NUMERICAL-ANALYSIS Course

Condition number

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The condition number (Conditionnement) is a measure of the dependence between the vector of parameters $X$ and the solution $b$.

\[ cond(A) = |||A||| * |||A^{-1}||| \]

Properties

  • if $A \in Gl_n(\mathbb{R})$ then $cond(A) \ge 1$
  • if $A \in Gl_n(\mathbb{R})$ then $cond(\lambda{A}) = cond(A)$
  • if $A, B \in Gl_n(\mathbb{R})$ then $cond(AB) \le cond(A) * cond(B)$

If you forgot, $A \in Gl_n(\mathbb{R})$ means that a matrix of reals $\mathbb{R}$ is invertible.


Notes

\[ \begin{split} cond_2(A) = \sqrt{\frac{\sigma_n}{\sigma_1}} \\ or \ if \ A \ is \ positive \ definite \\ cond_2(A) = \sqrt{\frac{\lambda_n}{\lambda_1}} \\\end{split} \]

Note that

  • $\sigma$: the smallest eigenvalue
  • $\lambda$: the biggest eigenvalue

\[ cond_p(A) = |||A||_p |||A^{-1}||_p \]