Linear equations, solution

The stability of x t) is determined by whether the solution wiU return back to Xg, yg following a perturbation. This will occur if the solutions x are stable, and this requires that the real part of A. be negative. This can also be regarded as a problem of finding the eigenvalues of the matrix [ j] from the Jacobean of the original linear equations. Solutions of x t) can be as shown in Figure 6-5. This pair of equations is stable if [(a 8) + 4 y] > 0 (A real). When we insert coefficients a, defined,... 

The Jacobian of the system is a square matrix, but importantly, because the residuals at any mesh point depend only on variables at the next-nearest-neighbor mesh point, the Jacobian is banded in a block-tridiagonal form. Figure 16.10 illustrates the structure of the Jacobian in the form used by the linear-equation solution at a step of the Newton iteration,... 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

Laplacian, symmetry of, 13 transformation properties of, 9 Legendre polynomials, 144 Linear equations, solutions of, 42 Linear momentum operator, symmetry of, 167... 

Solution of one non-linear equation Solution of a set of non-linear equations... 

This reduces the calculation at each step to solution of a set of linear equations. The program description and listing are given in Appendix H. 

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

The solution Xh(t) of the linearized equations of motion can be solved by standard NM techniques or, alternatively, by explicit integration. We have experimented with both and found the second approach to be far more efficient and to work equally well. Its handling of the random force discretization is also more straightforward (see below). For completeness, we describe both approaches here. 

C. Lanczos. Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bureau Standards, 49 33-53, 1952. 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

Equation (2.45) represents the weighted residual statement of the original differential equation. Theoretically, this equation provides a system of m simultaneous linear equations, with coefficients Q , i = 1,... m, as unknowns, that can be solved to obtain the unknown coefficients in Equation (2.41). Therefore, the required approximation (i.e. the discrete solution) of the field variable becomes detemfined. 

Iterative improvement of the solution of systems of linear equations... 

One is allowed to linearly eombine solutions of the Sehrodinger equation that have the same energy (i.e., are degenerate) beeause Sehrodinger equations are linear differential... 

The example demonstrates that not all the B-numbers of equation 5 are linearly independent. A set of linearly independent B-numbers is said to be complete if every B-number of Dis a product of powers of the B-numbers of the set. To determine the number of elements in a complete set of B-numbers, it is only necessary to determine the number of linearly independent solutions of equation 13. The solution to the latter is well known and can be found in any text on matrix algebra (see, for example, (39) and (40)). Thus the following theorems can be stated. 

Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P x)y = Q x). Its general solution is... 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

This method requires solution of sets of linear equations until the functions are zero to some tolerance or the changes of the solution between iterations is small enough. Convergence is guaranteed provided the norm of the matrix A is bounded, F(x) is Bounded for the initial guess, and the second derivative of F(x) with respect to all variables is bounded. See Refs. 106 and 155. 

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). 

Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of us be computed. Thus the gaussian integration formulas are useful because of the economy they offer. See references on numerical solutions of integral equations. 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

According to the Floquet theorem [Arnold 1978], this equation has a pair of linearly-independent solutions of the form x(z,t) = u(z, t)e p( 2nizt/p), where the function u is -periodic. The solution becomes periodic at integer z = +n, so that the eigenvalues e we need are = ( + n). To find the infinite product of the we employ the analytical properties of the function e z). It has two simple zeros in the complex plane such that... 

The practical way of calculating 2 is different from that used in the derivation of (4.18). Since 2 is invariant with respect to canonical transformations, it is preferable to seek it in the initial coordinate system. Writing the linearized equation for deviations from the instanton solution 6Q,... 

Similar treatment of the Knox equation does not predict that values of H(min) should be independent of the solute diffusivity neither does it predict that (uopt) should vary linearly with solute diffusivity. Consequently, the relationships shown in Figures 5... 

The principal use of the inverse matrix is in solution of linear equations or the application of transformations. If... 

A linearized, acoustic approach was found satisfactory for the description of the near-piston region for low piston Maeh-numbers by Guirao et al. (1976) and Gorev and Bystrov (1985). The linearized equations, however, provided a single solution at the location of the leading shock. 

Solution of Sets of Simultaneous Linear Equations 71. Least Squares Curve Fitting 76. Numerical Integration 78. Numerical Solution of Differential Equations 83. 

Because of round off errors, the Regula Falsa method should include a check for excessive iterations. A modified Regula Falsa method is based on the use of a relaxation factor, i.e., a number used to alter the results of one iteration before inserting into the next. (See the section on relaxation methods and Solution of Sets of Simultaneous Linear Equations. )... 

Gauss-Siedel method is an iterative technique for the solution of sets of equations. Given, for example, a set of three linear equations... 

The coefficients CK for a solution to the Schrodinger equation (Eq. II. 1) may now be determined by the variation principle (Eq. II.7) which leads to an infinite system of linear equations... 

---