Matlab iiISolving non-linear algebra problems

Justin DawberSeptember 22, 2011

Expectations/PreRequisites

• Introduction to MatLab I & II (or equal experience)• MatLab as a calculator• Anonymous Functions• Function and Script m-files• MatLab Basics• Element-wise Operations• 2-D and 3-D Plotting• Code Cells and Publishing

Anonymous Function Review

• Used in calling functions indirectly• >> Sin = @sin; % The variable ‘Sin’ points to

the function ‘sin’• >> Sin(pi) % Evaluates the sine of pi• (Not the most useful example, more later)

• Can be used to create ‘anonymous functions’• >> myfun = @(x) 1./(x.ˆ3 + 3*x - 5)• >> myfun(3)

M-file review

• Scripts: No input and no output arguments. Contain a series of commands that may call other scripts and functions

• Functions: Accept input and returns output arguments. Usually called program routines and have a special definition syntax.

• Code Cells: Defined breaks in code that will help breakdown solution process

What defines “non-linear”

• Any Equation where the output is not directly proportional to the input (or does not obey the superposition principle)

• Simplest Examples

• Polynomials

• Single Variable Equations • Non-linear Multivariate Equations

• The terms make this non-linear • Multivariate linear equations are also possible

HUMPS (Built-in)

• HUMPS is a built-in Matlab function that has a strong maxima at 0.3

• For those that want to know: 604.0)9.0(

101.0)3.0(

1)( 22

xx

xhumps

Introducing fsolve

• Solver for systems of non-linear equations• Requires Optimization Toolbox

• Single Dimension System• Use fsolve with an anonymous function

• Steps to a solution• Define the Anonymous Function• Plot the function to visualize initial guess• Call fsolve with function and guess

• solution = fsolve(@f, initial_guess)

Lets See It!

• We will do the following:• Review a script m-file in Matlab for HUMPS

• Start with a complete srcipt • Review Code cells to break-up the code• Plot the function to visualize the guess• Iterate through the cells• Review the initial guess and solutions

• Switch to MatLab

nla_humps_comp.m

An Example: Finding Equilibrium Concentrations

• For the reaction H2O + CO <=> CO2 + H2• Given the Equilibrium constant, K = 1.44• For an equimolar feed of H2O and CO, compute

the equilibrium conversion

• Solution• C_H2O = (C_H20)0 * (1-Xe)• C_CO = (C_H20)0 * (1-Xe)• C_H2 = (C_H20)0 * Xe• C_CO2 = (C_H20)0 * Xe• K = C_CO2*C_H2/(C_H20*C_CO)

Lets Try It:

• We will do the following:• Write a script m-file in Matlab for the Equilibrium

Conversion• Start with a skeleton script • Use Code cells to break-up the code• Plot the function to visualize the guess

• Review a common syntax problem for element-wise operations

• Iterate through the cells• Review the initial guess and solutions

• Switch to MatLab

nla_equilibrium_comp.m

2-Dimensional System of non-linear equations

• What do we have in this case?• 2 surfaces

• What are we trying to find?• The point in x and y where both surfaces are

zero

• What is different about this case?• Hard to visualize• Two initial guesses required• Requires a Function-Function m-file

• Also know as sub-functions or function in a function

The multi-dimensional function m-file

• Use sub-functions (function-function)• Primary function – call fsolve• Secondary or sub-function – define the multi-

variate systemfunction mainclear all; close all; clc;

%% We can make some plots to help identify initial guessesx = 0:0.1:2; y=x;[X,Y] = meshgrid(x,y);

hold onsurf(X,Y,2*X - Y - exp(X) + 2) % first functionsurf(X,Y,-X + 2*Y - exp(Y) + 2) % second functionzlabel('z')view(69,8)

%%initial_guesses = [1.3 0.9];[X,fval,exitflag] = fsolve(@myfunc,initial_guesses)

function z = myfunc(X)% set of equations to be solvedx = X(1);y = X(2);z = [2*x - y - exp(x) + 2; -x + 2*y - exp(y) + 2];

Lets see it:

• We will do the following:• Review a script m-file in Matlab for two

stretched HUMPS surfaces• Start with the complete script • Review each of the Code cells • Plot the function to visualize the guess• Review the initial guess• Call fsolve and review solutions

• Switch to MatLabnla_1v_2d.m

An example: 2-Dimensional system

• Find a solution to the following system:

Lets see it:

• We will do the following:• Review a script m-file in Matlab for the

preceding example• Start with the complete script • Review each Code cell• Plot the function• Review the initial guess• Call fsolve and review solutions

• Switch to MatLab

nla_2d.m

n-Dimensional Systems Example

• fsolve also works for the n-dimensional case• No way to graph this case• Must have knowledge of the problem to specify and initial guess

