MATH2089 Chap.2 Linear Systems and Conditioning
Linear Systems and Conditioning
Linear systems combine a mathematical sensitivity question with an algorithmic solve. The condition number estimates how data perturbations can affect a solution, while LU and Cholesky factorisations organise the computation into triangular solves.
This chapter distinguishes a small residual from a small solution error, explains when Cholesky is eligible and shows why sparse or banded structure matters for memory, fill in and repeated right hand sides. Before computing, inspect the matrix. Its size, symmetry, definiteness, sparsity and bandwidth can change both the eligible method and the cost.
A general nonsingular system can be handled by pivoted LU, while a real symmetric positive-definite matrix admits a Cholesky factorisation. Symmetry alone is not enough: a failed positive pivot signals that the Cholesky assumptions should be reconsidered rather than forced. Factorisation separates an expensive matrix-dependent stage from cheaper right-hand-side solves.
With A = LU, solve Ly = b by forward substitution and then Ux = y by back substitution. If several vectors b share the same A, reuse the factors. For Cholesky, the two systems use a triangular factor and its transpose. Keep the intermediate vector visible so substitution order and arithmetic can be checked. Conditioning belongs to the problem, not to one implementation.
A large condition number warns that small relative perturbations in data can produce much larger relative changes in the solution. Scaling and norm choice should be stated when a bound is used. Stability belongs to the algorithm and describes how rounding or local errors propagate.
A stable solve can still return few trustworthy digits for an ill-conditioned system, whereas an unstable algorithm can spoil a well-conditioned problem. Compute a residual r = b minus Ax with the original matrix. A small scaled residual supports backward accuracy: the computed vector nearly solves a nearby system. It does not alone guarantee a small forward error.
Combine it with conditioning before deciding how many digits deserve to be reported. Sparse and banded matrices contain structural information that dense arithmetic ignores. Their local nonzero patterns can reduce storage and work, although factorisation may create fill-in. Node ordering in a discretised problem can therefore influence cost without changing the equations.
A complete solution states which matrix property licensed the method, shows the factorisation and solve order, reports a residual check, interprets sensitivity, and explains what is reused when further right-hand sides arrive.
What this chapter covers
- 01
Vector and matrix norms
- 02
Condition numbers and sensitivity
- 03
LU factorisation and pivoting
- 04
Forward and back substitution
- 05
Cholesky eligibility and reuse
- 06
Sparse and banded matrix structure
Worked example · free
Solve from LU factors
- +1Forward substitution gives y1 equal to 7.
- +1The second lower triangular equation gives y2 equal to 4.
- +1Back substitution gives x2 equal to 1.
- +1The first upper triangular equation gives x1 equal to 2; verify with the original system.
Key terms
- Condition number
- A norm based measure of potential relative error amplification in a problem.
- LU factorisation
- A decomposition into lower and upper triangular factors used for repeated solves.
- Cholesky factorisation
- A specialised triangular factorisation for real symmetric positive definite matrices.
Linear Systems and Conditioning FAQ
Does a tiny residual prove the solution is accurate?
Not by itself. An ill conditioned matrix can permit a small residual while the solution changes substantially. Interpret the residual together with the condition number and the scale of input uncertainty.
When should Cholesky be used?
Use Cholesky when the matrix is real, symmetric and positive definite. Those conditions justify the factorisation and its reduced work; symmetry alone is not enough.
Why reuse a factorisation?
The factorisation depends on the matrix, while triangular solves depend on the right hand side. Reusing the factors saves the dominant work when many load cases share one matrix.
Exam move
For each system, write matrix properties before choosing a solver. Practise one LU solve and one Cholesky solve by hand, then compute a residual and attach a sensitivity sentence. Add a small sparse grid example so the link between local coupling, matrix structure and later PDE systems is clear. Use a second right-hand side to rehearse factor reuse and keep the intermediate forward-substitution vector visible.
Create one example with a tiny residual but a large condition number, then explain why the residual alone cannot certify accurate solution digits. For method selection, say exactly which property licenses Cholesky and what pivoting changes in LU. Sketch a banded or sparse pattern before counting storage.
A complete timed answer should include eligibility, solve order, a scaled residual and a conditioning conclusion, because those four lines expose both algebraic and numerical understanding.
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