Here we focuses on Fixed-Point Iterative Technique for solving nonlinear Equations in Numerical Analysis. It is one of the opened-iterative techniques for finding roots called Fixed-Point of Non-linear Equations.
A short presentation on the topic Numerical Integration for Civil Engineering students.
This presentation consist of small introduction about Simpson's Rule, Trapezoidal Rule, Gaussian Quadrature and some basic Civil Engineering problems based of above methods of Numerical Integration.
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
Beyond Floating Point – Next Generation Computer Arithmeticinside-BigData.com
John Gustafson from the National University of Singapore presented this talk at Stanford.
“A new data type called a “posit” is designed for direct drop-in replacement for IEEE Standard 754 floats. Unlike unum arithmetic, posits do not require interval-type mathematics or variable size operands, and they round if an answer is inexact, much the way floats do. However, they provide compelling advantages over floats, including simpler hardware implementation that scales from as few as two-bit operands to thousands of bits. For any bit width, they have a larger dynamic range, higher accuracy, better closure under arithmetic operations, and simpler exception-handling. For example, posits never overflow to infinity or underflow to zero, and there is no “Not-a-Number” (NaN) value. Posits should take up less space to implement in silicon than an IEEE float of the same size. With fewer gate delays per operation as well as lower silicon footprint, the posit operations per second (POPS) supported by a chip can be significantly higher than the FLOPs using similar hardware resources. GPU accelerators, in particular, could do more arithmetic per watt and per dollar yet deliver superior answer quality.”
Sign up for our insideHPC Newsletter: http://insidehpc.com/newsletter
Watch the video presentation: http://wp.me/p3RLHQ-gjH
Here we focuses on Fixed-Point Iterative Technique for solving nonlinear Equations in Numerical Analysis. It is one of the opened-iterative techniques for finding roots called Fixed-Point of Non-linear Equations.
A short presentation on the topic Numerical Integration for Civil Engineering students.
This presentation consist of small introduction about Simpson's Rule, Trapezoidal Rule, Gaussian Quadrature and some basic Civil Engineering problems based of above methods of Numerical Integration.
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
Beyond Floating Point – Next Generation Computer Arithmeticinside-BigData.com
John Gustafson from the National University of Singapore presented this talk at Stanford.
“A new data type called a “posit” is designed for direct drop-in replacement for IEEE Standard 754 floats. Unlike unum arithmetic, posits do not require interval-type mathematics or variable size operands, and they round if an answer is inexact, much the way floats do. However, they provide compelling advantages over floats, including simpler hardware implementation that scales from as few as two-bit operands to thousands of bits. For any bit width, they have a larger dynamic range, higher accuracy, better closure under arithmetic operations, and simpler exception-handling. For example, posits never overflow to infinity or underflow to zero, and there is no “Not-a-Number” (NaN) value. Posits should take up less space to implement in silicon than an IEEE float of the same size. With fewer gate delays per operation as well as lower silicon footprint, the posit operations per second (POPS) supported by a chip can be significantly higher than the FLOPs using similar hardware resources. GPU accelerators, in particular, could do more arithmetic per watt and per dollar yet deliver superior answer quality.”
Sign up for our insideHPC Newsletter: http://insidehpc.com/newsletter
Watch the video presentation: http://wp.me/p3RLHQ-gjH
NUMERICA METHODS 1 final touch summary for test 1musadoto
MY FINAL TOUCH SUMMARY FOR TEST 1
ON 6TH MAY 2018
TOPICS AND MATERIALS COVERED
1. Class lecture notes (Basic concepts, errors and roots of function).
2. Lecture’s examples.
3. Past Years Examples.
4. Past Years examination papers.
5. Tutorial Questions.
6. Reference Books + web.
Number systems - Efficiency of number system, Decimal, Binary, Octal, Hexadecimalconversion
from one to another- Binary addition, subtraction, multiplication and division,
representation of signed numbers, addition and subtraction using 2’s complement and I’s
complement.
