Algebra and abstract structures · Maths EE idea · Solid

When solving linear systems goes wrong

A research question to start from

How do rounding errors grow when Gaussian elimination solves an ill-conditioned linear system, and how does partial pivoting reduce them?

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Why it works as a maths EE

Exact algebra meets finite precision; you can measure error and explain it with condition numbers.

Mathematics you would need

Gaussian elimination
Matrix norms and condition numbers
Floating-point rounding
Hilbert matrices

Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.

One possible line of attack

Solve systems with known solutions using limited precision.
Measure error with and without pivoting.
Relate error to the condition number.
Discuss when the method is unreliable.

Scope and difficulty

Solid. Solid; keep the matrices small enough to analyse.

Pitfalls

Software output with no analysis.
Undefined norms.

Where to start reading

Search a library catalogue or a university's open lecture notes for: condition number Hilbert matrix; partial pivoting rounding error. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).

Make it your EE

Plan this EEOpens the free EE planner with this question as a starting point. Sharpen the questionSix quick checks for a focused research question. Self-check a draftCheck a section against each criterion, free.