Introduction to Numerical Methods

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Learn approximate methods of solving mathematical models!

853 students enrolled

What Will I Learn?

- Goals: By end of this course, you will be able to use approximate methods for the most used mathematical procedures.
- Objectives: In this course, you will learn the numerical methods for the following mathematical procedures and topics - Differentiation, Nonlinear Equations, Simultaneous Linear Equations, Interpolation, Regression, Integration, and Ordinary Differential Equations. Calculation of errors and their relationship to the accuracy of the numerical solutions is emphasized throughout the course.

Requirements

- Differential Calculus, Integral Calculus, Ordinary Differential Equations, High School Algebra

Description

**About**: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral). Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns).

**Materials Included**: Textbook Chapters, Video Lectures, Quizzes, Solutions to Quizzes

**How Long to Complete**: About 40 hours of lectures need to be watched and estimated time to read textbook and do quizzes is 80 hours. It is a typical 15-week semester length course.

**Course Structure**: For each section, you have video lectures, followed by a textbook chapter, a quiz and solutions to quizzes.

Who is the target audience?

- Juniors in STEM (Science, Technology, Engineering and Mathematics) fields

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Curriculum For This Course

274 Lectures

10:59:00
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CHAPTER 01.01: Introduction to Numerical Methods
7 Lectures
00:00

Learn the four steps of solving an engineering problem.

This video teaches you the steps of solving an engineering problem- define the problem, model the problem, solve, and implementation of the solution.**For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at **http://youtu.be/-PE02R1c68s

Video Lecture : Steps of Solving an Engineering Problem: Part 1 of 2

ImportContent

Learn the four steps of solving an
engineering problem.

This video teaches you the steps of solving an engineering problem- define the problem, model the problem, solve, and implementation of the solution.

**For a better experience, see the same video with close
captioning, helpful annotations, and possible corrections in any
browser at **http://youtu.be/L0kqSdgzxSI

Video Lecture : Steps of Solving an Engineering Problem: Part 2 of 2

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Learn the different mathematical procedures for which numerical methods are used.

This video teaches you the mathematical procedures for which numerical methods are used. An example is given for each category.

**For a better experience, see the same video with close
captioning, helpful annotations, and possible corrections in any
browser at **http://youtu.be/1rEDN542Fho

Enumerating Use of Numerical Methods for Mathematical Procedures: Part 1 of 2

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Learn the different mathematical procedures for which numerical methods are used.

This video teaches you the mathematical procedures for which numerical methods are used. An example is given for each category.

Enumerating the Use of Numerical Methods for Mathematical Procedures:Part 2 of 2

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After reading this chapter, you should be able to:

1. understand the need for numerical methods, and

2. go through the stages (mathematical modeling, solving and implementation) of solving a particular physical problem.

Textbook Chapter 01.01 : Introduction to Numerical Methods

6 pages

Multiple choice test for Chapter 01.01 : Introduction to Numerical Methods

Quiz Chapter 01.01 : Introduction to Numerical Methods

2 pages

Solution to multiple choice test for Chapter 01.01 : Introduction to Numerical Methods

Quiz Solution Chapter 01.01 : Introduction to Numerical Methods

6 pages

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Chapter 01.02 : Measuring Errors
7 Lectures
00:00

Learn the definition of true error.
The explanation is then followed by an example.

Video Lecture: True Error : Definition & Example

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Learn the concept of relative true error to judge how much error you have in a numerical method

Video Lecture: Relative True Error : Definition & Example

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Learn how to calculate approximate error as this is the way you measure the error in a numerical method

Video Lecture: Approximate Errors : Definition & Example

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Learn the concept of relative approximate error to gauge the relative error in a numerical method.

Video Lecture: Relative Approximate Error : Definition & Example

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After reading this chapter, you should be able to:

1. find the true and relative true error,

2. find the approximate and relative approximate error,

3. relate the absolute relative approximate error to the number of significant digits at least correct in your answers, and

4. know the concept of significant digits.

