Floating-point comparisons

Program 4.13 Using the ROUND Function with Floating-Point Comparisons... 

Notice how the lab value variable has been rounded to an arbitrary precision to the ninth decimal place. This gets around the floating-point comparison problem, while at the same time it avoids rounding the lab value variable to a precision where meaningful data are lost. 

At times you need to perform some form of floating-point number comparison. The following example illustrates such a comparison where you are setting laboratory value flags to indicate whether a lab test is above or below normal. 

You should read Technical Support Note TS-230 Dealing with Numeric Representation Error in SAS Applications to learn more about SAS floating-point numbers and storage precision in SAS. Another good resource for rounding issues is Ron Cody s SAS Functions by Example (SAS Press, 2004). In short, whenever you perform comparisons on numbers that are not integers, you should consider using the ROUND function. 

Fig. 4.4. Comparison of the computing effort, expressed in thousands of floating point operations (Aflop), required to factor the Jacobian matrix for a 20-component system (Nc = 20) during a Newton-Raphson iteration. For a technique that carries a nonlinear variable for each chemical component and each mineral in the system (top line), the computing effort increases as the number of minerals increases. For the reduced basis method (bottom line), however, less computing effort is required as the number of minerals increases.

Plane waves have infinite lateral extent and, for this reason, cannot be simulated on a computer because of floating-point overflow. If the lateral extent is constrained, as in Problem 6.11 of Jackson [5], longitudinal solutions appear in the vacuum, even on the U(l) level without vacuum charges and currents. This property can be simulated on the computer using boundary conditions, for example, a cylindrical beam of light. It can be seen from a comparison of Eqs. (625) and (629) that if the Lorenz condition is not used, there is no increase... 

The estimated full commercial cost of an MVP-9500/8 system based on a Vector MZ with 5M bytes of disk space, printer and terminal is around 10,400, compared with around 25,000 for a PDP11/34 with similar peripherals and a floating point processor. This makes the vector processor option around eight times more cost effective than the PDP-11 PROVIDED THAT THE CALCULATION UNDER CONSIDERATION CAN MAKE EFFICIENT USE OF A VECTOR PROCESSOR. A similar comparison of the MVP-9500/128 and AP120B puts the former ahead by a factor of two in cost effectiveness. 

Bit by bit comparison except for floating point variables whose fractions both evaluate as zero. (Refer to text for consequences in SECS program.)... 

When Statements 1 and 2 are used to calculate the values of two small numbers, x and y , one of them ( y ) is incorrectly rounded off to zero, whereas the smaller of the two ( x ) still retains nonzero value. Thereafter, all subsequent comparisons of x and y lead to unexpected and incorrect results. For example, the results of comparison in Statements 4 and 5 are definitely wrong, whereas the comparison in Statement 3 may or may not lead to an incorrect conclusion. There are some techniques to overcome these issues. For example, Matlab provides a variable eps that indicates the smallest floating point increment possible in a given precision. Statements 9 and 10 use eps in the context of comparing the difference of two numbers, as well as in deciding whether inequality comparisons can be made, and produce more predictable results. 

The performance of various methods for evaluation of ERls can be easily assessed by comparing their floating-point operations (FLOP) counts. Such a comparison for the (pp pp) integral block, based on published data, is presented in Table 2 for different degrees of contraction. It is obvious that none of the methods is optimal for the entire range of M. 

Wf loat-equal Check for exact equality comparison between floating point values. The usage of equal comparator on floats is usually misguided due to the inherent computational errors of floats. 

The minimum number of floating point operations are based on very special optimizations, which are only possible for the benchmark system used, i.e. a serial chain with rigid bodies connected by revolute joints. These well-known results show that a multibody system program can be made more efficient by choosing an adequate formalism. As Fig. 1 shows, the eflficiency of a formalism depends strongly on the system it is applied to. Unfortunately, there are only a few comparisons for some formalisms on only some benchmark problems. To find an optimal formalism for a given problem, it would therefore be helpful to have a better comparison of formalisms on a wider variety of systems and to have an environment to program and modify comfortably formalisms and to apply them to a particular system. 

The establishment of a floating-point computation in a security function requires the implementation of a security technique. The introduction of a partial redundancy of the floating-point computation requires a diversity, which might, for example, use different libraries. Therefore, it is necessary to accept certain errors during comparison. 

From tha preceding remarks, the cause of ice— solidified or crystallized water—floating in water will be readily understood. It has been shown that at the point of solidification the liquid has the same, if not a lesser density than at 40° but, in passing to the Solid state, the gravity is much further reduced, as well by the arrangement which its particles assume, as by the expulsion of the gases dissolved in the water, and which, before they can escape, are enveloped and compressed within the solid crystal. From both, and perhaps other conjoint causes, the density of ice at a temperature of 32° is less than that of water at 212°, and hence the former floats in the latter. At the normal degree at which chemists are accustomed to compare the densities of bodies, namely 60s, and a barometric pressure of the atmosphere of thirty inches, pure water is taken as the standard of comparison, and is expressed by unity, or 1000—compared to this ice has a density of 0-916. 

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