Summary and Supplement

8/6/2019 Summary and Supplement

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University of Jordan

Faculty of Engineering and Technology

Engineering Numerical Methods 0904301

Lecture 02: Numerical Solution of Nonlinear AlgebraicLecture 02: Numerical Solution of Nonlinear Algebraic

Equations in One VariableEquations in One Variable

Dr.Dr. YousefYousefZurigatZurigat (([email protected]))

Dr. Ali AlDr. Ali Al--Matar (Matar ([email protected]))

2004/20052004/2005

Numerical Methods 002: Numerical Solution of Nonlinear

Algebraic Equations in One Variable

2

2004:Dr.AliAl-Matar([email protected]),ChemicalEngineeringDept.,UniversityofJordan

2004:Dr.YousefZurigat([email protected]),MechanicalEngineeringDept.,UniversityofJordan

Introduction and Definitions

The general form for an algebraic equation in single

variable is:

This form is encountered frequently in many engineering

areas:

Equations of state in thermodynamics.

Force and moment balances in statics and dynamics. Friction factors and drag coefficients in fluid mechanics.

Electrical engineering circuits.

Vibration analysis.

( ) 0f x =


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:Dr.AliAl-Matar([email protected]),ChemicalEngineeringDept.,Universityo

fJordan

2004

:Dr.YousefZurigat([email protected]),MechanicalEngineeringDept.,University

ofJordan

Hierarchy

f(x) = 0

Explicit Implicit

Polynomials

Transcendental: involves trig.,

exp., log. & special functions

Form of Function unknown.

Its value can be calculated

3 2( ) 0 f Z Z Z Z = + + + =

( )

1

( ) / 2 ln Re / 8 0g f f f B Ak= + =

( ) (1 e ) 0DC tm

D D f C vC mg

= =

10

2 f ff + =



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Root Finding Methods

Root Finding

Real roots for

algebraic or

transcendental equations

All real and

complex roots of

a polynomial

Usually looking for

an approximation to

one single real root.

Usually looking for an

approximation to a root.

Sometimes can get exact

roots when the

polynomial is factorable

(order 2,3, and 4).


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Iterative Root Finding Methods

Root-Finding Methods

Bracketing Open

Initial Guesses Bracket the rootAn initial guess or two that

do not necessarily bracket the root

Incremental-Search

Newton-Raphson(Newton's)

Secant Fixed-pointiteration

Bisection Regula-Falsi(False-Position)



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Relative

Error

Absolute

Error

Function

Value

Termination

Criteria

Termination Criteria

( ) 0rf x =

1i ix x + 1i i

i

x x

x+


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Bracketing Methods for Root Finding

Incremental Search

Bisection

Regula Fasli



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Incremental-Search Method

Disadvantages:

1. Fails if f(x) is tangent to thex-axis.

2. Inefficient

Advantages

1. Used to find the variousintervals over the entirerange ofx in which the realroots of the function arelocated.

2. Good applied withinteractive interface.

3. Often precedes othermethods and is thusemployed in conjunctionwith them.

Locate an interval [a,b]

where the function changes

sign.

Divide [a,b] into a number of

subintervals.

Search each subinterval to

locate the sign change.

The process is repeated andthe root estimate is refined by

dividing the subinterval into

finer increments.


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4 .2 4 .2 1 4 .2 2 4 .2 3 4 .2 4 4 .2 5 4 .2 6 4 .2 7 4 .2 8 4 .2 9 4 .3

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

x

sin(10 x)+cos(3 x)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

sin(10 x)+cos(3 x)

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

sin(10 x)+cos(3 x)



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Bisection Method

1. Locate an interval [xl,xu] where thefunction changes sign.

2. An estimate of the rootxrisdetermined by halving theinterval i.e.

3. The resulting two intervals ([xl,xr]& [xr,xu]) are checked for signchange and step 2 is repeated forthis interval with sign change.

( ) ( ) 0l u f x f x

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