Using MATLAB Built-In Functions

MATLAB offers a large selection of built-in functions that you can use to analyze data. As we discussed in the previous chapter, by built-in funaions we mean standard functions such as the sine or cosine of an angle, as well as formulas that calculate the total value, the average value, or the standard deviation of a set of data points. The MATLAB functions are available in various categories, including mathematical, trigonometric, statistical, and logical functions. In this chapter, we will discuss some of the common functions. MATLAB offers a Help menu that you can use to obtain information on various commands and functions. 

The Help button is marked by a question mark located to the left of the current directory. You can also type help followed by a command name to learn how to use the command. 

Some examples of commonly used MATLAB functions, along with their proper use and descriptions, are shown in Table 15.7. Refer to Example 15.2 when studying Table 15.7. 

The following set of values will be used to introduce some of MATLAB s built-in functions. Mass = [102 115 99 106 103 95 97 102 98 96]. When studying Table 15.7, the results of the executed functions are shown under the Result of the Example column. 

More examples of MATLAB s Functions are shown in Table 15.8. 

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The Hemiite polynomials usually are defined by an integral equation. None the less, they are typically computed by annother series expansion. For the computation of the Hermite polynomials we use the built in functions of Matlab which are very accurate and computationally fast. 

The following MATLAB built-in functions will be used in this example ... 

NOTE Regarding the definition of each MATLAB built-in function used in this section, please refer, first, to Sec. 4.7 and, second, to MATLAB Product Help. In this section, the differences from the previous section will be pointed out. 

Integrate the function x In x (MATLAB x. log(x)) between 0 and 2 using the MATLAB built-in function quad. Compare this answer with that of Problem 6.6 Why is it smaller although the range of x is wider than that of Problem 6.6 ... 

This can be directly found using the MATLAB built-in function ... 

MATLAB has its own function, roots.m, for calculating all the roots of a polynomial equation of the form in Eq. (1.11). This function accomplishes the task of finding the roots of the polynomial equation [Eq. (1.11)] by first converting the polynomial to the companion matrix A shown in Eq. (1.56), It then uses the built-in function eig.m, which calculates the eigenvalues of a matrix, to evaluate the eigenvalues of the companion matrix, which are also the roots of the polynomial Eq. (1.11) ... 

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. 

In this book we shall use MATLAB codes and explain more involved features of MAT-LAB as we encounter them. MATLAB has a built-in help menu typing help format at the prompt, or help, for example, will show the syntax and variations of these two commands format and backslash . Whenever a student encounters a MATLAB command that is not self explanatory, we suggest using this built-in help function of MATLAB. 

We note that fzerotryl is a MATLAB function m file that is stored in its folder with the extension. m as fzerotryl.m. Our program code is annotated with comments following the % symbol. Anything that follows after a % symbol on a line of code is not executed in MATLAB. More on MATLAB files, their storage, creation, etc. is given in Section 1.2.5. Specific built-in MATLAB functions and their use, such as fzero in the code below, should always be scrutinized by our students for their input/output syntax etc. using the built-in help MATLAB command help fzero for example. 

Hint Linearize the equations and find the associated matrix eigenvalues by using the built-in MATLAB function eig.]... 

The purpose of this appendix is to provide a viable code to integrate the BzzMath library into Madab. The guidelines for implementing a MEX function are available to allow user-created functions and hbraries written in C++ (or Fortran) to be used into Matlab as built-in functions. 

Needless to say that within MATLAB we have all the algebraic, statistical functions predefined along with plotting capabilities. Before briefly commenting on the basics, it is interesting to point out the use of help within the command window. We use the help function followed by any built-in function and MATLAB returns a small tutorial on the use of that function, the arguments, and the options (i.e., try help ode45 ). 

To solve equations (7-9), an approach with direct solution of the system (8) at each step of the iterative process by the pseudo- (or generalized) inversion method is used. It is based on singular value decomposition (SVD). It is well-known that the SSVDC procedure in the Linpack library is used to calculate SVD [12]. Paper [13] presents a standard SVD procedure in Fortran-IV used in the present paper. The current MATLAB system versions have a built-in function svd(A) implementing this decomposition for an arbitrary nxm matrix A. The calculation scheme of the SVD procedure is in decomposing the matrix (9) at each step of the iterative process into the product of three matrices. 

The user may use any of the MATLAB built-in library functions that are already defined and can be used in any command statement. MATLAB has a miscellany of functions. Some of these are standard functions, including trigonometric functions, and so on, and others are user-defined functions and third-party functions. All of these enable the user to easily carry out complex computational tasks. Examples are shown in Table 1.1. 

NOTE The solution Z is 1 x 1 struct that has three symbolic variables yl, y2, and y3. After defining t as a row vector, the MATLAB built in eval function is used to evaluate the symbolic expression of y into numbers as a function f. 

The eigenvalues and eigenvectors of matrix A can be calculated using the techniques developed in Chap. 2 or simply by applying the built-in function eig in MATLAB. This is demonstrated in Example 5.2. 

Next in the function are several sections dealing with building the matrix of coefficients and the vector of constants according to what is discussed in the method of solution. Finally, the function calculates all the u values by matrix inversion method using the built-in MATLAB function inv. The outputs of elUptic.m are the vectors of jc and y and the matrix of values. 

The following MATLAB session shows a variety of matrix operations. It can be seen again that the sanicolon is used to suppress printing when placed at the end of a command, and it also indicates the end of a row (and the beginning of a new one) in a matrix. The apostrophe is the transposition operator, and the built-in function inv takes the inverse of a matrix. If the inverse does not exist, an error message appears. 

The built-in function minsearch is based on a rather unsophisticated algorithm. There are more robust unconstrained minimization functions available in some of the MATLAB Toolboxes, but unfortunately, these are not standard. For simple problems, minsearch often works well enough. It is used here to minimize the sum of squares between fictitious data (program generated data) and a function in which the regression coefficients appear nonlinearly. 

MATLAB has a built-in root finder for scalar equations f(x) = 0 in one real variable x that are in standard form. The built-in MATLAB function is fzero. The use of fzero hinges on a user-defined function, such as the function f inside the following fzero tester, called fzerotryl, that we apply to our previously studied third degree polynomial. 

Method used MATLAB s fzero built-in root finding function... 

In the previous sections we have used MATLAB commands that were typed in on the desktop command line, as well as MATLAB commands that invoke special stored MATLAB files, called m files. When starting to work with MATLAB, we advise our users to create a special folder for their personal MATLAB m files once and to continue to use this folder to store personal m files. To access the stored m files from the MATLAB desktop, the user has to point the small desktop window called Current Directory to the personal m file folder. Then all stored m files, as well as all built in MATLAB functions can be accessed, called upon, and used from the command line. All stored personal... 

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