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1Copyright © 2005 by yshong
2
Engineering Applications with Engineering Applications with Computers IComputers I
(Aspect in Numerical (Aspect in Numerical Methods)Methods)
YUNG-SHAN HONG, Ph.D., PE.
Office: E723
Tel: 26215656 ext. 3260
Instructor:
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Objective:
This course covers a variety of numerical methods and their applications in various engineering problems. Emphasis is placed on the solution of solving nonlinear equation, matrix analysis of linear and nonlinear equations, eigen-value problems, curve fitting, numerical integration and differentiations as well as interpolation methods.
Pre-knowledge of Engineering Mathematics and programming skills with computer language (s) are strongly required.
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Outline and Schedule:
Introduction (2 hrs)
Mathematical modeling and engineering problem solving (2hrs)
Error and definition (2hrs)
Roots of equations (1) - bracketing methods (2hrs)
Roots of equations (2) - open methods (2hr)
Systems of nonlinear equations (2hrs)
Linear algebraic equations - mathematical and numerical method (3hrs)
Eigenvalue problems (3hrs)Copyright © 2005 by yshong
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Outline and Schedule:
Least squares regression (2hrs)
Interpolation - Lagrange and Newton approach (2hr)
Interpolation - spline function (2hrs)
Numerical integration - general (2hrs)
Numerical integration - double integral (2hrs)
Numerical solution of ordinary differential equations (2hrs)
Numerical solution of partial differential equations (2hrs)
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Grading:
Ordinarily expression 20%
Homework (3~4 times) 20%
Mid term exam 30%
Final term exam 30%
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Textbook:
Chapra, S.C. and Canale, R.P.(2002), “Numerical methods for engineers – with programming and software applications”, Fourth Edition, McGRAW-Hill.
Reference: Gerad, C.F. and Wheatley, P.O.(1999), “Applied numerical analysis”, Sixth Edition, Addison-Wesley.
Schilling, R.J. and Harris, S.L.(1999), “Applied numerical methods for engineers – using Matlab and C”, Brooks/Cole.
林聰悟、林佳慧 (1997), “ 數值方法與程式” , 圖文技術服務。
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About the authors:
Steve Chapra teaches in the Civil and Environmental Engineering Department at Tufts University.
Dr. Chapra received engineering degrees from Manhattan College and the University of Michigan. Before joining the faculty at Tufts, he worked for the Environmental Protection Agency and the National Oceanic and Atmospheric Administration, and taught at Texas A&M University and the University of Colorado. His general research interests focus on surface water-quality modeling and advanced computer applications in environmental engineering.
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About the co-authors:
Raymond P. Canale is an emeritus professor at the University of Michigan.
During his over 20-year career at the university, he taught numerous courses in the area of computers, numerical methods, and environmental engineering. He also directed extensive research programs in the area of mathematical and computer modeling of aquatic ecosystems. He has authored or coauthored several books and has published over 100 scientific papers and reports.
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Why you should study numerical methods ?
Numerical methods are extremely powerful problem-solving tools. They are capable of handling large systems of equations, nonlinearities, and complicated geometries that are not uncommon in engineering practice and often impossible to solve analytically.
During your careers, you may often have occasion to use commercially available prepackaged that involve numerical methods. The intelligent use these programs is often predicated on knowledge of the basic theory underlying the methods.
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Many problems cannot be approached using prepackaged programs. If you are conversant with numerical methods and are adept at computer programming, you can design your own programs to solve problems without having to buy expensive software.
Numerical methods are an efficient vehicle for learning to use computers. Because numerical methods are for the most part designed for implementation on computers, they are ideal for this purpose. You will also learn to control the errors of approximation that are part of large-scale numerical calculations.
Numerical methods provide a vehicle for you to reinforce your understanding of mathematics. Because one function of numerical methods is to reduce higher mathematics to basic arithmetic operations.
