Use of the Digital Computer

Both the numerical and the analytical methods discussed in this chapter can be tedious to carry out, especially with large collections of precise data. Fortunately, the modem digital computer is ideally suited to carry out the repetitive arithmetic operations that are involved. Once a program has been written for a particular computation, whether it be numerical integration or the least-squares fitting of experimental data, it is only necessary to provide a new set of data each time the computation is to be calculated. 

Although the details of computer programming are beyond the scope of this text, the student unfamiliar with the subject is urged to consult one of the many books available (8). Many programs designed to carry out the calculations described in this section are available commercially for use on desktop personal computers. 

Many calculations needed to produce the tables of data and results in this text can conveniently be carried out without programming with the use of a spreadsheet, several of which are available for personal computers. 

---

Hall, L. H. (ed.), Structure of Crystalline Polymers , Elsevier, New York, 1984. The book emphasizes the experimental techniques used to study polymer structure, including use of the digital computer, wide angle X-ray diffraction electron microscopy, neutron scattering and X-ray scattering. 

It is clear that to develop an explicit algebraic expression for (tj-) or (n) would be exceedingly cumbersome and, as already stated, in this modern day of the digital computer, is unnecessary. A simple program can be written, using equations (6), (7), (8) and (9), that searches for the time (tr) that allows the equivalence defined in... 

The method is quite effective, but is not widely used now because of the ubiquity of digital computers. Zuman and Patel - 36. show circuit designs for some kinetic schemes. Williams and Bruice made good use of the analog computer in their study of the reduction of pyruvate by 1,5-dihydroflavin. In this simulation eight rate constants were evaluated variations in these parameters of 5% yielded discemibly poorer curve fits. 

The integral P can rarely be evaluated analytically, and the use of a digital computer is required. 

Simulation by means of the digital computer has become an extremely useful technique (see Section 3.7) that goes far beyond classical interpolation/ extrapolation. The reasons for this are fourfold ... 

This is simpler than 7.1-15, in that many of the terms in the lOth-order polynomial equation which will result on expanding the determinant will now be equal to zero. Nevertheless, the basic, awkward fact is that a lOth-order equation still has to be solved. This is not a task to be confronted with pleasurable anticipation without the use of a digital computer it would be a protracted, tedious job. Fortunately, in this case and all others in which the molecule possesses symmetry, the secular equation can be factored—that is, reduced to a collection of smaller equations—by using the symmetry properties in the right way. The method of symmetry factoring will now be explained and illustrated. 

Step 5 in Table 8-6 involves the computation of the optimum point. Quite a few techniques exist to obtain the optimal solution for a problem. We describe several classes of methods below. In general, the solution of most optimization problems involves the use of a digital computer to obtain numerical answers. Over the past 15 years, substantial progress has been made in developing efficient and robust computational methods for optimization. Much is known about which methods are most successful. Virtually all numerical optimization methods involve iteration, and the effectiveness of a given technique can depend on a good first guess for the values of the variables at the optimal solution. After the optimum is computed, a sensitivity analysis for the objective function value should be performed to determine the effects of errors or uncertainty in the objective function, mathematical model, or other constraints. 

The development of ab initio methods, which has come to be known as quantum chemistry, is one of the outstanding cumulative intellectual and technical achievements of the past fifty years. Fifty years ago, at the time of the publication of Herzberg s classic book [14], quantum mechanics had been applied to the calculation of the wave functions of the very simplest molecules, but for most systems the problems were considered intractable. At the turn of the millenium we have come a long way, but many difficult problems remain. Progress has been closely related to the development of the digital computer, and the technical achievements in the use of computers to solve problems in quantum mechanics have been impressive. There has always, however, been an accompanying and continual need for intellectual advances to make use of the technology. This section describes the nature of the problems to be solved, and some of the methods which have been developed to tackle them. 

By use of a digital computer, develop and present a printout of the data of effective interest versus nominal interest compounded continuously as given in Table 2. 

This is the principle which we will invoke in every case to set up a functional equation. It appears in a form that is admirably suited to the powers of the digital computer. At the same time, every device that can be employed to reduce the number of variables is of the greatest value, and it is one of the attractive features of dynamic programming that room is left for ingenuity in using the special features of the problem to this end. 

Data Acquisition and Processing System. The data acquisition and processing system used in conjunction with the transducer mentioned above is one example of use of small digital computers in data acquisition, signal analysis, computation and experimental control in viscoelastic measurements. We will restrict the description to the DAPS used in conjunction with the MLR apparatus. Excellent description of a more general use of this method is given by Birnboim et al. 17). 

This Report constitutes a manual for the use of a digital computer program to solve complex chemical composition problems. The circumstances of the program s development at RAND over the last 15 years reflect a phase in the development of applied mathematics to which RAND has made an important contribution. A new mathematical discipline arose after World War II as a result of theoretical mathematicians being drawn into the war effort. "Mathematical programming," concerned with optimization, represented a blending of applied and theoretical mathematics. RAND, with many of its staff from the ranks of operations researchers, became a leader in the new discipline, particularly in its branch of linear programming. 

Hie preceding example shows that accurate representation of fuel-cyde performance requires such detailed examination of the time-varying dianges in power distribution and fuel composition at many points in a nuclear reactor as to necessitate use of a digital computer. The purpose of this section is to develop an approximate procedure for calculating fuel-cycle performance that, although complicated, can be carried out by hand calculation. The FWR described in Sec. 4.1 will be used as example, except that its rated electric output is taken as 10S4 MWe instead of 1060 MWe. 

Today, computer simulation is used extensively to analyze the dynamics of chemical processes or aid in the design of controllers and study their effectiveness in controlling a given process. Analog and digital computers have been used for this purpose, with the emphasis having shifted almost entirely in favor of the digital computers. 

As Kc increases, we need an iterative, trial-and-error, numerical procedure to find the roots of the characteristic equation. Such a solution is feasible through the use of a digital computer. Table 15.1 shows how the locations of the four roots change with the value of Kc. These results have been transferred in Figure 15.7, which displays the four branches of the root locus for the closed-loop reactor system. 

The molecules in the sample absorb at their characteristic frequencies, and hence the radiation intensity / (x) that reaches the detector is modified by the presence of the sample. The Fourier transform of /a(x) is the absorption spectrum of the sample, and this transform yields percent transmission versus wavenumber (cm ). This type of spectroscopy is single-beam, so a background spectrum is required in most cases. Also it requires the use of a digital computer to calculate the Fourier transform of /a(x). 

---