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Interpolation and Polynomial Approximation











































The following data are given for a polynomial P (x) of unknown degree.

   x      0     1     2      3
_______________________
P(x)     4     9    15    18


Determine the coefficient of x^3 in P(x) if all fourth-order forward differences are 1.

Comments for Interpolation and Polynomial Approximation

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Mar 25, 2013
Polynomial Approximation
by: Staff


Answer

Part I


I worked out the equation for you two ways: using the (1) Newton's Form, and the (2) Vander monde Approach

the coefficient of x³ in P(x) = -⅔


The complete 3rd degree polynomial function passing through the four points is:

3rd degree polynomial function passing through the four points:  (0, 4); (1, 9); (2, 15) and (3, 18)




A graph of the function is shown below:

3rd degree polynomial function passing through the four points:  (0, 4); (1, 9); (2, 15) and (3, 18)




Deriving the Equation

Newton’s form

The advantage of using this method is that you can compute the coefficients easily, by hand.

With four data points, the formula is:


Formula for Newton Polynomial:  four data points:  (x_0, y_0); (x_1, y_1); (x_2, y_2) and (x_3, y_3)





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Mar 25, 2013
Polynomial Approximation
by: Staff


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Part II

n = 4. The curve must go through four points.

when x = 0, P(x) = 4

Formula for Newton Polynomial:  (x_0, y_0), compute the coefficient c_0






when x = 1, P(x) = 9

Formula for Newton Polynomial:  (x_1, y_1), compute the coefficient c_1




when x = 2, P(x) = 15

Formula for Newton Polynomial:  (x_2, y_2), compute the coefficient c_2





when x = 3, P(x) = 18

Formula for Newton Polynomial:  (x_3, y_3), compute the coefficient c_3






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Mar 25, 2013
Polynomial Approximation
by: Staff

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Part III

The final equation is:

Final Derivation of Newton Polynomial function passing through the four points:  (0, 4); (1, 9); (2, 15) and (3, 18)




Vander monde Approach

The standard form of an Interpolating Polynomial for four data points is the following third degree polynomial.

Interpolating Polynomial:  standard form for 4 data points.





Determine the value of the coefficients a₁,a₂, a₃, and a₄ as follows:

Evaluate the function for the data point (0, 4)

Interpolating Polynomial:  evaluate for the data point (0, 4).






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Mar 25, 2013
Polynomial Approximation
by: Staff


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Part IV

Evaluate the function for the data point (1, 9)

Interpolating Polynomial:  evaluate for the data point (1, 9).





Evaluate the function for the data point (2, 15)

Interpolating Polynomial:  evaluate for the data point (2, 15).





Evaluate the function for the data point (3, 18)

Interpolating Polynomial:  evaluate for the data point (3, 18).






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Mar 25, 2013
Polynomial Approximation
by: Staff


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Part V

Determine the value of the coefficients a₁,a₂, a₃, and a₄ by solving the following system of four linear equations:

Four equations, four unknown coefficients.




The final derivation of the equation is:

Finalize the Interpolating Polynomial.







Thanks for writing.

Staff
www.solving-math-problems.com


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