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ChE 211, Process Simulation
Exam 3 Solution
Instructions:
Open note
NOT open book (this section isn’t covered in the book anyway)
NOT open computer
Where it is appropriate, write your answers on the screen-shot. Feel free to
change the units if needed, or to fill in a value that you think will show up
after your initial selection.
1)
Sketch what is happening when we use the Regula-Falsi method to find the
root of a non-linear equation. (10 pts)
2)
What steps do you need to go through to use the fixed point iteration method
to solve a of non-linear equation? (15 pts)
1. Starting with your equation in the form f(x)=0, rewrite as x=g(x).
There are very likely multiple options for what g(x) will actually
look like.
a. Examine options for g(x)
b. Calculate g’(x) at your initial guess.
c. If abs(g’(x))<1, it should converge, pick a g(x) accordingly.
2. plot y=g(x) and y=x. The intersection occurs at the root we are
searching for.
3. pick a point
4. draw a line in the x direction from the point to y=x, then in the y
direction back to y=g(x), this becomes our new point.
5. repeat step four until the change in x is within tolerance.
3)
Fill out the guess values and constraints for a Mathcad solve block to solve
the following system of equations: (15 points)
z3-4x2+2y=cos(36x)
sin(y)=20z-x
tan(z)+tan(x)=y4
4)
Rearrange the equations in problem 3 to be in an ‘implicit’ form. (10 pts)
F(x,y,z)=z3-4x2+2y-cos(36x)=0
F(x,y,z)=20z-x-sin(y)=0
F(x,y,z)=tan(z)+tan(x)-y4
5)
What method is the following section of matlab code using to solve a nonlinear equation? (10 pts)
Function Xsol=mysterymethod(Fun,a,b,imax,tol)
%Bwa ha ha ha. Ha ha.
% a and b are the x values for the initial guesses, imax is
the maximum number of
%iterations, and tol is the desired tolerance.
Toli=10
%set iteration tolerance high enough that
the loop will complete one time
Fa=Fun(a); Fb=Fun(b);
For itr = 1:imax
xNS=b-Fb*(a-b)/(Fb-Fa); %extrapolate the intercept of
a straight line between a and b
if Fun(xNS)=0 % check to see if we stumbled onto an
exact solution
Xsol=xNS;
Break;
end
if toli<tol
Xsol=xNS;
break
%stop iterations because tolerance is met
end
if itr == imax
Xsol=’solution not found within maximum
iterations’; %check that we aren’t going over on
iterations
Break
end
if abs(Fun(Fa))< abs(Fun(Fb)) %develop new guesses.
toli=abs(b-xNS)/b;
b=xNS;
else
toli=abs(a-xNS)/a;
a=xNS;
end
end
end
Secant
6)
Why do the initial guesses matter when we are using numerical methods?
(10 pts)
Aside from the obvious needing a place to start, getting too far away affects the
amount of time it takes to converge on a solution, and if there are multiple
possible roots/answers, a numerical method usually converges on the answer
closest to the initial guess. With some methods, the initial guesses might
cause the method to not converge (Newton’s has some of these issues).
7)
What is a limitation on the type of function we can use bisection or regulafalsi to solve? (10 pts)
The function must cross the x-axis
8)
Write a Matlab function to contain the following system of equations: (20
pts)
6y3-5x2+30e-6x=0
20x-ey=0
function fxy=fun(x,y)
fxy(1)=6*y^3-5*x^2+30*exp(-6*x)
fxy(2)=20*x-exp(y)