Numerical Methods Contents - SAM

Local quadrature error (for f ∈ C 2 ([a,b])):
10000
9000
❸
(local mesh refinement)
∫ x k
f(t) dt − 1 2 (f(x k−1) + f(x k ))
x k−1
≤ (x k − x k−1 ) 3 ∥ ∥f ′′∥ ∥ L ∞ ([x k−1 ,x k ]) .
➣ Do not use equidistant mesh !
Refine M, where |f ′′ | large !
f(t)
8000
7000
6000
5000
4000
3000
2000
1000
f(t) = 1
10 −4 +t 2
S := {k ∈ {1,...,m}: EST k ≥ η · 1 m∑
EST
m j } , η ≈ 0.9 . (10.6.3)
j=1
new mesh: M ∗ := M ∪ {p k : k ∈ S} .
Then continue with step ❶ and mesh M ← M ∗ .
Makes sense, e.g., for “spike function” ✄ 0
−1 −0.5 0 0.5 1
✬
Goal:
Tool:
Policy:
✫
Equilibrate error contributions of all mesh intervals
Local a posteriori error estimation
(Estimate contributions of mesh intervals from intermediate results)
Local mesh refinement
t
Fig. 117
✸
✩
Ôº½¿
Ôº½ ½¼º
Adaptive multigrid quadrature → [13, Sect. 9.7]
11 i f ( ( e r r _ t o t > r t o l ∗abs ( I ) ) and ( e r r _ t o t > a b s t o l ) )
½¼º
✪
10 %
Non-optimal recursive MATLAB implementation:
Code 10.6.2: h-adaptive numerical quadrature
1 function I = adaptquad ( f ,M, r t o l , a b s t o l )
2 h = d i f f (M) ; %
3 mp = 0.5∗(M( 1 : end−1)+M( 2 : end ) ) ; %
4 f x = f (M) ; fm = f (mp) ; %
5 t r p _ l o c = h . ∗ ( f x ( 1 : end−1)+2∗fm+ f x ( 2 : end ) ) / 4 ; %
6 simp_loc = h . ∗ ( f x ( 1 : end−1)+4∗fm+ f x ( 2 : end ) ) / 6 ; %
7 I = sum( simp_loc ) ; %
8 e s t _ l o c = abs ( simp_loc −t r p _ l o c ) ; %
9 e r r _ t o t = sum( e s t _ l o c ) ; %
Idea: local error estimation by comparing local results of two quadrature formulas
Q 1 , Q 2 of different order → local error estimates
heuristics: error(Q 2 ) ≪ error(Q 1 ) ⇒ error(Q 1 ) ≈ Q 2 (f) − Q 1 (f) .
Now: Q 1 = trapezoidal rule (order 2) ↔ Q 2 = Simpson rule (order 4)
12 r e f c e l l s = find ( e s t _ l o c > 0.9∗sum( e s t _ l o c ) / length ( h ) ) ;
13 I = adaptquad ( f , sort ( [M,mp( r e f c e l l s ) ] ) , r t o l , a b s t o l ) ; %
14 end
Comments on Code 10.6.1:
Given: mesh M := {a = x 0 < x 1 < · · · < x m = b}
❶
❷
(error estimation)
For I k = [x k−1 ,x k ], k = 1,...,m (midpoints p k := 2 1(x k−1 + x k ) )
EST k := ∣ h k
6 (f(x k−1) + 4f(p k ) + f(x k )) − h k
} {{ } 4 (f(x k−1) + 2f(p k ) + f(x k )) ∣ . (10.6.1)
} {{ }
Simpson rule
trapezoidal rule on split mesh interval
(Termination)
Simpson rule on M ⇒ preliminary result I
m∑
If
k=1
Ôº½ ½¼º
EST k ≤ RTOL · I (RTOL := prescribed tolerance) ⇒ STOP (10.6.2)
• Arguments: f ˆ= handle to function f, M ˆ= initial mesh, rtol ˆ= relative tolerance for termination,
abstol ˆ= absolute tolerance for termination, necessary in case the exact integral value = 0,
which renders a relative tolerance meaningless.
• line 2: compute lengths of mesh-intervals [x j−1 ,x j ],
• line 3: store positions of midpoints p j ,
• line 4: evaluate function (vector arguments!),
• line 5: local composite trapezoidal rule (10.3.2),
• line 6: local simpson rule (10.2.4),
• line 7: value obtained from composite simpson rule is used as intermediate approximation for
integral value,
• line 8: difference of values obtained from local composite trapezoidal rule (∼ Q 1 ) and ocal
simpson rule (∼ Q 2 ) is used as an estimate for the local quadrature error.
Ôº½ ½¼º