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The simplest form of the formula for Steffensen's method occurs when it is used to find a zero of a real function; that is, to find the real value that satisfies () =.Near the solution , the derivative of the function, ′, is supposed to approximately satisfy < ′ <; this condition ensures that is an adequate correction-function for , for finding its own solution, although it is not required ...
Sequential minimal optimization (SMO) is an algorithm for solving the quadratic programming (QP) problem that arises during the training of support-vector machines (SVM). It was invented by John Platt in 1998 at Microsoft Research. [1] SMO is widely used for training support vector machines and is implemented by the popular LIBSVM tool.
The conjugate gradient method with a trivial modification is extendable to solving, given complex-valued matrix A and vector b, the system of linear equations = for the complex-valued vector x, where A is Hermitian (i.e., A' = A) and positive-definite matrix, and the symbol ' denotes the conjugate transpose.
In the case of gradient descent, that would be when the vector of independent variable adjustments is proportional to the gradient vector of partial derivatives. The gradient descent can take many iterations to compute a local minimum with a required accuracy , if the curvature in different directions is very different for the given function.
The matrix–vector multiplication can be done in () arithmetical operations where is the average number of nonzero elements in a row. The total complexity is thus O ( d m n ) {\displaystyle O(dmn)} , or O ( d n 2 ) {\displaystyle O(dn^{2})} if m = n {\displaystyle m=n} ; the Lanczos algorithm can be very fast for sparse matrices.
The result of fitting a set of data points with a quadratic function Conic fitting a set of points using least-squares approximation. In regression analysis, least squares is a parameter estimation method based on minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each ...
There are two approximations in this what is called first approximation estimate: reduction to the average of the vector field and negligence of () terms. Uniformity with respect to the initial condition x 0 {\displaystyle x_{0}} : if we vary x 0 {\displaystyle x_{0}} this affects the estimation of L {\displaystyle L} and c {\displaystyle c} .
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.