Every method in this topic — Arnoldi, GMRES, conjugate gradients — was motivated by matrix-vector products being cheap: rather than . This capstone note makes that promise concrete: it shows how a sparse matrix is actually stored so that a matrix-vector product only touches the nonzero entries, surveys where sparsity arises naturally in quantitative finance, and closes both this topic and the unit by tying the whole numerical-linear-algebra toolkit (Stage 4) back to the practical problems that motivate it.
Learning objectives. By the end of this note you should be able to:
- explain why dense storage is wasteful for a sparse matrix;
- describe the compressed sparse row (CSR) storage format and its matrix-vector product cost;
- identify structured, naturally sparse problems arising in quantitative finance;
- summarise the full Stage 4 toolkit (Topics 8–10) as a coherent "solve or find eigenvalues, at any scale" pipeline.
In this lesson
- Why dense storage is wasteful
- Sparse storage formats
- Where sparsity arises naturally
- Closing the unit: the Stage 4 pipeline
Before you start
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