Contents of the presentation:
- ABOUT ME
- Bisection Method using C#
- False Position Method using C#
- Gauss Seidel Method using MATLAB
- Secant Mod Method using MATLAB
- Report on Numerical Errors
- Optimization using Golden-Section Algorithm with Application on MATLAB
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Polynomials are very important mathematical tool for Engineers. In this lecture we will discuss about how to deal with Polynomials in MATLAB and one of its application, Curve Fitting.
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I am Joshua M. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a master's in Statistics from, Michigan State University, USA. I have been helping students with their assignments for the past 6 years. I solve assignments related to Statistics. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
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I am Bianca H. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Master in Statistics from, the University of Nottingham, UK. I have been helping students with their assignments for the past 7 years. I solve assignments related to Statistics. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
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Polynomials are very important mathematical tool for Engineers. In this lecture we will discuss about how to deal with Polynomials in MATLAB and one of its application, Curve Fitting.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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Numerical method-Picards,Taylor and Curve Fitting.Keshav Sahu
Here i have given some topics which is related to numerical method and computing.I covered picards method, Taylors series method, Curve fitting of method of least square and fitting a non leaner curve.
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Numerical Method Analysis: Algebraic and Transcendental Equations (Non-Linear)Minhas Kamal
Numerical Method Analysis- Solution of Algebraic and Transcendental Equations (Non-Linear Equation). Algorithms- Bisection Method, False Position Method, Newton-Raphson Method, Secant Method, Successive Approximation Method.
Visit here for getting code implementation- https://github.com/MinhasKamal/AlgorithmImplementations/blob/master/numericalMethods/equationSolving/NonLinearEquationSolvingProcess.c
Created in 2nd year of Bachelor of Science in Software Engineering (BSSE) course at Institute of Information Technology, University of Dhaka (IIT, DU).
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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I am Stacy W. I am a Probability Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
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You can also call on +1 678 648 4277 for any assistance with Probability Assignments.
Numerical method-Picards,Taylor and Curve Fitting.Keshav Sahu
Here i have given some topics which is related to numerical method and computing.I covered picards method, Taylors series method, Curve fitting of method of least square and fitting a non leaner curve.
I am Gill H. I am a Programming Assignment Expert at programminghomeworkhelp.com. I hold a Ph.D. in Electronics Engineering from, the University of Texas, USA. I have been helping students with their homework for the past 8 years. I solve assignments related to Programming.
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I am Watson A. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Liberty University, USA
I have been helping students with their homework for the past 6 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
Numerical Method Analysis: Algebraic and Transcendental Equations (Non-Linear)Minhas Kamal
Numerical Method Analysis- Solution of Algebraic and Transcendental Equations (Non-Linear Equation). Algorithms- Bisection Method, False Position Method, Newton-Raphson Method, Secant Method, Successive Approximation Method.
Visit here for getting code implementation- https://github.com/MinhasKamal/AlgorithmImplementations/blob/master/numericalMethods/equationSolving/NonLinearEquationSolvingProcess.c
Created in 2nd year of Bachelor of Science in Software Engineering (BSSE) course at Institute of Information Technology, University of Dhaka (IIT, DU).
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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Sparse data formats and efficient numerical methods for uncertainties in nume...Alexander Litvinenko
Description of methodologies and overview of numerical methods, which we used for modeling and quantification of uncertainties in numerical aerodynamics
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.2: Regression
(Slides) Efficient Evaluation Methods of Elementary Functions Suitable for SI...Naoki Shibata
Naoki Shibata : Efficient Evaluation Methods of Elementary Functions Suitable for SIMD Computation, Journal of Computer Science on Research and Development, Proceedings of the International Supercomputing Conference ISC10., Volume 25, Numbers 1-2, pp. 25-32, 2010, DOI: 10.1007/s00450-010-0108-2 (May. 2010).
