Using Built-In Functions

Equations. The only one not covered under the ODE system is the Euler 

ODE45 Runge-Kutta fourth-order method with fifth-order error prediction. ODE23s Rosenbrock. Implicit low-order method for stiff problems. ODE113 Adams method. Multistep method. 

ODElSs Gear backward difference formulas for differential equations and differential and algebraic equations. Multistep method. 

All of them share the same syntax as we can see in Example 4.3.3. We write ode, followed by the name of the file where the function is defined using NAMEFUNCTION or NAMEFUNCTION, a comma before including the vector with the initial and final value of the independent variable and finally the initial values of the dependent variables as a vector. 

In order to solve partial differential equations, MATLAB has the pdepe tool. This function uses odelSs to solve the differential equation after a few transformations. The key issue is that we need to provide the proper files for the function to operate. In fact MATLAB solves an equation of the form given by the following equation  

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Complex calculations may be carried out in Excel using built-in functions. Eurther applications of complex calculations will be shown later. 

Here is a short-cut that simplifies the calculation of 5d and 57. Enter the values for Dj into your calculator, and use its built-in functions to find the standard deviation. Divide this result by Vd to obtain Sq. You can use the same approach to calculate 57. 

The seven ways (r, through r7) for calculating correlation as the square root of the ratio of the explained variation over the total variation between X (concentration of analyte data) and Y (measured data) are described using many notational forms. For example, many software packages provide built-in functions capable of calculating the coefficient of correlation directly from a pair of X and Y vectors as given by rx (Equation 59-7). 

Alternatively, Matlab s built-in function norm can be used to determine normalisation coefficients and perform the same task. An example for column-wise normalisation of a matrix X with orthogonal columns is given below. It is worthwhile to compare X with equation (2.15) the subspace command can be used to determine the angle between the vectors (in rad) and reconfirm orthogonality. ... 

A large number of chemical/biological processes will be presented, modeled and efficient numerical techniques will be developed and programmed using MATLAB 2. This is a sophisticated numerical software package. MATLAB is powerful numerically through its built-in functions and it allows us to easily develop and evaluate complicated numerical codes that fulfill very specialized tasks. Our solution techniques will be developed and discussed from both the chemical/biological point of view and the numerical point of view. 

MATLAB comes as one main body of built-in functions and codes, and there are many additional specialized MATLAB toolboxes for various applications. As this book is primarily directed towards undergraduate and beginning graduate students, we have restricted ourselves deliberately to using the main body of MATLAB only in our codes and none of its many toolboxes. 

The last part of this section lists a few standard MATLAB operations, functions, and commands, collected into groups, together with short descriptions. This may help our readers to more easily find and use built-in MATLAB functions in their own MATLAB program codes. Please note that our MATLAB function descriptions below are very few and very short by necessity. The user should use the help. . . command to find the full length MATLAB reference guide entry for each MATLAB function when the need arises. This will help our readers use the full power and functionality of MATLAB commands and will enable them to browse for and find related built-in MATLAB functions. 

One such construct is for a "one-shot. Some PLCs provide this as a built-in function, but here it will be presented in terms of separate components. The one-shot is generated by the third rung of ladder logic in Fig. 8-60a. But first examine the fourth rung. The input LL drives the output coil LL1. This coil provides the state of level switch LL on the previous scan of ladder logic. This is used in the third rung to produce the one-shot. Output coil OneShot is energized if... 

Function procedures augment Excel s library of built-in functions. A function macro is used in a worksheet in the same way as, for example, the SORT function. It is entered in a single cell of a worksheet, performs a calculation and returns a single result (or an array result) to the cell in which it is located. For example, a custom function macro named ALPHA can be used to calculate aj, the fraction of an acid-base species in one of its protonated forms HjX at a particular pH. The function takes three arguments the pH of the solution, the range of pKa values of the weak acid and the coefficient j. This function is useful in constructing distribution diagrams, titration curves, and so on. 

A custom function is used in a worksheet formula in exactly the same way as any of Excel s built-in functions. You can enter it in a formula by using Paste Function, or by typing it. The workbook containing the custom function must be open. 

