Numerical methods linear equations

Dennis J E and Schnabel R B 1983 Numerical Methods for Unconstrained Optimization and Non-linear Equations (Englewood Cliffs, NJ Prentice-Hall)... 

The approximate method developed is constructive in the following sense. If A is a linear operator, then the equation (1.105) is linear too and, therefore, it can be solved by standard numerical methods. 

Unlike (2.166), the constructed iterative equation (2.167) is linear, which allows us to apply the standard numerical methods to solve it. 

These Uj may be solved for by the methods under Numerical Solution of Linear Equations and Associated Problems and substituted into Eq. (3-78) to yield an approximate solution for Eq. (3-77). 

The solution of the Laplace equation is not trivial even for relatively simple geometries and analytical solutions are usually not possible. Series solutions have been obtained for simple geometries assuming linear polarisation kinetics "" . More complex electrode kinetics and/or geometries have been dealt with by various numerical methods of solution such as finite differencefinite elementand boundary element. ... 

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. 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

The solution phase has been characterized in the past by a concentration on methods to obtain analytic solutions to the mathematical equations. These efforts have been most fruitful in the area of the linear equations such as those just given. However, many natural phenomena are nonlinear. While there are a few nonlinear problems that can be solved analytically, most cannot. In those cases, numerical methods are used. Due to the widespread availability of software for computers, the engineer has quite good tools available. 

A set of linear equations can be solved by a variety of procedures. In principle the method of determinants is applicable to any number of equations but for large systems other methods require much less numerical effort. The method of Gauss illustrated here eliminates one variable at a time, ends up with a single variable and finds all the roots by a reverse procedure. 

Although K appears linearly in both response equations, rx in (2.12) and rx and r2 in (2.13) appear nonlinearly, so that nonlinear least squares must be used to estimate their values. The specific details of how to carry out the computations will be deferred until we take up numerical methods of unconstrained optimization in Chapter 6. 

J.E.Dennis, R.B. Schnabel Numerical methods for unconstrained optimization and non-linear equations, Prentice Hall Series in computational mathematics, New York, (NY, USA), 1983... 

Some of the simple linear equations that we will simulate can, of course, be solved analytically by the methods covered in Part III to obtain general solutions. The nonlinear equations cannot, in general, be solved analytically, and computer simulation is usually required to get a solution. Keep in mind, however, that you must give the computer specific numerical values for parameters, initial conditions, and forcing functions. And you will get out of the computer specific numerical values for the solution. You cannot get a general solution in terms of arbitrary, unspecified inputs, parameters, etc., as you can with an analytic solution... 

In our numerical model, Eq.(2.8) was transformed into a six-point finite-difference equation using the alternative direction implicit method (ADIM). At the edges of the computational grid (—X,X) radiation conditions were applied in combination with complex scaling over a region x >X2, where —X X j) denotes the transverse computational window. For numerical solution of the obtained tridiagonal system of linear equations, the sweep method" was used. 

The numerical values of a, b, and c can be found by direct substitution in the algebraic expressions if care is taken to carry an apparently excessive number of significant figures through the calculations, which involve taking small differences between large numbers. Alternatively, the determinants in Equations (A.9)-(A.ll) can be evaluated by methods described in the references, or the linear equations, (A.6)-(A.8) can be solved by matrix methods (2). 

Since the exact solution of the Hartree-Fock equation for molecules also proved to be impossible, numerical methods approximating the solution of the Schrodinger s equation at the HF limit have been developed. For example, in the Roothan-Hall SCF method, each SCF orbital is expressed in terms of a linear combination of fixed orbitals or basis sets ((Pi). These orbitals are fixed in the sense that they are not allowed to vary as the SCF calculation proceeds. From n basis functions, new SCF orbitals are generated by... 

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. 

Detonation, Nonlinear Theory of Unstable One-Dimensional. J.J. Erpenbeck describes in PhysFluids 10(2), 274-89(1969) CA 66, 8180-R(1967) a method for calcg the behavior of 1-dimensional detonations whose steady solns are hydrodynamic ally unstable. This method is based on a perturbation technique that treats the nonlinear terms in the hydro-dynamic equations as perturbations to the linear equations of hydrodynamic-stability theory. Detailed calcns are presented for several ideal-gas unimol-reaction cases for which the predicted oscillations agree reasonably well with those obtd by numerical integration of the hydrodynamic equations, as reported by W. Fickett W.W. Wood, PhysFluids 9(5), 903-16(1966) CA 65,... 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

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