Numerical Methods Contents - SAM

Code 2.4.8: Finding outeps in MATLAB
1 >> format hex ; eps , format long ; eps
2 ans = 3cb0000000000000
3 ans = 2.220446049250313e−16
2.5.1 Vector norms and matrix norms
Remark 2.4.9 (Addingeps to 1).
✸
Norms provide tools for measuring errors. Recall from linear algebra and calculus:
eps is the smallest positive
number ∈ M for which 1˜+ǫ ≠
1 (in M):
Code 2.4.10: 1 + ǫ in MATLAB
1 >> f p r i n t f ( ’ %30.25 f \ n ’ ,1+0.5∗eps )
2 1.0000000000000000000000000
3 >> f p r i n t f ( ’ %30.25 f \ n ’ ,1−0.5∗eps )
4 0.9999999999999998889776975
5 >> f p r i n t f ( ’ %30.25 f \ n ’ ,(1+2/ eps ) −2/eps ) ;
6 0.0000000000000000000000000
Definition 2.5.1 (Norm).
X = vector space over field K, K = C, R. A map ‖ · ‖ : X ↦→ R + 0 is a norm on X, if it satisfies
(i) ∀x ∈ X: x ≠ 0 ⇔ ‖x‖ > 0 (definite),
(ii) ‖λx‖ = |λ|‖x‖ ∀x ∈ X,λ ∈ K (homogeneous),
(iii) ‖x + y‖ ≤ ‖x‖ + ‖y‖ ∀x,y ∈ X (triangle inequality).
In fact 1˜+eps = 1 would comply with the “axiom” of roundoff error analysis, Ass. 2.4.2:
1 = (1 +eps)(1 + δ) ⇒ |δ| =
eps
∣1 +eps∣ ≤ eps ,
2
eps = (1 + 2
∣ ∣∣∣
)(1 + δ) ⇒
eps |δ| = eps
2 +eps∣ ≤ eps .
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Examples: (for vector space K n , vector x = (x 1 , x 2 ,...,x n ) T ∈ K n )
name : definition MATLAB function
√
Euclidean norm : ‖x‖ 2 := |x 1 | 2 + · · · + |x n
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| 2 norm(x)
1-norm : ‖x‖ 1 := |x 1 | + · · · + |x n | norm(x,1)
∞-norm, max norm : ‖x‖ ∞ := max{|x 1 |,...,|x n |} norm(x,inf)
Simple explicit norm equivalences:
for all x ∈ K n
!
Do we have to worry about these tiny roundoff errors ?
YES (→ Sect. 2.3):
• accumulation of roundoff errors
• amplification of roundoff errors
✄ back to Gaussian elimination/LU-factorization with pivoting
△
Definition 2.5.2 (Matrix norm).
‖x‖ 2 ≤ ‖x‖ 1 ≤ √ n ‖x‖ 2 , (2.5.1)
‖x‖ ∞ ≤ ‖x‖ 2 ≤ √ n ‖x‖ ∞ , (2.5.2)
‖x‖ ∞ ≤ ‖x‖ 1 ≤ n ‖x‖ ∞ . (2.5.3)
Given a vector norm ‖·‖ on R n , the associated matrix norm is defined by
M ∈ R m,n ‖Mx‖
: ‖M‖ := sup
x∈R n \{0} ‖x‖ .
2.5 Stability of Gaussian Elimination
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Issue: Gauge impact of roundoff errors on Gaussian elimination with partial pivoting !
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✎ notation: ‖x‖ 2 → ‖M‖ 2 , ‖x‖ 1 → ‖M‖ 1 , ‖x‖ ∞ → ‖M‖ ∞
sub-multiplicative: A ∈ K n,m ,B ∈ K m,k : ‖AB‖ ≤ ‖A‖ ‖B‖