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Quantitative Finance · Glossary

What is Condition number, backward error, backward stability?

Also known as: condition number · backward error · backward stability

Definition 25.5 Quantitative Methods · Chapter 25 — Floating Point and Numerical Linear Algebra

The condition number of a problem bounds the relative change of its answer per relative change of its data; for solving Ax=bAx = b it is κ(A)=∥A∥∥A−1∥\kappa(A) = \lVert A\rVert\lVert A^{-1}\rVert, the ratio of extreme singular values in the 2-norm. The backward error of a computed answer is the smallest perturbation of the data for which it is exact. An algorithm has backward stability if its backward error is always of the order of the unit roundoff.

Least squares on an exactly consistent system of 500 observations and five regressors whose design has condition number : relative error of the coefficients from the normal equations (which square ) and from a QR factorisation. Data: the chapter’s tutorial, seeded.
Figure 25.4. Least squares on an exactly consistent system of 500 observations and five regressors whose design has condition number κ\kappa: relative error of the coefficients from the normal equations (which square κ\kappa) and from a QR factorisation. Data: the chapter’s tutorial, seeded.
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