For Excel s built-in functions, the Formula Palette (Step 2 of Paste Function) provides help information about each argument as you begin to enter it. There s no way to provide similar information about arguments in the Formula Palette for custom functions. But since the same description text appears in the Step 1 and the Step 2 dialog boxes, you can provide information about the arguments in the description. Unfortunately the limit for Description is 255 characters. You can provide line breaks in the text by using Chr(lO) or Chr(13), but the Formula Palette can only display two lines of text. 

Your custom functions will now be indistinguishable from Excel s built-in functions. By giving them names using lower-case letters, you can distinguish them from Excel s built-in functions. 

Metal-doped silica gels exhibit a wide range of optical properties which allow them to be used for optical and optoelectronic triplications. The preparation is done via the sol-gel process. Emission and absorption maxima as well as quantum yield can be adjusted by built-in functionalized silanes [2] or adsorbed semiconductor or metal colloids, respectively [3,4]. 

Although the Find/Replace function works well for these data, it is not as generally useful as the built-in functions for manipulating strings of alphabetical characters and numbers. These functions are called string functions. We will use string functions to strip the parenthetical uncertainties from the data in column D and produce a column of numerical atomic masses. 

In this spreadsheet exercise, we learn to calculate the mean of a data set. First, we define formulas to calculate the mean, and then we use the built-in functions of Excel to accomplish the task. 

Excel has built-in functions to compute many of the quantities that are of interest to us. Now we shall see how to use them to calculate the mean, or in Excel s syntax, the average. Click on cell C13 and type... 

Spreadsheet Summary In the first exercise in Chapter 3 of Applications of Microsoft Excel in Analytical Chemistry, we use Excel to perform the t test for comparing two means assuming equal variances of the two data sets. We first manually calculate the value of t and compare it with the critical value obtained from Excel s function TINV(). We obtain the probability from Excel s TDIST() function. Then, we use Excel s built-in function TTEST() for the same test. Finally, we employ Excel s Analysis ToolPak to automate the t test with equal variances. 

Spreadsheet Summary In Chapter 3 of Applications of Microsoft Excel in Analytical Chemistry, we use two Excel functions to perform the F test. First, we use the built-in function FTEST(), which returns the probability that the variances in two data arrays are not significantly different. Then we use the Analysis ToolPak for the same comparison of variances. 

Linear least-squares analysis is quite easy with Excel. This type of analysis can be accomplished in several ways by using the equations presented in this chapter, by employing the basic built-in functions of Excel, or by using the regression data analysis tool. Because the built-in functions are the easiest of these options, we explore them in detail here and see how they may be used to evaluate analytical data. 

The chi-square distribution is used to perform statistical tests on the sample variance. It is highly asymmetric for small values of n, but becomes more symmetric and similar to a normal distribution as n becomes large, such as 20 or 30. The cumulative distribution function of the chi-square distribution is listed in Table 3.4 as a function of v and a, where v = - 1 is the number of degrees of freedom and a is the percentage of the distribution above the particular Microsoft Excel has built-in functions, CHIDIST and CHIINV, that compute a chi-square distribution [5, 6]. 

In summary, the current state of the art is to use FO-approximation for the first few runs to identify and correct any errors in the data set and control streams. Also, FO-approximation provides good initial estimates for more computationally intensive algorithms. Thereafter, conditional estimation (usually FOCE-I) is usually used for method development, unless the model is developed not using the built-in functions within NONMEM. In other words, if the user has to develop the model using user-written differential equations, even a modest size data set with the conditional estimation algorithms may lead to prohibitive run times. Therefore, with these user-written models FO-approximation is usually the algorithm of choice. Perhaps at the final step in model development then a conditional estimation algorithm is used to obtain more accurate parameter estimates. 

Octave and Matiab provide many convenient built-in functions. Some that we have used in this text include the matrix exponential, expm incomplete gamma function, gammai pseudo-random number generators, rand and randm and several others. Type help -i and select the menu item Functi on Index for a complete list of Octave s built-in functions,... 

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