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Chapter 1 MATHEMATICAL MODELING MAT 530
CHAPTER ONE
Mathematical
Modeling Process
At the end of this module, students should be able:
To define mathematical modeling
To explain what is a mathematical model
To describe various classification of mathematical models
To outline procedures involved in constructing a mathematical model
To solve a simple modeling problem
To apply the method of least squares
1.1 Introduction
Consider the problems of finding the shortest route from one location to a
destination in a busy city or deciding locations for bus stops in an urban
area so that greater number of passengers can be achieved. These are
some examples of complex real-world problems. In order to obtain the
required solution this person need to investigate necessary questions
about the observed world and give a simplified description of the
problem. He/she can setup equations using mathematical concepts and
language to test some ideas and then solved the model to make
prediction or decision. The developed process mentioned above is
known as mathematical modeling.
Definition 1.1
Mathematical modeling is the process of constructing
mathematical objects as a quantitative method to represent the
properties or behaviors of the real-world system, process or
phenomena. Mathematical object refers to sets of numbers,
variables or parameters and their symbols and functional
relationships in the form of equation or system of equations and
geometrical or algebraic structure.
Chapter 1 MATHEMATICAL MODELING MAT 530
Mathematical modeling cultivates the interaction between mathematics
and the real world by translating the real world problem into mathematical
formulation or model, solving the model and interpreting the solutions
obtained using the model so as to make the model useful to the real
world. The real world problem covers situations which may come from
diverse disciplines such as engineering, science, architecture, business,
sports science and computer science while the system can be a physical
system, a biological system, a financial system, a social system or an
ecological system. Mathematical modeling allows a modeler to
undertake experiments on mathematical representations of the real world
instead of undertaking experiments in the real world. Mathematical
models can take many forms such as dynamical systems, statistical
models, differential equations, or game theoretic models. Mathematical
modeling can be viewed as a process, as illustrated in the schematic
diagram of Figure 1.1.
Figure 1.1: Process of Mathematical Modeling
Figure 1.1 shows mathematical modeling as an iterative process or a
cycle (or loop) which starts and finishes with the real world problem. It
may start with the the upper left-handcorner, the real world problem. This
represents quantitative measurement or data of system of interest and
knowledge of how the system works such that properties, behaviors and
Given {some factors}, find {outputs}, such that {objective is
achieved}.
Test /
Analyze
Translate (Formulate)
Solve /
Analyze
Real-World Conclusion
or Prediction
Real-World
Problem
Mathematical
Solutions
Translate
(Interpret)
Data Collection and Analysis Making Assumptions
Construct Model
FINISH
START
Simulation ‘What-If’ Analysis
Validation and
Verification
Mathematical Model
Chapter 1 MATHEMATICAL MODELING MAT 530
relationship between factors (variables and parameters) could be
observed and problem pertaining to the system that needs to be modeled
identified. A mathematical models is characterized by assumptions about
variables (the things which change), parameters (the things which do not
change) and functional forms (the relationship between the two). Thus,
based on data and knowledge of the system gathered, a mathematical
model can be constructed.
Construction of a model requires: 1) a clear objective of the modeling –
which aspect of the system is involved, what do you wish the model to
produce and how accurate you want the model be; and 2) a precise idea
of the main factors involved and therefore, it may include taking a
simplified view of the system and considering the simplest practical
approach by neglecting certain factors and making assumptions. Real-
world problem could be simplified or transformed to a problem relatively
close to the original problem but all essential features of the problem
must be retained.
Mathematical modeling may encompass the following goals:
a) integrating various existing good models of various parts of the
system to represent the whole system of interest;
b) generalizing existing models of different systems through adoption,
adaptation or modification of these models to be applied to the
system of interest; and
c) simplifying or making approximation to general model since the
current models are difficult to compute or analyzed.
After a mathematical model is established, computational experiments or
computations are required to solve the model which is to generate results
or solutions. Based on the model developed, optimum or approximate
solutions maybe obtained. As the model is aimed to represent the real
system, results or solutions found based on the model should be able to
be interpreted as descriptive properties or behaviors concerning the
system, thus, enabling prediction or conclusion to be made about the real
system. The solutions, however, must first be tested (validated and
verified) against real data to determine the effectiveness of the model (to
Chapter 1 MATHEMATICAL MODELING MAT 530
test assumptions and for sensitivity analysis). If the solutions agree with
the real system (i.e., makes sense), then the model does indeed capture
correctly important aspects of the real-world situation and thus can be
accepted and used. In the case where the solutions found do not
substantiate the real-world phenomena, the modeling cycle is followed
through again with a revised formulation of model, new calculated results,
improved predictions, and so on. It is useful to note that the modeling
process may not necessarily results in solving the problem entirely but it
will furnish useful information regarding the situation under investigation.
The whole process of mathematical modeling can be divided into three
main activities: i) construct the model, ii) compute and analyze results,
and iii) test (validate and verify) results, as depicted in Figure 1.2 below.
Figure 1.2: Mathematical Modeling Process
Definition 1.2
A mathematical model is a system (S) containing a set of
questions (Q) relating to the system and a set of mathematical
statements (M) which can be used to answer the questions
(Velten, 2009).
A system refers to a set of interacting, interrelated, or interdependent
elements (or components) which are functionally related. When modeling,
very specific sets of questions must be addressed in order to transform
Test the Results
Construct the Model
Compute and Analyze Results
Apply the Model
Chapter 1 MATHEMATICAL MODELING MAT 530
data concerning the system into some mathematical statements. These
questions, which transpire the purpose of the model, are very focused and
challenging questions and can only be answered with a conclusion based
on the analysis and interpretation of evidence. In other words, the
questions invoke logical statement that progresses from what is known or
believed to be true to that which is unknown and requires validation.
Mathematical statements refer to relationships of numbers, variables and
parameters formulated using notations, symbols and operators. These
statements are used to solve the problem and provide meaningful
information useful for making conclusion, decision or prediction regarding
the system.
1.2 Definition of a model
A model is a representation of reality containing the essential structure of
some object or event in the real world. It can be a physical, symbolic
(mathematical), graphical, verbal or simplified version of a real world
system or phenomenon. Since most systems and phenomenon are very
complicated (due to their large number of components) and complex
(very highly interconnected) to be understood entirely, a model may
contain only those features that are of primary importance to the
modeler’s purpose. Thus, a model is usually a tradeoff between
generality, realism and precision. Models, therefore, may be necessarily
incomplete and may be changed or manipulated with relative ease. Due
to these, there is no model can be considered as the best model, only
better model.
