interpolation - copy.pdf

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Interpolation and curve fitting

In interpolation our aim is to find the curve which passes through our given data set. Equation

of the curve gives us an idea as to how the parameters represented in the plane (or space) are

related, and extension of that curve gives an idea of how the system might behave when some

of the parameters are changed. Engineering is all about approximation and finding best solution

as opposed to finding exact solution. Sometimes it might not be possible to obtain a curve

which passes exactly through our given dataset. In such cases, we need to find the curve i.e.

closest to these points. This is the domain of curve-fitting. Following fundamental theories of

algebra might be useful in our study and application of interpolation principles.

1. Values of n unknowns can be ascertained if there are minimum n variables.

2. N+1 data points can be accommodated by a polynomial of degree N, which has exactly

N+1 unknowns

3. A polynomial of nth degree is represented by the equation

4. A quadratic equation is represented by

By various experiments (P3/P1) values and corresponding NLR values given in the

table 1 is known.

Table 1

NLR P3/P1

0 0.25

30 0.625

50 1.25

65 1.94

75 2.62

85 3.5100 5.13

Now if we know that NLR speed at any instant is 75 and P1 is 91kPa we can refer table1

and realize that P3/P1 value is 2.62.


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Therefore P3=2.62*91=238.42kPa.

If we were to know the value of P3 when its value directlyfrom sensor is not available

or erroneous Then we can predict a possible value of P3 given NLR is known if we can

find a relation between P3 and NLR.As already stated an equation of degree 7 will give a

curve which passes through these points. However, due to practical limitations we willuse a second degree equation using C coding. Our algorithm is explained below

1. Obtaining the solution for a quadriatic equation given 3 points:

A quadriatic equation is given by

Which passes through (x1,y1),(x2,y2),(x3,y3).Our aim is to find a,b,c.

Substituting these points, we have

() ( ) ()

Eqn (1)-(2) gives

() ( ) () ()Eqn (1)-(3) gives

() ( ) () ()

From eqn 1

() ()( )

()

Substituting the value of a from (5) in eqn (4) we get,

(3)22()(232)

(22)(3)()(232) (6)

()()

(22) (7)


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Where b is known by (6)

Where a and b are known from (6) and (7)

Thus the equation of the curve

Which passes through (x1,y1) , (x2,y2) , ( x3,y3) is known using eqn(6) ,(7) and (8).

2.

Finding the possible equations for all the combination of three data points.

Equation of second degree requires three data points to be solved. There are seven data points.

Therefore 7C3=35 possible combinations of three data points and Thereby that many equations

are generated.

3.

Determining the accuracy of each equation.

Suppose the datapoints given are (x1,y1),(x2,y2)(xn,yn).

Where

is the average error in the kth curve fitting equation ()is the equation of the kth curve is the number of datapoints

(() ) is the difference between actual y value and predicted y value.i.e.f(x).Thus the summation gives the total error value for an equation

4.

Selecting the Best Curve

The curve with the best value is chosen as winner

(8)

Average Error (() = )


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Code

The procedure stated above is coded as given.

#include

#include

void main()

{

int i,j,k,l=0,N,M,m=0,L=0;

float x[20],y[20],Y,a[100],b[100],c[100],e[100],E;

for(i=0;i


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for(k=j+1;k


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for(i=0;i


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Comparision between measured and calculated values

0

1

2

3

4

5

6

0 30 50 65 75 85 100

P3/P1

P3/P1(calc)

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