Roundoff errors

An even better handling of the velocities is obtained by another variant of the basic Verlet integrator, known as the velocity Verlet algorithm. This is a Verlet-type algorithm that stores positions, velocities, and accelerations all at the same time t and minimizes roundoff errors [14]. The velocity Verlet algorithm is written... 

Since equation 8.4 can be trivially solved for Oi t - 1) (= 4 [aj t) A/)] 0/c ai t + 1)), we see that any pair of consecutive configurations uniquely specifies the backwards trajectory of the system. Moreover, this statement holds true for arbitrary (and, in particular, irreversible) functions < ). An important consequence of this, first pointed out by Fredkin [vich84a], is that a numerical roundoff in digital computers need not necessarily result in a loss of information. In particular, if the computation is of the form given by equation 8.4, where resulting dynamics will nonetheless be reversible and no information will be lost throughout the computation. ... 

In addition to the chemical inferences that can be drawn from the values of AS and AH, considered in Section 7.6, the activation parameters provide a reliable means of storing and retrieving the kinetic data. With them one can easily interpolate a rate constant at any intermediate temperature. And, with some risk, rate constants outside the experimental range can be calculated as well, although the assumption of temperature-independent activation parameters must be kept in mind. For archival purposes, values of AS and AH should be given to more places than might seem warranted so as to avoid roundoff error when the exponential functions are used to reconstruct the rate constants. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

The actual NBO lone-pair hybridizations are found to be 3.64, 0.87, and 0.25, respectively, nearly identical to the above estimates (within expected roundoff errors). 

As a consequence, the gradient of the objective function and the Jacobian matrix of the constraints in the nonlinear programming problem cannot be determined analytically. Finite difference substitutes as discussed in Section 8.10 had to be used. To be conservative, substitutes for derivatives were computed as suggested by Curtis and Reid (1974). They estimated the ratio /x of the truncation error to the roundoff error in the central difference formula... 

For reasons of roundoff errors due to water being the dominant species, a simplification is introduced for dilute solutions (Morel and Hering, 1993). One unknown and one equation are simultaneously eliminated from the set of conservation equations which make the recipe. The unknown species HzO is expressed as a function of the unknowns OH- and H+, which assigns OH- a — 1 H+ coefficient, and the OH- conservation equation (first column in Table 6.1) is left out. 

Because equilibrium constants may often be of widely different orders of magnitude, solving some problems may lead to roundoff errors and poor accuracy. 

Since the constant terms on the right-hand side of the previous equations vary by some 17 orders of magnitude, we may face serious roundoff errors in estimating/(x). One way of scaling the equations is to divide each conservation equation by the total amount of the corresponding component and each mass action relation by the corresponding equilibrium constant. If the choice of the initial estimate x<0) is not too awkward, we should obtain the six equations as differences between numbers of more similar magnitudes (ideally unity for all but the electroneutrality condition)... 

The A (K ) and the corresponding RE values are indicated with a precision that is not warranted by our actual knowledge. They are used as indicated primarily in order to facilitate recalculations without being continually bothered by roundoff errors.]... 

At this point, we note that there is no mechanism presently built into the relaxation methods to prevent undesirable high-frequency noise from growing with each iteration. Any spurious solution 6(x) satisfies Eq. (1) (see also Chapter 1, Sections V.A and V.B) for co beyond the band limit. If we know that the object 6 is truly band limited, with frequency cutoff co = 2, we can band-limit both data i and first object estimate d(1). The relaxation methods cannot then propagate noise having frequencies greater than Q into an estimate o(k). (One possible exception involves computer roundoff error. Sufficient precision is usually available to avoid this problem.)... 

First let us deal with deconvolution in general. We have a few admonitions to the reader of a literature report on a new method. They should ask, does the writer deal fairly with noise Even the most volatile of the linear methods can produce a reasonable restoration when noise is limited to roundoff error in the seventh significant figure of the data. A method s capability of yielding acceptable restorations in the presence of realistic noise is critical to its practicality. 