Models can be categorized into three classes on the basis of their degree
of abstraction, as the following:
a) Physical (Iconic) model: An iconic, 'look-alike' model, which is the
least abstract. Examples include a model of an airplane or an
architect’s model of a building.
b) Analogous model: A more abstract type of model in which the
model behaves like a real system but only having some resemblance
Chapter 1 MATHEMATICAL MODELING MAT 530
to what it represents. Chart (including flowchart), graph, map and
network diagram are considered as analogous models.
c) Symbolic or mathematical model: It is the most-abstract model
with no resemblance but only an approximation to what it represents.
The relationships between variables of the system are expressed
using a set of rules or operators according to their meaning (syntax)
rather that what they represent (semantic). Examples of these
models include mathematical equations or formula, financial
statement and set of accounts.
Definition 1.3
A model is called dynamic if at least one of its system parameters or
state variables depends on time.
Definition 1.4
A model is called static if the system parameters or state variables
do not change with respect to time.
A model can be a static or dynamic model. Figure 1.2, 1.3 and 1.4
illustrates some examples of the physical, analogous and symbolic model
in both static and dynamic forms.
a) A Static Physical Model
- A miniature replicate of a submarine b) A Dynamic Physical Model - Aerodynamics of a car in a wind tunnel
Figure 1.2: Physical Model
Chapter 1 MATHEMATICAL MODELING MAT 530
a) An Static Analogous Model
- A model of an airport city c) A Dynamic Analogous Model
- A Traffic Anomalous Events Model
Figure 1.3: Analogous Model
a) A Static Symbolic Model - A geometry model for right
triangles
b) A Dynamic Symbolic Model - Population Growth Models
Figure 1.4: Symbolic Model
1.3 Why Model?
To gain understanding.
A model represents characteristics and behaviors of a real-world system
concerned, thus, providing an improved understanding. It facilitates
understanding by focusing and analyzing necessary components of the
system. Through modeling we can comprehend how different parts of
the system are related and which factors are most important in the
system.
To predict or simulate. Very often we wish to know what a real-world
system will do in the future, but it is expensive, impractical, or impossible
Chapter 1 MATHEMATICAL MODELING MAT 530
to experiment directly with the system. Examples include nuclear reactor
design, space ight, extinction of species, weather prediction, drug efficacy
in humans, and so on. Using models, simulation of the real-world system
or phenomenon can be carried out, thus providing means to explain,
control, and predict events on the basis of controlled observations.
To aid in decision making. Testing using the model developed allows
the possibility of simulating the ‘what if scenario’ or performing the
sensitivity analysis. Hence, the model provides concrete justifications for
any decison made.
Mathematics has been describe in old adage as the “Queen of Science”
and “language of science". It been used to describe real phenomena
through the mathematical modeling process. Thus, it is not a surprise
that mathematical modeling has emerged to be a much needed tool in
this modern science and technology era. Cheaper and more advanced
computers enable mathematical models to play vital role in discovering
new knowledge and solving problems aside from becoming increasingly
cost-effective alternative to direct experimentation.
Despite the clear advantages of modeling, there is no definite steps for
constructing a “one for all” model. There is no model that can be applied
to all situations. Modeling is an art. It involves the creativity in utilizing a
sound mathematical knowledge together with the knowledge of the
system to create something useful to provide meaningful information and
interpretation of the system. Different expertise and perspectives yields
different models for the same system. Such scenario justifies that there
is no “best” or “perfect” model. A model is generally a trade-offs between
the following:
a) accuracy
b) flexibility
c) cost.
More accurate model requires higher cost and reduces flexibility. Higher
flexibility model, on the other hand, may jeopardize the accuracy of the
model and may comes at increased cost. In addition, expensive model is
Chapter 1 MATHEMATICAL MODELING MAT 530
not preferrable since it does not necessarily guarantees accuracy or
flexibility. Thus, modeling is always aimed at producing a sufficiently
accurate and flexible model preferrably at low cost.
Categorize the following models as one of the following: a) A static physical model d) An dynamic analogous model b) A dynamic physical model e) A static symbolic model c) An static analogous model f) A dynamic symbolic model
Universal Harmonics Model Malignancies Dynamic Pathway Model
Answer: _______________________
Answer: _______________________
A Hurricane Forecast Model A Volume Optimization Model
Answer: _______________________ Answer: _______________________
Warm up exercise
Chapter 1 MATHEMATICAL MODELING MAT 530
1.4 Classification of Mathematical Models
When studying or developing models, it is helpful to identify the type of
models that we are dealing with. Mathematical models can be classified
into four different types: a) deterministic model, b) stochastic models, c)
empirical models and d) mechanistic models
Definition 1.5
A deterministic model is a model which is represented by a
function that allows predictions of the dependent variable to be
made based on the independent variable(s).
Deterministic models have no components that are inherently uncertain,
i.e., no parameters in the model are characterized by probability
distributions. Deterministic models ignore random variation, and so
always predict the same outcome from a given starting point. In other
words, for fixed starting values, a deterministic model will always produce
the same result. Deterministic model is usually applied for situations
where the outputs are direct consequence of the initial conditions of the
problem.
Definition 1.6
A stochastic model is a model where randomness is present
and variables are not described by unique values, but rather by
probability distributions.
A stochastic model is mainly used for situations where a random effect
plays a vital role in the model formulation and solving. This model, which
is statistical in nature, will produce many different results depending on
the actual values that the random variables take in each realization.
A stochastic model predicts the distribution of possible outcomes.
Chapter 1 MATHEMATICAL MODELING MAT 530
Definition 1.7
A model is known as empirical model if it was constructed from
and based entirely on experimental data only, using no priori
information about the system.
For an empirical model, no account is taken of the mechanism by which
changes to the system occur and minimal information is used a priori in
the development of the model. The model just simply notes that some
phenomena occur and tries to account quantitatively for changes
associated with different conditions. Thus, an empirical model usually
provides a quantitative summary of these observed relationships among
a set of measured variables. Models developed using statistical
techniques are examples of empirical models.
In an empirical model, relationship between variables is determined by
inspecting the data on the variables and selecting the best fit
mathematical formula to represent the relationship. Therefore, it is a
compromise between accuracy of fit and simplicity of mathematics.
Definition 1.8
A model is known as mechanistic model if some of the
mathematical statements are based on priori information about
the system.
A mechanistic model uses a large amount of theoretical information by
attempting to explain phenomena that occur at a more detailed level of
structure. It generally describes what happens at one level in the
hierarchy by considering processes at lower levels. It takes account of
the mechanisms through which these changes occur and generates
prediction concerning the variables based on this knowledge. The model
is built on causal relationships represented by equations.
