You can free download Numerical Analysis. By Alamdar Hussain in pdf format. Interpolation with unequal intervals: Introduction ...
You can free download Numerical Analysis.By Alamdar Hussain in pdf format.
Interpolation with unequal intervals:
Introduction
so far we have studied the application of interpolation formulae,where the arguments are equally spaced .In the unit, we will study interpolation the values when the arguments are not equally spaced by introducing the concept of divided difference.
Let f(x0),f(x1).......................,f(xn) be entery the values of the function y=f(x) at the arguments x0,x1,x2,..............,xn which are not equally spaced.i.e(the interval of differencing is not constant).
Divided difference can now be defined as the difference between two successive values of the entry divided by the difference between the corresponding values of the argument .Therefore ,first divided difference f(x) for the arguments x0 and x1 is defined as
Relation Between Divided Differences and Forward Differences
1 Newton Divided Difference Formula.
2 Lagrange's Interpolation Formula.
3 Inverse Interpolation.
Interpolation with unequal intervals:
Introduction
so far we have studied the application of interpolation formulae,where the arguments are equally spaced .In the unit, we will study interpolation the values when the arguments are not equally spaced by introducing the concept of divided difference.
Let f(x0),f(x1).......................,f(xn) be entery the values of the function y=f(x) at the arguments x0,x1,x2,..............,xn which are not equally spaced.i.e(the interval of differencing is not constant).
Divided difference can now be defined as the difference between two successive values of the entry divided by the difference between the corresponding values of the argument .Therefore ,first divided difference f(x) for the arguments x0 and x1 is defined as
Relation Between Divided Differences and Forward Differences
1 Newton Divided Difference Formula.
2 Lagrange's Interpolation Formula.
3 Inverse Interpolation.
Newton Divided Difference Formula
Let f(x 0),f(x 1),f(x 2),.....................,f(x n).be the values of f(x) corresponding to the non-equally spaced arguments x0,x1,x2,..............................,xn.From the definition of divided differences,we have.
f(x)=f(x)- f(x0) ∕x-x0
f(x)=f(x)+(x-x0)f(x,xo) are continous.
Lagrange's Interpolation Formula
Let y=f(x) be a function which takes the (n+1) values y0,y1,y2,...............,yn corresponding to X=x0,x1,x2,................,xn.Now f(x) can be represented as a polynimal of the nth degree in x.
y=f(x)=(x-x1)(x-x2)........(x-xn)∕(x0-x1)(x-x2)........(x0-xn) y0 +(x-x0)(x-x1)........(x-xn)–(x1-x0)(x1-x2)........(x-xn) y1+.....................(x-x0)(x-x1)........(x-xn-1)—(xn-x0)(xn-x2)........(xn-xn-1).
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