Introduction to numerical methods - Strona główna AGHhome.agh.edu.pl/~zak/downloads/NM1.pdf · 2...
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Numerical Methods - Lecture 1 1
Numerical Methods
dr hab.inż. Katarzyna Zakrzewska, prof. AGH, Department of Electronics, AGH
e-mail: [email protected]://home.agh.edu.pl/~zak
Lecture 1.Introduction to numerical methods
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Numerical methods are a part of applied mathematics focused on the development of approximate methods for solving mathematical problems that can not be solved either by the exact methods or the exact methods have such a large computational complexity that are useless
Numerical methods are involved in constructing algorithms in which the input data, intermediate results and final results are represented by numbers
Introduction to numerical methods
Numerical Methods - Lecture 1
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Characteristics of numerical methods:• calculations are performed on approximate
numbers• solutions are expressed as approximate numbers• error in the numerical calculation should be
always controlled
Introduction to numerical methods
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References:
• Z. Fortuna, B. Macukow, J. Wąsowski, Metody numeryczne, Podręczniki Akademickie EIT, WNT Warszawa,1982, 2005
• L.O. Chua, P-M. Lin, Komputerowa analiza układów elektronicznych-algorytmy i metody obliczeniowe, WNT, Warszawa, 1981
• G.Dahlquist, A.Björck, Metody matematyczne, PWN Warszawa, 1983
• Autar Kaw, Luke Snyder
http://numericalmethods.eng.usf.edu
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Additional references:
• M.Wciślik, Wprowadzenie do systemu Matlab, Wydawnictwo Politechniki Świętokrzyskiej, Kielce, 2000
• S. Osowski, A. Cichocki, K.Siwek, Matlab w zastosowaniu do obliczeń obwodowych i przetwarzania sygnałów, Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa 2006
• W.H. Press, et al., Numerical recipes, Cambridge University Press, 1986
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Outline
• Solving engineering problems• Overview of typical mathematical procedures• Fixed and floating-point representation of numbers
Introduction to numerical methods
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How to solve an engineering problem?
Numerical Methods - Lecture 1
Problem Description
Mathematical Model
Solution of Mathematical Model
Application of the Solution
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Example of Solving an Engineering Problem
Bascule Bridge THG
the Bridge of Lions in St. Augustine, Florida
Autar Kawhttp://numericalmethods.eng.usf.edu
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Trunnion
Hub
Girder
Bascule Bridge THG
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Trunnion-Hub-Girder Assembly Procedure
Step 1. Trunnion immersed in dry-ice/alcohol (108 F, around -80 C)
Step 2. Trunnion warm-up in hub
Step 3. Trunnion-Hub immersed in dry-ice/alcohol
Step 4. Trunnion-Hub warm-up into girder
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A problem occurred!
After cooling, the trunnion got stuck in the hub
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Why did it get stuck?
Magnitude of contraction of the trunnion was expected to be 0.015” or more. Did it contract enough?
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Calculations
TDD Δ××=Δ α
FT o18880108 −=−−=Δ
Finin o//1047.6 6−×=α
0.01504")188)(1047.6)(363.12( 6
−=−×=Δ −D
"363.12=D
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Units
CFT o 07,2680 ≈=
cminD 03820.001504.0 ==Δ
cmD 4,31"363.12 ≈=
cm2.54 in1 =
FTCF
0
32⎟⎠⎞
⎜⎝⎛ +=
59T
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Is the formula used correct?
TDD Δ=Δ α
T(oF) α (μin/in/F)
-340 2.45
-300 3.07
-220 4.08
-160 4.72
-80 5.43
0 6.00
40 6.24
80 6.47
TDD Δ××=Δ α
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The correct model should account for varying thermal expansion coefficient α
dTTDDc
a
T
T
)(∫=Δ α
Problem solving
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Can you roughly estimate the contraction?
dTTDDc
a
T
T
)(∫=Δ α
Ta=80F; Tc=-108F; D=12.363”dTTDDc
a
T
T
)(∫=Δ α
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Estimating Contraction Accurately
dTTDDc
a
T
T
)(∫=Δ α
Change in diameter (ΔD) by cooling it in dry ice/alcohol is given by
0150.6101946.6102278.1 325 +×+×−= −− TTα
Ta = 80oFTc = -108oFD = 12.363"
"0137.0−=ΔD to small!!!Numerical Methods - Lecture 1
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So what is the solution to the problem?