http://www.springerlink.com/content/340228x165742104/
http://freshmeat.net/projects/sleef
Data-parallel architectures like SIMD (Single Instruction Multiple Data) or SIMT (Single Instruction Multiple Thread) have been adopted in many recent CPU and GPU architectures. Although some SIMD and SIMT instruction sets include double-precision arithmetic and bitwise operations, there are no instructions dedicated to evaluating elementary functions like trigonometric functions in double precision. Thus, these functions have to be evaluated one by one using an FPU or using a software library. However, traditional algorithms for evaluating these elementary functions involve heavy use of conditional branches and/or table look-ups, which are not suitable for SIMD computation. In this paper, efficient methods are proposed for evaluating the sine, cosine, arc tangent, exponential and logarithmic functions in double precision without table look-ups, scattering from, or gathering into SIMD registers, or conditional branches. We implemented these methods using the Intel SSE2 instruction set to evaluate their accuracy and speed. The results showed that the average error was less than 0.67 ulp, and the maximum error was 6 ulps. The computation speed was faster than the FPUs on Intel Core 2 and Core i7 processors.
Features of "Create Sheets":
• Create sheets automatically through an excel sheet.
• Sheets are created with your desired names and numbers.
• You can assign a specific Title Block for your company.
• Automatically use of the default title block if you don't write yours.
• You don't need to have Microsoft excel installed in your PC.
Features of "Print/Export Sheets":
• Print sheets automatically and export them to a PDF.
• A ‘sheets set’ are created and saved to your Revit model.
Application of GIS in Flood Hazard Mapping - GIS I Fundamentals - CEI40 - AGAAhmed Gamal Abdel Gawad
Contents of the presentation:
• Overview
• GIS Basics
• Water Resources Engineering
• GIS and Water Resources
• Flood Hazard Mapping
• Research Paper
• Flood mapping in ArcGIS
Contents of the presentation:
• GA – Introduction
• GA – Fundamentals
• GA – Genotype Representation
• GA – Population
• GA – Fitness Function
• GA – Parent Selection
• GA – Crossover
• GA – Mutation
• Research Paper
Contents of the presentation:
1. IDM OVERVIEW
- buildingSMART STANDARDS
- Data, Information & Knowledge
- What Is IDM?
- Why IDM?
- IDM Benefits
- IDM for BIM Users
- IDM for BIM Solution Providers
- IDM — Process Standard
- IDM — Requirements & Goal
- IDM — Improving the Construction Process
- IDM Components
2. PROCESS MAPS & EXCHANGE REQUIREMENTS
- Preview
- What is a Process map?
- Process Map Components
- Process Map Report
- What is an Exchange Requirement (ER) ?
- Exchange Requirement Form/template
- ER Form/template (Functional Part)
- Exchange Requirement Examples
3. IDM VS MVD
- Integrated Process Overview
- Overview of the IDM/MVD Method
- IDM Deliverables
- Exchange Requirements Model
- Model View Definition (MVD)
- MVD Concepts
- MVD Deliverables
- Implementation Guidance of Concepts
- Large Concepts vs Small Concepts
- MVD Life Cycle
4. IDM/MVD INTEGRATED DEVELOPMENT
- mvdXML
- IfcDoc DEMO
- Software Certification
- b-Cert Certification Process
- BIM Validation
- Validation Approaches
- Research Efforts
Contents of the presentation:
1. IFC OVERVIEW
- Back to The Idea of BIM
- Open BIM
- What Is IFC?
