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Mathematics Final Year Topic: Normal Equations and QR Factorisation in a Nearly Rank-Deficient Least-Squares Problem

This Mathematics final year project compares two least-squares solution approaches as a documented design matrix approaches rank deficiency.

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Why choose this project topic?

A focused study of least-squares stability gives you a specific question in numerical approximation. It compares two least-squares solution approaches as a documented design matrix approaches rank deficiency. The bounded comparison creates room to explain how your evidence supports an interpretation and where the method has limits.

How does matrix conditioning affect residual and coefficient accuracy when normal equations or QR factorisation are used?

Agree the apparatus or dataset, comparison range and feasible measurement schedule for least-squares stability with your supervisor.

Proposed project objectives

  1. 01Define the materials, variables and comparison conditions for least-squares stability.
  2. 02Construct a parameterised design with a known reference solution.
  3. 03Evaluate the measurements or model outputs in relation to this question: How does matrix conditioning affect residual and coefficient accuracy when normal equations or QR factorisation are used?

A suggested research approach

Construct a parameterised design with a known reference solution. Use consistent arithmetic, track rank decisions and compare residuals, coefficient errors and condition measures without interpreting every coefficient change as a fit failure. Agree the available resources and record uncertainty, deviations from the protocol and any observations that challenge the initial interpretation.

What you will need

  • A parameterised least-squares matrix family
  • QR and normal-equation references
  • Reference solutions and numerical-rank diagnostics

Keep your project scope clear

A small least-squares residual does not guarantee stable coefficients or identify a unique solution near rank deficiency.

Mathematics project chapter outline

Use this outline as a starting point. You can edit the chapter titles to match your department’s format during setup.

  1. Chapter 1Introduction
  2. Chapter 2Literature Review
  3. Chapter 3Theory and Methodology
  4. Chapter 4Results and Applications
  5. Chapter 5Summary, Conclusion and Recommendations

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