DON MARIANO MARCOS MEMORIAL STATE UNIVERSITY

Mid - La Union Campus

College of Engineering
Department of Mechanical Engineering

MODULE 1:

LESSONS: NUMERICAL METHODS: Discrete

Algebra, Accuracy and Errors
 MODULE 1

 INTRODUCTION

Many engineering problems are too time consuming to solve or may not be able
to be solved analytically. In these situations, numerical methods are usually
employed. Numerical methods are techniques designed to solve a problem using
numerical approximations. An example of an application of numerical methods is
trying to determine the velocity of a falling object. If you know the exact function
that determines the position of your object, then you could potentially differentiate
the function to obtain an expression for the velocity. More often, you will use a
machine to record readings of times and positions that you can then use to
numerically solve for velocity:

where f is your function, t is the time of the reading, and h is the distance to the next
time step.
Because your answer approximates the analytical solution, there is an inherent error
between the approximated answer and the exact solution.

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.

Numerical Methods and Engineering Practice Since the late 1940s the
widespread availability of digital computers has led to a veritable explosion in the use
and development of numerical methods. At first, this growth was somewhat limited
by the cost of access to large mainframe computers, and, consequently, many
engineers continued to use simple analytical approaches in a significant portion of
their work. The recent evolution of inexpensive personal computers has given us
ready access to powerful computational capabilities. There are several additional
reasons why you should study numerical methods:

1. 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
that are often impossible to solve analytically. As such, they greatly
enhance your problem-solving skills.

2. During your careers, you may often have occasion to use commercially
available prepackaged, or “canned,” computer programs that involve
 numerical methods. The intelligent use of these programs is often
predicated on knowledge of the basic theory underlying the methods.

3. Many problems cannot be approached using canned 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 or commission expensive software.

4. Numerical methods are an efficient vehicle for learning to use computers. It

is well known that an effective way to learn programming is to write
computer programs. Because numerical methods are for the most part
designed for implementation on computers, they are ideal for this purpose.
Further, they are especially well-suited to illustrate the power and the
limitations of computers. When you successfully implement numerical
methods on a computer and then apply them to solve otherwise intractable
problems, you will be provided with a dramatic demonstration of how
computers can serve your professional development. At the same time, you
will also learn to acknowledge and control the errors of approximation that
are part and parcel of large-scale numerical calculations.

5. 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, they get at
the “nuts and bolts” of some otherwise obscure topics. Enhanced
understanding and insight can result from this alternative perspective.

OBJECTIVES

By the end of this module, you should be able to:

 To understand what Numerical Methods is.
 Develop skills for solving problems related to Numerical Methods.
 To understand basic concepts of Numerical Methods.

Lesson 1:

Discrete Algebra

Algebra is frequently taught as a series of formulas and algorithms for students

to memorize (for example, the quadratic formula, solving systems of linear equations
by substitution, and so on), whereas geometry is frequently taught as a series of
definition > theorem > proof exercises that are frequently performed by rote. While
the subject matter is unquestionably significant, the content (at least at the
introductory level) does not encourage creative mathematical thinking. In contrast,
discrete mathematics requires pupils to think creatively and flexibly right away.
There are relatively few formulas to memorize; instead, several key principles
must be grasped and applied in a variety of contexts. What is discrete mathematics?
Most broadly, the term "discrete" contrasts with the term "continuous," hence discrete
mathematics can be defined as mathematics that deals with discrete rather than
continuous mathematical structures and objects. We define discrete mathematics as
disciplines containing discrete objects and structures (such as integers in
combinatorics), as opposed to courses concentrating on continuous objects (such as
real numbers in calculus). Discrete means different, distinct, or distinguished.
Discrete math is used in a variety of everyday situations. Cryptography, a
branch of computer science, is one of the topics under research. The field enables
users to create passwords for computers and other digital tokens such as credit or
debit cards. This is because information is normally transmitted in discrete bits.
Discrete math is seen as the opposite of continuous math. Calculus, integrations, and
other math ideas are examples of continuous mathematics. Discrete Mathematics is
concerned with the study of mathematical structures. It works with objects that can
have multiple different values. It is also known as Decision Mathematics or Finite
Mathematics.
It is the study of mathematical systems that are fundamentally discrete in
nature and does not imply the concept of continuity. Discrete mathematics is the
branch of mathematics concerned with objects that can only have distinct, isolated
values. The term "discrete mathematics" is thus used in contrast to "continuous
mathematics," which is the field of mathematics concerned with items that can vary
smoothly (such as calculus). Continuous things require real numbers, although
discrete objects can easily be described using integers.
Combinatorics is the study of how discrete items combine with each other and
the probabilities of different outcomes. Graph theory and computation theory are two
other topics of mathematics that are regarded to be part of discrete mathematics.
Discrete mathematics includes number theory topics such as congruences and
recurrence relations.

