Double precision value

Of course, one needs a complementary facility in order to regenerate the ar bels from the double precision value here it is... 

IlyperChem uses 16 bytes (two double-precision words) of storage for each electron repulsion integral. The first 8 bytes save thecom-pressed four indices and the second S bytes store the value of the integral. Each index lakes 16 bits. Thus the maximum number of basis fiinctions is 65,535. This should satisfy all users of IlyperChem for the foreseeable future. 

The random number generator in all simulations was RAN3 (Press et al., 1992). Computations were made in FORTRAN 90. To reduce cancellation errors, theoretical values of r for PAR were computed in double precision. 

The lack of knowledge of precise values of the roughness factor makes it difficult to compare data reported from different studies. This applies in particular to the double-layer capacity data, the values of surface concentration of the adsorbates, and the rates of electrochemical reactions. Therefore, the question of how to determine the real surface of the electrode is of cmcial importance. A survey of various methods for determining roughness was given by Trasatti and Petrii. For noble metal electrodes, the charges of hydrogen deposition and surface oxide formation can be utilized in real-surface determination." ... 

The derivatives of In g and B of a single conformation which correspond to the y. torsional angle, have been calculated numerically using the double precision declaration as real 16. For this purpose, nine values of B and Ing around y,(Tr + Ay, y, + 2Ay. . . and y,- - Ay, y, - 2Ay. . . ) have been calculated. The grid has been defined for Ay = 10 rad The resulting nine values have been fitted to four termed Taylor series which were derived. The numerical derivatives are identical (up to 15 decimal numbers) to those obtained analytically with the C algorithm shown in Table 3. 

In this module, the two kinds of real values, respectively single precision (6 decimals or a little better) and what used to be called double precision (here defined as 14 decimals or a little better), are given the names sgl and dbl, and the constant pi is made a parameter. As well, the (real) parameter small is set up. This is useful in those cases where something is tested for being close to zero. 

The parameter eg is a small positive number such as 10-8. A reasonable choice is around the order of the square root of machine precision, em, defined as the smallest number x such that the floating point value of (1 + x) is greater than the floating representation of 1. This precision-dependent (i.e., double versus single) and machine-dependent quantity is approximately the value of the unit roundoff, or 2 ( +D for binary computer arithmetic involving t binary digits (or bits) in the fractional part of the number. For example, for double-precision computations on a DEC VAX, t = 52 and m 10 16. As computational errors will enter from sources other than finite arithmetic, a suitable eg is then a number greater than or equal to 10 8. 

Rwork is a double precision work array of length Lrw used by DDAPLUS. A value Lrw=5000 suffices for many medium-sized problems. If the input value Lrw is too small, (for example, if Lrw=l), DDAPLUS will promptly terminate and state the additional length required. Some of the contents of Rwork are summarized here the user may access these via the program MAIN. 

Rpar is a double precision work array, and Ipar is an integer work array. These are provided so that DDAPLUS can pass special information to the user-provided subroutines. The parameters whose sensitivity coefficients are desired must be stored in the Rpar array in the locations specified by the elements l,.Nspar of the array Ipar. For example, the input Ipar(2)=3 would indicate that the second sensitivity parameter was stored in Rpar(3). The location Ipar(O) is reserved for a user-defined constant, such as the address (later in Ipar) of a list of pointers to other information. The location Rpar(O) may be used to store values of a user-defined continuation parameter. 

SQRW.Double precision array used when JWT= 1 to hold the values... 

PAR.Double precision vector of parameter values 0i in models... 

Double precision 8 bytes -1.79769313486232E308 to. 94065645841247E-324 for negative values 4.94065645841247E-324 to 1.79769313486232E308 for positive values... 

Fat fractals answer a fascinating question about the logistic map. Farmer (1985) has shown numerically that the set of parameter values for which chaos occurs is a fat fractal. In particular, if r is chosen at random between r and r = 4, there is about an 89% chance that the map will be chaotic. Farmer s analysis also suggests that the odds of making a mistake (calling an orbit chaotic when it s actually periodic) are about one in a million, if we use double precision arithmetic ... 

Since the values of x, are raised to the fourth power in this calculation, a sum such as 5 can be very much smaller or very much larger than the sum S. As the operations of determinant diagonalization demonstrate, this means that double precision arithmetic must be used in solving the normal equations. 

The cleaning of all the matrices by making equal to zero all quantities which in absolute value are smaller than 10-17 (in double precision) must be rendered more systematic. In this way we expect the accumulation of errors to diminish noticeably. The introduction of the Newton method as well as Ait.ken s extrapolation and other standard techniques to speed up convergency may be suitable in future application. However, before having recourse to these standard convergence techniques, we hope to find other more fundamental basic conditions - such as the spin equation which smoothed the oscillations away — in order to attain a complete control of the process. The most important question which remains open is the way in which the renormalizations of the p-RDM s is performed. Another possible improvement is to extend the spin-adaptation to the renormalization of the 4-RDM in order to make a thorough use of the partial traces of the different symmetries. 

Verify that repeated use of the Power and Root (or Powerl and Rootl) macros now does lead to computational errors with relative values smaller than 10 15. This final example illustrates that, while the spreadsheet always (and automatically) treats data as double precision, the mathematical operations in VBA must be told specifically to do so ... 

PPP ZDO integrals, values from Tom Peacock s book p97 Integer function ppptwod, j, val) integer i, j double precision val double precision r, two ... 

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