Numerical Methods Contents - SAM

Assuming
p = 1:
p > 1:
∥
C ∥e (0)∥ ∥p−1 < 1 (guarantees convergence):
∥
∥e (k)∥ ∥ ∥
!
≤ C k ∥ ∥ ∥e
(0) ∥ ∥ ∥ requires k ≥
log ρ
∥
∥e (k)∥ ∥ !
≤ C pk −1
p−1
log C , (3.3.9)
∥
∥e (0)∥ ∥p k requires p k log ρ
≥ 1 +
∥
∥
log C/p−1 + log( ∥e (0) ∥∥)
⇒
k ≥ log(1 + log ρ
log L 0
)/ log p ,
L 0 := C 1/p−1 ∥ ∥∥e (0) ∥ ∥ ∥ < 1 .
(3.3.10)
If ρ ≪ 1, then log(1 + log ρ
log L 0
) ≈ log | log ρ| − log | log L 0 | ≈ log | log ρ|. This simplification will be
made in the context of asymptotic considerations ρ → 0 below.
➣
W Newton = 2W secant ,
p Newton = 2, p secant = 1.62
3.4 Newton’s Method
log p
⇒ Newton
W Newton
secant method is more efficient than Newton’s method!
: log p secant
W secant
= 0.71 .
Non-linear system of equations: for F : D ⊂ R n ↦→ R n find x ∗ ∈ D: F(x ∗ ) = 0
Assume: F : D ⊂ R n ↦→ R n continuously differentiable
✸
Notice: | log ρ| ↔ No. of significant digits of x (k)
Measure for efficiency:
no. of digits gained
Efficiency :=
total work required
asymptotic efficiency w.r.t. ρ → 0 ➜ | log ρ| → ∞):
⎧
⎪⎨ − log C , if p = 1 ,
Efficiency |ρ→0 =
W
log p| log ρ| ⎪⎩
, if p > 1 .
W log | log ρ|
Example 3.3.12 (Efficiency of iterative methods).
max(no. of iterations), ρ = 1.000000e−08
10
9
8
7
6
5
4
3
2
1
C = 0.5
C = 1.0
C = 1.5
C = 2.0
0
1 1.5 2 2.5
p
Fig. 37
∥
Evaluation (3.3.10) for ∥e (0)∥ ∥ = 0.1, ρ = 10 −8
no. of iterations
7
6
5
4
3
2
1
=
| log ρ|
k(ρ) · W
Ôº¾ ¿º¿
(3.3.11)
(3.3.12)
Newton method
secant method
0
0 2 4 6 8 10
−log (ρ) 10 Fig. 38
Newton’s method ↔ secant method, C = 1,
∥
initial error ∥e (0)∥ ∥ = 0.1
Ôº¾ ¿º¿
3.4.1 The Newton iteration
Idea (→ Sect. 3.3.2.1):
Given x (k) ∈ D ➣
local linearization:
x (k+1) as zero of affine linear model function
F(x) ≈ ˜F(x) := F(x (k) ) + DF(x (k) )(x − x (k) ) ,
(
DF(x) ∈ R n,n ∂Fj n
= Jacobian (ger.: Jacobi-Matrix), DF(x) = (x))
.
∂x k j,k=1
Newton iteration: (↔ (3.3.1) for n = 1)
Terminology:
x (k+1) := x (k) −DF(x (k) ) −1 F(x (k) ) , [ if DF(x (k) ) regular ] (3.4.1)
−DF(x (k) ) −1 F(x (k) ) = Newton correction
Ôº¾ ¿º
Ôº¿¼¼ ¿º