1 Online Orientation and Course Introductions 08/26/2023 - 08/28/2023

Overview

In this pseudo-module, we will discuss course organization, policies, and mechanics. We will review the CS 417 course website structure, and get set up for the semester to come.

We will take a brief look at the major themes and areas of emphasis that we will discuss during the coming semester. This will include a review of prerequisite mathematics (mainly, Calculus) and programming (namely, principles from CS 250). We will discuss the basics of Python 3 with particular emphasis on:

2 Chapter 1: Machine Arithmetic & Related Matters

2.1 Real Numbers & Machine Numbers & Rounding 08/29/2023 - 09/06/2023

Overview

In mathematics the impact finite precision is oft dismissed as an artifact of rounding. In most–if not all cases–the solution is to use more decimal places or signifigant digits until precision errors are within an “acceptable threshold” (a nebulous phrase). In this module, we will discuss how to measure and analyze finite precision errors arise from representing a single numerical quantity.

Objectives

After this module students will be able to:

1. Discuss the impact of finite precision on machine representation of floating-point values.
2. Apply mathematical analysis to examine (i.e., quantify) the magnitudes that can be represented with a finite number of mantissa and exponent bits.
3. Compare the error that arises in finite representation using the well known absolute error and relative error formulae.
4. Explain the difference between absolute and relative error, and discuss how absolute and relative error are mathematically linked.

Relevance

When developing software of any size machine precision is a consideration. The impact may be something seemingly inconsequential (e.g., a rounding error of less than one cent). Machine precision can result in interesting impacts (e.g., a solar system simulation may compute planets spiraling into the Sun).

2.2 Rounding & Finite Arithmetic 09/07/2023 - 09/18/2023

Overview

In this module, we will continue our discussion of finite precision and its impact on arithmetic operations. The propagation of error through arithmetic operations will be our primary focus.

Objectives

After this module students will be able to:

1. Discuss the impact of finite precision on error propagation in arithmetic operations involving two operands.
2. Examine the impact of finite precision on error propagation in arithmetic operations involving three or more operands and one operation (e.g., addition, subtraction, multiplication, and division).
3. Examine the impact of finite precision on error propagation in arithmetic operations involving three or more operands and combinations of two or more operations).

Relevance

When developing software of any size rounding is a consideration. The impact may be something seemingly inconsequential (e.g., a rounding error of less than one cent). Machine precision can result in interesting impacts (e.g., a solar system simulation may compute planets spiraling into the Sun).

2.3 Condition of a Problem 09/19/2023 - 10/01/2023

Overview

In this module, we will continue our discussion of finite precision and its impact on numerical computations. The propagation of error and the resulting behavior on computations will be our primary focus.

Objectives

After this module students will be able to:

1. Discuss the sensitivity of computations to errors in input resulting from finite precision.
2. Quantify (i.e., examine/model) through *Condition Number*s the sensitivity of computations to errors in input resulting from finite precision.
3. Explain (through mathematical analysis) when a problem is ill-conditioned or well-conditioned.
4. Leverage the results of Condition Number analysis to develop alternative methods of computation when a problem is ill-conditioned.

Relevance

Error propagation can result in interesting impacts (e.g., a solar system simulation may compute planets spiraling into the Sun). The ability to quantify the impact of a small change in input on a computation is foundational when building (or selecting) numerical methods (and algorithms)

2.4 Unit Summary 10/02/2023 - 10/07/2023

Overview

This module serves as a review of Chapter 1. We will revisit finite precision, arithmetic error, and conditioning of problems. This summary module also serves to force another review of requisite principles from Calculus.

Objectives

After this module students will be able to:

1. Summarize the techniques discussed in this unit.
2. Combine the techniques discussed in this unit to examine non-trivial problems.
3. Utilize various principles from Calculus (e.g., differentiation, integration, limits, sequences, and series) in combination without becoming overwhelmed.

3 Chapter 2 Approximation

3.1 Introduction to Approximation 10/11/2023 - 10/16/2023

Overview

Approximation has numerous applications throughout all disciplines. This can be something as simple as plotting data in Excel and computing a line of best fit or something more domain specific (e.g., transforming raw data for use in A.I. applications. In this Module we will explore Least Squares Approximation at an introductory level in preparation for a more rigorous and thorough derivation in later modules.

Objectives

After this module students will be able to:

1. Locate and review web-based resources covering Least Squares Approximation at an introductory level.
2. Explain how approximation methods can be employed for trend analysis for simple data (e.g., grade projection and price analysis).
3. Manually (i.e., by hand on paper) build a solution for discrete data.

Relevance

To apply approximation methods to discrete (and later continuous) data, one must analyze (possibly) unfamiliar mathematical notation. Application of approximation methods requires a solid foundation in linear algebra.

3.2 Approximation Proper 10/17/2023 - 10/30/2023

Overview

The previous module discussed the XTX|XTY method for computing polynomial approximation functions for discrete data. This Module will formalize the process and discuss how to generalize Least Squares approximation to both discrete and continuous data sets.

Objectives

After this module students will be able to:

1. Explain the differences between the Riemann Integral and Riemann-Stieltjes Integral.
2. Apply the weighted L2-Norm, properties of inner products, derivatives, and integrals to explain derivations of Least Squares Approximation.
3. Explain the impact of basis functions.
4. Manually (i.e., by hand on paper) build a solution for continuous functions/data.
5. Discuss the motivation for a single Least Squares notation to reconcile the discrete and continuous cases.

Relevance

3.3 Unit Summary 10/31/2023 - 11/11/2023

Overview

This module serves as a review of Least Squares Approximation. We will revisit both the XTX|XTY and Ac = b methods and discuss how the two methods are equivalent for discrete points.

Objectives

1. Explain how the [XTX|XTY] and [Ac|b] notations are related.
2. Construct a mathematical proof to show [XTX|XTY] is equivalent to [Ac|b] for discrete data.

4 Chapter 3: Interpolation 11/12/2023 - 12/01/2023

Overview

5 Selected Topics - (Adjustments and Updates In Progress)

5.1 Chapter 4: Non-Linear Solvers 12/02/2023 - 12/08/2023

Overview

There exist many problems that can not be solved analytically. Such problems include finding zeroes (i.e., solutions), computing derivatives, and evaluating integrals. In this module we will discuss for first problem (i.e., computing numerical solutions to equations).

Objectives

After this module students will be able to:

1. Discuss the requirements for each of the Bisection method, False Position (Regula Falsi) method, Secant method, and Newton’s method.
2. Manipulate provided pseudocode and refine it into a form suitable for implementation in a selected language (e.g., C, C++, Python 3, or Rust).
3. Identify the considerations inherent in converting pseudocode (theory) into usable code (application)–with particular emphasis on the impact of finite precision.

Relevance

There exist various numerical methods for solving problems, including finding zeroes of a function, performing numerical integration, and performing numerical differentiation. The ability to select the appropriate method, interpret pseudocode, and design an implementation (using best an appropriate language) is foundational when building numerical software.

6 Future Work

Overview

Computational Methods and Software covers a broad range of topics. In this course, we covered a few selected topics. This section contains resources and references for topics not covered this semester.

Example Code

7 Special Events and Dates