- IFC Formats
- IFC Workflow
- Interoperability
- BIM Collaboration Format (BCF)
- Model View Definition (MVD)
- Data Modeling
- Modeling Language
- IFC Data Modeling (Schema)
- EXPRESS Schema
2. IFC DATA MODEL
- Inheritance Hierarchy
- Explicit vs Inverse Attributes
- Objectified Relationships
- Viewers
- Spatial Aggregate Hierarchy
- Geometric Representation Methods
- Relative Positioning
3. ATTRIBUTES & PROPERTIES
- It’s all about Data
- Data Mapping
- Attributes Categories
- Attributes in Revit
- Properties Classification
- IFC Property Sets:
- Revit Implementation
- Data Mapping files
4. IFC: THE NOW & THE FUTURE
- Preview
- Ifc Versions Evolution
- Ifc Certification
- ifcBridge Addition to IFC4.2
- Ifc5.0 Infrastructure & Better GIS Integration
- Ifc New Candidate Formats
- Brief Case Study: Ifcjson Format
الفصل الثالث عشر - القواعد المنفصلة - تصميم المنشآت الخرسانية المسلحةAhmed Gamal Abdel Gawad
المحاضرة الرابعة والأربعون - مقدمة عن الأعمدة
https://youtu.be/ceab-Qj2w90
المحاضرة الخامسة والأربعون - تحديد حالة تقييد الأعمدة
https://youtu.be/GmfdHby1TvA
المحاضرة السادسة والأربعون - الانبعاج في الأعمدة
https://youtu.be/4aiZBuzalfg
المحاضرة السابعة والأربعون - ملاحظات على الأعمدة
https://youtu.be/iMCm9eGfLT4
م. أحمد جمال عبد الجواد
المحاضرة السابعة والثلاثون - مقدمة عن الأعمدة
https://youtu.be/ceab-Qj2w90
المحاضرة الثامنة والثلاثون - تحديد حالة تقييد الأعمدة
https://youtu.be/GmfdHby1TvA
المحاضرة التاسعة والثلاثون - الانبعاج في الأعمدة
https://youtu.be/4aiZBuzalfg
المحاضرة الأربعون - ملاحظات على الأعمدة
https://youtu.be/iMCm9eGfLT4
المحاضرة الحادية والأربعون - تصميم الأعمدة القصيرة
https://youtu.be/V1roYDDKCLA
المحاضرة الثانية والأربعون - المقاطع المعرضة لعزوم وقوى
https://youtu.be/miOkArTNh7Y
المحاضرة الثالثة والأربعون - تصميم الأعمدة النحيفة
https://youtu.be/DUP4vb98_OE
م. أحمد جمال عبد الجواد
المحاضرة الرابعة والثلاثون - مقدمة عن السلالم وأنواعها
https://youtu.be/aGVJS-Zg7BM
المحاضرة الخامسة والثلاثون - اعتبارات ومصطلحات معمارية للسلالم
https://youtu.be/1z1QdAni-AI
المحاضرة السادسة والثلاثون - تصميم السلالم وتفاصيل تسليحها
https://youtu.be/lGK28FoQNvE
م. أحمد جمال عبد الجواد
الفصل الثامن - بلاطات القوالب المفرغة - تصميم المنشآت الخرسانية المسلحةAhmed Gamal Abdel Gawad
المحاضرة التاسعة عشر : مقدمة عن بلاطات القوالب المفرغة
https://youtu.be/LRdqk_C0QmM
المحاضرة العشرون : البلاطات المفرغة في الاتجاه الواحد
https://youtu.be/PTnOdpH9pgI
المحاضرة الحادية والعشرون : البلاطات المفرغة في الاتجاهين
https://youtu.be/94BX_1Qi5gY
المحاضرة الثانية والعشرون : الكمرات المدفونة
https://youtu.be/MoajLxlVmKg
المحاضرة الثالثة والعشرون : نمذجة البلاطات المفرغة على الريفيت
https://youtu.be/ess87oGrITk
م. أحمد جمال عبد الجواد
المحاضرة السادسة عشر : مقدمة عن البلاطات
https://youtu.be/SfrGZm-4vjA
