Convergence

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Convergence

• Numerical methods for finding roots of equations

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Numerical Methods for Finding Roots

• Simple equations
• Rearrange to solve
• Use Maple command solve
• More complicated equations
• solve does not work
• what does?
• not symbolic, but numerical
• Matlab fzero
• Maple fsolve

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Bisection Method (discussed later)
Other Methods (not discussed) Successive
Substitution Newtons Method Secant Method
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Two Member Frame Problem
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The buckling load for the Two Member Frame
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Cantilevered Beam Problem
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The deflection of the cantilevered beam
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The deflection of the cantilevered beam
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Bisection Method

• Iterative method
• f(x) 0

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We assume that the function f() is a continuous
function, so that somewhere between left and
right, there is some point at which f() will be
zero.
left
right
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Bisection Method
1.
2. Evaluate f(m) a. If the sign of f(m) is
the same as the sign of f(l) then the next left
endpoint is m. (The new value of l is m.) b.
If the sign of f(m) is the same as the sign of
f(r) then the new right endpoint is m. (The
new value of r is m). c. If f(m) is zero,
then the computation is finished.
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Bisection Method

• Bisection method is not fast
• Maple Matlab use Bisection Method to start
  calculations, then switch to more sophisticated
  methods of root finding

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Root finding in Maple

• fsolve
• f refers to floating point numbers

• Dont need equations in standard form
• Eq1tan(z)-z/(1(z2/4))
• fsolve(Eq1,z2..4)
• 3.828861865

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Root finding in Maple

• fsolve requires an interval

• Find this interval by plotting the function
• Plotting is very important to finding roots

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Root finding in Matlab

• fzero

• Uses standard form f(x)0

• Put f(x) into an m-file
• For beam equation, use beam.m

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beam.m
function f beam(a) 
f a.2.(a.24a6)3/2

• Now lets look at the main program

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Numerical Methods Summary

• Works with numbers, not symbols
• Always plot the function
• Maple command fsolve
• Matlab command fzero

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Numerical Evaluation of Integrals

• Maple cant always evaluate an integral
  symbolically
• Only a numerical value can be found
• Sometimes a numerical answer is preferred

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Integral examples

• Volume of a tank

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Integral examples
Probability of Motor Failure m number of
months to failure Probability distribution
given by
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Integral examples
What is the probability that a motor survives a
year of use?
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Trapezoid method
f(a)
y
f(b)
Areahftrap
hb-a
x
a
b
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Smaller values of h (larger values of N) give
better approximations of integrals.
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Evaluating integrals in Maple

• evalf
• Evaluates to get a floating point number
• Can use capital Int
• Sets up integral, but doesnt try to symbolically
  evaluate it
• Maple (and Matlab) use more sophisticated version
  of Trapezoid method
• Varies h based on smoothness of curve

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Volume of a tank (Maple)
for
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Numerical Integration in Matlab

• quad
• Qquad(fun,a,b)
• Quadrature means numerical integration

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Volume of a tank (Matlab)
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Volume of a tank (Matlab)
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Probability of Motor Failure (Matlab)
Motor.m function z Motor(m) z
13.765m./(m.221).(7/4)
Q quad(Motor, 0,12) Q 0.7361 (73.61 of
motors will fail, therefore 26.39 will survive
the whole year)
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Average Age of Rebuilt Motors
function z ExpectAge(m) z m.Motor(m)
quad(ExpectAge,0,12) .263912
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Pulling a square plug out of a sphere
Pull a 4x4 plug out of the center of the sphere
Uses the dblquad command in Matlab (see course
notes)
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Numerical Integration Summary

• Trapezoid rule

• Maple
• evalf(int,xa..b)
• Matlab
• quad(fun,a,b)