Matrix computations, MATLAB operations

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. 

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

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. 

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. 

C+ is the so-called pseudoinverse of C. It can be computed as C+ = (C C) 1 Cl. However, MATLAB provides a numerically superior method for the calculation of A by means of the back-slash operator ( ). Refer to the MATLAB manuals for details. The important point is that we are now in a position to write the residual matrix R, and thus ssq, as a function of the rate constants k only ... 

The rank of E is computed by standard methods, such as counting the number of independent rows after performing elementary row operations. We prefer here to employ MATLAB. Inputting E and computing its rank gives an answer of three. Since n = 3, rank(E) = n. The controllability matrix E has full row rank and the system is therefore controllable. 

The operation is the left division and / is right division. In this case, left division is used since the matrix A is to the left of vector b. You can learn about these operations in MATLAB by entering help slash. The right division B/A computes (A /B ) again using LU factorization and Guassian eliminination. 

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