View
218
Download
0
Category
Preview:
Citation preview
8/6/2019 Summary and Supplement
1/15
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 ((zurigat@ju.edu.jo))
Dr. Ali AlDr. Ali Al--Matar (Matar (aalmatar@ju.edu.jo))
2004/20052004/2005
Numerical Methods 002: Numerical Solution of Nonlinear
Algebraic Equations in One Variable
2
2004:Dr.AliAl-Matar(aalmatar@ju.edu.jo),ChemicalEngineeringDept.,UniversityofJordan
2004:Dr.YousefZurigat(zurigat@ju.edu.jo),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 =
8/6/2019 Summary and Supplement
2/15
Numerical Methods 002: Numerical Solution of Nonlinear
Algebraic Equations in One Variable
3
2004
:Dr.AliAl-Matar(aalmatar@ju.edu.jo),ChemicalEngineeringDept.,Universityo
fJordan
2004
:Dr.YousefZurigat(zurigat@ju.edu.jo),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 + =
Numerical Methods 002: Numerical Solution of Nonlinear
Algebraic Equations in One Variable
4
2004:Dr.AliAl-Matar(aalmatar@ju.edu.jo),ChemicalEngineeringDept.,UniversityofJordan
2004:Dr.YousefZurigat(zurigat@ju.edu.jo),MechanicalEngineeringDept.,UniversityofJordan
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).
8/6/2019 Summary and Supplement
3/15
Numerical Methods 002: Numerical Solution of Nonlinear
Algebraic Equations in One Variable
5
2004
:Dr.AliAl-Matar(aalmatar@ju.edu.jo),ChemicalEngineeringDept.,Universityo
fJordan
2004
:Dr.YousefZurigat(zurigat@ju.edu.jo),MechanicalEngineeringDept.,University
ofJordan
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)
Numerical Methods 002: Numerical Solution of Nonlinear
Algebraic Equations in One Variable
6
2004:Dr.AliAl-Matar(aalmatar@ju.edu.jo),ChemicalEngineeringDept.,UniversityofJordan
2004:Dr.YousefZurigat(zurigat@ju.edu.jo),MechanicalEngineeringDept.,UniversityofJordan
Relative
Error
Absolute
Error
Function
Value
Termination
Criteria
Termination Criteria
( ) 0rf x =
1i ix x + 1i i
i
x x
x+
8/6/2019 Summary and Supplement
4/15
Bracketing Methods for Root Finding
Incremental Search
Bisection
Regula Fasli
Numerical Methods 002: Numerical Solution of Nonlinear
Algebraic Equations in One Variable
8
2004:Dr.AliAl-Matar(aalmatar@ju.edu.jo),ChemicalEngineeringDept.,UniversityofJordan
2004:Dr.YousefZurigat(zurigat@ju.edu.jo),MechanicalEngineeringDept.,UniversityofJordan
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.
8/6/2019 Summary and Supplement
5/15
Numerical Methods 002: Numerical Solution of Nonlinear
Algebraic Equations in One Variable
9
2004
:Dr.AliAl-Matar(aalmatar@ju.edu.jo),ChemicalEngineeringDept.,Universityo
fJordan
2004
:Dr.YousefZurigat(zurigat@ju.edu.jo),MechanicalEngineeringDept.,University
ofJordan
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)
Numerical Methods 002: Numerical Solution of Nonlinear
Algebraic Equations in One Variable
10
2004:Dr.AliAl-Matar(aalmatar@ju.edu.jo),ChemicalEngineeringDept.,UniversityofJordan
2004:Dr.YousefZurigat(zurigat@ju.edu.jo),MechanicalEngineeringDept.,UniversityofJordan
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
Recommended