Table 1.1 shows an example of classification of some models according
to the four classes of models.
Chapter 1 MATHEMATICAL MODELING MAT 530
Table 1.1: Classification of Models
Empirical Mechanistic
De
term
inis
tic
Short term forecasts (nowcasts) of seasonal climate based on historical climate records (climatological database) of rainfall or temperature using regression analysis
A hypothetical study on predator-prey relationships of the larval salamanders and predacious aquatic invertebrates – a deterministic model with mechanistic explanantion to patterns observed
Sto
ch
as
tic
A mass flux balance model to estimate heavy-metals in agriculture soil. Normal or log normal probability distribution is used to represent the metal concentration.
A risk projection model in epidemiology (in the form of a stochastic mechanistic model) developed to estimate radiation-induced cancer risks to the end of life for Japanese atomic bomb survivors.
A model can also be classified according to the level of understanding on
which the model is based. The simplest explanation is to consider the
hierarchy of organizational structures within the system being modeled.
For example, in the case of animals, models can be distinguished
according to different levels of this hierarchical structure. For example, a
model concerning an animal can be classified as a molecular model,
cellular model or an individual model of the animal. This classification
can be illustrated as in Figure 1.6.
High Herd
Individual
Organs
Cells
Low Molecules
Figure 1.6: A Classification of Model for Animals based on Hierarchy
Aside from classifying models based on the categories mentioned,
models can be distinguished based on whether they are static or dynamic
models. As described in Definition 1.3 and Definition 1.4, a static model
Chapter 1 MATHEMATICAL MODELING MAT 530
does not account for the element of time whereas a dynamic model is
time dependent. The static model describes the steady-state or
equilibrium situations of the system at a specific time instant, where the
output does not change if the input is the same. In a dynamic model, the
output changes with respect to time. A resistor system is a static model
where the voltage is directly proportional to the current, independent of
time. On the other hand, a capacitor is modeled as dynamic model since
the voltage is dependent on the time and previous time history.
Dynamic models can be divided further into continuous and discrete
models, as described in the following definitions.
Definition 1.9
A model is called discrete if the quantities have varied at discrete
times or places (or that we only consider discrete variations even
if they may change continuously).
Definition 1.10
A model is called continuous if there is continuous variation in
quantities.
Discrete models use a discrete or combinatoric description such as
integer numbers, graphs and difference equations. On the other hand,
continuous models often involve real quantities, i.e., real numbers,
physical quantities and differential equations rather than the difference
equations. Differential equations denotes changes in time over an
infinitesmall interval whereas difference equations represent changes
over a finite interval (time jumps).
One further type of model, the system model, is worthy of mention. This
is built from a series of sub-models, each of which describes the essence
of some interacting components. The above method of classification then
refers more properly to the sub-models: different types of sub-models
may be used in any one system model.
Chapter 1 MATHEMATICAL MODELING MAT 530
Much of the modelling literature refers to ’simulation models’. Why are
they not included in the classification? The reason for this apparent
omission is that ’simulation’ refers to the way the model calculations are
done - i.e. by computer simulation. The actual model of the system is not
changed by the way in which the necessary mathematics is performed,
although our interpretation of the model may depend on the numerical
accuracy of any approximations.
1.5 Constructing Mathematical Models
Mathematical modeling is the art of transforming real world problem from a
specific application area into a structured mathematical formulation which is
solvable, thus the theoretical and numerical analysis can lead to useful
understanding, prediction, solution and interpretation related to the original
problem. Mathematical modeling has been successfully applied in wide areas of
applications, indispensable in some of these applications. The model developed
provides precise guide to solution of the problem, enables comprehensive
understanding of the system involved and allows the application of advanced
computing software and techniques. Mastering the skill in mathematical
modeling should be one of the expertise that must be acquired by students,
aside from the theoretical mathematical learning in class, to enable them to
handle real world challenging problem of our time.
Figure 1.t illustrates the flow of interaction of the four main components of the
mathematical modeling process. Problem statement defines the problem and
wishes to are to be achieved by the by the model. The mathematical theory
includes theory related to the application, mathematical theory to be employed
and theories discussed in literature. Mathematical solution methods underline
the mathematical approaches which could be utilized in generating the solutions
based on the mathematical model. Finally, computational experiments refer to
the execution of computer programs or simulation runs using certain data and
according to the mathematical model and solution techniques constructed.
Chapter 1 MATHEMATICAL MODELING MAT 530
Figure 1.7: Interaction of the Components of the Mathematical Modeling Process
The modeling process itself is usually an iterative process characterized by a
sequence of steps which sometimes required to be repeated. The steps include
the following:
a) define the problem and objectives of the model;
b) conduct a thorough literature review of the problem and existing models;
c) gather complete data through data collection, empirical observations or
experiments;
d) analyze data to identify the important factors (main variables, parameters
and constants) and determine properties and relationships between
these factors as well as their influence towards the behavior of the
system;
e) construct diagram, flowchart, network or computer visualization model to
get better understanding of the complex interaction of factors within the
system;
f) list all assumptions and ways to verify and validate the model;
g) transform the problem using abstraction or formulization into a
mathematical model. Start with a simple model as among models with
similar predictive power, the simplest one is the most desirable;
Mathematical Model
Problem Statement
Mathematical Theory
Mathematical Solution Methods
Computational Experiments
Chapter 1 MATHEMATICAL MODELING MAT 530
h) develop solution techniques that can be used to determine the solutions
of the model;
i) perform computational experiments,which can consist of simulations,
analytical and qualitative analysis using mathematical techniques;
j) analyze the results;
k) validate and verify results based on interpretation and comparison with
the real system using data set that have not been used to build the
model;
l) make conclusions, predictions and decisions based on j) and k);
m) carry out the changes in the real system based on the findings and
mechanisms proposed by the model; and
n) refine the model, if necessary, by repeating step a) to h) for enhanced
performance of the real system.
Steps : Constructing Mathematical Model
1. Identify the real problem
2. Formulate a mathematical model
i. Identify and classify the variables
ii. Determine interrelationships between the variables
4. Obtain the mathematical solution to the model
5. Interpret the mathematical solution
6. Verify the model
i. Does it address the problem?
ii. Does it make common sense?
iii. Test or compare solution with real world data
7. Write a report/Conclusion
8. Implement the model
9. Maintain the model
Chapter 1 MATHEMATICAL MODELING MAT 530
Modeling Approaches Construction of a mathematical model includes the following steps:
1. Identify the real problem
a. This concern with questions such as the following:
◦ How to classify the problem?
◦ Is there an underlying physical/scientific behavior to be taken
account?
