What Is a Condition Number?
by Nicholas J. Higham · nhigham.com
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.
More resources on Condition Numbers
Wolfram MathWorld
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.
Applied Numerical Linear Algebra
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.
LAPACK Users' Guide: How to Measure Errors
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.
Numerical Computing with MATLAB, Chapter 2: Linear Equations
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.