Topics in discrete mathematics are frequently studied in terms of algorithms,

implementations, and efficiency. Discrete mathematics is the mathematical language
of computer science; hence its importance has grown considerably in recent decades.
Discrete mathematics primarily studies countable sets such as formal languages,
integers, finite networks, and so on. It has grown in popularity in recent decades
because of its applications in computer science. It is utilized in programming
languages, software development, cryptography, algorithms, and other applications.
Some key concepts covered in discrete mathematics include set theory, graph theory,
logic, permutation, and combination.

Discrete Mathematics Topics

● Set Theory: Set theory is described as the study of sets, which are collections
of items organized into groups. The set of numbers or objects can be denoted
by the braces {} symbol. For example, the set of first 4 even numbers is
{2,4,6,8}

● Graph Theory: It is the analysis of graphs. The graph is a mathematical

structure that pairs the relationships between items. Graphs are a major topic
of study in Discrete Mathematics.

● Logic: Logic in mathematics is defined as the study of correct reasoning. There

are three types of logic gates. They are AND(∧), NOT(~), and OR(∨)
 ● Permutation: Permutation refers to the various arrangements that can be
constructed with a given number of sets by moving some or all of them in a
specific order at a time. For example, there are six permutations of the set
{5,6,7}, namely (5,6,7), (5,7,6), (6,5,7), (6,7,5), (7,5,6), and (7,6,5).

● Combination: Combination refers to the selecting of multiple objects and

taking some or all of them at the same time. The combination is unaffected by
the sequence of selection.

● Sequence: A Sequence is a series of integers arranged in a specific sequence,

according to certain principles. A sequence is a function that has the countable
set of natural numbers as its domain.

● Series: A series is the sum of the terms of a sequence. The result of adding all
the terms together: s1+s2+s3+s4… is the sum of the series.

Discrete Mathematics and Applications

Discrete mathematics is critical to the advancement and efficiency of modern
technology. Its ideas underpin the algorithms, data structures, and systems that
power our digital world. This branch of mathematics provides tools and concepts that
are essential for tackling complicated issues in computer science, information
security, network design, and other areas. Understanding its uses provides insight into
how technological improvements are achievable.

Today's technology relies on the ability to analyze, manage, and secure massive
volumes of data. Discrete Mathematics offers the theoretical underpinning for these
tasks through its different disciplines, such as combinatorics, graph theory, and
cryptography. For example, graph theory-based algorithms are used by search engines
to efficiently crawl and index the web. Similarly, data encryption and cybersecurity
measures use number theory and complexity theory ideas to prevent unwanted access
to information.

Discrete math themes and concepts are commonly employed in computer

science. Discrete math deals with discrete things rather than continuous ones. It
focuses on objects that can only accept distinct, separated values. There are no
fractions or decimals involved. For example, there may be a set of random integers or
graphs that are not continuous or discrete. However, some topics, like calculus or
Euclidean geometry, are not part of discrete mathematics because they are
continuous by nature.

Discrete Mathematics in the Real World

1. Railway Planning
- The railway department use graph theory techniques to determine the correct
and shortest route between two stations. They involve linear algebra ideas as
well as graph theory.

2. Cell Phone Communication

- Mobile phones and broadcast spectrum are operated efficiently using
information theory and linear algebra ideas. Graph theory and optimization
 approaches assist in assigning the correct frequency to the phones while
avoiding interference.

3. Password Creation Criteria

- Many internet services require a password. Users can utilize discrete math
methods to generate passwords and share encrypted information. Strong
algorithms can greatly benefit communication.

4. Online Delivery Services

- Many people now use internet purchasing or e-commerce websites. In addition,
individuals use popular applications to order food. However, the service
provider may be required to select the quickest and most efficient route for
any type of delivery. TSP, or Traveling Salesman Problem-like algorithms, assist
providers in determining the correct and best path between source and
destination.

