ECIV 301

Programming & Graphics

Numerical Methods for Engineers

Lecture 6

Roots of Equations

Bracketing Methods

Last Time - Accuracy and PrecisionAccuracy

Pre

cisi

on

Last Time - Truncation Errors

vi

ti ti+1

vi+1

True Slope

Approximate Slope

ii

ii

tt

tvtv

dt

dv

1

1

Truncation errors due to using approximation in place of exact solution

Last Time - Roundoff Errors

14.3

141592654.3

A=d2

1415.3

Last Time - Error Definition

Et=true value - approximation

True Error

t= (Et/True Value)100%

Relative True Error

Last Time - Error Definition

%100ionapproximat

erroreapproximata

Approximate Relative Error

Iteration Relative Error

%100ionapproximatcurrent

ionapproximatpreviousionapproximatcurrent a

Last Time - The Taylor Series

vi

ti ti+1

vi+1

vivi

tititi ti+1ti+1

vi+1vi+1

Predict value of a function at one point in terms of the function value and its

derivatives at another point

Last Time - Taylor’s Theorem

nn

i1ii

n

3i1i

i

2i1i

ii1iii1i

Rxx!n

xf

xx!3

xf

xx!2

xfxxxfxfxf

Error of Order (xi+1 – xi)n+1

Last Time - Numerical Differentiation

ii

iii

ffxf

xx

xx

1

1

Forward Difference

1

1

xx

xx

ii

iii

ffxfBackward Difference

11

11

xx

xx

ii

iii

ffxfCentral Difference

The Problem

vm

cg

dt

dv

tm

c

ec

gmv 1

Analytic Solution

The Problem

To design the parachute:

v=10 m/st=3 secm=64 kgg=9.81

c=?

tm

c

ec

gmv 1

CANNOT rearrange to solve for c

The Problem

)(1)( tvec

gmcf

tm

c

Define Function

0)( cfc must satisfy

c is the ROOT of the equation

Objectives

• Master methods to compute roots of equations

• Assess reliability of each method

• Choose best method for a specific problem

ClassificationMethods

Bracketing Open

• Graphical• Bisection Method• False Position

• Fixed Point Iteration• Newton-Raphson• Secand

Graphical Methods)(1)( tve

c

gmcf

tm

c

f(c)

-10

-5

0

5

10

15

20

25

0 50 100 150 200

f(c)

c

f(c)

v=10 m/st=3 secm=65 kgg=9.81

Graphical Methods

No Roots Even Number of Roots

Lower and Upper Bounds of interval yield values of same sign

Graphical MethodsLower and Upper Bounds of interval yield values of opposite sign

Odd number of Roots

Bisection Method

Choose Lower, xl and Upper xu guesses that bracket the root

f(c)

-10

-5

0

5

10

15

20

25

0 50 100 150 200

f(c)

xl xu

Bisection Method

Calculate New Estimate xr and f(xr)

f(c)

-10

-5

0

5

10

15

20

25

0 50 100 150 200

f(c)

xl xu

xr=0.5(xl+xu)

Bisection MethodCheck Convergence

ErrorAcceptablex

xxnewr

oldr

newr

%100

Root = newrx

If Error

Bisection MethodDefine New Interval that Brackets the RootCheck sign of

f(xl)*f(xr) and f(xu)*f(xr)

f(c)

-10

-5

0

5

10

15

20

25

0 50 100 150 200

f(c)

xl xu

Previous Guess

xu

Bisection MethodRepeat until convergence

f(c)

-10

-5

0

5

10

15

20

25

0 50 100 150 200

f(c)

xl xu

Previous Guess

xr=0.5(xl+xu)

Bisection - FlowchartLoop

xold=x

x=(xl+xu)/2

Error=100*abs(x-xold)/x

Sign=f(xl)*f(xr)

Sign

xu=x xl=xError=0

Error<Eall ROOT=xFALSE

<0 >0

Pseudo Code