Fzero

Instead of adapting the NewtonRaphson.m function we just use the Matlab function fzero which is a general routine for that kind of one-dimensional problem. ... 

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. 

We note that fzerotryl is a MATLAB function m file that is stored in its folder with the extension. m as fzerotryl.m. Our program code is annotated with comments following the % symbol. Anything that follows after a % symbol on a line of code is not executed in MATLAB. More on MATLAB files, their storage, creation, etc. is given in Section 1.2.5. Specific built-in MATLAB functions and their use, such as fzero in the code below, should always be scrutinized by our students for their input/output syntax etc. using the built-in help MATLAB command help fzero for example. 

Method used MATLAB s fzero built-in root finding function... 

Newt,fval,exitflag,output] = fzero( f,-2) 7. call fzero for f from x = -2... 

The first call of fzero inside fzerotryl takes 24 iterations to arrive at the real root x = -1.1304 of our trial polynomial p(x) x — 2x2 + 4 when starting at x0 = —2, while the second call converges after 9 iterations when looking for real roots of p inside the interval [—2, —1]. Please look up help fzero to learn more about this MATLAB function and how it was used. 

Note that MATLAB s fzero algorithm stops after 35 iterations when it is about 2.7% away from the true root 2 with an approximate start at 1.5. From the inclusion interval [1.91, 2.1] it also ends rather prematurely and off target as well. 

Adapt f zerotry2. m on p. 30 to various other polynomials of your choice. Use polynomials of degrees less than 7 that have some multiple roots, as well as no multiple roots. What happens to the complex roots of a polynomial under fzero ... 

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. 

Note that the above program works for both endothermic reactions (/ < 0) and exothermic reactions (/ > 0) and that only exothermic reactions can have multiple steady states. The built-in MATLAB root finder fzero finds the roots of a function / from a starting guess a if we call fzero( /,a,.. . ), i.e., if we attach the function handle to / and follow this with the appropriate list of parameters in MATLAB. 

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... 

Then uses MATLAB s fzero to find the solution xsol. 

Which usually takes more effort, finding a root inclusion interval or solving for / via fzero in MATLAB ... 

How close to the bifurcation limits does your bisection program succeed when the graphics solutions are used as starting points for fzero What are the sizes of the residues in the computed solutions near the bifurcation points Which of the proposed steady-state finders of part (a) or (b) do you prefer Be careful and monitor your hybrid algorithm s effort via clock and etime. 

Da = fzero( DaPe,start,optimset( dis ,[], tolx ,ltol),Pe,percent) err = DaPe(Da,Pe,percent) ... 

Pe(i) = fzero(SDaPe,start,optimset( Display , off , TolX ,ltol),... 

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