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Solutions of the problem in engineering:
INTRODUCTION
Analytical solution:
(closed form solution)
Ex. Determine
)(sin xdx
d
)(sin xdx
dat x=0
let x=0 1cos)(sin xxdx
d 900 x(o)
sinx
Ex. Determine AE
PL
P
1
?
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Numerical solution:(approximation solution)
Ex. Determine )2ln( 2xxedx
d x at x=10
let )2ln()( 2xxexf x
x
f(x)
10
1?
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Numerical method:
Data + Mathematical theory + computer program
Approximation
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Types of the problem:
(a) Solution of nonlinear equation (roots of equation)
0594 23 xxx
594)( 23 xxxxflet
x
f(x)
Ex.
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(b) Matrix analysis (solution of linear algebratic eqs.)
Ex.
2222121
1212111
puaua
puaua
2
1
2
1
2221
1211
p
p
u
u
aa
aa
u1
u2
Ex.
232
1043
yx
yx
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(c) System of nonlinear eqs.
Ex.
x1
x2
222221121
122121111
)()(
)()(
cxxaxxa
cxxaxxa
2
1
2
1
222121
212111
)()(
)()(
c
c
x
x
xaxa
xaxa
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(d) Curve fitting
Regression – Least squares regression
Interpolation & Extrapolation
x
y y
x
Regression Interpolation & Extrapolation
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(e) Integration technique
x
f(x)
a b
I
f(x)
b
a
dxxfI )( p(w)
½ space
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(f) Ordinary differential equation (ODE)
Because many physical laws are couched in terms of the rate of change of a quantity rather than the magnitude of the quantity itself.
Ex.
tdt
dy3
Difference scheme viewpoint ),(1 ytf
t
yy
t
y
dt
dy ii
Solve y as a function of t
),(1 ytftyy ii
y
t
f(t,y)yi+1yi
yRi+1
ti ti+1
Δt
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(f) Ordinary differential equation (ODE)
Additional data must be given:
Initial value problem
Boundary value problem
x1
f(x1
) x
?
x1
f(x1
) x
?
x2
f(x2
)
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(g) Partial differential equation (PDE)
The behavior of a physical quantity is couched in terms of its rate of change with respect to two or more independent variables.
Elliptic – solid mech., flow mech.potential
),(
2
2
2
2
yxpy
u
x
u
0),( yxp Laplace eqs. ( 滲流控制方程式 )
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(g) Partial differential equation (PDE)
Parabolic – consolidation, heat…
t
u
cz
u
v
1
2
2
Analytical sol.
0
[.......]m
u
2dr
vv H
tcT
Hyperbolic – wave eqs.
2
2
22
2 1
x
u
ct
u
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Motivation:
Numerical methods are techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations. Although there are many kinds of numerical methods, they have one common characteristic: they invariably involve large numbers of tedious arithmetic calculations.
It is little wonder that with the development of fast, efficient digital computers, the role of numerical methods in engineering problem solving has increased dramatically in recent years.
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Non-computer methods:
(1) Solutions were derived for some problems using analytical, or exact method.
Ex. 0422 xx
Exact sol.
ix 312
1642
Ex. 082sin/3 257 xxexx x
?Exact sol.
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(2) Graphical solutions were used to characterize the behavior of systems.
Ex.
01043
0522
2
yx
yx
1
2
x
yx ….y ….
x ….y ….
The results are not very precise. Graphical techniques are often limited to problems that can be described using three or fewer dimensions.
(3) Calculators and slide rules were used to implement numerical method manually.
The method used to simple engineering problems.
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Numerical method:
Data + Mathematical theory + computer program
Approximation
Complex engineering problems:
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The engineering problem-solving process :
Problem definition
Mathematical model
Numeric or graphic results
Implementation
DataTheory
Problem-solving tools:Computers, statistics,Numerical methods, graphics, etc.
Societal interfaces:Scheduling, optimization, communication, public interaction, ect.