5. Use of Discrete Math in Cryptography

- Cryptography is the study of how to create passwords and security procedures.
All of these notions are essentially dependent on discrete math, because
information is typically transmitted in bits and discrete forms.

- Number theory, one of the most fundamental concepts in discrete math,

enables mathematicians and password makers to generate and crack numerical
passwords. Mathematicians who understand Number Theory can create secure
passwords and complicated encryption algorithms.

6. Use of Discrete Math in Financial Organizations Like Banks

- A relational database is an essential component of any organization. It aids in
keeping track of personnel, resources, and customers. A relational database
aids in the identification of links between different types of information. For
example, if more than one member of the same family works for a company,
such as a bank, the company can connect both clients using relational
databases.

7. Information Routing with Discrete Math

- Everyday loads of information pass from source to destination on the internet.
Discrete mathematics includes a variety of topologies such as computational
topology, combinatorics, discrete topological space, finite topological space,
and many others. Topology is a mathematical field, but it also plays an
important part in real-time computing applications. These algorithms aid in
optimizing the routing path, allowing information to travel in the shortest
amount of time possible.

8. Logistic Management and Discrete Math

- Organizing commodities, services, and information is known as logistic
organization. Computer programmers employ discrete math principles and
theories to create and construct the finest logistic management algorithms.4
Discrete math concepts also assist programmers in writing algorithms and
speeding them up by lowering their number of steps.
Discrete mathematics has a wide range of practical applications. Some more
applications include Google maps, cell phone communication, digital image
processing, healthcare resource management, and many others.

Examples of Discrete Mathematics:

Example 1: Determine that in how many ways can three prizes be shared among 4
boys when
i) No one gets more than one prize.
ii) A boy can get any number of prizes.

Solution:
i) The first prize can be given in 4 ways as one cannot get more than one prize; the
remaining two prizes can be given in 3 and 2 ways respectively.
The total number of ways = 4 x 3 x 2 = 24.

ii) As there is no restriction, each prize can be given in 4 ways.

The total number of ways = 43 = 64.

Example 2: Find the sum of all four-digit numbers formed by using 2, 3, 6, 9 in which
no digit is repeated.

Solution:
If 2 occupies unit’s place, the remaining 3 digits can be arranged in 3!= 6 ways.
Similarly, if 2 occupies ten’s place, hundreds place, thousand’s place, in each of
these cases we get 3! numbers. Thus, the positional value contributed by 2 to the sum
when it occupies different values.

(3!)(2) + (3!)(20) + (3!)(200) + (3!)(2000) = 3!(2)(1111)

Similarly, the values contributed by 3, 6, 9 to the sum are 3! (3)(1111), 3! (6) (1111),
3! (9)(1111) respectively.

The required sum is 3!(1111)(2 + 3 + 6 + 9) = 1,33,320

Other Examples:
1. If a set B has n elements, then what is the total number of subsets of B. Justify
your answer.

Solution:

If a set B has “n” elements, then the total number of subsets of B is 2n.

For example, if B contains 5 elements, say B = {1, 2, 3, 4, 5}, then the total number
of subsets of B is 25 = 32.

2. If X and y are the two finite sets, such that n(X U Y) = 36, n(X) = 20, n(Y) = 28,
then find n( X ∩ Y).
Solution:

Given: n(X) = 20, n (Y) = 28, n (X U Y) = 36.

As we that, n(X U Y) = n(X) + n(Y) – n(X ∩ Y)

On rearranging the above formula, we get;

n(X ∩ Y) = n(X) + n(Y) – n(X U Y)

Now, substitute the given values in the above formula, we get;

n(X ∩ Y) = 20 + 28 – 36
n(X ∩ Y) = 48 – 36
n(X ∩ Y) = 12.

Hence, n(X ∩ Y) is 12.

3. Find the missing number in the sequence: 5, 10, 15, ___, 25, 30.

Solution:

Given sequence: 5, 10, 15, ___, 25, 30.

In the given sequence, each preceding term is added with 5 to get the succeeding
term.

I.e., 5 + 5 = 10
10 + 5 = 15
15 + 5 = 20
20 + 5 = 25
25 + 5 = 30.
Hence, the missing term in the given sequence is 20.

Therefore, the complete sequence is 5, 10, 15, 20, 25, 30.

4. Calculate the tenth term of the Arithmetic Progression 13, 8, 3, -2 …?

Solution:
Given AP: 13, 8, 3, -2…

Here, the first term, a = 13

Common difference, d = -5.

