LagrangePolynomials - Lagrange interpolating Polynomials...

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Lagrange interpolating Polynomials James Keesling 1 Determining the Coefficients of the Lagrange Interpolat- ing Polynomial by Linear Equations It is frequently the case that we will have certain data points, { ( x 0 , y 0 ) , ( x 1 , y 1 ) , . . . , ( x n , y n ) } , and will want to fit a curve through these points. In this chapter we will fit a polynomial of minimal degree through the points. We assume that the points { x 0 , x 1 , . . . , x n } are all distinct. In that case we can fit a polynomial of degree n (or possibly less) through the points. If we write the polynomial in the following form, then we can use the points to determine the coefficients. L ( x ) = a 0 + a 1 · x + a 2 · x 2 + · · · + a n · x n y 0 = a 0 + a 1 · x 0 + a 2 · x 2 0 + · · · a n · x n 0 y 1 = a 0 + a 1 · x 1 + a 2 · x 2 1 + · · · a n · x n 1 . . . y n = a 0 + a 1 · x n + a 2 · x 2 n + · · · a n · x n n We can solve these equations using matrices. The vector A = a 0 a 1 . . . a n is the set of coefficients to be determined. We let M = V [ x 0 , x 1 , . . . , x n ] be the Van- dermonde Matrix and B
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