Textbook Chapter 01.02 : Measuring Errors

7 pages

Multiple choice test for the chapter 01.02 : Measuring Errors

Quiz Chapter 01.02: Measuring Errors

2 pages

Solution to the multiple choice test for the chapter 01.02 : Measuring Errors

Quiz Solution Chapter 01.02 : Measuring Errors

7 pages

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Chapter 01.03 : Sources of Error
9 Lectures
00:00

Learn about one of the two sources of errors in numerical
methods: round-off error. Round off error is caused by
approximating the representation of numbers.

**For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at **http://youtu.be/-jrOrEc1k-k

Video Lecture : Round of Error : Definition & Example

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Learn the effect of carrying
limited number of significant digits for a real-life problem. We
regressed the data of coefficient of thermal expansion of steel as a
function of temperature to a second order polynomial using Excel.
We found that the general format of the regression model provided by
Excel creates large round-off errors because of carrying limited
significant digits. A solution to this issue is also presented.

**For a better experience, see the same video with close
captioning, helpful annotations, and possible corrections in any
browser at **http://youtu.be/_JVOce7bP1w

Video Lecture : Effect of Carrying Significant Digits

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Learn the definition of truncation error. Truncation error is caused by approximating mathematical procedures. Examples include using a finite step size for finding the derivative of a function, using a finite number of terms of an infinite Maclaurin series, using Riemann approximations with finite rectangles to find the value of a definite integral.

Video Lecture : Truncation Error : Definition

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Learn the concept of truncation error
through the example of an infinite Maclaurin series for exp(x).

Video Lecture : Truncation Error : Example : Series

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Learn the example showing the truncation error
through the mathematical procedure of integration.

Video Lecture : Truncation Error : Example : Integration

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Learn the concept of truncation error
through an example of differentiation.

Video Lecture : Truncation Error : Example : Differentiation

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After reading this chapter, you should be able to:

1. know that there are two inherent sources of error in numerical methods – round-off and truncation error,

2. recognize the sources of round-off and truncation error, and

3. know the difference between round-off and truncation error.

Textbook Chapter 01.03 : Sources of Error

6 pages

Multiple choice test for the chapter 01.03 : Sources of Error

Quiz Chapter 01.03 : Sources of Error

2 pages

Solution to the multiple choice test for the chapter 01.03 : Sources of Error

Quiz Solution Chapter 01.03 : Sources of Error

6 pages

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Chapter 01.04 : Binary Representation of Numbers
6 Lectures
00:00

Learn about binary representation of
numbers.

Video Lecture : Introduction to Binary Representation

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Learn a method to convert base-10 number
to base-2 number.

Video Lecture : Base-10 to Base-2 Conversion Method

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Learn another method to convert base-10 number
to base-2 number.

Video Lecture : Base-10 to Base-2 Conversion Another Method

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After reading this chapter, you should be able to:

1. convert a base-10 real number to its binary representation,

2. convert a binary number to an equivalent base-10 number.

Textbook Chapter 01.04 : Binary Representation of Numbers

7 pages

Multiple choice test for the chapter 01.04 : Binary Representation of Numbers

Quiz Chapter 01.04 : Binary Representation

2 pages

Solution to the multiple choice test for the chapter 01.04 : Binary Representation of Numbers

Quiz Solution Chapter 01.04 : Binary Representation

6 pages

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Chapter 01.05 : Floating Point Representation
11 Lectures
00:00

Learn the background of the concept of
floating point representation. We use the decimal system to start the
discussion.

Video Lecture : Background : Part 1 of 3

ImportContent

Learn the background of the concept of
floating point representation. We use the decimal system to start the
discussion.

Video Lecture : Background : Part 2 of 3

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Learn the background of the concept of
floating point representation. We use the decimal system to start the
discussion.

Video Lecture : Background : Part 3 of 3

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Learn via an example how a base-10 number is represented as floating point number in base-2.

Video Lecture : Example: Part 1 of 2

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Learn via an example how a base-10 number
is represented as floating point number in base-2.

Video Lecture : Example 2 of 2

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Learn via an example how a base-10 number
is represented as floating point number in base-2 with a biased exponent.