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CHAPTER 1
A SIMPLE MATHEMATICAL MODEL
A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms.
In a very general sense, it can be represented as a functional relationship of the form:
Dependent variable = f (independent variables, parameters, forcing functions)…………..(1.1)
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Dependent variable = f (independent variables, parameters, forcing functions)…………..(1.1)
Where the dependent variable is a characteristic that usually reflects the behavior or state of the system; the independent variables are usually dimensions, such as time and space, along which the system’s behavior is being determined; the parameters are reflective of the system’s properties or composition; and the forcing functions are external influences acting upon it.
AE
PL
: dependent variable
P: forcing functions
A,E,L: parametersCopyright © 2005 by yshong
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The following illustrates a physical problem how to represent by a mathematical model.
According Newton second law,
maF
Where F = net force acting on the body (N or kg-m/sec2)
m = mass of object (kg)
a = its acceleration (m/sec2)
…………………………(1.2)
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(1.2):
Where a = the dependent variable reflecting the system’s behavior
F = the forcing function (net froce)
m = a parameter represent a property of the system
m
Fa …………………………(1.3)
Note: this simple case there is no independent variable because we are not yet predicting how acceleration varies in time or space.
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To illustrate a more complex model of this kind, Newton’s second law can be used to determine the terminal velocity of a free-falling body near the earth’s surface. The falling body will be a parachutist. (Fig.1.2)
m
F
dt
dv ……….(1.4)
Fu
Fd
F: net force+: the object will accelerate
-: the object will decelerate
0: the object will remain at a constant level
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UD FFF ……….(1.5)
FD: the downward pull of gravity
FU: the upward force of air resistance
cvFU
mgFD ……….(1.6)
……….(1.7)
g: the gravitational constant ≈ 9.8 m/s2
c: drag coefficient = f(shape, surface roughness,….)
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From eqs.(1.4) through (1.7) combined:
m
cvmg
dt
dv
vm
cg
dt
dv
or
……….(1.9)
……….(1.8)
Type of eq. ? ODE
Eq.(1.9) is a differential equation that relates the acceleration of a falling object to the forces acting on it.
If the parachutist is initially at rest (v=0 at t=0), that is a initial value problem. Solve eq.(1.9) for
What type of problem ?
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)1()( )/( tmcec
gmtv ……….(1.10)
Note : v(t): the dependent variable
t= the independent variable
c,m= parameters
g= the forcing function
The following will illustrate the analytical solution and the numerical solution, respectively.
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Ex 1.1 analytical solution
Known: mass=68.1 kg, c=12.5 kg/s
Eq.(1.10) then
)1(39.53)( 18355.0 tetv
terminal velocity53.39
exact sol.
t(s)
v(m/s)
Eq.(1.10) is called an analytical, or exact solution because it exactly satisfies the original differential equation. Unfortunately, there are many mathematical models that cannot be solved exactly. In many of these cases, the only alternative is to develop a numerical solution that approximates the exact solution.Copyright © 2005 by yshong
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Ex 1.2 numerical solution
tv
t
tt
vv
t
v
dt
dv
ii
ii
0lim
1
1 ……….(1.11)
vm
cg
dt
dv ……….(1.9)
So eq.(1.9):
i
ii
ii vm
cg
tt
vv
1
1
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)( 11 iiiii ttvm
cgvv
When t=0, v=0, if step size (time step)=2
6.192)0(
1.68
5.128.90)2(
tv
322)6.19(
1.68
5.128.96.19)4(
tv
i=0
i=1
m/s
m/s
v(t=6), v(t=8), …………..Copyright © 2005 by yshong
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v (m/s)
t (s)
terminal velocity
exact sol.
numerical sol.
2 4 860
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Homework :
Problems 1.3, 1.4 and 1.5 (p.22)
Due : One week
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CHAPTER 2
PROGRAMMING AND SOFTWARE
pp. 25-49
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CHAPTER 3
APPROXIMATIONS AND ROUND-OFF ERRORS
How much error is present in our calculations and is it tolerable ?
