logo for solving-math-problems.com
leftimage for solving-math-problems.com

National Interpolating Polynomial

by Gilmar











































National Interpolating Polynomial

Given the following three data points:

   x    -1     0     1
__________________
F(x)    1    -1    -1

Determine the national interpolating polynomial of lowest degree possible to this data using

           a. Lagrange’s interpolation formula,

           b. Newton’s interpolation formula

           c. Quote the formula for the error applied to these special cases (since you are not given f(x), just leave any term’s in the error formula involving f(x) as they are)

Comments for National Interpolating Polynomial

Click here to add your own comments

Mar 21, 2013
Polynomial
by: Staff


Answer

Part I

I didn't work out the entire problem, but I did determine the function f(x) for you using the Vandermonde Approach.

The 2nd degree polynomial function passing through the three points is:

2nd degree polynomial function passing through the three points:  (-1, 1); (0, -1); and (1, -1)





Vandermonde Approach

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

Interpolating Polynomial:  standard form for 3 data points.




Once the values of the coefficients have been determined, the function can be used to evaluate other values of x (in addition to the three data points given).

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

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

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





--------------------------------------------------

Mar 21, 2013
Polynomial
by: Staff


--------------------------------------------------


Part II

Evaluate the function for the data point (0, -1)

Interpolating Polynomial:  evaluate for the data point (0, -1).





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

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





Solve the three equations for the coefficients a₁,a₂, and a₃

Three equations, three unknown coefficients.




since the value of a₁ = -1, substitute -1 for the coefficient a₁ in all equations

a₁ equals -1 in all three equations.





--------------------------------------------------

Mar 21, 2013
Polynomial
by: Staff


--------------------------------------------------


Part III

Solve for a₃ using addition / elimination

 Solve for a_3 using addition (elimination).





substitute -1 for a₁, and 1 for a₃

substitute -1 for a_1 and 1 for a_3.




solve for a₂

 Solve for a_2.





substitute the numerical values of all three coefficients for the letter values (a₁, a₂, and a₃) in the standard form of the original Interpolating Polynomial

Finalize the Interpolating Polynomial.









Thanks for writing.

Staff
www.solving-math-problems.com



Click here to add your own comments

Join in and write your own page! It's easy to do. How? Simply click here to return to Math Questions & Comments - 01.



Copyright © 2008-2015. All rights reserved. Solving-Math-Problems.com