◦ What kind of sources of facts and data are relevant?
◦ Is data available? Is itr primary or secondary data?
◦ What assumptions and simplifications about the problem can be
made?
◦ What kinds of models are applicable?
◦ What is the purpose and objective of the model to be developed?
◦ Will the solution based on the model be unique or not?
◦ Who will use the model?
b. Formulate a precise problem statement
In most problems we can develop the problem statement by examining:
◦ What is known or given?
◦ What is to be found, estimated or decided
◦ What are the conditions to be satisfied?
◦ What are the objectives to be achieved?
2. Formulate mathematical model
Various activities need to be carried as listed below:
a. Identify and list the relevant factors and/or relationships.
Factors are quantifiable and can be classified as variables, parameters
or constants, and each of these can be continuous, discrete or random.
◦ Continuous variables: takes all real values over an interval, e.g.
time, speed, length, density, etc.
◦ Discrete variables: takes on certain isolated values only, e.g. the
number of people, cars, houses, months, etc.
Chapter 1 MATHEMATICAL MODELING MAT 530
◦ Random variables: unpredictable, but governed by some
underlying statistical model, e.g. bus’s arrival time.
◦ Input Variables: quantities which determine subsequent
evaluations within the model. Input variables are to be known, or
give, or assumed, or can be considered to have an arbitrary value.
◦ Output Variables: quantities whish are consequences of given
values of input variables and parameters and cannot be given
arbitrary values. These represent outcomes from a model.
◦ Parameters: Quantities which are constant for a particular
application of model, but can have different values for another
application of the same model, e.g. fixed costs, the dimension of a
room, price of a ticket, etc.
◦ Constants: quantities whose values we cannot change, e.g. π,
gravity, speed of light, etc.
All the factors must have suitable algebraic symbols and represent
measurements with certain unit of measurement.
b. List the assumptions
Basic considerations for assumptions include:
i. whether or not to include certain factors,
ii. the relative magnitudes of certain effects of various factors (help
to identify the main factors), and
iii. the forms of relationship between factors – this is the heart of the
model.
Appropriate assumptions keep the model simple.
c. Collect data
Utilize data as much as possible.
i. At initial stage – analyze data to identify factors and their
relationships.
ii. In the model development stage – fix values of parameters and
constants that may occur in the model based on data.
iii. In the final stage – valicate model by comparing its solution or
prediction with real data
Chapter 1 MATHEMATICAL MODELING MAT 530
Challenges in data collection:
◦ Getting permission to collect data.
◦ Getting right instruments / equipments.
◦ Getting respondents to complete survey questionnaire.
◦ Getting the right and complete data.
d. Formulate model
i. Define variables representations and symbols.
ii. Assign units.
iii. Determine the relations and equations connecting the problem
variables.
iv. Formulate objective of the model.
v. Identify the form of outputs to be obtained.
vi. Specify constraints.
vii. Check model’s structure:
Given {values of input variables, parameters and constants},
find {values of output variables} such that {conditions are
satisfied or objective achieved}.
e. Obtain the mathematical solution of the model
Solving process may involve the following:
◦ Plot graphs
◦ Use Calculus approaches
◦ Apply numerical methods
◦ Apply optimization techniques
◦ Run simulations
◦ Execute computer programs
3. Analyze and Interpret solution
Analyze and examine results or solutions obtained.
a. Check whether values of variables have the correct sign and size?
b. Ask the following questions:
◦ Are solutions produced reasonable?
◦ Has data measurement and accuracy been achieved?
◦ Have the best solution found?
Chapter 1 MATHEMATICAL MODELING MAT 530
4. Compare with reality
a. Test results against real data.
b. Check the following:
◦ Do proposed predictions based on solution make sense?
◦ Do predictions agree with real data?
◦ Evaluate the model. Is the model efficient?
◦ Do results suggest more accuracy need to be achieved?
5. Write a report
The following questions can be considered in preparing the report.
◦ Who is the report for?
◦ What do the readers want to know?
◦ How detail should the report be?
◦ Are important features clearly described?
◦ Are the results stand out?
Figure 1.8: Mathematical Modeling Process
Identify the real problem
Formulate Model
Solve the Model
Interpret Solutions
Compare with Reality
Write a report
Validation
Problem Definition
Modeling
Translation
Conclusion
Chapter 1 MATHEMATICAL MODELING MAT 530
STEPS
1. Identify the Problem
Given {the box is to have square base and double thickness top and
bottom; cardboard costs RM1.50 per square meter}, find {the dimension of
the box} such that {the cost is minimized}.
Draw diagram to understand the problem better.
TOP TOP
SIDE SIDE SIDE SIDE SIDE
BOTTOM BOTTOM
Figure 1.9: Illustration for the Problem
Example 1: Most Economical Size Box
A company decides to produce its own box for packaging products manufactured by the company. It has been decided the box should hold 0.1m3 (0.1 cubic meters). The box should have a square base and double thickness top and bottom. Cardboard costs RM1.50 per square meter. You are given the task of designing the box and it is up to you to decide the most economical size. What is the minimum cost to produce such box.
h
w
w
h
w w
TOP
SIDE
BOTTOM
w
w
Chapter 1 MATHEMATICAL MODELING MAT 530
2. Formulate mathematical model
Assumption: Ignore the thickness of cardboard for this model.
Volume
The volume of the box is given to be 0.1m3. Thus,
(1.1)
Area of the 4 Sides
Area of double tops and bases
Total cardboard needed Area of Cardboard
Total Cost (C) Area of Cardboard (in m2) x price (in RM) per square meter
(1.2)
Let us use method of substitution to solve a system of two equations,
equations (1.1) and (1.2), with two unknowns, . From (1.1), rearrange the equation to get:
(1.3)
Substitute equation (1.3) into equation (1.2):
(1.4)
3. Solve the model
To find the dimension of the box which gives a minimum total cost,
differentiate equation (1.4) with respect to .
(1.5)
Set the derivative equals to 0 to get the value of .
m 0.37 m or cm 37 cm.
Chapter 1 MATHEMATICAL MODELING MAT 530
Substituting the value of in (1.3) gives us:
m 0.74 m or cm 74 cm
Find the second derivative of with respect to
(1.6)
Substituting into equation (1.6), we obtain,
Since
, we conclude that is minimum when meter and
meter. Substituting the values of and in equation (1.4), we
get the minimum cost, .
4. Analyze and Interpret the solution
Plotting the graph of equation (1.4) gives us the graph as shown in
Figure 1.10, where the point (0.37, 2.44) is the lowest point on the graph.
