Introduction to Numerical Methods
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# Introduction to Numerical Methods

Learn approximate methods of solving mathematical models!
4.5 (13 ratings)
853 students enrolled
Published 5/2013
English
Price: Free
Includes:
• 86 Supplemental Resources
• Access on mobile and TV
• Certificate of Completion
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.
View Curriculum
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
+
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
ImportContent

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
ImportContent

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/pugw_1EobbI

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

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
+
Chapter 01.02 : Measuring Errors
7 Lectures 00:00
Learn the definition of true error.  The explanation is then followed by an example.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/ej8WDza_pPA

Video Lecture: True Error : Definition & Example
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/33fS7V75C_Y

Video Lecture: Relative True Error : Definition & Example
ImportContent

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

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

Video Lecture: Approximate Errors : Definition & Example
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/e682T_4UjWY

Video Lecture: Relative Approximate Error : Definition & Example
ImportContent

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
+
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
ImportContent

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
ImportContent

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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/pfV9kqh4Mpg

Video Lecture : Truncation Error : Definition
ImportContent

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

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

Video Lecture : Truncation Error : Example : Series
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/wDyc1L1rP34

Video Lecture : Truncation Error : Example : Integration
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/hRvFyynk5-k

Video Lecture : Truncation Error : Example : Differentiation
ImportContent

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
+
Chapter 01.04 : Binary Representation of Numbers
6 Lectures 00:00
Learn about binary 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/X3pA8cU9CkY

Video Lecture : Introduction to Binary Representation
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/96MJVzVKoIE

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

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/hXhz80U8Fjo

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

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
+
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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/svFJXukm2uE

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.

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

Video Lecture : Background : Part 2 of 3
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/ZKYSPOZx9_4

Video Lecture : Background : Part 3 of 3
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/IGQZ49d-_lI

Video Lecture : Example: Part 1 of 2
ImportContent

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

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

Video Lecture : Example 2 of 2
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/vi5RXPBO-8E

Video Lecture : Biased Exponent : Example
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/atlaD7M30sY

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

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/b7u_oFlG4_M

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

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
+
Chapter 01.07 : Taylor Series Revisited
7 Lectures 00:00
Learn what a Taylor series is all about. We are just revisiting.

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

Video Lecture : Introduction
ImportContent

Learn the application of Taylor series through an example.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/UqiHOM7WwkI

Video Lecture : Example
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/T7RBItSYQHg

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

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/E852dYK2Ib8

Video Lecture : Maclaurin Series of sin(x)
ImportContent

After reading this chapter, you should be able to

1. understand the basics of Taylor’s theorem,
2. write transcendental and trigonometric functions as Taylor’s polynomial,
3. 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,
4. calculate errors and error bounds of approximating a function by Taylor series, and
5. 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
+
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
+
Chapter 02.02 : Differentiation of Continuous Functions
12 Lectures 00:00
Learn the forward divided difference to approximate the first derivative of a function.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/KNFjPIqJAJ4

Video Lecture : Forward Divided Difference : Part 1 of 2
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/1EV0fKPiQMk

Video Lecture : Forward Divided Difference : Part 2 of 2
ImportContent

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

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

Video Lecture : Backward Divided Difference : Part 1 of 2
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/6DWEVJrvRC4

Video Lecture : Backward Divided Difference : Part 2 of 2
ImportContent

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

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

Video Lecture : Central Divided Difference
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/6wWOv6pZm0k

Video Lecture : Higher Order Derivative Divided Difference: Theory
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/lRR7wi5sP2I

Video Lecture : Higher Order Derivatives : Example
ImportContent

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/AN3ILGZ0e18

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

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

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/5qXwg_gCrAk

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

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
+
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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/MJ7Yj2Mpcb4

Video Lecture: Divided Difference Approach
ImportContent

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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/fHJyJVFIaf8

Video Lecture: Polynomial Interpolation Method
ImportContent

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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/95jPedy1aMA

Video Lecture: Newton Divided Difference Polynomial Method : Theory
ImportContent

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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/E8fQ7de5z_s

Video Lecture: Newton Divided Difference Polynomial Method : Example
ImportContent

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
+
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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/b4L5-pce71U

Video Lecture : The Quadratic Equation Formula
ImportContent

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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/1dsjamqX8Sw

Video Lecture : Derivation of the Quadratic Formula
ImportContent

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.

For a better experience, see the same video with close captioning, helpful annotations, and possible corrections in any browser at http://youtu.be/XAU92V0qRP8

Video Lecture : Example of Quadratic Equation
ImportContent

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
23 More Sections
 4.5 Average rating 0 Reviews 42 Students 1 Course
A Global Teacher

Autar Kaw is a professor of mechanical engineering and Jerome Krivanek Distinguished Teacher at the University of South Florida. He is a recipient of the 2012 U.S. Professor of the Year Award from the Council for Advancement and Support of Education (CASE) and Carnegie Foundation for Advancement of Teaching.

Professor Kaw received his BE Honors degree in Mechanical Engineering from Birla Institute of Technology and Science (BITS) India in 1981, and his degrees of Ph.D. in 1987 and M.S. in 1984, both in Engineering Mechanics from Clemson University, SC. He joined University of South Florida in 1987.

Professor Kaw’s main scholarly interests are in engineering education research, open courseware development, bascule bridge design, fracture mechanics, composite materials, computational nanomechanics, and the state and future of higher education. His research has been funded by National Science Foundation, Air Force Office of Scientific Research, Florida Department of Transportation, Research and Development Laboratories, Wright Patterson Air Force Base, and Montgomery Tank Lines.

Professor Kaw has written several books on subjects such as composite materials, numerical methods, computer programming, matrix algebra, and engineering licensure examination.

Since 2002, under Professor Kaw's leadership, he and his colleagues from around the nation have developed, implemented, refined and assessed online resources for open courseware in Numerical Methods. This courseware annually receives more than a million page views, 900,000 views of the YouTube lectures and 150,000 annual visitors to the "numerical methods guy" blog.

Professor Kaw's work has appeared in the St. Petersburg Times, Tampa Tribune, Chance, Oracle, and his work has been covered/cited in Chronicle of Higher Education, Inside Higher Education, Congressional Record, ASEE Prism, Tampa Bay Times, Tampa Tribune, Campus Technology, Florida Trend Magazine, WUSF, Bay News 9, Times of India, NSF Discoveries, Voice of America, and Indian Express.

Professor Kaw is a Fellow of the American Society of Mechanical Engineers (ASME) and a member of the American Society of Engineering Education (ASEE). He has also been a Maintenance Engineer (1982) for Ford-Escorts Tractors, India, and a Summer Faculty Fellow (1992) and Visiting Scientist (1991) at Wright Patterson Air Force Base.