Mathcad

Before posing the problem for this computer project, we shall introduce another vei y useful piece of microcomputer software by repeating the integration of Eq. (l-36a) with Mathcad (Appendix A). Like other software of this kind, there is a short learning process before mathcad can be used with ease. Once one has entered the equation of interest, mathcad solves it with a click on the = sign. In the present example, the constants of (Eq. l-36a) are entered followed by the desired integral... 

Note that the constants must be defined equal to their numerical values (defined = is on the keyboard). These definitions must be above the integral you wish to solve. Mathcad operates top down. Mathcad produces the same value for the integral that we obtained from TableCurve. This calculation is redundant with the calculations already performed in this section to inboduce new software by solving a problem for which we already know the answer. 

Write a short BASIC program to solve for AB above. Solve for BA. Do AB and BA commute Solve the same problem using Mathcad. 

Both problems can be solved by hand, by writing a short BASIC program, or by Mathcad as follows ... 

Note that in Mathcad, both matrices must be defined above the problem to be worked. In Mathcad, the symbol A = means matrix A is set equal to. ... 

Notice that the matrix A does not have to be squared before entering it into the Mathcad equation. Mathcad does the work of squaring A as part of the solution of the matrix equation, (keystroke translates as = in Mathcad.)... 

Invert A in Problem 1. Systematic methods exist for inverting matrices and will be discussed in the next chapter. For now, use Mathcad if it is available to you. 

Solve the same two problems with Mathcad. Is there a noticeable difference between the two sets Mathcad uses a variant on the Gaussian substitution method called LU Factorization (Kreyzig, 1988). 

Solve, by hand, by Mathcad, and with youi QBASIC programs... 

Procedure. Use Mathcad, QLLSQ, or TableCurve (or, preferably, all three) to determine a value of the ionization energy of hydrogen from the wave numbers in Table 3-4 taken from spectroscopic studies of the Lyman series of the hydrogen spectrum where ni = 1. 

To set up the problem for a microcomputer or Mathcad, one need only enter the input matrix with a 1.0 as each element of the 0th or leftmost column. Suitable modifications must be made in matrix and vector dimensions to accommodate matrices larger in one dimension than the X matrix of input data (3-56), and output vectors must be modified to contain one more minimization parameter than before, the intercept otq. 

Alternative procedure Mathcad. Follow the procedure above except that where QMOBAS is indicated, use Mathcad instead. Enter the Huckel molecular orbital matrix, modified by subtracting xl, with some letter name. For example, call the modified matrix A. Type the command eigenvals(A) = with the name of the modified HMO matrix in parentheses. Mathcad prints the eigenvalues. The command eigenvecs(A) yields the eigenvectors, which are useful in ordering the energy spectrum. 

Using QMOBAS, TMOBAS, or Mathcad and the method from Computer Project 6-2, calculate the energy separation between the HOMO and LUMO in units of p for all compounds in Table 6-1 and enter the results in Table 6-2. Enter the observed energy of ultraviolet radiation absorbed for each compound in units of cm . The reciprocal wavelength is often used as a spectroscopic unit of energy. 

Let us look at the Mathcad output for a Huckel matrix in more detail. We select the matrix for ethylene to preserve simplicity. 

In the Mathcad calculation of eigenvalues and eigenvectors of the Huckel matrix for ethylene (j g), the eigenvalues are given in the order upper followed by lower. The matr ix E for this order is... 

In this case Mathcad has already arranged the eigenvectors in their proper order as can be verified by the following queries ... 

Use Mathcad to calculate the first approximation to the SCF energy of the helium atom... 

Using as many methods as are available to you for comparison (Mathcad, QBASIC, and TRUE BASIC), determine the self-consistent field (SCF) energies of the He atom and of the ions Li+, Be " ", and B +. Fill in the SCF column of Table 8-1. 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

Fig. 4. Mathcad simulations (Cp = 5000 mg/L) as a function of Reynolds number for a NaCl solution (a) concentration polarization (CP), and (b) (-...

In the presented work an algorithm for the primary radiation filter optimization has been developed and realized in the Mathcad envelope which provides a minimal detection limit of a critical element both at the given X-ray tube power and at the given maximal acceptable count rate. 

Here, since the measurements were done in an integral reactor, calculation must start with the Conversion vs. Temperature function. For an example see Appendix G. Calculation of kinetic constants starts with listed conversion values as vX and corresponding temperatures as vT in array forms. The Vectorize operator of Mathcad 6 tells the program to use the operators and functions with their scalar meanings, element by element. This way, operations that are usually illegal with vectors can be executed and a new vector formed. The v in these expressions indicates a vector. 

Multiple regression analysis can be executed by various programs. The one shown in the Appendix is from Mathcad 6 Plus, the regress method. Taking the log of the rates first and averaging later gives somewhat different result. 

Taking the upward direction as positive, when both Fy < 0 and Fs<0, represents a co-current downflow operation, if Fs < 0, but Fy > 0 means a counter-current operation. The rest of the equation defining the system can be seen in Berty (1997). Integration was done by the Romberg method as is used in Mathcad PLUS 6 (1996) software. 

Mathcad Plus 6, of MathSoft, Inc. was used) Independent Variables on log scale... 

I should mention that matrix diagonalization can be easily done with any commercially available mathematics package such as Mathcad. 

This article complements the three lectures on Interferometry at the CNRS-/NATO/EC Summer school Optics in Astrophysics in Carggse, Corse, in September 2002, The original lectures make extensive use of the features of PowerPoint, including animated text and figures. Therefore the original PowerPoint files, as well as PDF versions thereof, are available for download from the web sites of the Summer school and of the author. Several animations were prepared using MathCAD to produce animations in avi format, each of which is several MB in size. These animations are separately available for download at the same web sites. 

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