Roundness error

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

More digits are retained in such presentations than are required to express the experimental precision in order that rounding errors be minimized. 

But in order for a matrix to have a multiple root, it is necessary that its elements satisfy a certain algebraic relation to have a triple root they must satisfy two relations, and so forth for roots of higher order. Thus, if a matrix is considered as a point in 2-space, only those matrices that lie on a certain algebraic variety have multiple roots. Clearly, if the elements of a matrix are selected at random from any reasonable distribution, the probability that the matrix selected will have multiple roots is zero. Moreover, even if the matrix itself should have, the occurrence of any rounding errors would almost certainly throw the matrix off the variety and displace the roots away from one... 

As a practical matter, for keeping rounding errors as small as possible, it is to be observed that since the dements just below the diagonal... 

Regarding x3 = x 0 as a new starting value, repeat to form x3 = x"Q and continue. It can be shown that if the initial iteration is of order k, that of the iteration that produces x0,x 0,Xq, is of the order 2k — 1. Evidently the transformation could be applied again, using the terms x0, x 0, and x2 of the derived sequence to produce the initial term of a new derived sequence. However, new sources of rounding error are introduced in this process, and the finally accepted approximation should be a result < ( ) of substituting a into the basic iteration formula (2-17). 

A note on good practice Note how, to minimize rounding errors, we carry out the calculation in a single step. However, to help guide you through the calculation, we often give intermediate numerical results in the examples. 

A note on good practice Note that although, as usual, we have left the numerical calculation to a single, final step, the same is not true for the units canceling units docs not introduce rounding errors and clarifies each step. 

STRATEGY Begin by writing the chemical equation for the complete oxidation of octane to carbon dioxide and water. Then calculate the theoretical yield (in grams) of CO, by using the procedure in Toolbox L.l. To avoid rounding errors, do all the numerical work at the end of the calculation. To obtain the percentage yield, divide the actual I mass produced by the theoretical mass of product and multiply by 100%. 

STRATEGY We expect a positive entropy change because the thermal disorder in a system increases as the temperature is raised. We use Eq. 2, with the heat capacity at constant volume, Cv = nCV m. Find the amount (in moles) of gas molecules by using the ideal gas law, PV = nRT, and the initial conditions remember to express temperature in kelvins. Because the data are liters and kilopascals, use R expressed in those units. As always, avoid rounding errors by delaying the numerical calculation to the last possible stage. 

A note on good practice F.xponential functions are very sensitive to rounding errors, so it is important to carry out the numerical calculation in one step. A common error is to forget to express the enthalpy of vaporization in joules (not kilojoules) per mole, but keeping track of units will help you to avoid that mistake. 

A note on good practice Exponential functions (inverse logarithms, e ) are very sensitive to the value of x, so carry out all the arithmetic in one step to avoid rounding errors. 

The values for K, listed here have been calculated from pK, values with more significant figures than shown so as to minimize rounding errors. Values for polyprotic acids—those capable of donating more than one proton—refer to the first deprotonation. 

This is identical within rounding error to the exit concentration in Example 10.9. 

It is necessary to point out that calculations by these formulae may induce accumulation of rounding errors arising in arithmetic operations. As a result we actually solve the same problem but with perturbed coefficients A-i, Bi, Ci, Xj, Xj and right parts Fi, /Ij, /jj. If is sufficiently large, the growth of rounding errors may cause large deviations of the computational solution yi from the proper solution j/,-. 

The trivial example shows how instability may arise in the process of calculations of yi by the formula t/i+i — qyi, q > One expects that, for any y, there exists a number Uq such that overflow occur for t/ = q y n = (, thus causing a abnormal termination. An important obstacle in dealing with this problem is that yi satisfies the equation yi y = qyi + rj with a rounding error rj. Indeed, for the error S yy = iji — yi the equation is valid ... 

We quote below the results of computations for problem (3) with j/q = 1 and j/j =82, where is the smallest root to the quadratic equation (4). Once supplemented with those initial conditions, the exact solution of problem (3) takes the form j/, = i s (A = 0). Because of rounding errors, the first summand emerged in formula (5). This member increases along with increasing i, thus causing abnormal termination in computational procedures. 

Because of rounding errors, is determined with some error e still subject to the approved decomposition... 

In both cases 1 — rS < 1 and, therefore, for a solution of problem (15)-(16) estimate (14) is valid, but r > 2/A. Because of rounding errors, the computational process is unstable for large j the growth of its solution causes abnormal termination in the computer realization of the algorithm. 

In this regard, rounding errors can be treated as possible perturbations of the right-hand side of equation (1) at every step. The iteration scheme (14) with parameters (29), (41) or (42) becomes unstable with respect to the right-hand side by exactly the same reasoning as before the norm of the operator Sk = E — Tf.A for the transition from the k — l)th iteration to the fcth iteration may exceed 1 for negative values of tf., since... 

Re-ordering of iteration parameters. With regard to scheme (14) one interesting problem arises in connection with re-ordering of the iteration parameters r so as to minimize as much as possible the influence of rounding errors and to avoid large intermediate values dependent on n. 

Murtagh (M9) pointed out that rounding errors and storage limitations restrict the applicability of such techniques to networks of approximately 100 pipe sections or less. As an alternative he proposed to solve the following dual problem ... 

It is convenient to set Ap1 = 1, L = d - -dq = 1. Rounding errors are suppressed by replacing die intensity by 1/s2 (Porod s law) for big arguments (s > 8). A smooth phase transition zone (in all the example curves dz = 0.1) is considered by multiplication with exp (2nsdz/3)2 j. From this one-dimensional scattering intensity die correlation function is obtained by Fourier transformation. 

Yes, there is the same number of P atoms in 31.6 g of pure phosphorus, regardless of whether the phosphorus is in the form of P4 or P2. The difference is due to rounding error only. 

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