Numerical Methods Contents - SAM

and hence
In conlcusion,
∥
∥I − J −1
k J −J −1
k+1∥ = k F(x(k+1) )∆x (k)
∥ 2 ∥
∥ ∥∆x (k) ∥∥ 2
=
(I − J −1 (∆x (k) ) T
k J)∆x(k) ∥
∥
∥ ∥ ∥∆x (k) ∥∥ 2
∥ ∥ 2 2 2 2
∥
≤ ∥I − J −1
k J ∥∥2
.
(3.4.10) gives the ‖·‖ 2 -minimal relative correction of J k−1 , such that the secant condition (3.4.8)
holds.
△
Fehlernorm
10 0
10 1
10 −2
10 0
10 −4
10 −6
10 −1
10 −8
10 −2
10 −10
1 2 3 4 5 6 7 8 9 10 11
Iterationsschritt
Konvergenzmonitor
Here:
convergence monitor
=
quantity that displays difficulties in the
convergence of an iteration
∥
∥J −1
k−1 F(x(k) ) ∥
µ :=
∥
∥∆x (k−1)∥ ∥ Heuristics: no convergence whenever µ > 1
✸
Remark 3.4.19. Option: damped Broyden method (as for the Newton method, section 3.4.4)
△
Ôº¿¾ ¿º
Ôº¿¾ ¿º
Example 3.4.18 (Broydens Quasi-Newton Method: Convergence).
• In the non-linear system of the example 3.4.1, n = 2 take x (0) = (0.7.0.7) T and J 0 = DF(x (0) )
The numerical example shows that the
method is:
slower than Newton method (3.4.1), but ✄
Normen
10 −2
10 −4
10 −6
10 −8
Implementation of (3.4.11): with Sherman-Morrison-Woodbury Update-Formula
⎛
⎞ ⎛
⎞
J −1
k+1 = ⎜ J −1
⎝I − k F(x(k+1) )(∆x (k) ) T ⎟
∥
∥
∥∆x (k) ∥∥ 2
⎠J −1 ⎜
+ 2 ∆x(k) · J −1
k = ⎝I + ∆x(k+1) (∆x (k) ) T ⎟
∥
k F(x(k+1) ∥
)
∥∆x (k) ∥∥ 2 ⎠J −1
k
2
(3.4.12)
that makes sense in the case that
∥
∥J −1
k F(x(k+1) ∥
) ∥ < ∥∆x (k)∥ ∥ ∥2
2
"simplified Quasi-Newton correction"
10 0 Iterationsschritt
better than simplified Newton method
(see remark. 3.4.5)
10 −10
10 −12
10 −14
Broyden: ||F(x (k) )||
Broyden: Fehlernorm
Newton: ||F(x (k) )||
Newton: Fehlernorm
Newton (vereinfacht)
0 1 2 3 4 5 6 7 8 9 10 11
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Ôº¿¾ ¿º