1. ENGG 1801 Engineering Computing
MATLAB Lecture 7: Tutorial Weeks 11-13
Solution of nonlinear algebraic equations (II)

2. Outline of lecture Solving sets of nonlinear equations
Multivariable Newton’s method
Example (2 equations in 2 unknowns)
Solving example problem in Matlab
Functions
Conclusions

3. Sets of Nonlinear Equations
Equation sets can be large
100’s (1000’s) of equations
Initial solution estimate can be a challenge
Fewer solution options
Plotting and direct substitution are not options
Most widely used approach ?
Multivariable Newton’s method

4. Multivariable Newton’s Method
Single variable algorithm
Each iteration solves a linear approximation to function
Multivariable algorithm
Each function approximated by a linear equation
Each iteration solves a set of linear equations
Scalar equation
Vector-matrix equation

5. Example (I) Solve the pair of equations:
Elements of the Jacobian matrix
Solution is:
x1 = 4 x2 = 2

6. Example (II) Use as initial estimate for solution (3, 3)
Next estimate obtained from:
Functions evaluated at old point
New point = (3.5714, )
Jacobian evaluated at old point
Old point = (3, 3)

7. Solution in Matlab counter = 1; error = 10; xold = [3;3];
while error > 1e-3 & counter < 10
fold(1) = xold(1) - xold(2)^2; fold(2) = xold(1)*xold(2) - 8;
J(1,1) = 1; J(1,2) = -2*xold(2); J(2,1) = xold(2); J(2,2) = xold(1);
xnew = xold - inv(J)*fold'
error = max(abs(xnew(1)-xold(1)),abs(xnew(2)-xold(2)));
counter = counter + 1;
xold = xnew;
end

8. Advice on Iterative Methods
Follow one cycle through code by hand
Initially use modest convergence criterion
Put in a ‘counter’ ( to prevent infinite loop )
Check final solution
Be prepared for multiple solutions
Initial guess has a big impact:
On the final solution obtained
On the time taken to converge to solution

9. Functions Breaking up complex calculations into simpler blocks.
Main program
a = 2; b = 3;
[answer, diff] = my_function(a,b)
Separate file, my_function.m; outputs calculated inside function using input arguments (in1 & in2)
function [answ, differ] = my_function(in1,in2)
answ = in1 + in2;
differ = in1 – in2;
One to one correspondence
between inputs, outputs in
calling statement and in
function

10. Conclusions Solution of nonlinear equation sets ?
Very common problem in engineering
Built-in Matlab functions ( e.g. fsolve from MATLAB’s Optimization Toolbox)
User supplied Jacobian speeds convergence
If unavailable → Matlab gets by finite differencing
User has to supply initial estimate of solution
Make your own functions to split up a big problem into simpler pieces.