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Numerical differentiation. Use of numerical analysis to estimate derivatives of functions. Finite difference estimation of derivative. In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function.
Definition. The Laplace operator is a second-order differential operator in the n -dimensional Euclidean space, defined as the divergence ( ) of the gradient ( ). Thus if is a twice-differentiable real-valued function, then the Laplacian of is the real-valued function defined by: {\displaystyle \Delta f=\nabla ^ {2}f=\nabla \cdot \nabla f}
The second derivative of a quadratic function is constant. In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object ...
In mathematics and computer algebra, automatic differentiation (auto-differentiation, autodiff, or AD), also called algorithmic differentiation, computational differentiation, [1][2] is a set of techniques to evaluate the partial derivative of a function specified by a computer program. Automatic differentiation exploits the fact that every ...
Symmetry of second derivatives. In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate function. … n {\displaystyle f\left (x_ {1},\,x_ {2},\,\ldots ,\,x_ {n}\right)} does not change the result if some continuity conditions ...
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.
v. t. e. 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 of the Taylor series of the function.