Root-finding scheme

One approach is to solve the equation x = P tanh x numerically, using the Newton-Raphson method or some other root-finding scheme. (See Press et al. (1986) for a friendly and informative discussion of numerical methods.)... 

The flow rate eould be eomputed when the pressure drop is constant. Eq. 5.60 is nonlinear in flow rate, and thus a root-finding scheme is necessary to find qg when Ap is given. 

Write the appropriate driver routine to integrate the equations using either the DASSL [46] (DAE) or VODE (ODE) [49] software. You will need to iteratively solve for the laboratory-referenced molar fluxes N], N2 using a multidimensional root-finding routine, such as MNEWT / FDJAC [319], a Newton-Raphson scheme. 

The algorithm for computing H is analogous to the one used in the univariate case. Using a root-finding (or minimization) scheme, we must vary the components of E until the objective function is minimized. The components of the objective function found from the extended optimal moments are... 

Newton s method is a classic iterative scheme for solving a nonlinear system /(x) = 0 or for minimizing the multivariate function /(x). These root-finding and minimization problems are closely related since obtaining the minimum of a function f(x) can be formulated as solving for the zeros of / (x) for which f (x) > 0. (See historical note below on the method s name.)... 

Boley proceeds to find solutions for short times after melting in powers of the square root of the dimensionless post-melting time and indicates a numerical scheme for continuing the solution. The results are in good agreement with those of Landau for equivalent conditions. 

In contradiction to the scheme shown is the finding (137) that the source of the nitrogen of the pyrrolidine ring of nicotine is the S-amino group of ornithine, not the a-amino group as required above. No nitrogen from ornithine-2-440-(x-i5N was incorporated by a sterile root culture of N. tabacum ornithine-2-i40- -i5iJ resulted in a specific incorporation of... 

Here is another application of the Taylor series method. Suppose that you want to find the values x = x that cause a function to equal zero, fix) = 0. Such values of x are called the roots of the equation. The Newton method is an iteratixe scheme that works best when you can make a reasonable first guess X, and when the function does not have an extremum between the guess and... 

There is a special recipe, or method, for irranediately obtaining just tiie eigenvalues of a matrix. The recipe is to subtract an unspecified parameter—call it X—from the diagonal elements and then set the determinant formed from the elements of the matrix equal to zero. This generates a polynomial in the parameter X. The roots of that polynomial are the eigenvalues of the matrix. Here is an example of finding fhe eigenvalues of a 3 x 3 matrix, G, by this scheme. 

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