# ME205: Numerical Methods for Engineers

Unit 5: Interpolation   Interpolation is the process of inferring the value of a discretely sampled function at a value(s) of the independent variable(s) that is within the domain of sampled values. Several schemes have been used for interpolation, but most rely upon polynomial fitting.*

In engineering practice, interpolation is useful for interpreting experiment or process observations, making the best use of data, and predicting outcomes for underexplored conditions.

*You may find the material in this unit repetitive; it is, but the repetition provides different historical and utilitarian perspectives on the process of interpolation.

This unit will take you approximately 14 hours to complete.

☐    Subunit 5.1: 2 hours

☐    Subunit 5.2: 3 hours

☐    Subunit 5.3: 3 hours

☐    Subunit 5.4: 3 hours

☐    Subunit 5.5: 3 hours

Unit5 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Define interpolation.
• Define and use direct interpolation to approximate data and find derivatives.
• Define and use Newton’s divided difference method of interpolation.
• Define and use Lagrange interpolation.
• Define and use spline interpolation.

5.1 Introduction to Interpolation   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter Definition of Interpolation” and “Textbook Chapter History of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter Definition of Interpolation” (PDF) and “Textbook Chapter History of Interpolation” (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 “Polynomial Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Polynomial Interpolation” (YouTube)

Instructions: Under the “Digital Audiovisual Lectures” heading, click on the YouTube links to watch both videos: “Uniqueness of Polynomial Interpolant: Part 1 of 2” and “Uniqueness of Polynomial Interpolant: Part 2 of 2” (run time: approximately 18 minutes).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge on Background of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge on Background of Interpolation” (PDF)

Instructions: Attempt all of the questions on the multiple choice exam.  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).

5.2 Direct Method of Interpolation   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of the Direct Method of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of the Direct Method of Interpolation” (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 “Direct Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Direct Interpolation” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” select the YouTube link after each title to launch the video.  Watch all four videos (about 35 minutes): “Linear Interpolation;” “Quadratic Interpolation;” “Cubic Interpolation: Part 1 of 2;” and “Cubic Interpolation: Part 2 of 2.”

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of the Direct Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of the Direct Method” (PDF)

Instructions: Please work through each question on the 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).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Industrial Engineering Example on Direct Method of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s “Industrial Engineering Example on Direct Method of Interpolation” (PDF)

Instructions: Follow the example, and try to reproduce the calculations.

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).

5.3 Newton’s Divided Difference Method   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of the Newton’s Divided Difference Method of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of the Newton’s Divided Difference Method of Interpolation” (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 “Newton’s Divided Difference Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Newton’s Divided Difference Interpolation” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” click on the YouTube link after each lecture title to launch the video.  Please view all 9 videos in their entirety (approximately 69 minutes).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of the Newton’s Divided Difference Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of the Newton’s Divided Difference Method” (PDF)

Instructions: Please complete the entire 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).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Industrial Engineering Example on Newton’s Divided Difference Method of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s “Civil Engineering Example on Newton’s Divided Difference Method of Interpolation” (PDF)

Instructions: Follow the example, and try to reproduce the calculations as you read.

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).

5.4 Lagrangian Interpolation   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Lagrange Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Lagrange Method” (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 “Lagrange Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Lagrange Interpolation” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” click on the YouTube link after each title to launch a video.  Please watch all 6 videos in their entirety (about 47 minutes).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge on Background of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of the Lagrange Method” (PDF)

Instructions: Complete the 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).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example on Lagrange Method of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example on Lagrange Method of Interpolation” (PDF)

Instructions: Follow the example, and try to reproduce the calculations.

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).

5.5 Spline Interpolation   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Spline Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Spline Method” (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 “Spline Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Spline Interpolation” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” click on the YouTube link after each title to access the videos.  Please watch all 6 videos in their entirety (about 47 minutes).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge on Background of Interpolation” Link: University of South Florida: Holistic Numerical Methods Institute’s Test Your Knowledge of the Spline Method” (PDF)

Instructions: Please complete the entire 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).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example on Spline Method of Interpolation” Link: University of South Florida:  Holistic Numerical Methods Institute’s “Mechanical Engineering Example on Spline Method of Interpolation” (PDF)

Instructions: Follow the example, and try to reproduce the calculations as you read.

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).

5.6 An Exercise in Interpolation   - Assessment: The Saylor Foundation’s “ME205: Unit 5.6 Exercise” Link: The Saylor Foundation’s “ME205: Unit 5.6 Exercise” (PDF)

Instructions: Please perform this exercise.  When you are done, check your work against The Saylor Foundation’s “ME304: Unit 5.6 Exercise Solution Guide" (PDF).  This exercise should require less than 2 hours to complete.