ENGG 1801 Engineering Computing

1
ENGG 1801Engineering Computing

• MATLAB Lecture 7 Tutorial Weeks 11-13
• Solution of nonlinear algebraic equations (II)

2
Outline of lecture

• Solving sets of nonlinear equations
• Multivariable Newtons method
• Example (2 equations in 2 unknowns)
• Solving example problem in Matlab
• Functions
• Conclusions

3
Sets of Nonlinear Equations

• Equation sets can be large
• 100s (1000s) of equations
• Initial solution estimate can be a challenge
• Fewer solution options
• Plotting and direct substitution are not options
• Most widely used approach ?
• Multivariable Newtons method

4
Multivariable Newtons 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, 2.0952)
Jacobian evaluated at old point
Old point (3, 3)
7
Solution in Matlab

• counter 1
• error 10
• xold 33
• while error gt 1e-3 counter lt 10
• fold(1) xold(1) - xold(2)2 fold(2)
  xold(1)xold(2) - 8
• J(1,1) 1 J(1,2) -2xold(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
  MATLABs 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.