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Two 2x2 systems can have dramatically different condition numbers

Example

The diagonal system $A_{1} = diag (1, 4)$ is well conditioned, while the nearly singular system

$A_{2} = (\begin{array}{cc} 1 & 1 \\ 1 & 1 + ε \end{array}), ε = \frac{1}{100},$

is already severely ill conditioned, even though both matrices have the same $2 \times 2$ shape and entries of the same order.

Facts & Assumptions

Given: The matrices $A_{1} = diag (1, 4)$ and $A_{2} = (\begin{array}{cc} 1 & 1 \\ 1 & 1 + ε \end{array})$ with $ε = 1 / 100 > 0$ .

[L1]

$κ_{p} (A) = ∥ A ∥_{p} ∥ A^{- 1} ∥_{p}$ for $p \in {1, \infty}$ (The condition number kappa_p(A) = ||A||_p ||A^{-1}||_p of a nonsingular linear system).

[L2]

The induced $1$ -norm is the maximum column sum and the induced $\infty$ -norm the maximum row sum (The induced 1-norm is the maximum column sum and the induced infinity-norm is the maximum row sum).

[L3]

For a nonsingular $2 \times 2$ matrix $M = (\begin{array}{cc} a & b \\ c & d \end{array})$ , $M^{- 1} = (a d - b c)^{- 1} (\begin{array}{cc} d & - b \\ - c & a \end{array})$ .

Verification

technique · direct

1.1L2algebra

For $A_{1}$ , the column sums and row sums are ${1, 4}$ , so [L2] gives $∥ A_{1} ∥_{1} = ∥ A_{1} ∥_{\infty} = 4$ ; its inverse is $diag (1, 1 / 4)$ , with column and row sums ${1, 1 / 4}$ , so $∥ A_{1}^{- 1} ∥_{1} = ∥ A_{1}^{- 1} ∥_{\infty} = 1$ .

1.2L2algebra

For $A_{2}$ , the column sums are ${2, 2 + ε}$ and the row sums ${2, 2 + ε}$ , so [L2] gives $∥ A_{2} ∥_{1} = ∥ A_{2} ∥_{\infty} = 2 + ε$ .

1.3L2L3algebra

By [L3], $A_{2}^{- 1} = ε^{- 1} (\begin{array}{cc} 1 + ε & - 1 \\ - 1 & 1 \end{array})$ , whose column sums and row sums are both ${(2 + ε) / ε, 2 / ε}$ , the first being larger; so [L2] gives $∥ A_{2}^{- 1} ∥_{1} = ∥ A_{2}^{- 1} ∥_{\infty} = (2 + ε) / ε$ .

2.1L1step 1.1algebra

By [L1] and step 1.1, $κ_{1} (A_{1}) = κ_{\infty} (A_{1}) = 4 \cdot 1 = 4$ : for every right-hand side $b \neq 0$ a relative perturbation of $b$ changes the solution by at most four times as much.

2.2L1step 1.2step 1.3algebra

By [L1], $κ_{1} (A_{2}) = κ_{\infty} (A_{2}) = (2 + ε)^{2} / ε$ , and at $ε = 1 / 100$ this is $(201 / 100)^{2} \cdot 100 = 201^{2} / 100 = 40401 / 100$ .

3.1step 2.1step 2.2∎

The two systems of the same shape therefore have condition numbers $4$ and $40401 / 100 \approx 404$ : the nearly singular one amplifies relative data perturbations about a hundred times more, and the contrast comes from the small determinant $ε$ , not from large entries.

Depends on

Used by

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Dependency tree · two levels

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Sources

L. N. Trefethen and D. Bau III, Numerical Linear Algebra, Lecture 12 (standard reference, not scraped)