By examining the graph, the width ( ) could be anywhere between 0.35 m
to 0.38 m without affecting the minimum cost very much, that is
.
Figure 1.10: Graph of Cost (C) vs Width (w)
0.3, 2.54 0.4, 2.46
0
1
2
3
4
5
6
7
0 0.2 0.4 0.6 0.8 1 1.2
Co
st p
er
Bo
x (i
n R
M)
Width or length of the box, w (in m)
(0.37, 2.44)
Chapter 1 MATHEMATICAL MODELING MAT 530
Conclusion
Based on the mathematical model, the following can be recommended:
Length or width ( ) m,
Height ( ) m,
Cost ( )
But any width between 0.35 m and 0.38 m would be fine, since the cost
will not differ much.
Thus, a more practical solution may be:
Length or width ( ) m,
Height ( ) m,
Cost ( )
Refinement of the model
Improvements of this model may include the following:
Include cost of glue or staples
Include wages for assembly workers
Include wastage when cutting box shape from cardboard.
Should the design be changed to optimize use of cardboard?
Chapter 1 MATHEMATICAL MODELING MAT 530
1.6 Method of Least Squares The Method of Least Squares is a procedure to determine the best fit line to
data; the proof uses simple calculus and linear algebra. The basic problem is to
find the best fit straight line given that, for , the pairs
are observed. The method easily generalizes to finding the best fit of the
form:
where is not necessarily a linear function, but must be a linear
combination of these functions.
Given data, , we may define the error associated to
saying by
(1.7)
The goal is to find values of and that minimize the error. In multivariable
calculus we learn that this requires us to find the values of (a; b) such that
.
.
Differentiating and setting the derivatives equals to 0 gives:
and dividing by 2 yields:
.
.
We may rewrite these equations as:
.
.
Chapter 1 MATHEMATICAL MODELING MAT 530
It can be shown that solving the equations simultaneously yields:
.
.
From (1.12) and (1.13), we found that values of and which minimize the error
(defined in (1.7) satisfy the following matrix equation:
This, we have
Since the matrix A is invertible, this implies:
Define
,
Chapter 1 MATHEMATICAL MODELING MAT 530
= -1.03
120.88 Alternatively,
The equation of the best fit line is:
x y 1 26 92 676 8464 2392 2 30 85 900 7225 2550 3 44 78 1936 6084 3432 4 50 80 2500 6400 4000 5 62 54 3844 2916 3348 6 68 51 4624 2601 3468 7 74 40 5476 1600 2960
Example 2: Curve Fitting using the Method of Least Squares
Comsider the dat given below:
x 26 30 44 50 62 68 74
y 92 85 78 80 54 51 40
Find the best fit line using least squares method that describes the linear relationship between y and x.
Solution
Chapter 1 MATHEMATICAL MODELING MAT 530
Figure 1.10: Best Fit Line using Trendline in Excel
Identify the expected function : bxaey
Identify the unknown : a, b
y = -1.0344x + 120.88 R² = 0.9249
0
10
20
30
40
50
60
70
80
90
100
0 20 40 60 80
y
Example 3: Method of Least Squares and Linearization
A rectangular house is to be built with exterior walls that are 8 feet high. One
wall of the house will face north. The total enclosed area of the house will be
1500 square feet. Annual heating costs for the house are determined as
follows: each square foot of exterior wall with a northern exposure adds RM4 to
the annual heating cost; each square foot of exterior wall with an eastern or
western exposure adds RM2 to the annual heating cost; each square foot of
exterior wall with a southern exposure adds RM1 to the annual heating cost.
For what value of L is the annual heating cost minimized, and what is the
annual heating cost for this choice of L? The values given for the heating costs
per square foot can only be known approximately. How does the solution
depend on the given values for the heating costs?
Solution
Solution
Chapter 1 MATHEMATICAL MODELING MAT 530
Linearize the function :
bx
bx
aelnyln
aey
bxaln
)1elnelnbxaln
elnalnyln bx
Transform the data and/or unknown :
ylnY alna0 ba1
Identify normal equation
n
1i
2i
n
1i
i
n
1i
i
xx
xn
1
0
a
a =
n
1iii
n
1ii
Yx
Y
Determine the respective values:
i xi xi2 yi Yi xi Yi
1 0.4 0.16 800 6.6846 2.6738 2 0.8 0.64 975 6.8824 5.5059 3 1.2 1.44 1500 7.3132 8.7759 4 1.6 2.56 1950 7.5756 12.1209 5 2.0 4.00 2900 7.9725 15.9449 6 2.3 5.29 3600 8.1887 18.8340
3.8xn
1i
i
09.14yn
1i
i
6170.44Y
n
1i
i
8555.63Yxn
1i
ii
Substitute the respective values in the normal equation.
09.143.8
3.86
1
0
a
a =
8555.63
6170.44
Solve for the coefficients: 0a and 1a
3037.6a0 and
8187.0a1
Substitute to the transform unknown :
5906.546
ea
3037.6alna
3037.6
0
8187.0ba1
The expected function is x8187.0e5906.546y
Chapter 1 MATHEMATICAL MODELING MAT 530
kAxy
Taking natural logarithm both sides:
ln y = l n A + k ln x
Y Xaa 10
The necessary transformation for the data and unknowns are:
Y yln X xln Alna0 1ak
Rewrite
ln y = ln A + k ln x
as
Y = a0 + a1 X
=ROUND(LN(C4);4)
Example 4: Least Squares Technique
Use least-squares procedure to fit y = Axk to the following data.
xi 1.00 1.15 1.40 1.43 1.60 2.00
yi 4.33 4.58 4.98 5.06 5.28 5.80
Solution
Chapter 1 MATHEMATICAL MODELING MAT 530
Hence, the required normal equations:
6
1i
2i
6
1ii
6
1ii
xx
x6
1
0
a
a =
6
1iii
6
1ii
Yx
Y
9620.09971.1
9971.16
1
0
a
a =
3333.3
6358.9
Solving for a0 and a1:
a0 = 1.4649
Since a1 = 0.4239 then k = a1 = 0.4239
But ln A = a0 then
A = 0ae
= e1.4649
= 4.327
Hence, the expected function is y = Axk = 4.327x0.4239
Find the values of a and b of the form y = aex + be-x to the following data:
X 0 0.5 1.0 1.5 2.0 2.5 Y 5.02 5.21 6.49 9.54 16.02 24.53
Warm up exercise
Chapter 1 MATHEMATICAL MODELING MAT 530
1.7 Continuous Least-Squares Polynomials
The previous section considered the problem of least-squares
approximation to fit a collection of discrete data. The other approximation
problem concerns the approximation of functions or fitting continuous
data.