Two major forms of numerical error:
Round-off error
Truncation error
Inherent errorCopyright © 2005 by yshong
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The concept of a significant figure.See Fig.3.1 (p.51)
Accuracy and precision
Accuracy refers to how closely a computed or measured value agrees with the true value.
True value = 2.83
Precision refers to how closely individual computed or measured values agree with each other.
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Fig.3.2
Increasing accuracyIn
crea
sing
pre
cisi
on
(a) (b)
(c) (d)
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Numerical methods should be sufficiently
accurate or unbiased to meet the
requirements of particular engineering
problem. They also should be precise
enough for adequate engineering design.
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Error definitions
(1) True error Et (absolute error)
Et = true value - approximation
Ex. Two approaches to measure length of the two objects.
Approach (a) : Object (a) true length=1m, measured error=1cm
Approach (b) : Object (b) true length= 0.1m, measured error=1cm
What is better approach ?
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(2) Relative error t
%100
valuetrue
errortruet
Ex. Two approaches to measure length of the two objects.
Approach (a) : Object (a) true length=1m, measured error=1cm
Approach (b) : Object (b) true length= 0.1m, measured error=1cm
Approach (a) : t=1%
Approach (b) : t=10%
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(3) The approximation percent relative error a
%100
%100
)(
)1()(
mi
mi
mi
a
u
uu
ionapproximatcurrent
ionapproximatpreviousionapproximatcurrent
……….(3.5)
m : iteration number
i : point, position
Iterative approach characteristic
value
Cal. number
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Truncation error (Chapter 4)
Truncation errors are those that result from using an approximation in place of an exact mathematical procedure.
For example, in Chap. 1 we approximated the derivative of velocity of a falling parachutist by a finite-divided-difference eq. of the form.
ii
ii
tt
vv
t
v
dt
dv
1
1 ……….(4.1)
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A truncation error was introduced into the numerical solution because the difference eq. only approximates the true value of the derivative.
In order to gain insight into the properties of such errors, we now turn to a mathematical formulation that is used widely in numerical methods to express functions in an approximate fashion – the Taylor series.
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Taylor series
c is between [a, b], nth-order derivatives are existence for f(x), then f(x) at c can be to express following eq. using Taylor series.
dttftxn
xR
xRn
cxcfcxcfcxcfcfxf
x
c
nnn
n
nn
)()(!
1)(
)()!1(
))((.....
!2
))((
!1
))(()()(
)(
1)1(2
Rn(x) = remainder term
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If c=0, f(x) series expressing to call Maclaurin’s series,
dttftxn
xR
xRn
xfxfxffxf
xnn
n
n
nn
0
)(
1)1(2
)()(!
1)(
)()!1(
)0(.....
!2
)0(
!1
)0()0()(
If (n-1)th-oder approximate, then Rn(x) refers to truncation error
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Ex. Use fourth-order Maclaurin series expansions to approximate the function
xexf )(
Predict the function’s value at x=1.
Sol: let f(x)=ex, f’(x)=f’’(x)=f’’’(x)=f(4)(x)=ex,
∴ f(0)=1, f’(x)=f’’(x)=f’’’(x)=f(4)(x)=1
∵Maclaurin expansion series:
.....
