Solution , algebraic iterative

As shown in [231,232], the eigenvalues of RHP can be obtained very easily, for arbitrary/ and g, through an algebraic iterative procedme, which involves the Cardano-solution for cubic equations [242]. These findings open the way to studying the dynamics of arbitrarily large, finite RHP theoretically. 

The components of the momentum equation are usually solved one by one in a sequential manner, meaning that the set of algebraic equations for each component of the momentum is solved in turn, treating the grid point values of its dominant velocity as the sole set of unknowns. Due to the non-linearity and coupling of the underlying partial differential equations, (12.171)-(12.173) cannot be solved directly as the coefficients, a, and the source term, S, depend on the unknown solution An iterative solution method is required. 

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

Fast iterative root-finding algorithms do away with the necessity of algebraically solving for buried variables, an undertaking that often does not yield closed solutions (a solution is closed when the equation has the form x = f a,b,c,...) and x does not appear in the function/). 

An iterative solution method for linear algebraic systems which damps the shortwave components of the iteration error very fast and, after a few iterations, leaves predominantly long-wave components. The Gauss-Seidel method [85] could be chosen as a suitable solver in this context. 

Multigrid methods have proven to be powerful algorithms for the solution of linear algebraic equations. They are to be considered as a combination of different techniques allowing specific weaknesses of iterative solvers to be overcome. For this reason, most state-of-the-art commercial CFD solvers offer the multigrid capability. 

Thus Y1 is obtained not as the result of the numerical integration of a differential equation, but as the solution of an algebraic equation, which now requires an iterative procedure to determine the equilibrium value, Xj. The solution of algebraic balance equations in combination with an equilibrium relation has again resulted in an implicit algebraic loop. Simplification of such problems, however, is always possible, when Xj is simply related to Yi, as for example... 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

Digital simulation is a powerful tool for solving the equations describing chemical engineering systems. The principal difficulties are two (1) solution of simultaneous nonlinear algebraic equations (usually done by some iterative method), and (2) numerical integration of ordinary differential equations (using discrete finite-difference equations to approximate continuous differential equations). 

One of the most common problems in digital simulation is the solution of simultaneous nonlinear algebraic equations. If these equations contain transcendental functions, analytical solutions are impossible. Therefore, an iterative trial-and-error procedure of some sort must be devised. If there is only one unknown, a value for the solution is guessed. It is plugged into the equation or equations to see if it satisfies them. If not, a new guess is made and the whole process is repeated until the iteration eonverges (we hope) to the right value. 

The nonlinear character of the algebraic equations, Eqs. (3.59) and (3.61), implies the need for an iteration solution. 

By making the substitution for H and after some algebra taking the limit, he obtained a set of simultaneous nonlinear equations for the Fourier coefficients. Howard solved these equations by an efficient iterative method. His solutions are comparable to the best that we have come to expect from the positive-constrained restoring methods. 

All of the above conventions together permit the complete construction of the secular determinant. Using standard linear algebra methods, the MO energies and wave functions can be found from solution of the secular equation. Because the matrix elements do not depend on the final MOs in any way (unlike HF theory), the process is not iterative, so it is very fast, even for very large molecules (however, fire process does become iterative if VSIPs are adjusted as a function of partial atomic charge as described above, since the partial atomic charge depends on the occupied orbitals, as described in Chapter 9). 

Fig. 15.7 Conceptual illustration of the behavior of a Newton iteration on a nonlinear, stiff system of algebraic equations. A contour map of a norm of the residual vector F is plotted. The curvature represents nonlinear behavior, and the elongation represents disparate scaling, or stiffness. The desired solution of the problem is represented by the X the current iteration is marked by a dot. The elliptical contours represent residuals of the local linearization at the current iterate.

Because of large scale disparity the numerical solution to the locally linear problem at every iteration (Eq. 15.46) is highly sensitive to small errors. In other words very small variations in the trial solution y(m> or the Jacobian J cause very large variations in the correction vector Ay(m). From the linear algebra perspective, scale disparity can be measured by the condition number1 of the Jacobian matrix. As the condition number increases the... 

I will return to this diagram near the end of the chapter, particularly to amplify the meaning of error removal, which is indicated by dashed horizontal lines in Fig. 7.1. For now, I will illustrate the bootstrapping technique for improving phases, map, and model with an analogy the method of successive approximations for solving a complicated algebraic equation. Most mathematics education emphasizes equations that can be solved analytically for specific variables. Many realistic problems defy such analytic solutions but are amenable to numerical methods. The method of successive approximations has much in common with the iterative process that extracts a protein model from diffraction data. 

Associated with numerical problems is the concept of stability. A numerical scheme is stable when a solution is reached even with large time-steps (unsteady problems) or iteration steps (algebraic system of equations iteratively solved). Therefore the size of the time-step or of the iteration-step is dictated by stability requirements. It must be kept in mind that stability does not mean accuracy an implicit scheme of a dynamic problem is unconditionally stable but the solution obtained with large values of the time step may not be realistic. 

Note that in this case, the equation is solved for q, an algebraic solution, whereas solving for T (as required for the determination of TD24) results in a transcendental equation, since the heat release rate is an exponential function of temperature. This would require an iterative procedure. This heat release rate may serve as a reference for the extrapolation ... 

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