One solution is to immerse the trunnion in liquid nitrogen which has a boiling point of -321F as opposed to the dry-ice/alcohol temperature of -108F.
"0244.0−=ΔD
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Revisiting steps to solve a problem
1) Problem statement: trunnion stuck in the hub
2) Modeling: a new model
3) Solution: a) trapezoidal rule or b) regression and integration.
4) Implementation: cool the trunnion in liquid nitrogen.
dTTDDc
a
T
T
)(∫=Δ α
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Mathematical Procedures
• Nonlinear Equations• Differentiation• Simultaneous Linear Equations• Curve Fitting
– Interpolation– Regression
• Integration• Ordinary Differential Equations• Other Advanced Mathematical Procedures:
– Partial Differential Equations– Optimization– Fast Fourier Transforms
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Nonlinear Equations
How much of the floating ball is under water?
010993.3165.0 423 =×+− −xx2R=0.11m
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010993.3165.0)( 423 =×+−= −xxxf
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Differentiation
What is the acceleration at t=7 seconds?
dtdva =t.
t v(t) 89
500010161016ln2200 4
4
−⎟⎟⎠
⎞⎜⎜⎝
⎛−×
×=
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Time (s) 5 8 12V(m/s) 106 177 600
dtdva=
Numerical Methods - Lecture 1
Differentiation
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Simultaneous Linear Equations
Find the velocity profile, given:
,)( 2 cbtattv ++=
Three simultaneous linear equations:106525 =++ cba
125 ≤≤ t
177864 =++ cba60012144 =++ cba
Time (s) 5 8 12
V (m/s) 106 177 600
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Interpolation
What is the velocity of the rocket at t=7 s?
Time (s) 5 8 12
V (m/s) 106 177 600
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Regression
Thermal expansion coefficient data for cast steel
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Regression
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Integration
∫=Δfluid
room
T
T
dTDD α
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Ordinary Differential Equations
How long does it take a trunnion to cool down?
),( ahAdtdmc θθθ
−−= roomθθ =)0(
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What you need to know to create your own computing algorithms?
• the size of your computer's memory• the execution speed of arithmetic and logic operations• the acceptable range of numbers during the
calculations• the accuracy of basic arithmetic operations performed
on real numbers
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Numbers representation in a computer memory
The numbers are stored as• fixed-point numbers• floating-point numbers
The computer works in the binary system, and communicates with the outside world in the decimal system, therefore conversion procedures are needed.
This is a source of errors.
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How a Decimal Number is Represented21012 10610710710510276.257 −− ×+×+×+×+×=
In the binary system we use two digits: 0 and 1, called bits
Computer arithmetic
Base 2
1875.11)21212020(
)21212021()0011.1011(
104321
0123
2
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
×+×+×+×+
×+×+×+×=
−−−−
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Convert base 10 integer to binary representation
Quotient Remainder11/2 55/2 22/2 11/2 0
01 a=
11 a=
20 a=
31 a=
Hence
2
2012310
)1011()()11(
== aaaa
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http://numericalmethods.eng.usf.edu
Numerical Methods - Lecture 1
Start
Input (N)10
i = 0
Divide N by 2 to getquotient Q & remainder R
ai = R
Is Q = 0?