Consider a function y = f(x), which is continuous in [a, b]. Let
mm
ii
2210m xa...xa...xaxaa)x(P be a polynomial which
represent the function y = f(x), then the total least squares error is defined
as follows
∫b
a
2m dx(x)P- f(x) E
∫b
a
2mm
ii
2210 dxxa...xa...xaxaa- f(x)
We seek to minimize the sum of error squares. From the calculus of
functions of several variables, a necessary condition for the values a0, a , …,
and am to minimize E such that
0=a
E...=
a
E...=
a
E=
a
E
mi10 ∂
∂
∂
∂
∂
∂
∂
∂ mi0
Hence,
∫b
a
2mm
ii10
0
0dxxa...xa...xaa- f(x)a
(1)
∫b
a
2mm
ii10
1
0dxxa...xa...xaa- f(x)a
(2)
)(0dx......- f(x)
.
.
.
)(0dx......- f(x)
.
.
.
∫
∫
b
a
2
10
b
a
2
10
mxaxaxaaa
ixaxaxaaa
m
m
i
i
m
m
m
i
i
i
Chapter 1 MATHEMATICAL MODELING MAT 530
Simplifying (1) yields
∫b
a
mm
ii10 0dx)1(xa...xa...xaa- f(x)2
b
a
b
a
mm
b
a
ii
b
a
1
b
a
0
b
a
b
a
mm
b
a
ii
b
a
1
b
a
0
b
a
mm
ii10
dx)x(fdxxa...dxxa...xdxadxa
dx)x(fdxxa...dxxa...xdxadxa
0dx)x(fxa...xa...xaa∫
Simplifying (2) to (m) using similar procedures yield the normal equations
b
a
b
a
1mm
b
a
1ii
b
a
21
b
a
0 dx)x(xfdxxa...dxxa...dxxaxdxa
b
a
ib
a
imm
b
a
i2i
b
a
i21
b
a
i0 dx)x(fxdxxa...dxxa...dxxadxxa
: .
b
a
mb
a
m2m
b
a
mii
b
a
m21
b
a
m0 dx)x(fxdxxa...dxxa...dxxadxxa
In matrix form the normal equations can be represented as:
b
a
m
b
a
i
b
a
b
a
m
i
1
0
b
a
m2b
a
imb
a
1mb
a
m
b
a
mib
a
i2b
a
1ib
a
i
b
a
1mb
a
1ib
a
2b
a
b
a
mb
a
ib
a
b
a
dx)x(fx
.
.
dx)x(fx
.
.
dx)x(xf
dx)x(f
a.
.
a.
.
a
a
dxx...dxx...dxxdxx
..................
dxx...dxx...dxxdxx
..................
dxx...dxx...dxxxdx
dxx...dxx...xdxdx
Chapter 1 MATHEMATICAL MODELING MAT 530
Example 5:
Derive the normal equation in matrix form to approximate y = f(x) with a
straight line on the interval [a, b].
Let xaa)x(P 101
0=a
E=
a
E
10 ∂
∂
∂
∂
∫b
a
21 dx(x)P- f(x) E
∫b
a
210 dxxaa- f(x)
b
a
b
a
1
b
a
0
b
a
b
a
1
b
a
0
b
a
10
b
a
10
b
a
210
0
dx)x(fxdxadxa
dx)x(fxdxadxa
0dx)x(fxaa
0dx)1(xaa- f(x)2
0dxxaa- f(x)a
∫
∫
∫
Solution
Steps : Fitting Continuous Least Squares Polynomial Determine the degree m of the polynomial : m Write the general expression of the polynomial to be fitted
Pm(x) = mm
2210 xa ... xa x a a ++++
Write the normal equations in matrix form.
Compute : b
a
idxx for m2i0 and 1
0
i dx)x(fx for mi0
Solve for the coefficients a0, a1…am. Thus, the least-squares function is:
f(x) = mm
2210 xa ... xa x a a ++++
Solve for a0 and a1. Thus, the least-squares line is f(x) = a0 + a1x.
Chapter 1 MATHEMATICAL MODELING MAT 530
∫
∫
∫
b
a
10
b
a
10
b
a
210
1
0xdx)x(fxaa
0dx)x(xaa- f(x)2
0dxxaa- f(x)a
b
a
b
a
21
b
a
0
b
a
b
a
21
b
a
0
dx)x(xfdxxaxdxa
dx)x(xfdxxaxdxa
b
a
b
a
1
0b
a
2b
a
b
a
b
a
dx)x(xf
dx)x(f
a
a
dxxxdx
xdxdx
Example 6: Find the least squares polynomial approximation of degree
one to y = x 2 + 4x + 4 on the interval [0,2].
2
0
2
0
1
02
0
22
0
2
0
2
0
dx)x(xf
dx)x(f
a
a
dxxxdx
xdxdx
3
683
56
a
a
3
82
22
1
0
997.5
336.3
a
a
1
0
Hence, x997.5336.3)x(P1
Solution
Chapter 1 MATHEMATICAL MODELING MAT 530
Example 7:
Find the least squares polynomial approximation of degree two to f(x) =
ex on the interval [ 0, 1]
Determine the degree m of the polynomial
m = 2
Write the general expression of the polynomial to be fitted
P2(x) = 2210 xa x a a
Write the normal equations in matrix form.
1
0
x2
1
0
x
1
0
x
2
1
0
1
0
41
0
31
0
2
1
0
31
0
21
0
1
0
21
0
1
0
dxex
dxxe
dxe
a
a
a
dxxdxxdxx
dxxdxxxdx
dxxxdxdx
Compute : 1
0
idxx for 4i0 and 1
0
i dx)x(fx for mi0
7183.0
1
7183.1
a
a
a
2.025.03333.0
25.03333.05.0
3333.05.01
2
1
0
Solve for the coefficients a0, a1…am.
8449.0
8445.0
0140.1
a
a
a
2
1
0
Thus, the least-squares function is:
P2(x) = 0.8449x 0.8445x 1.0140 2
2.7 Legendre Polynomial
The previous section considered the problem of least-squares
approximation to fit a collection of discrete data. The other approximation
problem concerns the approximation of functions or fitting continuous data.
In this section we shall learn to fit data in continuous form using sequences
of Legendre polynomials.
Solution
Chapter 1 MATHEMATICAL MODELING MAT 530
Definition
The functions listed below are called Legendre polynomials and are
defined for
-1 x 1:
P0(x) = 1
P1(x) = x
1- 3x2
1)x(P 2
2
3x - 5x2
1)x(P 3
3
3+30x - 35x8
1)x(P 24
4
15x+70x - 63x8
1)x(P 35
5
.
