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Nonlinear Algebra
Published in Brian Vick, Applied Engineering Mathematics, 2020
The bisection method is one type of incremental search methods where the interval containing the root is refined by dividing into halves and retaining the subinterval containing the root. This process is repeated until some desired accuracy criterion is met. In order to start the search, an interval that brackets or surrounds the root must be located. In general, if f(x) is real and continuous in the interval from xl to xu, and f(xl) and f(xu) have opposite signs (i.e., f(xl)f(xu)<0), then there exists at least one real root between xl and xu. The method is depicted graphically in Figure 6.5.
Solving systems of algebraic equations
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
The bisection method brackets an interval in which a root of the function f(x) must lie, then repeatedly bisects the interval until a root is found within the desired precision. The initial guesses for the lower and upper limits, xmin and xmax, of the interval must give function values of opposite signs, since then a zero crossing is guaranteed to lie between them. In the code below, the default value of the desired precision is set to tol = 1e-5; this can be changed as desired.
Roots of Equations
Published in Bilal M. Ayyub, Richard H. Mccuen, Numerical Analysis for Engineers, 2015
Bilal M. Ayyub, Richard H. Mccuen
Multiple roots pose difficulties for the previous methods. The bisection method has difficulties with multiple roots because the function does not change sign at even multiple roots. The Newton–Raphson and secant methods have difficulties because the derivative f′(x) at a multiple root is zero. Since f(x) reaches zero at a faster rate than f′(x) as x approaches the multiple root, it is possible to check for the condition f(x) = 0 and terminate the computations before reaching f′(x) = 0.
An adaptive regularization algorithm for recovering the rate constant distribution from biosensor data
Published in Inverse Problems in Science and Engineering, 2018
Y. Zhang, P. Forssén, T. Fornstedt, M. Gulliksson, X. Dai
As noted in Remark 5, for a given noise level for the measurement data, the regularization parameter can be computed by solving a nonlinear equation . According to Lemma 1, it can be done by the bisection method or a more efficient quadratic convergent method – the Newton iteration method if in the iteration process. However, for our real-world problem, is sometimes very small in the simulation process. Moreover, empirically, the peaks of the rate constant distribution are located in the region , i.e. the scale of association rate constant is , while the scale of the disassociation rate constant is . Therefore, the root of generalized discrepancy function possibly varies from to . The original bisection method may be very slow due to the long interval . Hence, in this work, we develop an accelerated bisection algorithm (see Algorithm 2 in the Appendix 1), which will be applied to the real-world problem.
Online profile monitoring for surgical outcomes using a weighted score test
Published in Journal of Quality Technology, 2018
Liu Liu, Xin Lai, Jian Zhang, Fugee Tsung
On theL: Control limit L is determined by the IC ARL. The simulation method based on bisection can be used to find the control limit. The bisection method in mathematics is a root-finding method that repeatedly bisects an interval, then selects a subinterval in which a root must lie for further processing. Specifically, we can apply it to numerically solve the equation f(L) = ARL(L) − ARL0 = 0 for the real variable L, where ARL(L) is the ARL when the control limit is L and ARL0 is the IC ARL we desired. f(L) is a continuous function defined on an interval [a, b] where f(a) and f(b) have opposite signs. In this case a and b are said to bracket a root since, by the intermediate value theorem, the continuous function f must have at least one root (control limit) in the interval (a, b).