Main Idea
Solving an interpolation problem leads to a problem in linear algebra where we have to solve a matrix. Using a standard monomial basis for our interpolation polynomial we get the Vandermonde matrix. By choosing another basis, the Lagrange basis, we get the much simpler identity matrix = δi,j which we can solve instantly: the Lagrange basis inverts the Vandermonde matrix.
This construction is the same as the Chinese Remainder Theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears.
Read more about this topic: Lagrange Polynomial
Famous quotes containing the words main and/or idea:
“So long as war is the main business of nations, temporary despotism—despotism during the campaign—is indispensable.”
—Walter Bagehot (1826–1877)
“To me, literature is a calling, even a kind of salvation. It connects me with an enterprise that is over 2,000 years old. What do we have from the past? Art and thought. That’s what lasts. That’s what continues to feed people and given them an idea of something better. A better state of one’s feelings or simply the idea of a silence in one’s self that allows one to think or to feel. Which to me is the same.”
—Susan Sontag (b. 1933)