By Seidenberg A.

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**Extra resources for A new decision method for elementary algebra**

**Sample text**

10) They are of degree n and have the property Li (xk ) = δi,k . 11) The interpolating polynomial is given in terms of Lagrange polynomials by n n n p(x) = fi Li (x) = i=0 fi i=0 k=0,k=i x − xk . 15) − xk ) becomes the first form of the barycentric interpolation formula Li (x) = ω(x) ui . 16) The interpolating polynomial can now be evaluated according to n p(x) = n fi Li (x) = ω(x) i=0 fi i=0 ui . 17) 18 2 Interpolation Having computed the weights ui , evaluation of the polynomial only requires O(n) operations whereas calculation of all the Lagrange polynomials requires O(n2 ) operations.

22 2 Interpolation Fig. 40). Its roots xi are given by the x values of the sample points (circles). 3 Spline Interpolation Polynomials are not well suited for interpolation over a larger range. Often spline functions are superior which are piecewise defined polynomials [186, 228]. 42) pi (x) = yi + xi+1 − xi s(x) = pi (x) where xi ≤ x < xi+1 . 43) The most important case is the cubic spline which is given in the interval xi ≤ x < xi+1 by pi (x) = αi + βi (x − xi ) + γi (x − xi )2 + δi (x − xi )3 .

2) f (x) = sin(x) xi = 0, 3π π , π, , 2π. 41) whereas the error increases rapidly outside the interval 0 < x < 2π (Fig. 3). 22 2 Interpolation Fig. 40). Its roots xi are given by the x values of the sample points (circles). 3 Spline Interpolation Polynomials are not well suited for interpolation over a larger range. Often spline functions are superior which are piecewise defined polynomials [186, 228]. 42) pi (x) = yi + xi+1 − xi s(x) = pi (x) where xi ≤ x < xi+1 . 43) The most important case is the cubic spline which is given in the interval xi ≤ x < xi+1 by pi (x) = αi + βi (x − xi ) + γi (x − xi )2 + δi (x − xi )3 .