I.e., d = 8 – 13 = -5
3 – 8 = -5
-2 -3 = -5.
And, n = 10.
The formula to find the nth term of Arithmetic Progression is Tn = a + (n-1)d

Now, substitute the given values in the formula, we get

T10 = 13 + (10 – 1)(-5)

T10 = 13 + (9)(-5)
T10 = 13 – 45
T10 = -32

Hence, the tenth term of given Arithmetic Progression 13, 8, 3, -2… is -32.

5. In the group of students, there are 6 boys and 4 girls. Out of 10 students, 4
students must be selected. Find out how many ways the students can be selected
such that at least one boy should be selected?

Solution:
Total number of students = 10
Number of boys = 6
Number of girls = 4.

So, here we must select 4 students. In that, at least one boy should be selected.

Hence, there are four cases:

Case 1: All four students are boys:
Thus, the number of ways = 6C4 = 15

Case 2: 3 Boys and 1 girl are selected:

The number of ways to select 3 boys and 1 girl = 6C3 × 4C1 = 20 × 4 = 80.

Case 3: 2 boys and 2 girls are selected.

The number of ways to select 2 boys and 2 girls = 6C2 × 4C2 = 15 × 6 = 90.

Case 4: 1 boy and 3 girls are selected.

The number of ways to select 1 boy and 3 girls = 6C1 × 4C3 = 6 × 4 = 24.

Therefore, the number of ways to select 4 students such that at least one boy to be
selected = 15 + 80 + 90 + 24 = 209.

Accuracy and Precision

The errors associated with both calculations and measurements can be
characterized by their accuracy and precision.
 Accuracy refers to how closely a computed or measured value agrees with the
true value. Accuracy refers to how closely a computed or measured value

agrees with the true value. Sometimes the true value could be the theoretical
value. Accuracy is expressed as % error. It is the % difference between the
measured and the known true value. Accuracy refers to how close your result is
to the true correct value.
 Precision refers to how closely individual computed or measured values agree
with each other. Precision is a description of a level of measurement that
yields consistent results when repeated. It is associated with the concept of
“random error”, a form of observational error that leads to measurable values
being inconsistent when repeated.
These concepts can be illustrated graphically using an analogy from target
practice. The bullet holes on each target in Fig. 3.2 can be thought of as the
predictions of a numerical technique, whereas the bull’s-eye represents the truth.
 Inaccuracy (also called bias) is defined as systematic deviation from the truth.
Thus, although the shots in Fig. 3.2c are more tightly grouped than those in
Fig. 3.2a, the two cases are equally biased because they are both centered on the
upper left quadrant of the target.
 Imprecision (also called uncertainty), on the other hand, refers to the
magnitude of the scatter.
Therefore, although Fig. 3.2b and d are equally accurate (that is, centered on
the bull’s-eye), the latter is more precise because the shots are tightly grouped.

FIGURE 3.2 An example from marksmanship illustrating the concepts of

accuracy and precision. (a) Inaccurate and imprecise; (b) accurate and imprecise;
(c) inaccurate and precise; (d) accurate and precise
  Numerical methods should be sufficiently accurate or unbiased to meet the
requirements of a particular engineering problem.
 They also should be precise enough for adequate engineering design.
We will use the collective term error to represent both the inaccuracy and the
imprecision of our predictions. With these concepts as background, we can now
discuss the factors that contribute to the error of numerical computations.

Example in Accuracy and Precision:

A student, in three titrations, used 18.00 ml, 17.80 ml, and 18.90 ml of base to
neutralize 10.00 ml acid.
a. What is the precision?
1. First, we will find the mean (average value) by adding all the measured values and
dividing them by 3.

2. Second, we will find the deviation of each measurement.

i. Deviation of the first measurement:
(18.00– 18.23) ml = - 0.23 ml
ii. Deviation of the second measurement:
(17.80 – 18.23) ml = - 0.43 ml
iii. Deviation of the third measurement:
(18.90 – 18.23) ml = + 0.67 ml
3. Next, we will find standard deviation:

4. To calculate % deviation:

b. If the true value is 15.77 ml, calculate the accuracy of the experiment.
Order of Accuracy
We consider a numerical approximation of an exact value u. The approximation
depends on a small parameter h , which can be for instance the grid size or time step
in a numerical method. We denote the approximation by u ˜ h. The numerical method
has order of accuracy p if there is a number C independent of h such that.
|u ˜ h −u|≤C h p ,
at least for sufficiently small h . Hence, the larger the order of accuracy, the
faster the error is reduced as h decreases. We say that the convergence rate of the
method is h p . The number C typically depends on the exact solution u and possibly on
other parameters in the numerical scheme. What is important is that it does not
depend on h. Often the error u ˜ h−u depends smoothly on h . Then there is an error
coefficient D such that
p p+1
u ˜ h−u=D h +O(h )

Note that this is not equivalent to (1) since the error may be a non-smooth
function of h . We will get back to this issue in Section 4 below. For now, however, we
will assume (2) holds.

