# ME205: Numerical Methods for Engineers

Unit 1: Introduction to Machine-Based Numerical Computations   This unit deals with the way numbers are represented by machine and the implications of that representation for numerical algorithms. As a simple example, consider the precision with which the difference between two numbers represented as 0.10437 and 0.10416 can be determined. If the numerical algorithm is sensitive to such limited precision, then it may not take many additional operations for the limited precision to cause problems.*

*This unit covers quantifying errors or precision, sources of those errors or limited precision, representation of numbers by the binary system on machines, the representation of floating point or decimal numbers, and the propagation of errors through an algorithm. Each subunit is accompanied by a short quiz to help you review.

Time Advisory: This unit will take you approximately 17 hours to complete.

☐    Subunit 1.1: 2 hours

☐    Subunit 1.2: 3 hours

☐    Subunit 1.3: 4 hours

☐    Web Media: 1 hour

☐    Assessment: 1 hour

☐    Subunit 1.4: 4 hours

☐    Web Media: 1 hour

☐    Assessment: 1 hour

☐    Subunit 1.5: 4 hours

☐    Web Media: 1 hour

☐    Assessment: 1 hour

Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Quantify absolute and relative errors.
• Distinguish between round-off and truncation errors.
• Interconvert binary and base-10 number representations.
• Define and use floating-point representations.
• Quantify how errors propagate through arithmetic operations.

1.1 Preliminaries   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Introduction to Numerical Methods” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Introduction to Numerical Methods” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

1.2 Numerical Errors   1.2.1 Measuring Errors   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Measuring Errors”

``````Link: University of South Florida: Holistic Numerical Methods
Institute’s “[Textbook Chapter of Measuring
(PDF)

is attributed to the University of South Florida and the original
version can be found [here](http://numericalmethods.eng.usf.edu/)
(HTML).
``````

1.2.2 Sources of Error   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Sources of Error” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Sources of Error”(PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Sources of Error” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Sources of Error” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” click on the links to YouTube for all 6 lectures: “Round-Off Error: Definition and Examples,” “Effect of Carrying Significant Digits,” “Truncation Error: Definition,” “Truncation Error: Example: Series,” “Truncation Error: Example: Integration,” and “Truncation Error: Example: Differentiation” (run time: approximately 42 minutes).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Introduction to Sources of Error” Link: University of South Florida:  Holistic Numerical Methods Institute’s “Test Your Knowledge of Introduction of Sources of Error” (PDF)

Instructions: Please attempt all the questions on this multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

1.3 Binary System   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Binary Representation of Numbers” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Binary Representation of Numbers” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Binary Representation of Numbers” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Binary Representation of Numbers” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” select the YouTube links for the following lectures: “Introduction to Binary Representation,” “Base-10 to Base-2 Conversion Method,” and “Base-10 to Base-2 Conversion another Method” (run time: approximately 25 minutes).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Introduction to Binary Representation of Numbers” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Introduction of Binary Representation of Numbers” (PDF)

Instructions: Please attempt all of the questions on this multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

1.4 Floating Point Numbers   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Floating Point Representation of Numbers” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Floating Point Representation of Numbers” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Floating Point Representation of Numbers” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Floating Point Representation of Numbers” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” please select the links to YouTube and view all 8 video lectures.  The total run time is approximately 57 minutes.

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Introduction to Floating Point Representation of Numbers” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Introduction of Floating Point Representation of Numbers” (PDF)

Instructions: Please attempt all of the questions on this multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

1.5 Error Propagation   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Propagation of Errors” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Propagation of Errors” (PDF)