MATLAB linear equations

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

Due to the special structure of MATLAB, readers should be familiar with the mathematical concepts pertaining to matrices, such as systems of linear equations, Gaussian elimination, size and rank of a matrix, matrix eigenvalues, basis change in n-dimensional space, matrix transpose, etc. For those who need a refresher on these topics there is a concise Appendix on linear algebra and matrices at the end of the book. 

Solving linear equations Ax = b is done in MATLAB via the backslash A b command. For example, let us consider a system of 4 linear equations in 4 unknowns... 

Our two command lines below first generate the coefficient matrix A and the right hand side vector b for (1.1), followed by the MATLAB backslash linear equations solver that computes the solution vector x. This is followed by a simple verification of the error inherent in the residual vector A-x — b for our numerical solution x. This error is nearly zero since in MATLAB the number -1.3323e-15 describes the real number —1.3323 10-15. 

Here is a simple MATLAB function that performs this task by solving AX = B via MATLAB s built-in backslash linear equations solver. 

The MATLAB code linearnoneqicol.m implements and solves the two linear equations... 

A number of standard computer programs easily handle problems of this type such as spreadsheet packages, Matlab, Mathcad, Polymath, and so on as well as symbolic manipulators such as Mathematica, Maple, Derive, etc. Most statistic packages and equation solvers will also solve linear equations and have a simple user interface. 

A Solve the following set of linear equations using a spread sheet program, MathCad, Matlab, or a related package ... 

If the parameters enter the equation linearly, then the minimization problem reduces to a set of linear equations which are solved easily by Excel and MATLAB. The effectiveness of the curve fit is often reported as values of the linear correlation coefficient squared, r. The linear correlation coefficient is defined as (Press et al., 1986, p. 484) ... 

In this section, we will show how you can use MATLAB s symbolic solvers to obtain solutions to a set of linear equations. 

L, U] = LU(A) LU decomposition of matrix A. Typically used within MATLAB to solve systems of linear equations. You need to provide [L, U] so that both matrices are stored. 

We use to include a comment within the code. We solve the linear equation. MATLAB typically uses LU decomposition for this ... 

Numerical Solution of Linear equations using MATLAB ]... 

Tliis problem focuses on using Polymath, an ordinary differential equation (ODE) solver, and also a non-linear equation (NLE) solver. Hiese equation solvers will be used extensively in later chapters. Infonnation on how to obtain and load the Polymath Software is given in Appendix E and on the DVD-ROM. (a) There are initially 400 rabbits (x) and 200 foxes (y) on Farmer Oat s property. Use Polymath or MATLAB to plot the concentration of foxes and rabbits as a function of time for a period of up to 600 days. The predator-prey relationships are given by the following set of coupled ordinary differential equations ... 

The values obtained from the function Gauss.m may be verified easily in MATLAB by using the original method of solution of the set of linear equations in matrix form, that is, T =... 

We consider a simple mass balance problem to demonstrate the use of MATLAB to solve a system of linear equations. For the separation system of Figure 1.9, we know the inlet mass flow rate (in kilograms per hour) and the mass fractions of each species in the inlet (stream 1) and each outlet (streams 2, 4, and 5). We wish to compute the mass flow rates of each outlet stream. 

In general, non-linear problems cannot be resolved explicitly, i.e. there is no equation that allows the computation of the result in a direct way. Usually such systems can be resolved numerically in an iterative process. In most instances, this is done via a truncated Taylor series expansion. This downgrades the problem to a linear one that can be resolved with a stroke of the brush or the Matlab / and commands see The Pseudo-Inverse (p.ll 7). 

We have encountered Excel s LINEST as a tool for linear regression. Unfortunately, LINEST cannot be generalised from vectors to matrices. To deal with matrices, we do not have an option but to use equations (4.59) and (4.61). It is possible to do so, but not as convenient as in Matlab. 

There are numerous other examples of two-box models. For instance, a two-box epilimnion/hypolimnion model was discussed in Chapter 21, and additional examples are given as problems at the end of this chapter. We must remember that as long as these models are linear, their solutions can be constructed with the help of Box 21.6. They always consist of the sum of not more than two exponential functions and are thus fairly simple. This situation changes drastically if we allow the differential equations to become nonlinear. A system of two or more nonlinear differential equations rarely can be solved analytically, yet the available computer tools (such as MATLAB) make their solution easy. 

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. 

The MATLAB backslash command solves all linear systems of equations, with rectangular or square matrices alike. If we instead want to solve the linear system Ax = b for our earlier matrix... 

Compared to solveadiabxy. m for the adiabatic CSTR case in Section 3.1, the above MATLAB function solveNadiabxy. m depends on the two extra parameters Kc and yc that were defined following equation (3.9). It uses MATLAB s built-in root finder fzero.m. As explained in Section 3.1, such root-finding algorithms are not very reliable for finding multiple steady states near the borders of the multiplicity region. The reason - as pointed out earlier in Section 1.2 - is geometric the points of intersection of the linear and exponential parts of equations such as (3.16) are very shallow, and their values are very hard to pin down via either a Newton or a bisection method, especially near the bifurcation points. 

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

In two later sections, we will deal with numerical integration, which is required to solve the differential equations for complex mechanisms. Before that, we will describe nonlinear fitting algorithms that are significantly more powerful and faster than the direct-search simplex algorithm used by the MATLAB function fminsearch. Of course, the principle of separating linear (A) and nonlinear parameters (k) will still be applied. 

The systematic approach described above for solving radiation heal transfer problems is very suitable for use with today s popular equation solvers such as lili.V, Mathcad, and Matlab, especially when there are a large number of surfaces, and is known as the direct melhod (formerly, the matrix method, since it resulted in matrices and the solution required a knowledge of linear algebra). The second method described below, called the network method, is based on Ihe electrical network analogy. 

Solutions of linear ODEs can also be found using the software tool MATLAB. To demonstrate this, let us consider the three-heated-tank process studied in Chapter 1. The process is described by three linear ODEs [Eqs. (1.10), (1.1 1), and ( 1.12)]. If flow rate F, volume V (assuming equal volumes in the three tanks), and physical properties p and Cp are all constants, these three equations are linear and can be converted into perturbation variables by inspection. 

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