n = i(N)10 = (an. . .a0)2
STOP
Integer N to be converted to binary format
i=i+1,N=Q
No
Yes
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Number Number afterdecimal Number beforedecimal
0.375 0.3750.75 0.751.5 0.51.0 0.0
10 −= a
20 −= a
31 −= a
41 −= a
2
2432110
)0011.0()()1875.0(
== −−−− aaaa
21875.0 ×2375.0 ×275.0 ×
25.0 ×
Converting a base-10 fraction to binary representation
Hence
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39http://numericalmethods.eng.usf.edu
Start
Input (F)10
Multiply F by 2 to getnumber before decimal,S and after decimal, T
ai = R
Is T =0?
n = i(F)10 = (a-1. . .a-n)2
STOP
Fraction F to beconverted to binaryformat
No
Yes
T=−= F1,ii
1i −=
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( ) ( )210 ?.?1875.11 =
210 )1011()11( =
210 )0011.0()1875.0( =and
we have
210 )0011.1011()1875.11( =
Since
Decimal Number to Binary
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Different approach
( )101875.11
( )
( )2
0123
013
13
310
101121212021
222122
3211
=×+×+×+×=
++=
++=
+=
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( )
( )2
4321
43
310
0011.21212020
22
0625.021875.0
=×+×+×+×=
+=
+=
−−−−
−−
−
( ) ( )210 0011.10111875.11 =
Numerical Methods - Lecture 1
Different approach
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The problem of accuracy
Number Number after
decimal
Numberbefore
decimal0.6 0.61.2 0.20.4 0.40.8 0.81.6 0.6
Example: Not all fractional decimal numbers cannot be represented exactly
23.0 ×26.0 ×22.0 ×24.0 ×28.0 ×
10 −= a21 −= a
30 −= a40 −= a
51 −= a
28125.0)01001.0()()3.0( 225432110 ==≈ −−−−− aaaaaThe accuracy depends on the computer words length. Rounding off and chopping off lead to errors.
Numerical Methods - Lecture 1
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Floating Point Representation
wNMx ×=M - mantissa
W - exponent
N=2, 10
The notation floating point number is represented by two groups of bits:
I – mantissa M, fractional part ½<M<1
II - exponent W , an integer, W determines the range of the numbers represented in the computer
Numerical Methods - Lecture 1
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wNMx ×=
Example:
If in binary representation M defines 5 bits and W defines 3bits, the first bit represents the sign of a number ("-" is 1),then:
10)0(1101)1(=xM W
1021101,0 +×−=x)1021(2
161
80
41
21 ⋅+⋅+×⎟
⎠⎞
⎜⎝⎛ +++−=x
in the decimal representation -3,25Numerical Methods - Lecture 1
Floating Point Representation
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wNMx ×=
In this notation, only certain positive number in the range from 0.0625 to 7.5, negative numbers from -0.0625 to -7.5 and the number 0 can be represented
There are some numbers that are not present in this representation
)0011(0011,0=x
The nearest number (for M = 5 and W = 3) is
Number of x = 0.2 (in decimal) in binary notation has an infinite expansion:
001100,0=x
we have 0,1875
This is a source of input errors
Numerical Methods - Lecture 1
Floating Point Representation
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47
Fixed-point representation
The fixed-point representation of the number allocated to n+2 bits (1 bit for the sign and n +1 bits for the absolute value of the number) has the following structure:
∑=
⋅=n
k
kkbsnumber
02
where:
s=1 or s=-1 (the sign of the number)
bk takes the value 0 or 1 (the absolute value of the number)
Numerical Methods - Lecture 1
sign the absolute value of the number
bits
weights
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On n+2 bits integers in the following range can be stored:
Fixed numbers are a subset of the integers.
[-2n+1+1;2n+1-1]
Overflow?
Numerical Methods - Lecture 1
Fixed-point representation
sign the absolute value of the number
bits
weights
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49
High-level programming languages offer several types of fixed-point numbers:
Integer - 16 bits
LongInt – 32 bits
ShortInt – 8 bits
Numerical Methods - Lecture 1
Fixed-point Representation
sign the absolute value of the number
bits
weights