!4
)0(
!3
)0(
!2
)0(
!1
)0()0()( 4
)4(32
x
fx
fx
fx
ffxf
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Expressing to fourth-order
4324 24
1
6
1
2
11)( xxxxxf
70833.224
65
24
1
6
1
2
111)1(4 f
But 71828.2)( 1
1
exf
x
∴ truncation error= 2.71828-2.70833= 0.00995
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In a similar manner, the complete Taylor series expansion:
nn
iii
n
iii
iii
iiiii
Rxxn
xf
xxxf
xxxf
xxxfxfxf
11
)1(
31
)3(2
111
)()!1(
)(
...)(!3
)()(
!2
)())(()()(
………..(4.5)
If we simplify the Taylor series,
))(()()( 11 iiiii xxxfxfxf Refer to first-order approximation
2111 )(
!2
)())(()()( ii
iiiiii xx
xfxxxfxfxf
Refer to second-order approximation
……
xi+1-xi=h refer to step sizeCopyright © 2005 by yshong
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Ex. 4.1 Use zero ~ fourth-order Taylor series expansions to approximate the function:
2.125.05.015.01.0)( 234 xxxxxf
from xi=0 with h=1. That is, predict the function’s value at xi+1=1
Sol: true value f(1)=0.2
zero-order: 2.1)0()()( 1 fxfxf ii Truncation error=0.2-1.2=-1
first-order: 95.025.02.11)0()0()1( fff 25.0)0( f∵
Truncation error=0.2-0.95=-0.75
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second-order: 45.0)1( f Truncation error=0.2-0.45=-0.25
f(x)
x
Zero order
first order
second order
xi=0 xi+1=1
1.2
0.95
0.45
)()( 1 ii xfxf
hxfxfxf iii )()()( 1
21 !2
)()()()( h
xfhxfxfxf i
iii
f(xi+1)
f(xi)
How order Taylor series expansion can be no truncation error ?
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In general, the nth-order Taylor series expansion will be exact for an nth-order polynomial. For other differentiable and continuous functions, such as exponentials and sinusoids, a finite number of terms will not yield an exact estimate. Each additional term will contribute some improvement, to the approximation. Only if an infinite number of terms are added will the series yield an exact result.
EX. 4.2
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Round-off error (Chapter 3)
Round-off errors originate from the fact that computers retain only a fixed number of significant figures during a calculation. Number such as p, e, or cannot be expressed by a fixed number of significant figures. Therefore, they cannot represented exactly by the computer.
In addition, because computers use a base-2 representation, they cannot precisely represent certain exact base-10 numbers. The discrepancy introduced by this omission of significant figures is called round-off error.
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Base-2: 00 01 10 11 100 101 110 111 1000 1001
Base-10: 0 1 2 3 4 5 6 7 8 9
Ex. 3253 is represented base-10:
3253103105102103 0123
Ex. 110.11 is represented base-2:
1021012 )75.6(2121202121
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But, ex. (0.2)10
8 numbers represented:
...212121212020
...)110011001100.0()2.0(874321
210
199219.0
00390625.00078125.00625.0125.0
21212121
)00110011.0()2.0(8743
210
To get a decimal point at sixth number
Round-off error = 0.2 - 0.199219 = 0.000781
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63
ROOTS OF EQUATIONS (Part 2, p.105)
Ex. 0tan xLxL xexf x )(
Such as f(x) cannot be solved analytically. In such instance, the only alternative is an approximate solution technique.
One method to obtain an approximate solution is to plot the function and determine where it crosses the x axis. This point, which represents the x value for which f(x) = 0, is the root.
f(x)
x
root
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Although graphical method are useful for obtaining rough estimates of roots, they are limited because of their lack of precision.
An alternative approach is to use trial and error. This “technique” consists of guessing a value of x and evaluating whether f(x) is zero.
Such this methods are obviously inefficient and inadequate for the requirements of engineering practice.
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65
Ex. )1()( )/( tmcec
gmtv
Such computations can be performed directly because v is expressed explicitly as a function of time.
However, suppose we had to determine the drag coefficient for a parachutist of a given mass to attain a prescribed velocity in a set time period.
Ex. vec
gmcf tmc )1()( )/(
There is no way to rearrange the equation so that c is isolated on one side of the equal sign. In such cases, c is said to be implicit.
