, backward andcentral differences.
In terms of backward differences [^ ,^ ,...,^0
n ]! yn n n y x^ x^
x n h
In terms of central differences
2 0 1 2
2 2 1 (1/ 2) 0 1 2 1
2 1 [^ ,^ ,...,^ ]^
(2^ )! [^ ,^ ,...,^
m m m m m m ] (^) m (2 1)! y m y x^ x^ x^
m^ h y y x^ x^ x^
(^) m h
Following the basic definitionof divided differences, wehave for any
x 0 0 0 0 0 1 1 0 1 0 1 0 1 2 2 0
1 2 0 1 0 1
0 ( )^ (^ ) [ ,
]
[ ,^ ]^ [^ ,^ ]^ (^
) [ ,^ ,^ ]
[ ,^ ,^ ]^ [^ ,^ ,^
]^ (^ ) [ ,^ ,^ ,
]
[ ,^ ,...,^ ]^ [^
,^ ,...,^ ]^ (^ ) [ ,
,...,^ ]
n^ n^
n^ n y x^ y^ x^ x^ y x xy x x^ y x^ x^ x^ x
y x x^ x y x x^ x^ y x^ x^
x^ x^ x^ y x x^
x^ x y x x^ x^ y x
x^ x^ x^ x^
y x x^ x ^ ^ ^
^ ^ ^
^ ^ ^
^ ^
^ docsity.com
Multiplying thesecond Equation by (
x^ –^ x ),^0 third by ( x^ –
x )( x^ –^ x )^01 and so on,and the last by( x^ –^ x )( x^ –^0
x ) … ( x^ –^ x^1
) andn- adding the resultingequations, we obtain
Error Term inInterpolation Formulae ( ) ( )^ ( ), x y x^ P^ x n^0
0 ( )^ (^ )(^
)^ (^ ) [ ,^ ,...,
] n n x^ x^ x^ x^ x^
x^ x^ y x x^ x
^ ^ ^
We know, if^ y^ (^ where,
x ) is approximated
by a polynomial
P(x)^ of degree n^
n
then the error is given by
Alternatively it is also expressed as
0 ( )^ ( ) [ ,
,...,^ ]^
( ) n x^ x y x x
x^ K^
x ^ ^
^
Now, consider a function
F^ ( x ), such
that^ ( )^
( )^ ( )^
( ) n F x^ y x^
P^ x^ K^
x ^ ^
^
Determine the constant
K^ in such a way that
F ( x ) vanishes for
x^ =^ x ,^ x , …,^ x^0 1 n^
and also
for an arbitrarily chosen point
, which is
different from the given (
x n + 1) points.