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Chap. 11Numerical Differentiation and Integration
Computer Theory and Formal Methods LAB HWANG Dae-Yon ([email protected])
SIM Jae-Hwan ([email protected])YANG Jin-Seok ([email protected])
May. 18, 2005
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Computer Theory and Formal Methods Lab.
Contents
11.1 DIFFERENTIATION 11.1.1 First Derivatives 11.1.2 Higher Derivatives 11.1.3 Richardson Extrapolation
11.2 BASIC NUMERICAL INTEGRATION 11.2.1 Trapezoid Rule 11.2.2 Simpson Rule 11.2.3 Midpoint Rule 11.2.4 Other Newton-Cotes Open Formulas
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11.1 DIFFERENTIATION
11.1.1 First Derivatives
Forward difference formula
Backward difference formula
Central difference formula
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Example 11.1
Forward, Backward and Central DifferencesData Points(x0, y0) = (1,2) (x1, y1) = (2,4) (x2, y2) = (3,8) (x3, y3) = (4,16) (x4, y4) = (5,32)
Forward
Backward
Central
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Three-point difference formula
Three-point forward difference formula
Three-point backward difference formula
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Discussion
Taylor polynomial
Forward : h = xi+1 – xi
Backward : h = xi-1 – xi
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Discussion (cont’)
Central : h = xi+1 – xi = xi – xi-1
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General Three-Point Formula
Based on Lagrange interpolation polynomial(x1 , y1) , (x2 , y2) , (x3 , y3)
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General Three-Point Formula(2)
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General Three-Point Formula(3)
If h = xi+1 – xi = xi – xi-1
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General Three-Point Formula(4)
If h = xi+1 – xi = xi – xi-1
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11.1.2 Higher Derivative
Formula for higher derivative can be founded by Differentiating the interpolating polynomial repeatedly. Using Taylor expansions.
For example, Three equally spaced abscissas xi-1,xi, xi+1 Formula for the second derivative is
1 12
1( ) [ ( ) 2 ( ) ( )]i i i if x f x f x f x
h with truncation error O(h2)
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11.1.2 Higher Derivative- Derivation of Second-Derivative Formula
If we assume that fourth derivative is continuous on [x-h, x+h], we can
write the error term as for some point
From the Taylor polynomial
2 3 4(4)
2( ) ( ) ( ) ( ) ( ) ( )2! 3! 4!
h h hf x h f x hf x f x f x f
2 3 4(4)
1( ) ( ) ( ) ( ) ( ) ( )2! 3! 4!
h h hf x h f x hf x f x f x f
where
1x x h and
2x h x . adding gives4
2 (4) (4)1 2( ) ( ) 2 ( ) ( ) [ ( ) ( )]
4!
hf x h f x h f x h f x f f
or 2
1( ) [ ( ) 2 ( ) ( )]f x f x h f x f x h
h with truncation error O(h4)
2(4) ( )
12
hf x h x h
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11.1.2 Higher Derivative- Derivation of Second-Derivative Formula
Table 11.1 Centered difference formulas, all O(h2)
1 1
1( ) [ ( ) ( )]
2i i if x f x f xh
1 12
1( ) [ ( ) 2 ( ) ( )]i i i if x f x f x f x
h
2 1 1 23
1( ) [ ( ) 2 ( ) 2 ( ) ( )]
2i i i i if x f x f x f x f xh
Example 11.3 Second DerivativeEstimate the second derivative at x2 = 3, using point (x1, y1) = (2, 4),
(x2, y2) = (3, 8), and (x3, y3) = (4, 6); for this example, h=1
(4)2 1 1 24
1( ) [ ( ) 4 ( ) 6 ( ) 4 ( ) ( )]i i i i i if x f x f x f x f x f x
h
(3) [ (4) 2 (3) (2)] [16 2(8) 4] 4f f f f
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11.1.2 Higher Derivative- Partial Derivatives
Partial derivative of a function of two variables General point as (xi, yj ) The value of the function u(x, y) at that point as u i, j
The spacing in the x and y directions is the same, h Using subscripts to indicate partial differentiation
1, 1,
1 1[ ]
2 2x i j i ju u uh h -1 0 1
i-1 i i+1
j
1, , 1,2 2
1 1[ 2 ]xx i j i j i ju u u u
h h 1 -2 1
i-1 i i+1
j
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11.1.2 Higher Derivative- Partial Derivatives
For the mixed second partial derivative and higher derivatives, the schematic form is especially convenient.The Laplacian operatorThe bi-harmonic operator
2xx yyu u u
4 2xxxx xxyy yyyyu u u u
2
1
4xyu h 2
2
1u
h
44
1u
h
-1 0 1
0
-1
00
1 0
1
1-41
1i-1 i i+1
j-1
j
j+1
i-1 i i+1j-1
j
j+1
1 -8 20 -8 1
2 -8 2
1
2 -8 2
1
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11.1.3 Richardson Extrapolation
Method of improving the accuracy of a low order approximation formula A(h) whose error can be expressed as
To apply Richardson extrapolation, we form approximations to A separately using the step size h and h/2
2 42 4( ) ...A A h a h a h
4 ( / 2) ( )
3
A h A hA
To continue the extrapolation process, consider4 6 8