Binary codes - BCD code, Excess 3 code, Gray code, Alphanumeric code, Error detection
codes, Error correcting code.Deepak john,SJCET-Pala
Chapter 2.1 introduction to number systemISMT College
Binary Number System, Decimal Number System, Octal Number System, Hexadecimal Number System, Conversion, Binary Arithmetic, Signed Binary Number Representation, 1's complement, 2's complement, 9's complement, 10's complement
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
2. CISE301_Topic1 2
Lecture 1
Introduction to Numerical Methods
What are NUMERICAL METHODS?
Why do we need them?
Topics covered in CISE301.
Reading Assignment: Pages 3-10 of textbook
3. CISE301_Topic1 3
Numerical Methods
Numerical Methods:
Algorithms that are used to obtain numerical
solutions of a mathematical problem.
Why do we need them?
1. No analytical solution exists,
2. An analytical solution is difficult to obtain
or not practical.
4. CISE301_Topic1 4
What do we need?
Basic Needs in the Numerical Methods:
Practical:
Can be computed in a reasonable amount of time.
Accurate:
Good approximate to the true value,
Information about the approximation error
(Bounds, error order,… ).
5. CISE301_Topic1 5
Outlines of the Course
Taylor Theorem
Number
Representation
Solution of nonlinear
Equations
Interpolation
Numerical
Differentiation
Numerical Integration
Solution of linear
Equations
Least Squares curve
fitting
Solution of ordinary
differential equations
Solution of Partial
differential equations
6. CISE301_Topic1 6
Solution of Nonlinear Equations
Some simple equations can be solved analytically:
Many other equations have no analytical solution:
31
)1(2
)3)(1(444
solutionAnalytic
034
2
2
−=−=
−±−
=
=++
xandx
roots
xx
solutionanalyticNo
052 29
=
=+−
−x
ex
xx
7. CISE301_Topic1 7
Methods for Solving Nonlinear Equations
o Bisection Method
o Newton-Raphson Method
o Secant Method
8. CISE301_Topic1 8
Solution of Systems of Linear Equations
unknowns.1000inequations1000
haveweifdoWhat to
123,2
523,3
:asitsolvecanWe
52
3
12
2221
21
21
=−==⇒
=+−−=
=+
=+
xx
xxxx
xx
xx
10. CISE301_Topic1 10
Methods for Solving Systems of Linear
Equations
o Naive Gaussian Elimination
o Gaussian Elimination with Scaled
Partial Pivoting
o Algorithm for Tri-diagonal Equations
11. CISE301_Topic1 11
Curve Fitting
Given a set of data:
Select a curve that best fits the data. One
choice is to find the curve so that the sum
of the square of the error is minimized.
x 0 1 2
y 0.5 10.3 21.3
12. CISE301_Topic1 12
Interpolation
Given a set of data:
Find a polynomial P(x) whose graph
passes through all tabulated points.
xi 0 1 2
yi 0.5 10.3 15.3
tablein theis)( iii xifxPy =
13. CISE301_Topic1 13
Methods for Curve Fitting
o Least Squares
o Linear Regression
o Nonlinear Least Squares Problems
o Interpolation
o Newton Polynomial Interpolation
o Lagrange Interpolation
14. CISE301_Topic1 14
Integration
Some functions can be integrated
analytically:
?
:solutionsanalyticalnohavefunctionsmanyBut
4
2
1
2
9
2
1
0
3
1
2
3
1
2
=
=−==
∫
∫
−
dxe
xxdx
a
x
15. CISE301_Topic1 15
Methods for Numerical Integration
o Upper and Lower Sums
o Trapezoid Method
o Romberg Method
o Gauss Quadrature
16. CISE301_Topic1 16
Solution of Ordinary Differential Equations
only.casesspecial
foravailablearesolutionsAnalytical*
equations.thesatisfiesthatfunctionais
0)0(;1)0(
0)(3)(3)(
:equationaldifferentitheosolution tA
x(t)
xx
txtxtx
==
=++
17. CISE301_Topic1 17
Solution of Partial Differential Equations
Partial Differential Equations are more
difficult to solve than ordinary differential
equations:
)sin()0,(,0),1(),0(
022
2
2
2
xxututu
t
u
x
u
π===
=+
∂
∂
+
∂
∂
18. CISE301_Topic1 18
Summary
Numerical Methods:
Algorithms that are
used to obtain
numerical solution of a
mathematical problem.