Multiplying (1.104) by X we have (X X)u = 0, and thus there exists an affine linear dependence of the form (1.103) among the columns of Y if and only if the matrix X X has a i = 0 eigenvalue. It is obvious that Xmin will equal not zero, but some small number because of the roundoff errors. 

Do2,. To obtain a similar upper bound on xmin in the case (i), when there are only roundoff errors present, Box at al. (ref. 21) suggested to assume that the rounding error is distributed uniformly with range -0.5 to +0.5 of the last digit reported in the data. The rounding error... 

These partial derivatives provide a lot of information (ref. 10). They show how parameter perturbations (e.g., uncertainties in parameter values) affect the solution. Identifying the unimportant parameters the analysis may help to simplify the model. Sensitivities are also needed by efficient parameter estimation procedures of the Gauss - Newton type. Since the solution y(t,p) is rarely available in analytic form, calculation of the coefficients Sj(t,p) is not easy. The simplest method is to perturb the parameter pj, solve the differential equation with the modified parameter set and estimate the partial derivatives by divided differences. This "brute force" approach is not only time consuming (i.e., one has to solve np+1 sets of ny differential equations), but may be rather unreliable due to the roundoff errors. A much better approach is solving the sensitivity equations... 

Since the actual data contains noise and computational roundoff errors, additional nonzero eigenvalues (noise eigenvalues) will be generated by the computation. The theory shows that the eigenvalues can be grouped into two sets a set which contains the factors or components together with an error contribution and a secondary set composed entirely of error. 

Unfortunately it is not going to be easy to test experimentally or even to simulate on the computer. The reason is the extreme sensitivity of states of high n to external perturbations. In the laboratory, stray electrical fields, which cannot be completely avoided (or black-body radiation) will cause ionization of these states. Even on the computer, numerical roundoff errors will act as external noise. 

In order to evaluate this equation, we need a value for cfe. Kontro, et al. [Kontro et al., 1992] used the Probability Distribution Function (PDF) of floating point roundoff error in multiplication ... 

Figure 6.2. Since the peak values Tmi for this model as well as the first onset temperature Tol are clear from the data, they were pinned at their correct values. Estimates of C = 2W/K, A = 1 C-1, To2 = 225°C, C2 = 8 W/K, and D2 = 1 °C 1 were used as seed values for the program. Within about 10 minutes, the program settled on values for the coefficients which matched the coefficients used to create the simulated data, within single precision roundoff error. Each term in the sum, representing the deconvoluted endotherms, is plotted in the figure as dotted lines.

Further Hylleraas-type calculations with basis sets of increasing size and sophistication, culminating with the work of Pekeris and coworkers in the 1960 s (see Accad, Pekeris, and Schiff [26]) showed that nonrelativistic energies accurate to a few parts in 109 could be obtained by this method, at least for the low-lying states of helium and He-like ions. However, these calculations also revealed two serious numerical problems. First, it is difficult to improve upon this accuracy of a few parts in 109 without using extremely large basis sets where roundoff error and numerical linear dependence become a problem. Second, as... 

The global error results from the addition of discretization and roundoff errors. Therefore, at least theoretically, there is an optimum step size for which the global error is minimum. However, for most computers and for stable algorithms (see below), in general, round-off errors may be neglected so that, essentially, discretization errors will be considered. 

When n = 1/2, the result is Ji/2(x) = (2/jrx)l/2sin x. From the recurrence relations, it can be found that J.1/2 = (2/jtx),/2cos x. The recurrence relation Jn+1 = (2n/x)J - Jn j can the be employed to discover all the other functions of half-integral index. Numerical calculations using recurrence equations are easily impaired by roundoff error, since the error can propagate through successive recurrences. 

In the basic descent algorithm [Al], we iterate until convergence. How exactly do we evaluate the optimality cf an approximate minimum x Furthermore, how do we ensure that computations will not continue unnecessarily (a) when no further progress can be realized or (b) beyond attainable accuracy The accuracy depends on machine precision and cumulative roundoff errors, in addition to algorithmic details. 

When E2 > 0, it is convenient, in order to avoid roundoff errors, to use a modification of the classical quadratic equation solution, such as ... 

---