Science:Math Exam Resources/Courses/MATH307/April 2012/Question 05 (b)

MATH307 April 2012

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Question 05 (b)
You are given a set of 100 data points (xn, yn) with xn increasing. (b) Write down the matrix equation that is satisfied by the coefficients of this polynomial. Is the numerical solution of this equation likely to be accurate, and why?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From part (a), we know that we require a polynomial of degree 99, of the form: $p(x)=a_{1}x^{99}+a_{2}x^{98}+...+a_{99}x+a_{100}$ Since $(x_{i},y_{i})$ for $1\leq i\leq 100$, then there are 100 linear equations for the coefficient vector ${\underline {a}}=[a_{1},a_{2},...,a_{100}]^{T}$ Therefore, the matrix equation that is satisfied is: ${\begin{bmatrix}x_{1}^{99}&x_{1}^{98}&...&x_{1}^{2}&x_{1}&1\\x_{2}^{99}&x_{2}^{98}&...&x_{2}^{2}&x_{2}&1\\&&&&&\\\vdots &\vdots &&\ddots &&\vdots \\&&&&&\\x_{100}^{99}&x_{100}^{98}&...&x_{100}^{2}&x_{100}&1\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\...\\...\\...\\a_{99}\\a_{100}\\\end{bmatrix}}={\begin{bmatrix}y_{1}\\y_{2}\\...\\...\\...\\y_{99}\\y_{100}\end{bmatrix}}$ However, since the Vandermonde matrix not very well conditioned (because Vandermonde matrices have large condition numbers, an error in the data points can cause a relatively large error in the solution), it is unlikely that the solution is accurate

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