المحاضرة السابعة عشر : البلاطات المصمتة في الاتجاه الواحد
https://youtu.be/TM7V8n-LcCI
المحاضرة الثامنة عشر : البلاطات المصمتة في الاتجاهين
https://youtu.be/FIMKygfs9bQ
م. أحمد جمال عبد الجواد
الفصل الثالث - طريقة إجهادات التشغيل - تصميم المنشآت الخرسانية المسلحةAhmed Gamal Abdel Gawad
حل أمثلة الفصل الثالث :
http://www.mediafire.com/?7a15r4o7dxa3u4t
المحاضرة السادسة : تحليل المقاطع بطريقة إجهادات التشغيل
http://youtu.be/u7kiYBuKVuM
المحاضرة السابعة : تصميم المقاطع بطريقة إجهادات التشغيل
http://youtu.be/EVIJ72Rs3pw
م. أحمد جمال عبد الجواد
الفصل الثاني - المقاطع تحت تأثير عزوم الانحناء - تصميم المنشآت الخرسانية المسلحةAhmed Gamal Abdel Gawad
حل أمثلة الفصل الثاني :
https://www.mediafire.com/?2x3mo52dv9jo6mc
المحاضرة الرابعة : المقاطع تحت تأثير عزوم الإنحناء
http://youtu.be/f5-kOqI3yGQ
المحاضرة الخامسة : تحليل المقاطع قبل التشرخ
http://youtu.be/Y1ikllWCgIU
م. أحمد جمال عبد الجواد
الفصل الأول - مقدمة في الخرسانة المسلحة - تصميم المنشآت الخرسانية المسلحةAhmed Gamal Abdel Gawad
حل أمثلة الفصل الأول :
https://www.mediafire.com/?krb5ubl78obirna
المحاضرة الأولى : مقدمة في الخرسانة المسلحة
http://youtu.be/f5-kOqI3yGQ
المحاضرة الثانية : حديد التسليح
http://youtu.be/Y1ikllWCgIU
المحاضرة الثالثة : أنظمة الوحدات وطرق ومتطلبات التصميم
http://youtu.be/QiDRIFP0Ias
م. أحمد جمال عبد الجواد
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
CW RADAR, FMCW RADAR, FMCW ALTIMETER, AND THEIR PARAMETERSveerababupersonal22
It consists of cw radar and fmcw radar ,range measurement,if amplifier and fmcw altimeterThe CW radar operates using continuous wave transmission, while the FMCW radar employs frequency-modulated continuous wave technology. Range measurement is a crucial aspect of radar systems, providing information about the distance to a target. The IF amplifier plays a key role in signal processing, amplifying intermediate frequency signals for further analysis. The FMCW altimeter utilizes frequency-modulated continuous wave technology to accurately measure altitude above a reference point.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Scientific Computing II Numerical Tools & Algorithms - CEI40 - AGA
1. 9-Month Program, Intake40, CEI Track
SCIENTIFIC COMPUTING II
NUMERICAL TOOLS & ALGORITHMS
Ahmed Gamal Abdel Gawad
2. CONTENTS
ABOUT ME
Bisection Method using C#
False Position Method using C#
Gauss Seidel Method using MATLAB
Secant Mod Method using MATLAB
Report on Numerical Errors
Optimization using Golden-Section Algorithm
with Application on MATLAB
3. TEACHING ASSISTANT AT MENOUFIYA UNIVERSITY.
GRADE: EXCELLENT WITH HONORS.
BEST MEMBER AT ‘UTW-7 PROGRAM’, ECG.
ITI 9-MONTH PROGRAM, INT40, CEI TRACK STUDENT.
AUTODESK REVIT CERTIFIED PROFESSIONAL.
BACHELOR OF CIVIL ENGINEERING, 2016.
LECTURER OF ‘DESIGN OF R.C.’ COURSE, YOUTUBE.
