Difference between revisions of "Scilab/C4/Interpolation/English"

Welcome to the Spoken Tutorial on “Numerical Interpolation”

• Develop Scilab code for different Numerical Interpolation algorithms
• Calculate new value of function from given data points

• Ubuntu 12.04 as the operating system
• and Scilab 5.3.3 version

• basic knowledge of Scilab
• and should know Numerical Interpolation

To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.

• constructing new data points
• within the range of
• a discrete set of known data points.

We can solve interpolation problems using numerical methods.

• We pass a polynomial of degree N – 1 through N points.
• Then, we find the unique N order polynomial y of x
• which interpolates the data samples.

We have to find the value of natural logarithm of nine point two.

Let us solve this problem using Lagrange interpolation method.

Lagrange(x0, x,f, n)

• X zero is the unknown interpolation point.

• x is the vector containing the data points.

• f is the vector containing the values of the function at correspoding data points.

• And n is the order of the interpolating polynomial.

m = n + 1;

N = ones(1,m);

The order of the interpolating polynomial determines the number of nodes created.

for j = 1:m

for k = 1:m

if (k<>j) then

N(j) = N(j)*(x0 - x(k))

D(j) = D(j)*(x(j) - x(k))

end

end

• we apply Lagrange interpolation formula
• to find the value of the numerator and denominator.

L(j) = N(j)/D(j);

y = y + L(j)*f(j);

end

disp(L','L')

disp(f,'f(x)')

We use L to find the value of the function y at the given data point.

Finally we display the value of L and f of x.

x=[9.0,9.5,11.0]

On the console type,

x equal to open square bracket nine point zero comma nine point five comma eleven point zero close square bracket.

Press Enter

f=[2.1972,2.2513,2.3979]

f equal to open square bracket two point one nine seven two comma two point two five one three comma two point three nine seven nine close square bracket

Press Enter

x0=9.2

x zero equal to nine point two

Press Enter

n=2

Type n equal to two

Press Enter

y = Lagrange(x0, x,f, n)

y equal to Lagrange open paranthesis x zero comma x comma f comma n close paranthesis

Press Enter.

It uses lesser number of computation than Lagrange method.

In spite of this, the same interpolating polynomial, as in Lagrange method, is generated.

We are given

• the data points and
• the corresponding values of the function
• at those data points.

We have to find the value of the function at x equal to three.

Open the file Newton underscore Divided dot sci on Scilab Editor.

Newton_Divided(x,f,x0)

• x is a vector containing the data points,

f is the corresponding function *value and

• x zero is the unknown interpolation point.

n = length(x);

a(1) = f(1);

for k = 1 : n - 1

D(k, 1) = (f(k+1) - f(k))/(x(k+1) - x(k));

end

for j = 2:n-1

for k = 1:n-j

D(k, j) = (D(k+1, j-1) - D(k, j-1))/(x(k+j) - x(k))

end

end

disp(D, 'The Divided Difference Table')

Df(1) = 1;

c(1) = a(1);

for j = 2 : n

Df(j)=(x0 - x(j-1)).*Df(j-1);

c(j) = a(j).*Df(j);

end

IP = sum(c);

• We sum the coefficient list
• to find the value of the function
• at given data point.

Press Enter.

x=[2,2.5,3.25,4]

Type

x equal to open square bracket two comma two point five comma three point two five comma four close square bracket

Press Enter.

f=[0.5,0.4,0.3077,0.25]

f equal to open square bracket zero point five comma zero point four comma zero point three zero seven seven comma zero point two five close square bracket

Press Enter.

x0=3

Press Enter.

IP = Newton_Divided(x,f,x0)

i p equal to Newton underscore Divided open parenthesis x comma f comma x zero close parenthesis

Press Enter.

In this tutorial,

we have learnt to develop Scilab code for interpolation methods.

We have also learnt to find the value of a function at new data point.

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