Help with finding and plotting interpolating polynomials

Question

Steven 2012-4-20

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I already plot using lagrange and newton polynomial but I need help with finding and plotting the interpolating polynomials by system of equations. It doesn't say anywhere in my textbook on how to do this so I need help.

Here is the data I'm given for the problem:

Given the (fictitious) gasoline price data over 10 years,

Year, x 1986 1988 1990 1992 1994 1996

Price (¢),y 113.5 132.2 138.7 141.5 137.6 144.2

a simple interpolating polynomial is proposed as:

y(x)= c0 + c1x + c2x^2 + c3x^3 + c4 x^4 + c5 x^5

Then I am asked to find it by using:

a) System of equations

b) Lagrange polynomial

c) Newton polynomial

Please I need help for the system of equations part.

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Walter Roberson 2012-4-20

duplicate is at http://www.mathworks.com/matlabcentral/answers/35883-help-with-an-interpolation-problem

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Answer 1

Richard Brown 2012-4-20

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I presume you mean c0 + c1x + c2x^2 + c3x^3 + c4 x^4 + c5 x^5

The system of equations method means treating c0 ... c5 as unknowns, plugging each data point into the equation, and forming a matrix (Vandermonde) equation to solve for the resulting linear equations in c0 ... c5. That should hopefully be enough to get you back underway

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Steven 2012-4-20

so what do the c's correspond to?

Walter Roberson 2012-4-20

Suppose you had x = 1, 2, 3, and y = 5, 8, 13. That would correspond to y = x^2 + 4, or y = 1 * x^2 + 0*x + 4, or c0 = 4, c1 = 0, c2 = 1. In order of descending coefficients, [1 0 4] . Now, does the 1 or the 4 correspond to one of the values [1 2 3] or [5 8 13]? No --the [1 0 4] are simply coefficients that make the polynomial work.

Extending this example with three values to the more generalized form,

y1 = c2*x1^2 + c1*x1 + c0

y2 = c2*x2^2 + c1*x2 + c0

y3 = c2*x3^2 + c1*x3 + c0

where x1, x2, x3 and y1, y2, y3 are known, then

let D = ((-x2+x1) * (-x3+x2) * (-x3+x1))

and then

c0 = ((x2*y3-x3*y2)*x1^2+(-x2^2*y3+x3^2*y2)*x1+x2*y1*x3*(-x3+x2)) / D

c1 = ((-y3+y2)*x1^2+(y3-y1)*x2^2+x3^2*(y1-y2)) / D

c2 = ((y3-y2)*x1+(-y3+y1)*x2-x3*(y1-y2)) / D

You can see that there is a _relationship_ between the x/y values and the c* coefficients, but you can also see that the c* are certainly not either the x or y values directly.

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