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Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute residuals (LAR), or least absolute values (LAV), is a statistical optimality criterion and a statistical optimization technique based on minimizing the sum of absolute deviations (also sum of absolute residuals or sum of absolute errors) or the L 1 norm of such values.
Examples of such mappings are the absolute value for integers, the degree for univariate polynomials, and the norm for Gaussian integers above. [ 150 ] [ 151 ] The basic principle is that each step of the algorithm reduces f inexorably; hence, if f can be reduced only a finite number of times, the algorithm must stop in a finite number of steps.
The Earth's orbit is known with an absolute precision of a few meters and a relative precision of a few parts in 100 billion (1 × 10 −11). Historically, observations of Venus transits were crucial in determining the AU; in the first half of the 20th century, observations of asteroids were also important.
In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
In mathematical analysis, the maximum and minimum [a] of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, [b] they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function.
Illustration of gradient descent on a series of level sets. Gradient descent is based on the observation that if the multi-variable function is defined and differentiable in a neighborhood of a point , then () decreases fastest if one goes from in the direction of the negative gradient of at , ().
The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that