ABOUT ME
4. Bisection Method C#
static double Bisection(double x1, double x2, int maxIterations, double tolerance, out int count)
{
double f1 = Function(x1);
double f2 = Function(x2);
double xm = 0.0;
double fm;
if (f1 * f2 > 0.0) throw new InvalidOperationException("No Bracket");
count = 0;
for (int i = 0; i < maxIterations; i++)
{
count++;
xm = (x1 + x2) / 2;
fm = Function(xm);
if (Math.Abs(fm) <= tolerance) break;
if (f1 * fm > 0)
{
x1 = xm;
f1 = fm;
}
else
{
x2 = xm;
f2 = fm;
}
}
return xm;
}
5. False Position Method C#
static double FalsePosition(double x1, double x2, int maxIterations, double tolerance, out int count)
{
double f1 = Function(x1);
double f2 = Function(x2);
double xp = 0.0;
double fp;
double s;
if (f1 * f2 > 0.0) throw new InvalidOperationException("No Bracket");
count = 0;
for (int i = 0; i < maxIterations; i++)
{
count++;
s = (f2 - f1) / (x2 - x1);
xp = x1 - f1 / s;
fp = Function(xp);
if (Math.Abs(fp) <= tolerance) break;
if (f1 * fp > 0)
{
x1 = xp;
f1 = fp;
}
else
{
x2 = xp;
f2 = fp;
}
}
return xp;
}
6. Gauss Seidel
function[x,nit] = gseidel(A,b,nmax,tol)
% Function to run gseidel method
[nr, nc] = size(A);
if (nc ~= nr), error('A is NOT Square'); end % check square matrix
x = zeros(nr,1); % Vector of inital values of x
for k = 1:nr
x(k) = b(k)/A(k,k); % Initial values of x
end
err = zeros(nr,1); % Vector of errors
errmax = 1; % Initial value for errmax > tolerance
nit=0.0; % No of iterations
while (errmax > tol && nit < nmax)
xold = x; % Set xold to the previous values of x
nit = nit + 1; % Increse No of iterations by 1
for k = 1:nr
sum = A(k,:)*x; % Calculate the sum term
sum = sum - A(k,k)*x(k); % Exclude akk and xk from calculations
x(k) = (b(k) - sum)/A(k,k); % Calculate x new values
err(k) = abs(x(k)) - abs(xold(k)); % Record vector of errors
end
errmax = max(abs(err));
end
end
7. Secant Mod
function [xr,nit]= secantmod(func,xo,deltax,kmax,etol)
% Secant method to find root of function “func” using
% one starting point xo and small perturbation ?x for
% max iterations kmax
xv1 = xo;
xv2 = xo + deltax;
nit = 0;
for k = 1:kmax
nit = nit +1;
vf1 = func(xv1);
vf2 = func(xv2);
vsec = (vf2 - vf1) / deltax;
if (abs(vsec) <= 10^(-15)),error('Zero Secant Slope');end
xnew = xv1 - vf1/vsec;
vfnew = func(xnew);
if abs(vfnew) <= etol
xr = xnew;
break
end
xv1 = xnew;
xv2 = xnew + deltax;
end
end
8. Report on Numerical Error
Truncation Error
The word 'Truncate' means 'to shorten'. Truncation error refers to
an error in a method, which occurs because some number/series
of steps (finite or infinite) is truncated (shortened) to a fewer
number. Such errors are essentially algorithmic errors and we can
predict the extent of the error that will occur in the method. For
instance, if we approximate the sine function by the first two non-
zero term of its Taylor series, as in sin 𝑥 = 𝑥 −
1
6
𝑥3 for small x,
the resulting error is a truncation error. It is present even with
infinite-precision arithmetic, because it is caused by truncation of
the infinite Taylor series to form the algorithm.
9. Report on Numerical Error
Roundoff Error
A roundoff error, also called rounding error, is the difference
between the result produced by a given algorithm using exact
arithmetic and the result produced by the same algorithm using
finite-precision, rounded arithmetic. Rounding errors are due to
inexactness in the representation of real numbers and the
arithmetic operations done with them. This is a form of
quantization error. When using approximation equations or
algorithms, especially when using finitely many digits to
represent real numbers (which in theory have infinitely many
digits), one of the goals of numerical analysis is to estimate
computation errors. Computation errors, also called numerical
errors, include both truncation errors and roundoff errors.
11. Report on Numerical Error
Accuracy and Precision
Measurements and calculations can be characterized with regard
to their accuracy and precision. Accuracy refers to how closely a
value agrees with the true value. Precision refers to how closely
values agree with each other. The following figures illustrate the
difference between accuracy and precision. In the first figure, the
given values (black dots) are more accurate; whereas in the
second figure, the given values are more precise. The term error
represents the imprecision and inaccuracy of a numerical
computation.
13. Report on Numerical Error
Real world example: Patriot missile failure due to
magnification of roundoff error
On 25 February 1991, during the Gulf
War, an American Patriot missile
battery in Dharan, Saudi Arabia, failed
to intercept an incoming Iraqi Scud
missile. The Scud struck an American
Army barracks and killed 28 soldiers.
