Solve command, MATLAB

Nonlinear algebraic equations can be solved using MATLAB, too. First, you have to define the problem to solve by writing a file called an m-file then, you check it finally, you issue a command to solve it. These steps are analogous to the steps used in Excel. You can use MATLAB most effectively if you learn to use the Command Window and learn to create m-files and save them properly. See Appendix B for additional details. 

In MATLAB, the solve command is used to obtain solutions to symbolic a braic equations. The basic form of the solve command is solve ( eqxil, eqxi2, . . . f eqn ). As shown below, we define each equation fost and then use the solve command to obtain the solution. 

The script file kalfild.m solves, in forward-time, the discrete solution of the Kalman filter equations, using equations (9.74), (9.75) and (9.76) in a recursive process. The MATLAB command Iqed gives the same result. 

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... 

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... 

When examining the command fzero( / , xO) in MATLAB, the f defines which problem to solve, the xO is your best guess of the solution, and fzero tells MATLAB to vary x, starting from xO until the/is zero. In Excel s Goal Seek, the analogous steps were to make a cell zero by varying the value of another cell. Goal Seek becomes fzero, a cell with an equation becomes / and another cell becomes xO. 

There are two main ways to solve multiple equations in MATLAB. The hrst is to use the fsolve command. The second way is to use the optimization routines and the fminsearch command. Both are illustrated here. 

We will first use the MATLAB matrk left division operator to solve this problem. The operator solves the problem usii the Gauss elimination. We then solve the problem using the inv command. 

MATLAB can solve ordinary differential equations symbolically with or without boundary conditions or initial value parameters. The dsolve command is used for this purpose. Within dsolve, the letters D(/ are used to indicate a derivative, where i is the order of differentiation andj is the dependent variable. D implicitly specifies a first-order derivative, D2 signifies a second-order derivative, and... 

Note that the Matlab ODE solver is given three arguments the function evaluating the right-hand side of the differential eqnation, the time interval to integrate over, and the initial conditions. The first line of code (opts) sets the relative error tolerance to le-4 (default is le-3). This line is followed by a command which calls the ode45 solver, and specifies that the differential equation is specified in the m-file. Fmther, the command also specifies that the ODE should be solved in the interval 0-1 and at the initial condition yo = 1 - The third line is just used to plot the solution in a graph. 

A MATLAB Command Window session where the initial guess for the concentrations is given and f solve is called appears below. The solution vector for the concentrations is essentially identical to that of Example 9.4. 

The optional MATLAB PDF toolkit (doc pdetool), created by tiie developers of FEMLAB (www.comsol.com), has tools for forming meshes and solving simple PDEs in two dimensions, pdetool opens a graphical nser interface (GUI), in which we can draw the domain, mesh it, specify boundary conditions and PDE parameters, solve, and plot the solution. As tutorials are provided on die use of the GUI, here our focus is upon use of the command-line interface to access the functions of the PDE toolkit directly. 

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