• Solve the set of equations:• 0 = K1-(yCO*yH2^3)/(yCH4*yH2O)*p^2;• 0 = K2 - ((yCO^2)*(yH2^2))/(yCO2*yCH4);• 0 = (2*yCH4 + yH2 + yH2O)*n - nH2f;• 0 = (0.5*yCO + yCO2 + 0.5*yH2O)*n - nO2f;• 0 = (yCH4 + yCO + yCO2)*n - nCf;• 0 = yCO + yCO2 + yCH4 + yH2 + yH2O - 1;

• Given• K1 = 25.82;• K2 = 19.41;• p = 1;• nCf = 1; nH2f=0.5; nO2f = 0.5;

Lets see it:

• We will do the following:• Review a script m-file in Matlab for the n-Dim

example• Start with the complete script • Review each Code cell• Cannot plot the function• Postulate on the initial guess• Call fsolve and review solutions

• Switch to MatLab

nla_multidim.m

Thing to consider when using fsolve

• No Solutions• Multiple Solutions

• Depends on the initial guess

• Infinite Solutions – coincidence• The nature of Numerical Solvers – Know your

tolerances

Lets take a look at one of each:

No solution example

• Translated HUMPS• Let’s slide the HUMPS graph up 50• It no longer crosses the X-axis• We can attempt to solve it in the same way

• Lets see how fsolve handles it?

Lets See it:

• We will do the following:• Run the earlier script for the 1-D humps

example with the graph translated +50• Start with the complete script • Run the script• Confirm the translation of +50• Review the output from fsolve

• Switch to MatLabnla_humps_translated_comp.m

Multiple Solution example

• Back to the earlier HUMPS example• Two different guesses yield two different solutions

• As you can see, two Zeros. A guess around -.4 will return the lower zero, while a guess near 1.2 will yield the high one.

Infinite Solutions Example

• Back to a 2-D fsolve example• Solve the system of equations:

• sin(x) + sin(y) - 1.5• -sin(x) - sin(y) + 1.5

Lets See it:

• We will review the graph of the two surfaces in the preceding example• View graph from different angles• Call fsolve with multiple initial guesses

• Switch to Matlabnla_2d_Coincidence.m

A little bit about Numerical Solvers - Tolerances

• Numerical Solvers search for a solution starting from and initial guess• Several search algorithms can be used• Termination criteria

• Solver terminates when the value of the function is within a certain range close to 0

• The solver is unaware of the scale of the problem• If you are looking for a value in ml, but the

problem is in m3 the solver may stop at ~0.003 m3 … but this is 3 L!

• Lets look at an example of this and how to correct it

Tolerance Concern Example

• Solve this Equation = 0• Given

• CA0= 2000 mol/m3

• V = .010 m3

• v = .0005 m3/s• k = .00023 m3/mol/s

• After plotting we will see solution is near 500• Guess = 500

• Call fsolve• Guess is exactly right?!? - Something must be wrong

• Scaling is the problem

Tolerance Concern Example (cont.)

• How to Proceed?• Option One – Change the tolerance

• Optimset• options = optimset('TolFun',1e-9);• Be careful not to set tolerance too tight (1e-15 =

too tight)• Then call fsolve with options

• fsolve(f,cguess,options)

• Results are now much more accurateTolerance_Concern.m

Tolerance Concern Example (cont.)

• How to proceed?• Option 2 - Scale the input units and guess:

• CA0= 2 mol/L• V = 10 L• v = .5 L/s• k = .23 L/mol/s• Guess = .5

• Call fsolve with default tolerances• Results now more accurate

Lets see it:

• We will do the following:• Review a script m-file in Matlab for the

preceding tolerance concern example• Start with the complete script • Review each Code cell• Iterate through code cells• Review solutions using different methods

• Switch to MatLab

Polynomials in MATLAB

• Defining Polynomials• How to get f(x) = Ax2+Bx+C into MatLab• Simple! --- >>f = [A B C]

• Finding Roots of a polynomial f• Also Simple --- >>roots(f)

• An example: The Van Der Waal equation:• V3 – ((pnb+nRT)/p)V2 + (n2a/p)V – n3ab/p• Coefficients [1 -(pnb+nRT/p) (n2a/p) – n3ab/p]

Lets try it:

• We will compare the solutions to the above polynomial using fsolve and roots• Start with a complete script m-file• Define Polynomial as Anonymous Function• Define Polynomial as coefficient vector• [a b c d]• Find solution using roots() and fsolve()• roots is an analytical solution

• All solutions• fsolve is a numerical solution

• Only the Closest Solution

• Switch to Matlab nla_cubic_polynomial.m

Summary

• fsolve(@f,initial_guess) – non-linear solver (From the optimization toolbox)• Remember to consider the no/multiple/infinite solution cases• Remember to set your tolerances or scale you problem to ensure accuracy

• Especially important when using the units package (more on that later)• Provides only the solution closest to the initial guess

• Quality of initial guess directly related to quality of solution• Intuition about problem is beneficial• Use graphical output to aid in choosing guess

• Optimset can set various options for solve• >>help optimset for more info

• roots() – Solves polynomial defined by their coefficients. • Provides all solutions