--- title: Condition Numbers description: Condition numbers measure how sensitive a mathematical function or system is to small changes in input. Learners will understand how to evaluate error propagation and assess the stability of numerical algorithms. category: mathematics subcategory: numerical-linear-algebra difficulty: beginner, intermediate, advanced url: /subject/condition-numbers --- # Condition Numbers Condition numbers measure how sensitive a mathematical function or system is to small changes in input. Learners will understand how to evaluate error propagation and assess the stability of numerical algorithms. ## Available Resources 1 Books • 3 Websites • 1 Papers ## Books ### 1. Applied Numerical Linear Algebra **Author:** James W. Demmel The standard graduate text that makes conditioning and backward stability the spine of the whole subject rather than a preliminary chapter, and the natural depth step past the Higham post. SIAM textbook treating perturbation theory, condition numbers and backward stability as the organizing frame for linear systems, least squares, eigenvalue and singular value problems. Chapters open with error analysis before algorithms, and include Matlab-based exercises and LAPACK references. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0898713897?tag=edmonddante07-20 **Tags:** numerical-linear-algebra, condition-numbers, backward-stability, perturbation-theory, eigenvalue-problems, lapack ## Websites ### 1. LAPACK Users' Guide: How to Measure Errors **Author:** Anderson, Bai, Bischof, Demmel, Dongarra et al. Where the theory becomes something a learner actually encounters in code: the guide defines the condition number, states the error bounds derived from it, and documents RCOND and the condition-estimation routines. Nothing else connects the definition to what a solver reports back to you. Section of the LAPACK Users' Guide defining how errors in scalars, vectors, matrices and subspaces are measured, giving the norm-based condition number and explaining RCOND, the reciprocal condition number LAPACK routines return to avoid overflow. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.netlib.org/lapack/lug/node75.html **Tags:** condition-numbers, error-bounds, lapack, numerical-linear-algebra, rcond ### 2. What Is a Condition Number? **Author:** Nicholas J. Higham Defines the condition number of a general function through its Jacobian, then specializes to the matrix case. Covers well- versus ill-conditioned problems, polynomial root conditioning, and why exact condition numbers are costly to compute. The best short definition anywhere: it gives both the abstract Jacobian-based definition and the concrete matrix case in one page, distinguishes absolute from relative conditioning, and notes that computing the condition number is as expensive as computing the function itself, which is exactly the motivation for condition estimation. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://nhigham.com/2020/03/19/what-is-a-condition-number/ **Tags:** condition-numbers, numerical-analysis, ill-conditioning, numerical-stability ### 3. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Papers ### 1. Numerical Computing with MATLAB, Chapter 2: Linear Equations **Author:** Cleve Moler The best free source that ties the condition number to a concrete algorithm the reader has just built and to code they can run. Chapter 2 of Numerical Computing with MATLAB, free from MathWorks. Builds Gaussian elimination, LU factorization and pivoting, then section 2.9 derives matrix norms and condition numbers and section 2.8 shows how roundoff errors propagate into the computed solution. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/lu.pdf **Tags:** gaussian-elimination, lu-factorization, condition-numbers, matrix-norms, roundoff-error, matlab --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/condition-numbers