Application's of Differentiation, Integration, Error Analysis, Curve Fitting , Maxima and Minima

1. ASSIGNMENT
Topic : Application of Numerical Methods in CSE
Assigned to : Md. Mehedi Hasan
Department of General Education Development
Assigned by-
Sanjana Mun; 163-15-8443
Sharmin Akter; 163-15-8436
Iffat Firozy; 163-15-8432
Umme Fatema Tuj Asha; 163-15-8395
Jannatul Nayem Himel; 163-15-8538
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2. INDEX
APPLICATION OF NUMERICAL METHODS IN CSE
(1) ERROR ANALYSIS
(2) N-R METHOD
(3) INTERPOLATION
(4) DIFFERENTIATION AND MAX. MIN
(5) CURVE FITTING
(6) INTEGRATION.
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3. SHARMIN AKTER
163-15-8434

4. 1
ERROR ANALYSIS
Error in solving an engineering or science problem can arise due to several
factors. First, the error may be in the modeling technique. A mathematical
model may be based on using assumptions that are not acceptable. For
example, there are two kinds of numbers such as exact and approximate
numbers, such the exact numbers are 1,2,5, ……
3
7
,
3
2
, ….. π …etc. and the
approximate numbers are representations of exact numbers to a certain degree
of accuracy. Thus, 3.1416 is an approximate number of π and 3.14159265 is
another approximate number of π. Second, errors may arise from mistakes in
programs themselves or in the measurement of physical quantities.
In CSE error analysis are mostly use in programming and many others side.
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5. 5
There are some types of program error,
Syntax errors: Syntax errors are due to the fact that the syntax of the Java or HTML or
any others languages is not respected.
Example-
Missing semicolon:
int a = 5 // semicolon is missing
Compiler message:
java:20: ';' expected
int a = 5
Semantic errors: Semantic errors indicate an improper use of Java statements.
Example 2: Type incompatibility:
int a = "hello"; // the types String and int are not compatible

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Logical errors: Logical errors are caused by the fact that the software specification
is not respected. The program is compiled and executed without errors, but does
not generate the requested result.
Example- Errors in the performed computation:
public static int sum (int a, int b) { return a - b;}// this method returns the wrong
value wrt the specification that requires// to sum two integers
Non termination:
String s = br.readLine();while (s != null) { System.out.println(s);} // this loop does
not terminate
There are many others errors, like compile time error, run time error etc. Error
analysis are use for finding error in a program and fixing the error.

7. UMME FATEMA TUJ ASHA
163-15-8395

8. 2
N-R METHOD
The Newton-Raphson method (also known as Newton's method) is a way to quickly
find a good approximation for the root of a real-valued function f(x)=0. It uses the
idea that a continuous and differentiable function can be approximated by a
straight-line tangent to it.
Suppose you need to find the root of a continuous, differentiable function f(x), and
you know the root you are looking for is near the point x=𝑥0 . Then Newton's
method tells us that a better approximation for the root is
𝑥1 = 𝑥0 −
𝑓(𝑥0)
𝑓′(𝑥0)
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9. This process may be repeated as many times as necessary to get the desired
accuracy. In general, for any x-value 𝑥 𝑛, the next value is given by
𝑥 𝑛+1 = 𝑥 𝑛 −
𝑓(𝑥 𝑛)
𝑓′(𝑥 𝑛)
Note: the term "near" is used loosely because it does not need a precise definition
in this context. However, 𝑥0 should be closer to the root you need than to any other
root (if the function has multiple roots).
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Advantage:
 Converge fast if it converges to the root compare to another method.
 Requires only one guess.
 Convergence to the root quadratically.
 Easy to convert to multiple dimensions.
 Can be to polish a root found by another methods.
Dis-advantage:
• Must find the derivative.
 Poor global convergence properties.
 It takes more computing time.
 It should never be used when the graph of f(x)=0 is nearly horizontal
where it crosses the x-axis.
 Dependent on initial guess o May be too far from local root o May
encounter a zero derivative o May loop indefinitely.

11. SHARMIN AKTER
163-15-8434

12. 3
INTERPOLATION
In the mathematical field of numerical analysis, interpolation is a method of
constructing new data points within the range of a discrete set of known data points.
Application of image processing:
There are 2 types of image processing. Rigid transformation and Nonlinear
transformation. One important application interpolation is the rigid transformation of
images. By rigid transformation we mean a linear transformation of the pixel
coordinates. Among rigid transformations, we find the subclass of affine
transformations. Here we use matlab for resizing, rotation and shear.
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13. 13
An example of nonlinear transformation 5_3 to make a function with
Input: an intensity image, a center point (x0,y0), a deformation parameter d.
Output: The transformed cropped image matrix.

