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Calculus. In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative.
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of ...
Numerical differentiation. 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.
The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. [1]: 26ff A partial derivative may be thought of as the directional derivative of the function along a coordinate axis.
Newton's method in optimization. A comparison of gradient descent (green) and Newton's method (red) for minimizing a function (with small step sizes). Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding ...
Mathematical optimization. Graph of a surface given by z = f (x, y) = − (x ² + y ²) + 4. The global maximum at (x, y, z) = (0, 0, 4) is indicated by a blue dot. Nelder-Mead minimum search of Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value. Mathematical optimization (alternatively ...
Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously.
The method of steepest descent is a method to approximate a complex integral of the form for large , where and are analytic functions of . Because the integrand is analytic, the contour can be deformed into a new contour without changing the integral. In particular, one seeks a new contour on which the imaginary part, denoted , of is constant ...