.
.
1- mm1m P 1+m
mP
1+m
1+2m)x(P
-
This set of Legendre polynomials is said to be orthogonal on [-1,1] with
respect to the weight function w(x) 1 .The criteria required is that these
functions are designed to satisfy the following orthogonality condition:
1
1
nm mnif1n2
2mnif0
dx)x(P)x(P
(details of orthogonality and weight function shall not be discussed here)
In general, the Legendre polynomials can also be derived by the formula:
n2
n
n
nn 1) - x(dx
d
!n2
1)x(P
Suppose y(x) is a function continuous on [-1,1]. Here, the approach to
finding the least-squares approximating polynomial f(x) to fit the function
y(x) (or, the continuous data) is done in a similar manner. Let f(x) be of
polynomial of degree m defined using sequences of Legendre polynomials
such that
f(x) = a0P0(x) + a1P1(x) + a2P2(x) + . . . + amPm (x) .
Chapter 1 MATHEMATICAL MODELING MAT 530
We seek to minimize the sum of error squares; i.e.
∫1
1-
2dxy(x)- f(x) L
∫1
1-
2mm1100 dxy(x)- (x)Pa + (x)Pa +(x)Pa
(1)
where, Pi (x) is a Legendre polynomial and a i is a constant coefficient.
From the calculus of functions of several variables, a necessary condition
for the values a0, a , …, am to minimize L is that
0,=a
L
k∂
∂for each k = , , …, m.
Hence, using (1),
0dx)x(y- )x(Pa )x(Pa )x(Pa a
L1
1-
2mm1100
k
∫
With the orthogonality property of Legendre polynomials, this term can be
simplified to:
0dx)x(P)x(y- )x(P)x(Pa
1
1-
2kkkk ∫
∫∫1
1-
k
1
1-
2kk dx)x(P)x(yx)x(P.a
∫1
1-
kk dx)x(P)x(ya1k2
2
Thus,
∫1
1-
kk dx)x(P)x(y2
1k2a
Notice that ak calculates the coefficients a0, a1, …, am with the condition that
x is defined for y(x) on the interval [-1,1].
Chapter 1 MATHEMATICAL MODELING MAT 530
Theorem
Suppose y(x) is continuous and defined on [-1,1], then y(x) can be
approximated by a least-squares polynomial f(x) of degree m, using
series of Legendre polynomials such that:
f(x) = a0P0(x) + a1P1(x) + a2P2(x) + . . . + amPm(x)
where, the coefficients a0, a1, …, am is be determined by
1
1
kk dx)x(P)x(y2
1k2a for k = , , , ,……….,m
Example 8:
Find the least squares polynomial approximation of degree one to y = x 2 +
4x + 4 on the interval [-1,1].
Identify the observed function :
y(x) = x 2 + 4x + 4
Determine the Legendre polynomials of degree 1:
f(x) = a0P0(x) + a1P1(x)
= a0 + a1x
where P0(x) = 1 and P1(x) = x
Determined the coefficients a0 and a1:
1
1
kk dx)x(P)x(y2
1k2a for k = 0,1
Solution
Steps : Fitting continuous data (or function) using Legendre polynomials if independent variable is defined on [-1,1] Identify the observed function: y(x) Determine the Legendre polynomials of degree m:
f(x)= a0P0(x) + a1P1(x) + a2P2(x) + . . . + amPm(x)
Determine the coefficients a0, a , …, am :
1
1
kk dx)x(P)x(y2
1k2a for k = , , , ,…m
Chapter 1 MATHEMATICAL MODELING MAT 530
3
13
dx)1)(4x4x(2
1
dx)x(P)4x4x(2
1)0(2a:0k
1
1
2
1
1
02
0
4
dxx)4x4x(2
3
dx)x(P)4x4x(2
1)1(2a:1k
1
1
2
1
1
12
1
Thus, the polynomial of degree one to fit y(x) is:
f(x) = a0P0(x) + a1P1(x)
= 3
13 + 4x
Other method which can be applied is
1
1
1
1
1
01
1
21
1
1
1
1
1
dx)x(xy
dx)x(y
a
a
dxxxdx
xdxdx
In the case where x (or the independent variable) is defined on [m,n], a
suitable linear transformation is required so that the interval range of the
independent variable is normalized to be on [-1,1].
Steps : Fitting continuous data (or function) using Legendre polynomials if independent variable is not defined on [-1,1] Transform x to t linearly: ]1,1[t]n,m[x
Let batx
Solve for a and b Rewrite y(x) in term of y(t) Determine the Legendre polynomials of degree m
f(t) = a0P0(t) + a1P1(t) + a2P2(t) + . . . + amPm(t)
Determine the coefficients a0, a , …, am :
1
1
kk dt)t(P)t(y2
1k2a for k = 0,1,2,3,..,m
The polynomial of degree m to fit y(t) is: f(t) = a0P0(t) + a1P1(t) + a2P2(x) + . . . + amPm(t)
Rewrite f(t) in term of f(x).
Chapter 1 MATHEMATICAL MODELING MAT 530
Example 9:
Find the least squares polynomial approximation of degree one to y = x 2
+ 4x + 4 on the interval [0,2]
Since ]2,0[x and ]1,1[x then transformation is required.
Transform x to t linearly:
]1,1[t]2,0[x
Let . x = at + b
Solve for a and b
When x = 0: t = - 1 : 0 = a(- + b …
When x = 2: t = : = a + b …
Solving (1) and (2):
a=1 and b=1
Thus
x = t + 1 or t = x – 1
Rewrite y(x) in term of y(t):
y(x) = x 2 + 4x + 4, but x = t+1
y(t) = (t+1) 2 + 4(t+1) + 4
= t 2 + 6t +9
Determine Legendre polynomials of degree 1:
f(t) = a0P0(t) + a1P1(t)
= a0 + a1t
where P0(t)=1 and P1(t)=t
Determine the coefficients a0 and a1:
1
1
kk dt)t(P)t(y2
1k2a for k = 0,1
3
28dt)1)(9t6t(
2
1
dt)t(P)9t6t(2
1)0(2a:0k
1
1
2
1
10
20
Solution
Chapter 1 MATHEMATICAL MODELING MAT 530
6dt)t)(9t6t(2
3
dt)t(P)t(y2
1)1(2a:1k
1
1
2
1
111
The polynomial of degree 1 to fit y(t) is:
f(t) = a0P0(t) + a1P1(t)
= 3
28 + 6t
Rewrite f(t) in term of f(x):
f(x) = 3
28 + 6(x-1) since t = x – 1
= 3
10 + 6x
Example 10:
Find the least squares polynomial approximation of degree two to f(x) =
ex on the interval
a) [-1, 1] b) [ 0, 1]
No transformation required since ]1,1[x .
