Matlab operators

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. 

MATLAB operations rely on matrices thus, the first topic is how to input vectors and matrices. A row vector is written using a = [all al2 al3] with the elements separated by spaces or commas. Instead, for entering a column vector, we can either use the transpose function applied to a row vector a or directly with a = [all a21 a31] separating the rows with semicolons. 

C.l MATLAB Operations and Equation Solving C.1.1 Matrix Operations... 

The transpose of a matrix M is calculated using the command M. The inverse of a matrix M is calculated as inv(M). In MATLAB, the multiphcation of matrices A and B is denoted by A B, while their addition and subtraction are denoted by A -h B and A - B, respectively. Commands for element-by-element multiphcation and division are also available. For more functions and help on any MATLAB operation, type help. 

Table 10.1 enumerates the MATLAB operators. Most of these are the same as for Excel and VBA. A notable difference is the backslash operator (called left division), which is used primarily when solving sets of linear algebraic equations. Examples using this operator appear in the sequel. 

The colon ( ) is an important MATLAB operator. It occurs in several different contexts. The expression... 

This tutorial introduees the reader to matrix operations using MATLAB. All text in eourier font is either typed into, or printed into the eommand window. 

MATLAB supports every imaginable way that one can manipulate vectors and matrices. We only need to know a few of them and we will pick up these necessary ones along the way. For now, we ll do a couple of simple operations. With the vector x and matrix a that we ve defined above, we can perform simple operations such as... 

Now using the MATLAB command line software, we can easily demonstrate this solution (for the multivariate problem we have identified) using a series of simple matrix operations as shown in Table 21-1 below ... 

Table 21-1 Matrix operations in MATLAB to compute equations 21-1-21-4...

For the next several chapters in this book we will illustrate the straight forward calculations used for multivariate regression. In each case we continue to perform all mathematical operations using MATLAB software [1, 2], We have already discussed and shown the manual methods for calculating most of the matrix algebra used here in references [3-6]. You may wish to program these operations yourselves or use other software to routinely make these calculations. 

With the advent of modern software tools, however, tools such as MATLAB and even the older language, APL, matrix operations can be coded directly from the matrix-math expressions, and then it becomes near-trivial to create and solve the matrix equations on-the-fly, so to speak, and calculate the coefficients for any derivative using any desired polynomial, and computed over any odd number of data points. 

The use of Equation (A. 17) for inversion is conceptually simple, but it is not a very efficient method for calculating the inverse matrix. A method based on use of row operations is discussed in Section A.3. For matrices of size larger than 3 X 3, we recommend that you use software such as MATLAB to find A 1. 

Matlab is a matrix oriented language that is just about perfect for most data analysis tasks. Those readers who already know Matlab will agree with that statement. Those who have not used Matlab so far, will be amazed by the ease with which rather sophisticated programs can be developed. This strength of Matlab is a weak point in Excel. While Excel does include matrix operations, they are clumsy and probably for this reason, not well known and used. An additional shortcoming of Excel is the lack of functions for Factor Analysis or the Singular Value Decomposition. Nevertheless, Excel is very powerful and allows the analysis of fairly complex data. 

The few short programs presented in this chapter may also serve as a very rudimentary introduction into Matlab. Readers not familiar with Matlab but otherwise proficient in an alternative language will be surprised at the almost complete lack of for. .. end loops. We also introduce matrix operations in Excel, assuming that the other, more common aspects of Excel are known to the reader. While there is a reasonable collection of matrix operations available in Excel, their usage is rather cumbersome. We believe that many readers will appreciate the short introduction into this aspect of Excel. 

Note that Matlab uses the quote ( ) as the transposition operator. 

In Matlab the standard mathematical operators for addition (+) and subtraction (-) can be used directly with matrices. As with transposition, Matlab automatically calls the appropriate functions to perform the operations. ... 

The capability of dragging results from one cell to others is a very useful property of Excel and becomes even more powerful in combination with the dollar operator ( ) correctly applied within the cell reference. Referring to the previous Matlab example, if the scalar element foi,2 (cell F3) is to be subtracted from matrix A (A3 C4) in Excel, putting the dollar operator ( ) in front of the column and row reference of the source cell containing the scalar foi,2 ( F 3), prevents "dragging-over" of the source cell F3 in both column and row direction. 

In Matlab the plus (+) and minus (-) operators cannot be directly applied to equations that involve vectors or matrices of different dimensions. In order to perform the same operation as in the former Excel example, column vector b ,3 must be replicated three times to match the dimensions of A. For this, the Matlab command repmat can be used. 

In Matlab the asterisk operator ( ) is used for the matrix product. If the corresponding dimensions match all individual scalar products, c xarj, are evaluated to form Y. 

As stated earlier, Matlab s philosophy is to read everything as a matrix. Consequently, the basic operators for multiplication, right division, left division, power (, /,, A) automatically perform corresponding matrix operations (A will be introduced shortly in the context of square matrices, / and will be discussed later, in the context of linear regression and the calculation of a pseudo inverse, see The Pseudo-Inverse, p.117). 

In contrast to Matlab, where the defaults are the matrix operators, in Excel the default is the element-wise operation. In fact, all basic operations (e-g- 0, 0> 0> 0> Q) and functions (e.g. EXP, LN, LOG) work element-wise in Excel. All... 

Initially, we develop Matlab code and Excel spreadsheets for relatively simple systems that have explicit analytical solutions. The main thrust of this chapter is the development of a toolbox of methods for modelling equilibrium and kinetic systems of any complexity. The computations are all iterative processes where, starting from initial guesses, the algorithms converge toward the correct solutions. Computations of this nature are beyond the limits of straightforward Excel calculations. Matlab, on the other hand, is ideally suited for these tasks, as most of them can be formulated as matrix operations. Many readers will be surprised at the simplicity and compactness of well-written Matlab functions that resolve equilibrium systems of any complexity. 

Matlab recognises the importance of linear regression calculations and introduced a very elegant and useful notation the / forward- and backslash operators, see p.l 17-118. 

The Matlab computations invoked by the back-slash and forward-slash / operators do not perform the calculation as given in equations (4.26) and (4.31). Here a short extract from the Matlab HELP ... 

Referring back to Matlab, it is very important to use the correct slash operator or / for the left and right pseudo inverse. Applying the wrong one will invariably result in an error message or worse, in a potentially undetected error. 

Overall, most of the requirements for a process spectrometer/analyzer are straightforward to implement, but they do require attention at the design level. Another important area, which is FTIR specific, is the user interface and the need to provide for industry standard data communications. Standard software packages do exist for process instrumentation. For prototype development, and even for the front-end interface in a stand-alone mode of operation, software products, such as National Instraments Lab View and the Mathworks MatLab, are also important instrumentation development tools. Note that National Instruments also provides important computer-based electronics and hardware that meet most of the computer interfacing, and system control and communications needs for modem instrumentation. For practical installations, a product known... 

One ideally suited software for engineering and numerical computations is MATL AET-7 1. This acronym stands for Matrix Laboratory . Rs operating units and principle are vectors and matrices. By their very nature, matrices express linear maps. And in all modern and practical numerical computations, the methods and algorithms generally rely on some form of linear approximation for nonlinear problems, equations, and phenomena. Nowadays all numerical computations are therefore carried out in linear, or in matrix and vector form. Thus MATLAB fits our task perfectly in the modern sense. 

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