Search results
Results From The WOW.Com Content Network
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 Taylor series of any polynomial is the polynomial itself.. The Maclaurin series of 1 / 1 − x is the geometric series + + + +. So, by substituting x for 1 − x, the Taylor series of 1 / x at a = 1 is
Taylor's theorem gives a precise bound on how good the approximation is. If f is a polynomial of degree less than or equal to d, then the Taylor polynomial of degree d equals f. The limit of the Taylor polynomials is an infinite series called the Taylor series. The Taylor series is frequently a very good approximation to the original function.
This is known as Opitz' formula. [2] [3] Now consider increasing the degree of to infinity, i.e. turn the Taylor polynomial into a Taylor series. Let be a function which corresponds to a power series.
For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as (+) = + ′ ()! + ()! + + ()! + (),. Where n! denotes the factorial of n, and R n (x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function.
For example, if a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy. In the case of a smooth function , the n th-order approximation is a polynomial of degree n , which is obtained by truncating the Taylor series to this degree.
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.
At each iteration the model is linearized by approximation to a first-order Taylor polynomial expansion about (,) (,) + (,) = (,) +. The Jacobian matrix , J , is a function of constants, the independent variable and the parameters, so it changes from one iteration to the next.