We need them when
No analytical solution
exists or it is difficult
to obtain it.
Solution of Nonlinear Equations
Solution of Linear Equations
Curve Fitting
Least Squares
Interpolation
Numerical Integration
Numerical Differentiation
Solution of Ordinary Differential
Equations
Solution of Partial Differential
Equations
Topics Covered in the Course
19. CISE301_Topic1 19
Number Representation
Normalized Floating Point Representation
Significant Digits
Accuracy and Precision
Rounding and Chopping
Reading Assignment: Chapter 3
Lecture 2
Number Representation and Accuracy
20. CISE301_Topic1 20
Representing Real Numbers
You are familiar with the decimal system:
Decimal System: Base = 10 , Digits (0,1,…,9)
Standard Representations:
21012
10510410210110345.312 −−
×+×+×+×+×=
partpart
fractionintegralsign
54.213±
21. CISE301_Topic1 21
Normalized Floating Point Representation
Normalized Floating Point Representation:
Scientific Notation: Exactly one non-zero digit appears
before decimal point.
Advantage: Efficient in representing very small or very
large numbers.
exponentsigned:,0
exponentmantissasign
104321.
nd
nffffd
±≠
±×±
22. CISE301_Topic1 22
Binary System
Binary System: Base = 2, Digits {0,1}
exponentsignedmantissasign
2.1 4321
n
ffff ±
×±
10)625.1(10)3212201211(2)101.1( =−×+−×+−×+=
23. CISE301_Topic1 23
Fact
Numbers that have a finite expansion in one numbering
system may have an infinite expansion in another
numbering system:
You can never represent 1.1 exactly in binary system.
210 ...)011000001100110.1()1.1( =
25. CISE301_Topic1 25
Significant Digits
Significant digits are those digits that can be
used with confidence.
Single-Precision: 7 Significant Digits
1.175494… × 10-38
to 3.402823… × 1038
Double-Precision: 15 Significant Digits
2.2250738… × 10-308
to 1.7976931… × 10308
26. CISE301_Topic1 26
Remarks
Numbers that can be exactly represented are called
machine numbers.
Difference between machine numbers is not uniform
Sum of machine numbers is not necessarily a machine
number
27. CISE301_Topic1 27
Calculator Example
Suppose you want to compute:
3.578 * 2.139
using a calculator with two-digit fractions
3.57 * 2.13 7.60=
7.653342True answer:
29. CISE301_Topic1 29
Accuracy and Precision
Accuracy is related to the closeness to the true
value.
Precision is related to the closeness to other
estimated values.
33. CISE301_Topic1 33
Can be computed if the true value is known:
100*
valuetrue
ionapproximatvaluetrue
ErrorRelativePercentAbsolute
ionapproximatvaluetrue
ErrorTrueAbsolute
t
−
=
−=
ε
tE
Error Definitions – True Error
34. CISE301_Topic1 34
When the true value is not known:
100*
estimatecurrent
estimatepreviousestimatecurrent
ErrorRelativePercentAbsoluteEstimated
estimatepreviousestimatecurrent
ErrorAbsoluteEstimated
−
=
−=
a
aE
ε
Error Definitions – Estimated Error
35. CISE301_Topic1 35
We say that the estimate is correct to n
decimal digits if:
We say that the estimate is correct to n
decimal digits rounded if:
n−
≤10Error
n−
×≤ 10
2
1
Error
Notation
36. CISE301_Topic1 36
Summary
Number Representation
Numbers that have a finite expansion in one numbering system
may have an infinite expansion in another numbering system.
Normalized Floating Point Representation
Efficient in representing very small or very large numbers,
Difference between machine numbers is not uniform,
Representation error depends on the number of bits used in
the mantissa.
38. CISE301_Topic1 38
Motivation
We can easily compute expressions like:
?)6.0sin(,4.1computeyoudoHowBut,
)4(2
103 2
+
×
x
way?practicalathisIs
sin(0.6)?computeto
definitiontheuseweCan
0.6
a
b
41. CISE301_Topic1 41
Maclaurin Series
Maclaurin series is a special case of Taylor
series with the center of expansion a = 0.