4 6 8( ) ...A B h b h b h b h Where B(h) is simply the extrapolated approximation to A, using step sizes h/2 and h/4, this would correspond to B(h/2). we get
16 ( / 2) ( )( )
15
B h B hC h
which has error O(h6)
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11.1.3 Richardson Extrapolation
The central difference formula can be written as2
41( ) ( ) [ ( ) ( )] ( ) ( )
2 6
hD h f x f x h f x h f x O h
h
We can also find f’(x) using one-half the previous value of h2
41( / 2) ( ) [ ( / 2) ( / 2)] ( ) ( )
24
hD h f x f x h f x h f x O h
h
Since the coefficient of the h2 term does not change , the two estimates can be combined to give
4 ( / 2) ( )
3
D h D hD
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11.1.3 Richardson Extrapolation- Example 11.4
Improved Estimate of the Derivative From Example 11.1 h=2 The approximation to f’(x2) is based on D(h)=7.5 and D(h/2)
4(6) 7.516.5 5.5
3D
The data in the example are points on the curve f(x)=2x. The actual value of f’(x) is (ln2)2x, which gives
(3) 5.54f
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11.1.3 Richardson Extrapolation- Discussion
Richardson extrapolation Forms a linear combination of approximation A(h) and A(h/2) Dominant error term, which depends on h2, cancels
2 42 4( ) ...A A h a h a h
2 4
2 4( / 2) ...4 16
h hA A h a a or
42
2 44 4 ( / 2) ...4
hA A h a h a
(11.2)
(11.3)
Subtracting eq. (11.2) from eq. (11.3) gives
43 4 ( / 2) ( ) ( )A A h A h O h or 414 ( / 2) ( ) ( )
3A A h A h O h
O(h2) O(h4)
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11.1.3 Richardson Extrapolation- Discussion
To continue the extrapolation, we write4 6 8
4 6 8( ) ...A B h b h b h b h where B(h) is simply the extrapolated approximation to A, using step size h, h/2.using step size h/2 and h/4, this would corresponding to B(h/2).
4 6 8 84 6 8( / 2) ( /16) ( / 64) ( / 2 ) ...A B h b h b h b h
Therefore,6 8
6 815 16 ( / 2) ( )A B h B h c h c h Define the second level extrapolated approximation to A as
16 ( / 2) ( )( )
15
B h B hC h
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Contents
Trapezoid Rule Example 11.5 Discussion about Trapezoid Rule
Simpson Rule Example 11.6 Example 11.7 Discussion about Simpson Rule
Midpoint Rule Example 11.8
Other Newton-Cotes Open Formulas
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Overview (1/2)
Numerical integration rules are very important. Functions may not have exact formulas for their antiderivatives (inde
finite integrals). An exact formula for the antiderivative dose exist, it may be difficult
to find.
A numerical integration rule has the form
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Overview (2/2)
Two basic types of Newton-Cotes formulas. “Closed” formulas : the endpoints values are used.
Trapezoid Rule Simpson Rule
“Open” formulas : the endpoints are not used. Midpoint Rule
Each of these formulas can be derived by approximating the function to be integrated by its Lagrange interpolating polynomial.
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Trapezoid Rule
This rule approximates the curve by the straight line that passes through the points (a,f(a)) and (b, f(b)).
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Trapezoid Rule – Example 11.5
Integral of Using the Trapezoid Rule
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Trapezoid Rule – Discussion (1/2)
It derived from the Lagrange form of linear interpolation of f(x) using the endpoints of the interval of integration.
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Trapezoid Rule – Discussion (2/2)
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Simpson Rule
Approximating the function to be integrated by a quadratic polynomial leads to the basic Simpon Rule:
The approximate integral is given by
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Simpson Rule – Example 11.6
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Simpson Rule – Example 11.7
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Simpson Rule – Discussion (1/2)
Simpson’s rule is found by integrating the Lagrange interpolating polynomial for f(x).
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Simpson Rule – Discussion (2/2)
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Midpoint Rule
If we use only function evaluations at points within the interval, the simplest formula is the midpoint rule.
This formula uses only one function evaluation (so n = 1),at the midpoint of the interval, xm=(a+b)/2.
Interpolating the function by the constant value f(xm) :
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Midpoint Rule – Example 11.8
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Other Newton-Cotes Open Formulas (1/2)
Using two function evaluations
Trapezoid Rule
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Other Newton-Cotes Open Formulas (2/2)
Using three function evaluations
Simpson Rule