Video Lecture : Biased Exponent : Example

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Learn how the IEEE-754 standard represents
a floating point number in single precision.

Video Lecture : IEEE-754 Single Precision Representation : Part 1 of 2

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Learn how the IEEE-754 standard represents
a floating point number in single precision.

Video Lecture : IEEE-754 Single Precision Representation : Part 2 of 2

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After reading this chapter, you should be able to:

1. convert a base-10 number to a binary floating point representation,

2. convert a binary floating point number to its equivalent base-10 number,

3. understand the IEEE-754 specifications of a floating point representation in a typical computer,

4. calculate the machine epsilon of a representation.

Textbook Chapter 01.05 : Floating Point Representation

8 pages

Multiple choice test for the chapter 01.05 : Floating Point Representation

Quiz Chapter 01.05 : Floating Point Representation

2 pages

Solution to the multiple choice test for the chapter 01.05 : Floating Point Representation

Quiz Solution : Chapter 01.05 : Floating Point Representation

6 pages

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Chapter 01.07 : Taylor Series Revisited
7 Lectures
00:00

Learn what a Taylor series is all about.
We are just revisiting.

Video Lecture : Introduction

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Learn the application of Taylor series
through an example.

Video Lecture : Example

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Learn how the Taylor series is used to
derive an expression for exp(x) as a series.

Video Lecture : Example to Derive Series for exp(x)

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Learn how the Maclaurin series for sin(x) is
derived by using Taylor's series.

Video Lecture : Maclaurin Series of sin(x)

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After reading this chapter, you should be able to

- understand the basics of Taylor’s theorem,
- write transcendental and trigonometric functions as Taylor’s polynomial,
- use Taylor’s theorem to find the values of a function at any point, given the values of the function and all its derivatives at a particular point,
- calculate errors and error bounds of approximating a function by Taylor series, and
- revisit the chapter whenever Taylor’s theorem is used to derive or explain numerical methods for various mathematical procedures.

Textbook Chapter 01.07 : Taylor Theorem Revisited

8 pages

Multiple choice test for the chapter 01.07 : Taylor Series Revisited

Quiz Chapter 01.07 : Taylors Series Revisited

2 pages

Solution to the multiple choice test for the chapter 01.07 : Taylor Series Revisited

Quiz Solution Chapter 01.07 : Taylors Series Revisited

7 pages

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Chapter 02.01 : Primer on Differential Calculus
3 Lectures
00:00

After reading this chapter, you should be able to:

1. understand the basics of differentiation,

2. relate the slopes of the secant line and tangent line to the derivative of a function,

3. find derivatives of polynomial, trigonometric and transcendental functions,

4. use rules of differentiation to differentiate functions,

5. find maxima and minima of a function, and

6. apply concepts of differentiation to real world problems.

Textbook Chapter 02.01 : Primer on Differentiation

24 pages

Multiple choice test for the chapter 02.01 : A Primer on Differentiation

Quiz Chapter 02.01 : A Primer on Differentiation

2 pages

Solution to the multiple choice test for the chapter 02.01 : A Primer on Differentiation

Quiz Solution Chapter 02.01 : Primer on Differentiation

6 pages

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Chapter 02.02 : Differentiation of Continuous Functions
12 Lectures
00:00

Learn the forward divided difference to
approximate the first derivative of a function.

Video Lecture : Forward Divided Difference : Part 1 of 2

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Learn the forward divided difference to
approximate the first derivative of a function.

Video Lecture : Forward Divided Difference : Part 2 of 2

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Learn the backward divided difference to
approximate the first derivative of a function.

Video Lecture : Backward Divided Difference : Part 1 of 2

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Learn the backward divided difference to approximate the first derivative of a function.

Video Lecture : Backward Divided Difference : Part 2 of 2

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Learn the central divided difference to approximate the first derivative of a function.

Video Lecture : Central Divided Difference

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Learn how you can used Taylor series to derive finite difference formulas for the second derivative of a function.

Video Lecture : Higher Order Derivative Divided Difference: Theory

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Learn how you can used Taylor series to derive finite difference formulas for the second derivative of a function.

