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Online optimization algorithms
Published in Xiaobiao Huang, Beam-based Correction and Optimization for Accelerators, 2019
With the initial three points, a < c < b, the golden section search then samples a new point t in one of the two subdivisions, say t ∈ (a, c). If f(t) < f(c), then [a, c] becomes the new bracket; if f(t) > f(c), [t, b] becomes the new bracket. It can iteratively proceed until the minimum is found with the desired tolerance. For the highest efficiency, the distance from the inside sample point to the bracket boundary is chosen to be 5−12≈0.618. For Brent’s method, the function values at point a, c, and b are used to construct a parabola and the location corresponding to the minimum of the parabola is used as the new sample point. The golden section method has linear convergence, while the inverse parabola interpolation has quadratic convergence. The latter is used more often for smooth functions. With noise in the function values, both the golden section search and inverse parabola interpolation methods could fail.
Non-Linear Systems
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
Brent’s method combines the bisection method, the secant method, and inverse quadratic interpolation. It is also known as the Wijngaarden-Dekker-Brent method. At every iteration, Brent’s method decides which method out of these three is likely to do best, and proceeds by doing a step according to that method. This gives a robust and fast method.
High-throughput XPS spectrum modeling with autonomous background subtraction for 3d 5/2 peak mapping of SnS
Published in Science and Technology of Advanced Materials: Methods, 2023
Tarojiro Matsumura, Naoka Nagamura, Shotaro Akaho, Kenji Nagata, Yasunobu Ando
Here, , and are fixed to be , and , respectively. This maximization is conducted by using the Brent’s method [34]. In CM-step for are updated from to by maximizing Equation (22) in which , and are fixed to be , and , respectively. In CM-step for , the parameter of are updated from to by maximizing Equation (22) for in which , and are fixed to be , and , respectively.