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In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.
The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin. The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations.
The initial guess will be x 0 = 1 and the function will be f(x) = x 2 − 2 so that f ′ (x) = 2x. Each new iteration of Newton's method will be denoted by x1 . We will check during the computation whether the denominator ( yprime ) becomes too small (smaller than epsilon ), which would be the case if f ′ ( x n ) ≈ 0 , since otherwise a ...
Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography.
If < , and =, where G 1 and G 2 are coprime polynomials, then there exist polynomials and such that = +, and < < . This can be proved as follows. Bézout's identity asserts the existence of polynomials C and D such that C G 1 + D G 2 = 1 {\displaystyle CG_{1}+DG_{2}=1} (by hypothesis, 1 is a greatest common divisor of G 1 ...
If the characteristic function of some random variable is of the form () = in a neighborhood of zero, where () is a polynomial, then the Marcinkiewicz theorem (named after Józef Marcinkiewicz) asserts that can be at most a quadratic polynomial, and therefore is a normal random variable. [33]
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
For an analytic function in variables one has (+) = ()!. In fact, for a smooth ... In fact, for a smooth enough function, we have the similar Taylor expansion ...