It turns out that the cause was an
inaccurate calculation of the time
since boot due to computer
arithmetic errors.
14. Optimization using Golden-
Section Algorithm
Euclid’s definition of the golden ratio is based
on dividing a line into two segments so that
the ratio of the whole line to the larger
segment is equal to the ratio of the larger
segment to the smaller segment. This ratio is
called the golden ratio.
15. Optimization using Golden-
Section Algorithm
The actual value of the golden ratio can be
derived by expressing Euclid’s definition as
𝑙1+𝑙2
𝑙1
=
𝑙1
𝑙2
Multiplying by
𝑙1
𝑙2
and collecting terms yields
∅2 − ∅ − 1 = 0
Where ∅ = 𝑙1/𝑙2 .The positive root of this
equation is the golden ratio:
∅ =
1+ 5
2
= 1.61803398874989
16. Optimization using Golden-
Section Algorithm
The golden-section search is similar in spirit to
the bisection approach for locating roots. Recall
that bisection hinged on defining an interval,
specified by a lower guess (xl) and an upper
guess (xu) that bracketed a single root. The
presence of a root between these bounds was
verified by determining that f (xl) and f (xu) had
different signs. The root was then estimated as
the midpoint of this interval:
𝑥 𝑟 =
𝑥 𝑢 + 𝑥𝑙
2
17. Optimization using Golden-
Section Algorithm
The key to making this approach efficient is the
wise choice of the intermediate points. As in
bisection, the goal is to minimize function
evaluations by replacing old values with new
values. For bisection, this was accomplished by
choosing the midpoint. For the golden-section
search, the two intermediate points are chosen
according to the golden ratio:
𝑥1 = 𝑥𝑙 + 𝑑
𝑥2 = 𝑥 𝑢 − 𝑑
where
𝑑 = (∅ − 1)(𝑥 𝑢 − 𝑥𝑙)
18. Optimization using Golden-
Section Algorithm
The function is evaluated at these two interior
points. Two results can occur:
1. If, as in Fig. 7.6a, f (x1)< f (x2), then f (x1) is
the minimum, and the domain of x to the
left of x 2, from xl to x2, can be eliminated
because it does not contain the minimum.
For this case, x2 becomes the new xl for the
next round.
2. If f (x2)< f (x1), then f (x2) is the minimum
and the domain of x to the right of x1, from
x 1 to xu would be eliminated. For this case,
x1 becomes the new xu for the next round.
20. Optimization using Golden-
Section Algorithm
function [x,fx,ea,iter]=goldmin(f,xl,xu,es,maxit)
% goldmin: minimization golden section search
% uses golden section search to find the minimum of f
if nargin<3,error('at least 3 input arguments required'),end
if nargin<4||isempty(es), es=0.0001;end
if nargin<5||isempty(maxit), maxit=50;end
phi=(1+sqrt(5))/2;
iter=0;
while(1)
d = (phi-1)*(xu - xl);
x1 = xl + d;
x2 = xu - d;
if f(x1) < f(x2)
xopt = x1;
xl = x2;
else
xopt = x2;
xu = x1;
end
iter=iter+1;
if xopt~=0, ea = (2 - phi) * abs((xu - xl) / xopt) * 100;end
if ea <= es || iter >= maxit,break,end
end
x=xopt;fx=f(xopt);
MATLAB Function
21. Optimization using Golden-
Section Algorithm
Use the following parameter values for your calculation: g =
9.81 m/s2, z0 = 100 m, v0 = 55 m/s, m = 80 kg, and c = 15 kg/s.
Example
22. Optimization using Golden-
Section Algorithm
Command Window
>> g=9.81;v0=55;m=80;c=15;z0=100;
>> z=@(t) -(z0+m/c*(v0+m*g/c)*(1-exp(-c/m*t))-m*g/c*t);
>> [xmin,fmin,ea,iter]=goldmin(z,0,8)
xmin =
3.8317
fmin =
-192.8609
ea =
6.9356e-05
iter =
29
Notice how because this is a maximization, we have
entered the negative of the equation. Consequently,
fmin corresponds to a maximum height of 192.8609.