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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2009 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Spring 2009 • Question 1

Polynomial Interpolation Let \(I=[a,b]\), let \(x_i\), \(i=0,\ldots ,n\), be \(n+1\) distinct points in \(I\), and let \(y_i\), \(i=0,\ldots ,n\), be \(n+1\) given real numbers. Show that there exists a unique polynomial \[ p(x)=\sum _{i=0}^n \alpha _i x^i \] such that \[ p(x_i)=y_i,\qquad i=0,\ldots ,n. \]

Raw LaTeX

{Polynomial Interpolation} Let \(I=[a,b]\), let \(x_i\),
\(i=0,\ldots,n\), be \(n+1\) distinct points in \(I\), and let \(y_i\),
\(i=0,\ldots,n\), be \(n+1\) given real numbers. Show that there exists a unique
polynomial
\[
  p(x)=\sum_{i=0}^n \alpha_i x^i
\]
such that
\[
  p(x_i)=y_i,\qquad i=0,\ldots,n.
\]

Topic Tags

Linear Programming and Optimization Polynomial Interpolation and Error Formulas