Numerical Methods Contents - SAM

f
• n = 1:
Trapezoidal rule
• n ≥ 8:
quadrature formulas with negative weights
3
> newtoncotes(8);
> trapez := newtoncotes(1);
̂Q trp (f) := 1 (f(0) + f(1)) (10.2.3)
2
( ∫ b
a
f(t) dt ≈ b − a
2 (f(a) + f(b)) )
2.5
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1.5 2 2.5 3 3.5 4
t
Fig. 111
✗
✖
1
28350 h (989f(0) + 5888 f(1 8 ) − 928 f(1 4 ) + 10496f(3 8 )
−4540f( 2 1) + 10496f(5 8 ) − 928f(3 4 ) + 5888 f(7 8 ) + 989 f(1))
✸
Negative weights compromise numerical stability (→ Def. 2.5.5) !
Alternative: If t j = Chebychev nodes (8.5.1) ➤ Clenshaw-Curtis rule
✔
✕
• n = 2:
Simpson rule
h
(
)
f(0) + 4f( 1 6
2 ) + f(1)
> simpson := newtoncotes(2);
( ∫ b
a
f(t) dt ≈ b − a (
f(a) + 4 f
6
( a + b
2
)
+ f(b)
) )
Ôº ½¼º¾
(10.2.4)
Remark 10.2.3 (Error estimates for polynomial quadrature).
Ôº½ ½¼º¾
∣ ≤ 1 ∥ n! (b − a)n+1 ∥ ∥f
(n) ∥L . ∞ ([a,b]) (10.2.5)
Quadrature error estimates directly from L ∞ -interpolation error estimates for Lagrangian interpolation
with polynomial of degree n − 1, see Thm. 8.4.1:
∫ f ∈ C n b
([a,b]) ⇒
f(t) dt − Q
∣
n (f) ∣
a
• n = 4:
Milne rule
(Separate estimates for Clenshaw-Curtis rules and analytic integrands)
△
> milne := newtoncotes(4);
1
(
)
90 h 7f(0) + 32 f( 1 4 ) + 12f(1 2 ) + 32 f(3 4 ) + 7 f(1)
( b − a
)
90 (7f(a) + 32f(a + (b − a)/4) + 12f(a + (b − a)/2) + 32f(a + 3(b − a)/4) + 7f(b))
• n = 6: Weddle rule
> weddle := newtoncotes(6);
10.3 Composite Quadrature
With a = x 0 < x 1 < · · · < x m−1 < x m = b
∫ b m∑
∫ xj
f(t) dt = f(t) dt . (10.3.1)
a
j=1
x j−1
Recall (10.2.5): for polynomial quadrature rule (10.2.1) and f ∈ C n ([a,b]) quadrature error shrinks
with n + 1st power of length of integration interval.
1
840 h (41 f(0) + 216 f(1 6 ) + 27f(1 3 ) + 272 f(1 2 )
Ôº¼
+27f( 2 3 ) + 216 f(5 6 ) + 41 f(1))
½¼º¾
Reduction of quadrature error can be achieved by
splitting of the integration interval according to (10.3.1),
using the intended quadrature formula on each sub-interval [x j−1 , x j ].
Note: Increasse in total no. of f-evaluations incurred, which has to be balanced with the gain in
accuracy to achieve optimal efficiency, cf. Sect. 3.3.3 and Sect. 10.6 for algorithmic realization.
Ôº¾ ½¼º¿