数值分析英文版课件 (14)

If f is continuous on [a, b], then the approximations in
∫ f ( x )w ( x ) dx ≈ ∑ A f ( x )
b a i =0 ni ni
n
( n ≥ 0)
(10 )
converge to the integral as n→∞.
6
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7.3.4 Summary (9)
As the polynomial gn (x) gn (x)=a0 + a1x + ... + an xn satisfies
( an ≠ 0 )
( g ,g ) = ∫
j k
b
a
0, j ≠ k ρ ( x )gj ( x ) gk ( x ) dx = A k , j=k
7.4.1 Recursive Trapezoid Rule 7.4.2 Romberg Algorithm 7.4.3 Analysis
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Topics for today
7.3 Gaussian Quadrature
7.3.1 Introduction 7.3.2 Gaussian Quadrature 7.3.3 Convergence and Error Analysis 7.3.4 Summary
Solution:
For the simplicity, we take the equal nodes x0=-1, x1=1, obviously, we can get the trapezoid formula
∫ f ( x )dx ≈ f ( −1) + f (1)
1 −1
And the accuracy degree is one.
Solve the equations, we get
3 3 A0 = A1 = 1, x0 = − , x1 = 3 3
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7.3.4 Summary (5)
From this, we can get the quadrature formula
3 3 ∫−1 f ( x )dx ≈ f − 3 +y (8)
From the definition, we can find that:
When the number of nodes is too much, it is quite difficult to solve. The orthogonal polynomial has to be used.
ρ ( x) ≡ 1
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7.3.4 Summary (11)
Some common orthogonal polynomials are: (1) Chebyshev polynomial
Tn ( x ) = cos ( n arccos x ) , n ≥ 0, x ∈ [ −1,1] is the nth orthogonal polynomial of the weighted 1 function ρ ( x ) = in interval [-1, 1] 2 1− x and satisfies the transmissive relations T0 ( x ) = 1, T1 ( x ) = x Tn +1 ( x ) = 2 xTn ( x ) − Tn −1 ( x )
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7.3.4 Summary (3)
But if we treat this problem in another way:
take x0, x1, A0, A1 as four pending parameters, choose from the definition of the accuracy,
then we get the following equations:
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7.3.4 Summary (4)
A0 + A1 = 2 A x + A x = 0 1 1 0 0 2 2 2 A0 x0 + A1 x1 = 3 3 3 A0 x0 + A1 x1 = 0
7.4 Romberg Integration
7.4.1 Recursive Trapezoid Rule 7.4.2 Romberg Algorithm 7.4.3 Analysis
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7.3.3 Convergence and Error Analysis (1)
Lemma 1 Lemma on Gaussian Quadrature Formula
7.3 Gaussian Quadrature
7.3.1 Introduction 7.3.2 Gaussian Quadrature 7.3.3 Convergence and Error Analysis 7.3.4 Summary
7.4 Romberg Integration
7.4.1 Recursive Trapezoid Rule （递推梯形法） 7.4.2 Romberg Algorithm 7.4.3 Analysis
2
Topics for today
7.3 Gaussian Quadrature
7.3.1 Introduction 7.3.2 Gaussian Quadrature 7.3.3 Convergence and Error Analysis 7.3.4 Summary
7.4 Romberg Integration
From the above definition, we can conclude that the orthogonality of the polynomial relies on the defined quadrature interval [a, b] and the defined weighted function ρ(x). ρ(x) should have some properties, such as ρ ( x ) ≥ 0, x ∈ [ a, b ] If there are no other specific indications, we all take the assumption
In a Gaussian quadrature formula, the coefficients are positive, and moreover, their sum is
∫ w ( x ) dx
b a
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7.3.3 Convergence and Error Analysis (4)
Theorem 3 Theorem on Gaussian Quadrature Convergence
then the absolute different answer will be obtained.
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7.3.4 Summary (3)
Take the quadrature function f(x) in turn as
1, x, x2, x3, put them into the enactive quadrature formula,
1
And it have 3 accuracy degree.
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7.3.4 Summary (6)
We use the same two nodes, but got the different accuracy degree, one is one, and the other is three. This indicates that the accuracy of the quadrature formula is determined by choosing the suitable quadrature nodes and coefficients.
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7.3.4 Summary (12)
(2) Legendre polynomial
L0 ( x ) = 1 n 1 dn 2 x − 1 ,( n = 1 L) ,2, Ln ( x ) = n n 2 ⋅ n! dx is the nth orthogonal polynomial sequence
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7.3.4 Summary (1)
Example 1
Find out the Gaussian Quadrature formula with the following form:
∫ f ( x )dx ≈ A f ( x ) + A f ( x )
1 −1 0 0 1 1
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7.3.4 Summary (2)
b a i =0 i i
2n For an f ∈ C [a, b] , we have
n −1
E=
f(
(ξ ) b q 2 x ω x dx ∫a ( ) ( ) ( 2n ) !
2n)
n −1 i =0
Where a<ξ<b and q ( x ) = ∏
( x − xi )
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Topics for today
15
7.3.4 Summary (7)
So we got the interpolation quadrature formula consisted of n+1 nodes:
∫
b
a
f ( x)dx ≈ ∑ Ak f ( xk )
k =0
n
(1)
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7.3.4 Summary (7)
Process
(
)
in interval [-1, 1]
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7.3.4 Summary (13)
Obviously ,Ln(x) is nth polynomial ,and can be proved that