Floating-point numbers

Basically, two different methods arc commonly used for representing a chemical struchiive in 3D space. Both methods utilize different coordinate systems to describe the spatial arrangement of the atoms of a molecule under con.sidcration. The most common way is to choose a Cartesian coordinate system, i.e., to code the X-, y-, and z-coordinates of each atom, usually as floating point numbers, For each atom the Cartesian coordinates can be listed in a single row. giving consecutively the X-, )> , and z-valnc.s. Figure 2-90 illustrates this method for methane. 

A lustrous metal has the heat capacities as a function of temperature shown in Table 1-4 where the integers are temperatures and the floating point numbers (numbers with decimal points) are heat capacities. Print the curve of Cp vs. T and Cp/T vs. T and determine the entropy of the metal at 298 K assuming no phase changes over the interval [0, 298]. Use as many of the methods described above as feasible. If you do not have a plotting program, draw the curves by hand. Scan a table of standard entropy values and decide what the metal might he. 

The format markers in Eile 4-lb are at intervals of five spaces each. Thus the entire file might be thought of as a 10 x 67 matrix with row 1 containing the integers 6 and 4 in columns 65 and 67 and zeros elsewhere. (In EORTRAN, a blank is read as a zero.) Row 5 has the floating point number 2. in columns 4 and 5. Both the 2 and the. (decimal point) occupy a column. Row 5 column 35 contains the integer 2 and so on. 

On a vector computer having vector registers that hold 64 floating-point numbers, this loop would be processed 64 elements at a time. The first 64 elements of Y would be fetched from memory and stored in a vector register. Each iteration of the loop is independent of the previous iteration, so this loop can be fliUy pipelined, with successive iterations started every clock cycle. Once the pipeline is filled, the result, X, will be produced one element per clock cycle and will be stored in another vector register. The results in the vector register will then be stored back into main memory or used as input to a subsequent vector operation. 

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. 

Data management in RS/1 is based on two-dimensional tables. Each cell of a table can contain data representing fixed or floating point numbers, dates, times, or free text. Cells in a particular column are not all constrained to the same type it is possible, for example, to include a note about some missing data in a column of numerical results. A user can work with many hundreds of tables. Tables are based on disk files, accessed through a kind of paging scheme, so there is no limit on table size. Some users work with tables containing hundreds of columns and tens of thousands of rows. 

Give a crude estimate of the relative errors of the columns of H floating-point numbers are stored to 7 digits. 

The TMS320C30 [Papamichalis and Simar, 1988] follows the basic architecture of the TMS-320 series. Unlike the DSP-32, it uses pipeline interlocks. Like the DSP-32, it features its own internal format for floating point numbers. Because of the four stage pipeline organization, it can perform a number of operations in parallel. It also features a delayed branch - something of a novelty in DSP processors. The TMS320C40 [Simar et al., 1992] has six parallel bidirectional I/O ports controlled by DMA on top of the basic TMS-320C30 architecture. These ports have been used for multiprocessor communication. 

Both types of calculations can require very large data arrays - to more than 10 floating point numbers, and future arrays may be even 1 to 2 order of magnitude larger. 

The second possibility for improving microcomputer floating point performance lies with the North Star Hardware Floating Point Board (3). This device executes floating point, add, subtract, multiply and divide with up to twelve decimal digits of precision. One byte of data is reserved for the exponent and the other six for the mantissa of each floating point number. 

Figure 2. The floating point number representation format used by the Am9511A.

The Am9511A internal stack can hold four floating point numbers in the case under consideration the top of stack holds say, Ax and the other three slots are empty. The Am9511A allows us to push the stack and duplicate Ax at the new top of stack, so that the stack now contains Ax, Ax and two empty slots. An FMUL operation to the Am9511A will result in the stack containing Ax2 and three empty slots. If we note that an interatomic distance = (Ax2 + Ay2 + Az2)35 then it can be seen that the extension... 

Use MOD(number,1) to return the decimal part of a floating-point number. Example =MOD(2.3333,2) returns 0.3333. =MOD(2.3333,3) returns... 

Three limits apply to floating-point numbers precision, accuracy, and magnitude. Floating-point numbers are allowed up to nine digits of precision. Go beyond nine, and your computer automatically rounds to the nearest nine-digit number. 

Magnitude is the final limit. It s the culprit in OVERFLOW errors. The operating system stores floating-point numbers in five bytes. What happens when all the bytes fill up The number is a little beyond 10 to the thirty-eighth power, a one followed by 38 zeros the computer cannot count any higher. 

In order to deal with roundoff errors due to the use of SP floating-point numbers on the GPU, Yasuda introduced a scheme in which the XC potential is approximated with a model potential // del which is chosen such that its matrix elements can be calculated analytically. This is done in DP on the CPU while the GPU is used for calculating the correction, that is, for the numerical quadrature of the matrix elements of Ai xc = Without the model potential, errors... 

Note that fsolve returns a floating point number with a decimal point. 

Traditional methods of simulation in hydrodynamics are based on the description of a fluid field obeying to partial differential equations. Finite difference, finite elements, spectral methods are generally used to approximate the equations and they are represented in the computer by floating point numbers. The implementation of the boundary conditions is the main difficulty of these methods. 

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