∑
∞
0
)(
3
)3(
2
)2(
'
)0(
!
1
)(
:writecanweconverge,seriestheIf
...
!3
)0(
!2
)0(
)0()0(
:)(ofexpansionseriesnMaclauriThe
=
=
++++
k
kk
xf
k
xf
x
f
x
f
xff
xf
42. CISE301_Topic1 42
Maclaurin Series – Example 1
∞.xforconvergesseriesThe
...
!3!2
1
!
)0(
!
1
11)0()(
1)0()(
1)0(')('
1)0()(
32∞
0
∞
0
)(
)()(
)2()2(
∑∑
<
++++===
≥==
==
==
==
==
xx
x
k
x
xf
k
e
kforfexf
fexf
fexf
fexf
k
k
k
kkx
kxk
x
x
x
x
exf =)(ofexpansionseriesnMaclauriObtain
44. CISE301_Topic1 44
Maclaurin Series – Example 2
∞.xforconvergesseriesThe
....
!7!5!3!
)0(
)sin(
1)0()cos()(
0)0()sin()(
1)0(')cos()('
0)0()sin()(
753∞
0
)(
)3()3(
)2()2(
∑
<
+−+−==
−=−=
=−=
==
==
=
xxx
xx
k
f
x
fxxf
fxxf
fxxf
fxxf
k
k
k
:)sin()(ofexpansionseriesnMaclauriObtain xxf =
46. CISE301_Topic1 46
Maclaurin Series – Example 3
∞.forconvergesseriesThe
....
!6!4!2
1)(
!
)0(
)cos(
0)0()sin()(
1)0()cos()(
0)0(')sin()('
1)0()cos()(
642∞
0
)(
)3()3(
)2()2(
∑
<
+−+−==
==
−=−=
=−=
==
=
x
xxx
x
k
f
x
fxxf
fxxf
fxxf
fxxf
k
k
k
)cos()(:ofexpansionseriesMaclaurinObtain xxf =
47. CISE301_Topic1 47
Maclaurin Series – Example 4
( )
( )
( )
1||forconvergesSeries
...xxx1
x1
1
:ofExpansionSeriesMaclaurin
6)0(
1
6
)(
2)0(
1
2
)(
1)0('
1
1
)('
1)0(
1
1
)(
ofexpansionseriesnMaclauriObtain
32
)3(
4
)3(
)2(
3
)2(
2
<
++++=
−
=
−
=
=
−
=
=
−
=
=
−
=
−
=
x
f
x
xf
f
x
xf
f
x
xf
f
x
xf
x1
1
f(x)
48. CISE301_Topic1 48
Example 4 - Remarks
Can we apply the series for x≥1??
How many terms are needed to get a good
approximation???
These questions will be answered using
Taylor’s Theorem.
49. CISE301_Topic1 49
Taylor Series – Example 5
...)1()1()1(1:)1(ExpansionSeriesTaylor
6)1(
6
)(
2)1(
2
)(
1)1('
1
)('
1)1(
1
)(
1atofexpansionseriesTaylorObtain
32
)3(
4
)3(
)2(
3
)2(
2
+−−−+−−=
−=
−
=
==
−=
−
=
==
==
xxxa
f
x
xf
f
x
xf
f
x
xf
f
x
xf
a
x
1
f(x)
50. CISE301_Topic1 50
Taylor Series – Example 6
...)1(
3
1
)1(
2
1
)1(:ExpansionSeriesTaylor
2)1(1)1(,1)1(',0)1(
2
)(,
1
)(,
1
)(',)ln()(
)1(at)ln(ofexpansionseriesTaylorObtain
32
)3()2(
3
)3(
2
)2(
−−+−−−
=−===
=
−
===
==
xxx
ffff
x
xf
x
xf
x
xfxxf
axf(x)
51. CISE301_Topic1 51
Convergence of Taylor Series
The Taylor series converges fast (few terms
are needed) when x is near the point of
expansion. If |x-a| is large then more terms
are needed to get a good approximation.