Errors
Errors can result prior to computation in the form of measurement errors or
assumptions in modeling. The focus of this blog post will be on understanding two
types of errors that can occur during computation: roundoff errors and truncation
errors.

Roundoff Error
Roundoff errors occur because computers have a limited ability to represent
numbers. For example, π has infinite digits, but due to precision limitations, only 16
digits may be stored in MATLAB. While this roundoff error may seem insignificant, if
your process involves multiple iterations that are dependent on one another, these
small errors may accumulate over time and result in a significant deviation from the
expected value. Furthermore, if a manipulation involves adding a large and small
number, the effect of the smaller number may be lost if rounding is utilized. Thus, it
is advised to sum numbers of similar magnitudes first so that smaller numbers are not
“lost” in the calculation.One interesting example that we covered in my Engineering
Computation class, that can be used to illustrate this point, involves the quadratic
formula. The quadratic formula is represented as follows:

Using a = 0.2, b = – 47.91, c = 6 and if we carry out rounding to two decimal places at
every intermediate step:
The error between our approximations and true values can be found as follows:

As can be seen, the smaller root has a larger error associated with it because
deviations will be more apparent with smaller numbers than larger numbers.
If you have the insight to see that your computation will involve operations with
numbers of differing magnitudes, the equations can sometimes be cleverly
manipulated to reduce roundoff error. In our example, if the quadratic formula
equation is rationalized, the resulting absolute error is much smaller because fewer
operations are required, and numbers of similar magnitudes are being multiplied and
added together:
Truncation Error
Truncation errors are introduced when exact mathematical formulas are represented
by approximations. An effective way to understand truncation error is through a
Taylor Series approximation. Let’s say that we want to approximate some function,
f(x) at the point xi+1, which is some distance, h, away from the basepoint xi, whose
true value is shown in black in Figure 1. The Taylor series approximation starts with a
single zero order term and as additional terms are added to the series, the
approximation begins to approach the true value. However, an infinite number of
terms would be needed to reach this true value.

Figure 1: Graphical representation of a Taylor Series approximation (Chapra, 2017)

The Taylor Series can be written as follows:

where Rn is a remainder term used to account for all of the terms that were not
included in the series and is therefore a representation of the truncation error. The
remainder term is generally expressed as Rn=O(hn+1) which shows that truncation error
is proportional to the step size, h, raised to the n+1 where n is the number of terms
included in the expansion. It is clear that as the step size decreases, so does the
truncation error.
The Tradeoff in Errors
The total error of an approximation is the summation of roundoff error and
truncation error. As seen from the previous sections, truncation error decreases as
step size decreases. However, when step size decreases, this usually results in the
necessity for more precise computations which consequently results in an increase in
roundoff error. Therefore, the errors are in direct conflict with one another: as we
decrease one, the other increases.

However, the optimal step size to minimize error can be determined. Using an
iterative method of trying different step sizes and recording the error between the
approximation and the true value, the following graph shown in Figure 2 will result.
The minimum of the curve corresponds to the minimum error achievable and
corresponds to the optimal step size. Any error to the right of this point (larger step
sizes) is primarily due to truncation error and the increase in error to the left of this
point corresponds to where roundoff error begins to dominate. While this graph
is specific to a certain function and type of approximation, the general rule and shape
will still hold for other cases.

Figure 2: Plot of Error vs. Step Size (Chapra, 2017)

Hopefully this was helpful to increase awareness of the types of errors that you may
come across when using numerical methods! Internalize these golden rules to help
avoid loss of significance:

 Avoid subtracting two nearly equal numbers.

 If your equation has large and small numbers, work with smaller
numbers first.
 Consider rearranging your equation so that numbers of a similar
magnitude are being used in an operation.

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Department of Mechanical Engineering

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