Search results
Results from the WOW.Com Content Network
Given two sets, A and T, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function: (, ()) is minimized. Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as:
One can, for example, modify the Hessian by adding a correction matrix so as to make ″ + positive definite. One approach is to diagonalize the Hessian and choose B k {\displaystyle B_{k}} so that f ″ ( x k ) + B k {\displaystyle f''(x_{k})+B_{k}} has the same eigenvectors as the Hessian, but with each negative eigenvalue replaced by ϵ > 0 ...
The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...
The rate of convergence is distinguished from the number of iterations required to reach a given accuracy. For example, the function f(x) = x 20 − 1 has a root at 1. Since f ′(1) ≠ 0 and f is smooth, it is known that any Newton iteration convergent to 1 will converge quadratically. However, if initialized at 0.5, the first few iterates of ...
It is impossible to count the number of steps of an algorithm on all possible inputs. As the complexity generally increases with the size of the input, the complexity is typically expressed as a function of the size n (in bits) of the input, and therefore, the complexity is a function of n. However, the complexity of an algorithm may vary ...
The function f is variously called an objective function, criterion function, loss function, cost function (minimization), [8] utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. A feasible solution that minimizes (or maximizes) the objective function is called an optimal solution.
Karatsuba's basic step works for any base B and any m, but the recursive algorithm is most efficient when m is equal to n/2, rounded up. In particular, if n is 2 k, for some integer k, and the recursion stops only when n is 1, then the number of single-digit multiplications is 3 k, which is n c where c = log 2 3.
[20] [21] Generally, such methods converge in fewer iterations, but the cost of each iteration is higher. An example is the BFGS method which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated line search algorithm, to find the "best ...