Video Lecture : Higher Order Derivatives : Example

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Learn about the accuracy of finite difference formulas for finding the derivative of a function.

Video Lecture : Accuracy of Finite Difference Formulas : Part 1 of 2

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Learn about the accuracy of finite difference formulas for finding the derivative of a function.

Video Lecture : Accuracy of Finite Difference Formulas : Part 2 of 2

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After reading this chapter, you should be able to:

1. derive formulas for approximating the first derivative of a function,

2. derive formulas for approximating derivatives from Taylor series,

3. derive finite difference approximations for higher order derivatives, and

4. use the developed formulas in examples to find derivatives of a function.

Textbook Chapter 02.02 : Differentiation of Continuous Functions

13 pages

Multiple choice test for the chapter 02.02 : Differentiation of Continuous Functions

Quiz Chapter 02.02: Differentiation of Continuous Functions

2 pages

Solution to the multiple choice test for the chapter 02.02 : Differentiation of Continuous Functions

Quiz Solution Chapter 02.02 : Differentiation of Continuous Functions

8 pages

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Chapter 02.03 : Differentiation of Discrete Functions
7 Lectures
00:00

Learn how you can find the derivative of a
discrete function by using divide difference method such as forward
divided difference, backward divided difference and central divided
difference.

This video teaches you how you can find
the derivative of a function by using divided difference method.

Video Lecture: Divided Difference Approach

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Learn how you can find the derivative of a
discrete function by using the polynomial interpolation method.

This video teaches you how you can find the derivative of a function by using polynomial interpolation method.

This video teaches you how you can find the derivative of a function by using polynomial interpolation method.

Video Lecture: Polynomial Interpolation Method

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Learn how to use Newton's divided difference polynomial method to find the derivative of a discrete function given at discrete data points.

This video teaches you how you can find the derivative of a function given at discrete data points by using Newton's divided difference polynomial method.

Video Lecture: Newton Divided Difference Polynomial Method : Theory

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Learn via an example how to use Newton's divided
difference polynomial method to find the derivative of a
discrete function given at discrete data points.

This video teaches you how you can find the derivative of a function given at discrete data points by using Newton's divided difference polynomial method with an example.

Video Lecture: Newton Divided Difference Polynomial Method : Example

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After reading this chapter, you should be able to:

1. find approximate values of the first derivative of functions that are given at discrete data points, and

2. use Lagrange polynomial interpolation to find derivatives of discrete functions.

Textbook Chapter 02.03 : Differentiation of Discrete Functions

7 pages

Multiple choice test for the chapter 02.03 : Differentiation of Discrete Functions

Quiz Chapter 02.03 : Differentiation of Discrete Functions

2 pages

Solution to the multiple choice test for the chapter 02.03 : Differentiation of Discrete Functions

Quiz Solution Chapter 02.03 : Differentiation of Discrete Functions

8 pages

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Chapter 03.01 : Solving Quadratic Equations Exactly
6 Lectures
00:00

Learn the formula for solving quadratic
equation.

This video teaches you the formula for solving quadratic equation.

This video teaches you the formula for solving quadratic equation.

Video Lecture : The Quadratic Equation Formula

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Learn how to derive the formula for solving a quadratic equation.

This video teaches you the derivation of the formula for solving a quadratic equation.

Video Lecture : Derivation of the Quadratic Formula

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Learn how a quadratic equation is solved using a physical example.

This video teaches you the solution for solving a quadratic equation with an example.

Video Lecture : Example of Quadratic Equation

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After reading this chapter, you should be able to:

1. find the solutions of quadratic equations,

2. derive the formula for the solution of quadratic equations,

3. solve simple physical problems involving quadratic equations.

Textbook Chapter 03.01 : Solution of Quadratic Equations

3 pages

Multiple choice test for the chapter 03.01 : Background of Nonlinear Equations

Quiz Chapter 03.01 : Background Nonlinear Equations

2 pages

Solution to the multiple choice test for the chapter 03.01 : Background of Nonlinear Equations

Quiz Solution Chapter 03.01 : Nonlinear Equations

6 pages

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About the Instructor