Equation fzero

Matlab does not include a routine of the kind of fzero for more than one variable. Only the function fsolve, which is part of the Optimisation Toolbox, can deal with systems of equations with several variables. Here we demonstrate the application of fsolve to the system of equations (3.70). 

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. 

Our first attempt involves MATLAB s built-in root finder fzero, which uses the bisection method and thereafter we introduce a new and more appropriate numerical method for solving equations with multiple roots. 

As in Section 3.1 for the adiabatic CSTR problem, we again start with a generic MATLAB fzero.m based root finder to try to settle the issues of multiplicity in the nonadiabatic CSTR case. The MATLAB m file solveNadiabxy. m below finds the values for y (up to three values if a lies in the bifurcation region) that satisfy equation (3.12) for the given values of a, / , 7, Kc, and yc using MATLAB s root finder fzero. 

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. 

This equation poses no problem at all for MATLAB s root-finder fzero, since F s zeros are always simple and the graph of F intersects the horizontal axis sufficiently steeply. Here is our MATLAB code colebrookplotsolve.m, which adapts itself automatically... 

The furnace equation (7.136) that describes the heat transfer in the side fired furnace is a single transcendental equation used to compute and iterate Tt o and Qr. It can be solved using fzero of MATLAB. 

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. 

MATLAB easily handles these kinds of problems. Basically you call a routine to integrate the ordinary differential equations (e.g., ode45). You construct a right-hand side function (m-file) to evaluate the right-hand side. The input variables are z and the three concentrations, and the output variables are the three derivatives. Take Ca and solve Eq. (8.46) using either fzero or fsolve. Then you have Ca,s at this location, z. You can evaluate the rates of reaction in Eq. (8.45) and put them in the output from the m-file. 

This same type of problem can arise with multiple concentrations or with one concentration and temperature. Then you would need to replace the fzero command with the fsolve command (from the Optimization Toolbox). You might need to experiment to provide good initial guesses for the concentrations and temperatures, because Eq. (8.46) would be turned into a set of nonlinear algebraic equations, and the solution is very sensitive to the temperature variable. 

This is an exaiq>le showing how to solve a nonlinear equation by y. calling the fzero function (bisection algorithm) in Matlab. 

Example 10.3 Solving a Single Nonlinear Equation Using fzero... 

The syntax for the fzero function is var = fzero( equation, init guess)... 

Once the bracket becomes sufficiently small that we feel that Newton s method or the secant method should be able to find the solution, we switch to one of those more efficient procedures. If this fails, we continue with bisection until the initial guess is sufficiently close for the iterative method to succeed. In MATLAB, the routine fzero takes such an approach. For further discussion of iterative methods to solve a single equation /(x) = 0, consult Press et al. (1992) and Quateroni et al. (2000). 

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