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  2. Taylor series - Wikipedia

    en.wikipedia.org/wiki/Taylor_series

    That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...

  3. Taylor's theorem - Wikipedia

    en.wikipedia.org/wiki/Taylor's_theorem

    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.

  4. Linearization - Wikipedia

    en.wikipedia.org/wiki/Linearization

    The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems , linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems . [ 1 ]

  5. Analytic function of a matrix - Wikipedia

    en.wikipedia.org/wiki/Analytic_function_of_a_matrix

    In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.

  6. Euler method - Wikipedia

    en.wikipedia.org/wiki/Euler_method

    It is the difference between the numerical solution after one step, , and the exact solution at time = +. The numerical solution is given by = + (,). For the exact solution, we use the Taylor expansion mentioned in the section Derivation above:

  7. Linear approximation - Wikipedia

    en.wikipedia.org/wiki/Linear_approximation

    Linear approximations in this case are further improved when the second derivative of a, ″ (), is sufficiently small (close to zero) (i.e., at or near an inflection point). If f {\displaystyle f} is concave down in the interval between x {\displaystyle x} and a {\displaystyle a} , the approximation will be an overestimate (since the ...

  8. Taylor expansions for the moments of functions of random ...

    en.wikipedia.org/wiki/Taylor_expansions_for_the...

    In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.

  9. Stencil (numerical analysis) - Wikipedia

    en.wikipedia.org/wiki/Stencil_(numerical_analysis)

    The finite difference coefficients for a given stencil are fixed by the choice of node points. The coefficients may be calculated by taking the derivative of the Lagrange polynomial interpolating between the node points, [3] by computing the Taylor expansion around each node point and solving a linear system, [4] or by enforcing that the stencil is exact for monomials up to the degree of the ...