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Zooming digital image using interpolation techniques:
Interpolation over Light Fields with Applications in Computer Graphics:

15. KHANDAKAR SANJANA AKTER MUN
ID: 163-15-8443

16. 4
DIFFERENTIATION
What is the use of differentiation in computer programming?
• Differentiation (calculus a well) is very important in computer science. It is perhaps
not as directly important to an average programmer, especially if that programmer is
writing, say, user interface code or a typical Web application.
• One major application of calculus in computer science is in the comparison of the
performance of various algorithms and the complexity of various problems. These are
often expressed using big O notation, which relies on the idea of the limits of ratios of
functions as a variable tends to infinity.
• On the programming side, I have seen programmers to write code that used the
intermediate value theorem and Newton's method to find the roots of a polynomial,
as part of a larger program. I have also heard to write code to draw functions of
graphs that used calculus concepts to determine which values of x to plot points for.
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17. • A few areas of programming off the top of my head that will require this
concepts at a greater or lesser extent:
• Numerical analysis for scientific software will use a lot.
• Physics engines for video games.
• Data analysis and prediction for business applications.
• Modeling software for things like biological systems, meteorology and
climatology, engineering applications, etc.
• Machine learning and artificial intelligence, including such things as natural
language processing, pattern recognition, etc.
• Image, video, and audio processing.
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18. Maxima and Minima
The terms maxima and minima refer to extreme values of a function, that
is, the maximum and minimum values that the function attains. Maximum
means upper bound or largest possible quantity. The absolute maximum
of a function is the largest number contained in the range of the function.
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19. APPLICATIONS
• There are numerous practical applications in which it is desired to find the maximum or
minimum value of a particular quantity. Such applications exist in economics, business, and
engineering. Many can be solved using the methods of differential calculus described above.
For example, in any manufacturing business it is usually possible to express profit as a
function of the number of units sold. Finding a maximum for this function represents a
straightforward way of maximizing profits. In other cases, the shape of a container may be
determined by minimizing the amount of material required to manufacture it. The design of
piping systems is often based on minimizing pressure drop which in turn minimizes required
pump sizes and reduces cost. The shapes of steel beams are based on maximizing strength.
•
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20. • Finding maxima or minima also has important applications in linear algebra
and game theory. For example, linear programming consists of maximizing (or
minimizing) a particular quantity while requiring that certain constraints be
imposed on other quantities. The quantity to be maximized (or minimized), as
well as each of the constraints, is represented by an equation or inequality.
The resulting system of equations or inequalities, usually linear, often contains
hundreds or thousands of variables. The idea is to find the maximum value of
a particular variable that represents a solution to the whole system. A
practical example might be minimizing the cost of producing an automobile
given certain known constraints on the cost of each part, and the time spent
by each laborer, all of which may be interdependent. Regardless of the
application, though, the key step in any maxima or minima problem is
expressing the problem in mathematical terms.
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21. IFFAT FIROZY RIMI
163-15-8432

22. 5
CURVE FITTING
Curve fitting is the process of constructing a curve, or mathematical function, that
has the best fit to a series of data points, possibly subject to constraints. Curve
fitting can involve either interpolation , where an exact fit to the data is required, or
smoothing, in which a "smooth" function is constructed that approximately fits the
data. A related topic is regression analysis, which focuses more on questions of
statistical inference such as how much uncertainty is present in a curve that is fit to
data observed with random errors. Fitted curves can be used as an aid for data
visualization , to infer values of a function where no data are available , and to
summarize the relationships among two or more variables. extrapolation refers to
the use of a fitted curve beyond the range of the observed data, and is subject to a
degree of uncertainty since it may reflect the method used to construct the curve
as much as it reflects the observed data.
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24. ALGEBRAIC FIT VERSUS GEOMETRIC FIT FOR CURVES
For algebraic analysis of data, "fitting" usually means trying to find the curve that
minimizes the vertical (y-axis) displacement of a point from the curve (e.g.,
ordinary least squares). However, for graphical and image applications geometric
fitting seeks to provide the best visual fit; which usually means trying to minimize
the orthogonal distance to the curve (e.g., total least squares), or to otherwise
include both axes of displacement of a point from the curve. Geometric fits are
not popular because they usually require non-linear and/or iterative calculations,
although they have the advantage of a more aesthetic and geometrically accurate
result.
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25. JANNATUL NAYEM HIMEL
163-15-8538

26. 6
INTEGRATION
Computer algebra systems that compute integrals and derivatives directly, either
symbolically or numerically, are the most blatant, but in addition, any software
that simulates a physical system that is based on continuous differential equations
necessarily involves computing derivatives and integrals.
Application: There are many applications of integral calculus especially in
computer graphics (lighting, raytracing...) and physics engines (basically all force
representations are based on calculus) but also in computer vision.
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27. In the video game: They are used often for video games, especially physics engines.
Physics engines define the physics in the game such as gravity, friction, etc.
For military software system: The military uses these integration visuals for
simulations, flight and artillery paths, maps, satellite images, etc. Architects use
them for graphing buildings, outlines, etc. Applications to Solve Problems They use
calculus for general problem-solving applications, simulations, and physics engines.
Integration of System integration: Integration is defined in engineering as the
process of bringing together the component sub-systems into one system (an
aggregation of subsystems cooperating so that the system is able to deliver the
overarching functionality) and ensuring that the subsystems function together as a
system. Technology as the process of linking together different computing systems
and software applications physically or functionally, to act as a coordinated whole.
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28. Design and analysis of algorithms application: The behavior of a
combinatorial algorithm on very large instances is often most easily
analyzed using calculus. This is especially true for randomized
algorithms; modern probability theory is heavily analytic. In the other
direction, sometimes one can design an algorithm for a discrete
problem by considering a continuous analogue, using calculus to solve
the continuous problem, and then discretizing to obtain an algorithm
for the original problem. The simplest example of this might be finding
an approximate root of a polynomial equation.
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