Given
y(x) = ex
and
f(x) = a0P0(x) + a1P1(x) + a2P2(x) polynomial of degree two
)1x3(2
1axaa
)x(Pa)x(Pa)x(Pa
221o
221100
since
)1x3(
2
1)x(Pandx)x(P,1)x(P 2
2110
Determined the coefficients a0 ,, a1 and a2:
1
1
kk dx)x(P)x(y2
1k2a for k = 0,1,2
Solution
Chapter 1 MATHEMATICAL MODELING MAT 530
1752.1
dxe2
1
dx)x(Pe2
1)0(2a:0k
1
1
x
1
1
0x
0
1036.1
dxxe2
3
dx)x(Pe2
1)1(2a:1k
1
1
x
1
1
1x
1
3578.0
dx)1x3(2
1e
2
5
dx)x(Pe2
1)2(2a:2k
1
1
2x
1
1
2x
1
Hence,
.x5367.0x1036.19963.0
)1x3(2
13578.0x1036.11752.1
)1x3(2
1axaa
)x(Pa)x(Pa)x(Pa)x(f
2
2
221o
221100
b)
Need a suitable linear transformation since ]1,1[x and ]1,0[x
Make a suitable linear transformation of x to t:
]1,1[t]1,0[x
i.e. x = at + b, solve for a and b
when x = 0, t = - 1 : 0 = a(-1) + b = - a + b …
when x = , t = : = a + b = a + b …
Solving (1) and (2):
2
1band
2
1a
Chapter 1 MATHEMATICAL MODELING MAT 530
Thus,
1x2t1tx2or2
1t
2
1t
2
1x
Write y(x) in the term of y(t):
2
1t
x
e)t(y
e)x(y
Write the least-squares polynomial in the new variable t in the form of
Legendre polynomials:
f(t) = polynomial of degree two (quadratic)
)1t3(2
1ataa
)t(Pa)t(Pa)t(Pa)t(f
221o
221100
Determine the coefficients a0 , a1 and a2:
1
1
kk dt)t(P)t(y2
1k2a for k = 0,1,2
7183.1
dte2
1
dt)t(Pe2
1)0(2a:0k
1
1
2
1t
1
10
2
1t
0
8452.0
dtte2
3
dt)t(Pe2
1)1(2a:1k
1
1
2
1t
1
11
2
1t
1
1398.0
0559.02
5
dt)1t3(2
1.e
2
5
dt)t(Pe2
1)2(2a:2k
1
1
22
1t
1
12
2
1t
2
Thus,
)t(Pa)t(Pa)t(Pa)t(f 221100
Chapter 1 MATHEMATICAL MODELING MAT 530
2
2
t8388.0t8452.06484.1
)1t3(2
1)1398.0(t8452.07183.1)t(f
Hence
2
2
x0968.13x7872.143097.1
)1x2(8388.0)1x2(8452.06484.1)x(f
Find the least squares polynomial approximation of degree two to:
a) f(x) = sin 2x on the interval [-1,1] and [0, 1]
b) f(x) = ln x on the interval [1,3]
Warm up exercise
Chapter 1 MATHEMATICAL MODELING MAT 530
Exercise 4
1. Fit a straight line to the given data.
a) X y b) x y c) x y -2 1 -6 -5.3 -3 -6
-1 2 -2 -3.5 1 -4
0 3 0 -1.7 3 -2
1 3 2 0.2 5 0
2 4 6 4.0 9 4
2. Fit a parabola to the given data.
a) X y b) x y c) x Y 2.0 5.1 -3 15 -2 2.80
2.3 7.5 -1 5 -1 2.10
2.6 10.6 1 1 0 3.25
2.9 14.4 3 5 1 6.00
3.2 19.0 2 11.50
3. The table below shows the time (in seconds) required for water to drain
through a hole in the bottom of a bottle as a function of depth (in meters) to which the bottle has been filled.
Depth 0.05 0.10 0.15 0.20 0.30 0.35 0.40 Time 65.99 120.28 166.69 207.85 279.95 313.04 344.24
a) Find the regression line to fit the data b) Estimate the time required for water to drain at depth of 2.5 meters. 4. Find the best-fit line for the given values.
x 5 6 10 14 16 20 22 28 28 36 38 y 30 22 28 14 22 16 8 8 14 0 4
5. Fit a least-squares quadratic function (parabola) to the given data.
x 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 p(x) 0.0 2.0 5.0 8.0 12.0 25.0 40.0 57.0
6. The following tabulated values are taken from an experiment. Complete
the table and calculate the sum of error squares.
x 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Observed y 0.0 2.0 5.0 8.0 12.0 25.0 40.0 57.0 Calculated y
Error
Chapter 1 MATHEMATICAL MODELING MAT 530
7. Find the values of A and B to fit a curve of the following form to the given
data
a) BAx
1y
b) B
x
Ay
8. The data given are for the solubility S of n-butane in anhydrous
hydrofluoric acid at high temperature T needed in the design of petroleum
refineries. Determine the constants and to fit the data to the model
TeS . Estimate the solubility when temperature is 95F.
9. One of the following data sets, (x,y1) and (x,y2) , follows an exponential
law bxAey and the other follows a power law bAxy . Which is which?
Fit the data to the respective model and calculate the sum square errors.
X 2.0 2.5 3.0 3.5 4.0 4.5 5.0
y1 14.79 27.75 47.09 74.07 109.99 156.10 213.69
y2 12.13 19.58 31.59 50.97 82.21 132.59 213.82
10. Find the least- squares polynomial approximation of degree one to the
function f(x) on the indicated interval.
a) f(x) = e2x ; [-1, 1] b) f(x) = sinx ; [-1, 1]
c) x
1)x(f ; [-3, -1] d) f(x) = cos 2x ; [0, 0.5] ,
e) f(x) = x3 – 1; [0,2] f) f(x) = ln x; [1, 2]
11. Find the least- squares polynomial approximation of degree two to the
function f(x) on the indicated interval.
a) f(x) = e-x ; [0, 1] b) f(x) = cos x; [-1, 1]
c) f(x) = x3 – x + 1; [0, 3] d) f(x) = sin2x; [0, ]
x -1 0.1 1 2 3
y 6.62 3.94 2.17 1.35 0.89
Temperature F 77 100 185 239 285
Solubility (weight %) 2.4 3.4 7.0 11.1 19.6
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