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Given a set of n+1 data points (xi,yi) where no two xi are the same we are trying to find a polynomial p of degree n with the property
The unisolvence theorem states that the polynomial p of degree n is uniquely defined by the n+1 data points.
Or to phrase it in terms of linear algebra: For n+1 interpolation nodes there exists a vector space isomorphism
where is the vector space of polynomials with degree n.
Suppose that the interpolation polynomial is given by
The statement that p interpolates the data points means that
If we substitute equation (1) in here, we get a system of linear equations in the coefficients . The system in matrix-vector form reads
We have to solve this system for to construct the interpolant .
The matrix on the left is commonly referred to as a Vandermonde matrix. Its determinant is nonzero, which proves the unisolvence theorem: there exists a unique interpolating polynomial.
We are trying to construct our unique interpolation polynomial in the vector space that is the vector space of polynomials of degree n. When using a monomial basis for we have to solve the Vandermonde matrix to construct the coefficients for the interpolation polynomial. This can be a very costly operation (as counted in clock cycles of a computer trying to do the job). By choosing another basis for we can simplify the calculation of the coeffiecients but then of course we have to do additional calculations when we want to express the interpolation polynomial in terms of a monomial basis.
One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients. The cost is O operations, while Gaussian elimination costs O operations. Furthermore, you only need to do a bit of extra work if an extra point is added to the data set, while for the other methods, you have to redo the whole computation.
Another method is to use the Lagrange form of the interpolation polynomial. The resulting formula immediately shows that the interpolation polynomial exists under the conditions stated in the above theorem.
The Bernstein form was used in a constructive proof of the Weierstrass approximation theorem by BernsteinSergei Natanovich Bernstein ( Russian: sometimes Romanized as Bernshtein ( March 5, 1880 October 26, 1968) was a Ukrainian mathematician who was born in Odessa, Ukraine and died in Moscow, USSR. His doctoral dissertation, submitted in 1904 to the Sorbonne and has nowadays gained great importance in computer graphics in the form of Bezier curves.
When interpolating a given function f by a polynomial of degree n at the nodes x0,...,xn we get the error
where
is the notation for divided differences. When f is n+1 times continously differentiable on the smallest interval I which contains the nodes xi then we can write the error in the Lagrange form as
for some in I. Thus the the remainder term in the Lagrange form of the Taylor theoremIn calculus, Taylor's theorem named after the mathematician Brook Taylor, who stated it in 1712, allows the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that p is a special case of interpolation error when all interpolation nodes xi are identical.
In the case of equally spaced interpolation nodes , it follows that the interpolation error is O. However, this does not necessarily mean that the error goes to zero as n → ∞. In fact, the error may increase without bound near the ends of the interval . This is called Runge's phenomenonIn the mathematical subfield of numerical analysis Runge's phenomenon is a problem which occurs when using polynomial interpolation with polynomials of high degree. It was discovered by Carle David Tolme Runge when exploring the error of polynomial interp.
The above error bound suggests choosing the interpolation points xi such that the product | ∏ (x − xi) | is as small as possible. The Chebyshev nodes achieve this.
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