or minima. For example, one can tell from looking at the graph that the point at −0.1 should have been at about −0.28. The way to do this in the algorithm is to use a single round of Newton's method. Since one knows the first and second derivatives of , one can calculate approximately how far a test point has to be moved so that the derivative will be zero. Calculating the derivatives of a polynomial is straightforward. One must also be able to calculate the first and second derivatives of f(x). Remez's algorithm requires an ability to calculate , , and to extremely high precision. The entire algorithm must be carried out to higher precision than the desired precision of the result. After moving the test points, the linear equation part is repeated, getting a new polynomial, and Newton's method is used again to move the test points again. This sequence is continued until the result converges to the desired accuracy. The algorithm converges very rapidly. Convergence is quadratic for well-behaved functions—if the test points are within of the correct result, they will be approximately within of the correct result after the next round. Remez's algorithm is typically started by choosing the extrema of the Chebyshev polynomial as the initial points, since the final error function will be similar to that polynomial. Main journals Journal of Approximation Theory Constructive Approximation East Journal on Approximations See also Estimation theory Fourier series Function approximation Orthonormal basis Padé approximant Schauder basis Kalman filter References N. I. Achiezer (Akhiezer), Theory of approximation, Translated by Charles J. Hyman Frederick Ungar Publishing Co., New York 1956 x+307 pp. A. F. Timan, Theory of approximation of functions of a real variable, 1963 C. Hastings, Jr. Approximations for Digital Computers. Princeton University Press, 1955. J. F. Hart, E. W. Cheney, C. L. Lawson, H. J. Maehly, C. K. Mesztenyi, J. R. Rice, H. C. Thacher Jr., C. Witzgall, Computer Approximations. Wiley, 1968, Lib. Cong. 67–23326. L. Fox and I. B. Parker. "Chebyshev Polynomials in Numerical Analysis." Oxford University Press London, 1968. W. J. Cody Jr., W. Waite, Software Manual for the Elementary Functions. Prentice-Hall, 1980, . E. Remes [Remez], "Sur le calcul effectif des polynomes d'approximation de Tschebyscheff". 1934 C. R. Acad. Sci., Paris, 199, 337–340. K.-G. Steffens, "The History of Approximation Theory: From Euler to Bernstein," Birkhauser, Boston 2006 . T. Erdélyi, "Extensions of the Bloch-Pólya