Numerical methods bisection method

The same regression analysis methodology has been appHed for analyzing the model for aggregation of surfactants at the interface. The non-Hnear Eq. 28 has been numerically solved by the bisection method. The surface tension predicted by Eq. 27 has been fitted to the experimental data by min-... 

The standard deviation has been determined as ct = j where v is the number of degrees of freedom in the fit. The parameters for the molecular interaction /3, the maximum adsorption Too, the equilibrium constant for adsorption of surfactant ions Ki, and the equilibrium constant for adsorption of counterions K2, are thus obtained. The non-linear equations for the Frumkin adsorption isotherm have been numerically solved by the bisection method. 

Our first attempt involves MATLAB s built-in root finder fzero, which uses the bisection method and thereafter we introduce a new and more appropriate numerical method for solving equations with multiple roots. 

Equation (13-14) is solved iteratively for V/F, followed by the calculation of values o(x,anAy, from Eqs. (13-12) and (13-13) and L from the total mole balance. Any one of a number of numerical root-finding procedures such as the Newton-Raphson, secant, false-position, or bisection method can be used to solve Eq. (13-14). Values of K, are constants if they are independent of liquid and vapor compositions. Then the resulting calculations are straightforward. Otherwise, the K, values must be periodically updated for composition effects, perhaps... 

Eigenfunction expansions as used in Ref. 168 are not accurate near the critical point. Instead, we developed a shooting point method in order to make a direct numerical integration of Eq. (110) with the condition Eq. (112). Real energies (bound and virtual) were found by bisection methods, and for complex energies it was necessary to combine the Newton-Raphson and grid methods. 

The pellet mass and heat balances are described by second order differential equations of the two point boundary value type. For this case the reaction is neither too fast nor highly exothermic and therefore the concentration and temperature gradients inside the pellet are not very steep. Therefore the orthogonal collocation method with one internal collocation point was found sufficient to transform the differential equation into a set of algebraic equations which were solved numerically using the bisectional method (Rice,... 

It can be solved numerically (using Newton-Raphson, bisectional methods, etc. see Appendix B), but for the sake of illustration, we solve it graphically (as shown in Fig. 3.9). 

At this point we can summarize the steps required to implement the Gauss-Newton method for PDE models. At each iteration, given the current estimate of the parameters, ky we obtain w(t,z) and G(t,z) by solving numerically the state and sensitivity partial differential equations. Using these values we compute the model output, y(t k(i)), and the output sensitivity matrix, (5yr/5k)T for each data point i=l,...,N. Subsequently, these are used to set up matrix A and vector b. Solution of the linear equation yields Ak(jH) and hence k°M) is obtained. The bisection rule to yield an acceptable step-size at each iteration of the Gauss-Newton method should also be used. 

The subsequent diagonalization of the symmetric tridiagonal matrix T is relatively straightforward and numerically inexpensive. It can be carried out by root-searching methods such as bisection if a small number of eigenvalues is of interest.18,19 When the entire spectrum is needed, on the other hand, one... 

Families of numerical integration methods have been suggested which use the standard (Gaussian) numerical quadrature techniques within each of a set of mutually exclusive polyhedra formed by the planes which bisect each bond emanating from a given atom. Integration proceeds outwards from the nucleus at the centre of each polyhedron with some suitable choice for tiic boundaries of the atoms on the periphery of the molecule. 

Finding the roots of the above equation can be numerically achieved by using the optimized method suggested by T. Dekker, employing a combination of bisection, secant, and inverse quadratic interpolation methods (Forsythe et al., 1976). 

The equation can only be solved by a tedious numerical technique which involves the calculation of dP /ax values fim a nuniber of x positions by a bisection linear interpolation tedmique as given by McCormick and Salvadori [44]. These values are inserted into a fourth-order Rnn -Kutta method to obtain the P versus ac data. 

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