55. CISE301_Topic1 55
Error Term - Example
( ) 0514268.82.0
)!1(
)(
)!1(
)(
1≥≤)()(
3
1
2.0
1
)1(
2.0)()(
−≤⇒
+
≤
−
+
=
=
+
+
+
ER
n
e
R
ax
n
f
R
nforefexf
n
n
n
n
n
nxn
ξ
ξ
?2.00at
expansionseriesTayloritsof3)(terms4firstthe
by)(replacedweiferrortheislargeHow
==
=
=
xwhena
n
exf x
56. CISE301_Topic1 56
Alternative form of Taylor’s Theorem
hxxwhereh
n
f
R
hRh
k
xf
hxf
hxx
nxfLet
n
n
n
n
n
k
k
k
+
+
=
=+=+
+
+
+
+
=
∑
andbetweenis
)!1(
)(
size)step(
!
)(
)(
:thenandcontainingintervalanon
1)(...,2,1,ordersofsderivativehave)(
1
)1(
0
)(
ξ
ξ
57. CISE301_Topic1 57
Taylor’s Theorem – Alternative forms
.andbetweenis
)!1(
)(
!
)(
)(
,
.andbetweenis
)(
)!1(
)(
)(
!
)(
)(
1
)1(
0
)(
1
)1(
0
)(
hxxwhere
h
n
f
h
k
xf
hxf
hxxxa
xawhere
ax
n
f
ax
k
af
xf
n
nn
k
k
k
n
nn
k
k
k
+
+
+=+
+→→
−
+
+−=
+
+
=
+
+
=
∑
∑
ξ
ξ
ξ
ξ
58. CISE301_Topic1 58
Mean Value Theorem
)()('
,,0forTheoremsTaylor'Use:Proof
)('
),(existstherethen
),(intervalopentheondefinedisderivativeitsand
],[intervalclosedaonfunctioncontinuousais)(If
abξff(a)f(b)
bhxaxn
ab
f(a)f(b)
ξf
baξ
ba
baxf
−+=
=+==
−
−
=
∈
59. CISE301_Topic1 59
Alternating Series Theorem
termomittedFirst:
n terms)firsttheof(sumsumPartial:
convergesseriesThe
then
0lim
If
S
:seriesgalternatinheConsider t
1
1
4321
4321
+
+
∞→
≤−
=
≥≥≥≥
+−+−=
n
n
nnn
n
a
S
aSS
and
a
and
aaaa
aaaa
60. CISE301_Topic1 60
Alternating Series – Example
!7
1
!5
1
!3
1
1)1(s
!5
1
!3
1
1)1(s
:Then
0lim
:sinceseriesgalternatinconvergentaisThis
!7
1
!5
1
!3
1
1)1(s:usingcomputedbecansin(1)
4321
≤
+−−
≤
−−
=≥≥≥≥
+−+−=
∞→
in
in
aandaaaa
in
n
n
62. CISE301_Topic1 62
Example 7 – Taylor Series
...
!
)5.0(
2...
!2
)5.0(
4)5.0(2
)5.0(
!
)5.0(
2)5.0(2)(
4)5.0(4)(
2)5.0('2)('
)5.0()(
2
2
222
∞
0
)(
12
2)(12)(
2)2(12)2(
212
212
∑
+
−
++
−
+−+=
−=
==
==
==
==
=
+
+
+
+
+
k
x
e
x
exee
x
k
f
e
efexf
efexf
efexf
efexf
k
k
k
k
k
x
kkxkk
x
x
x
5.0,)(ofexpansionseriesTaylorObtain 12
== +
aexf x
63. CISE301_Topic1 63
Example 7 – Error Term
)!1(
max
)!1(
)5.0(
2
)!1(
)5.01(
2
)5.0(
)!1(
)(
2)(
3
12
]1,5.0[
1
1
1
121
1
)1(
12)(
+
≤
+
≤
+
−
=
−
+
=
=
+
∈
+
+
+
++
+
+
+
n
e
Error
e
n
Error
n
eError
x
n
f
Error
exf
n
n
n
n
n
n
xkk
ξ
ξ
ξ
ξ