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66
Approach of Nonlinear equation solution:
Bracketing method (chap. 5) – bisection, false position
Open method (chap. 6) – one-point iteration, Newton-Raphson, secant method
Roots of polynomials (chap. 7) – Müller’s methos, Bairstow’s method
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Roots within the interval
Assumption a nonlinear equation f(x)=0 is a continue function. Two points are “a” and “b” on x-axis, then f(x) is whether solutions between a and b. According to follow as,
(1) If f(a)*f(b)=0, then f(x) has a solution.
(2) If f(a)*f(b)<0, then f(x) has a solution x=r between “a” and “b” to satisfy f(x)=0.
(3) If f(a)*f(b)>0, then ?
Ref. pp.114~115. fig.5.2 ~ fig.5.4.
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CHAPTER 5
BRACKETING METHODS
Bi-section method
a
b
f(a)
f(b)
x1
x2
x3
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False position method (linear interpolation method)
a
b
f(a)
f(b)
x1x2x3
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CHAPTER 6 OPEN METHODS
For the bracketing methods in the previous chapter, the root is located within an interval prescribed by a lower and an upper bound. Repeated application of these methods always results in closer estimates of the true value of the root. Such methods are said to be convergent because they move closer to the truth as the computation progresses.
In contrast, the open methods described in this chapter are based on formulas that require only a single starting value of x or two starting values that do not necessarily bracket the root. As such, they sometimes diverge or move away from the true root as the computation progresses. However, when the open methods converge, they usually do so much quickly than the bracketing methods.
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Secant method (ref. chap.6.3)
r0 r2r1 r3
x
y
f(x)
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Newton-Raphson method (ref. chap.6.2)
x
y f(x)
x0x2 x1
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Fixed-point iteration (ref. chap.6.1)f(x)=0, x=g(x)
Rewrite, y1=x, y2=g(x)
x
y
y1=x
y2=g(x)
x0 x1 x2
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CHAPTER 6.5 SYSTEMS OF NONLINEAR EQUATIONS
573
102
2
xyy
xyxEx.
Fixed-point iteration
Newton-Raphson method
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LINEAR ALGEBRAIC EQUATIONS
nnnnnn
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
...
.....
...
...
2211
22222121
11212111
Matrix form:
BxA BAx 1
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Mathematical background (ref. pp.219~230)
Diagonal matrix
Unit matrix
Upper triangular matrix
Lower triangular matrix
Transpose matrix
Symmetrical matrix
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Mathematical approach:
Numerical approach:
Inverse matrix method
Cramer’s method
Gauss elimination method
Gauss-Jordan elimination method
LU decomposition method
Jacobi’s iteration method
Gauss-Seidel iteration method
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78
EIGENVALUE PROBLEMS
(ref. chapter 27)
Engineering analysis:
Steady state (static equilibrium)
Eigenvalue problems (vibration, oscillating system, …)
Propagation problems (wave propagation, transient involve a lot of frequencies)
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79
Steady state (static equilibrium)-
Single frequency
Ex.
PKU K: stiffness
U: displacement
P: force
Solve the system of algebraic
The equation is nonhomogeneous
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Eigenvalue problems (vibration, oscillating system, …)
Solve the system of algebraic
The equation is homogeneous, and the U solution is not unique. (for P=0)
PKU
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81
CURVE FITTING, LEAST-SQUARE REGRESSION
(ref. chapter 17)
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INTERPOLATION
(ref. chapter 18)
Lagrange interpolation polynomial
Newton’s interpolation method
Spline interpolation (Spline function)
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NUMERICAL INTEGRATION
(ref. pp.569 ~ 612)
Rectangle integration
Trapezoidal integration
Simpson’s integration
Newton-cotes integration
Romberg integration
Double integralCopyright © 2005 by yshong
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NUMERICAL DIFFERENTIATION
(ref. chapter 23, pp.632 ~ 666)
Forward difference
Backward difference
Central difference
Difference scheme -
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Thank for your attention
Copyright © 2005 by yshong