Numerical Integration - UCSB MRSECghf/che230a/discussion3_numerical_integration.pdf · numerical...
Transcript of Numerical Integration - UCSB MRSECghf/che230a/discussion3_numerical_integration.pdf · numerical...
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Numerical Integration
(Quadrature)
Another application for our
interpolation tools!
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Lecture 11 2
Integration: Area under a curve
Curve = data or function
Integrating data • Finite number of data points—spacing
specified
• Data may be noisy
• Must interpolate or regress a smooth between points
Integrating functions • Non-integrable function (usually)
• As many points as you need
• Spacing arbitrary
Today we will discuss methods that work on both of these problems
( )b
af x dxArea =
Area
Area
x
y
x
y
a b
a b
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Lecture 11 3
Numerical Integration: The Big
Picture
Virtually all numerical integration methods rely on the following procedure: • Start from N+1 data points (xi ,fi), i = 0,…,N, or sample a specified
function f(x) at N+1 xi values to generate the data set
• Fit the data set to a polynomial, either locally (piecewise) or globally
• Analytically integrate the polynomial to deduce an integration formula of the general form:
Numerical integration schemes are further categorized as either:
• Closed – the xi data points include the end points a and b of the interval
• Open – the xi data points are interior to the interval
wi are the “weights”
xi are the “abscissas”, “points”, or
“nodes”
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Lecture 11 4
Further classification of
numerical integration schemes
Newton-Cotes Formulas • Use equally spaced abscissas
• Fit data to local order N polynomial approximants
• Examples: • Trapezoidal rule, N=1
• Simpson’s 1/3 rule, N=2
• Errors are algebraic in the spacing h between points
Clenshaw-Curtis Quadrature • Uses the Chebyshev abscissas
• Fit data to global order N polynomial approximants
• Errors can be spectral, ~exp(-N) ~ exp (-1/h), for smooth functions
Gaussian Quadrature • Unequally spaced abscissas determined optimally
• Fit data to global order N polynomial approximants
• Errors can be spectral, and smaller than Clenshaw-Curtis
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Lecture 11 5
Trapezoid rule: 1 interval, 2
points
( )b
af x dxArea =
x
f (x)
a b x
f (x)
a b
1
( ) ( ) ( )2
f a f b b a Area =
average height
1
( ) ( )2
f a f b
( )f b
( )f a
Approximate the function by a linear interpolant between
the two end points, then integrate that degree-1 polynomial
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Lecture 11 6
Trapezoid rule: 2 intervals, 3
points
( )b
af x dxArea =
x
f (x)
x0 x2 x
f (x)
a b
0 1 1 2
0 1 2
( ) ( ) ( ) ( )2 2
( ) 2 ( ) ( )2
h hf x f x f x f x
hf x f x f x
Area =
x1
We should be able to improve accuracy by increasing the number
of intervals: this leads to “composite” integration formulas
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Lecture 11 7
“Composite” Trapezoid rule: 8 →
n intervals, n +1 points
x
f (x)
x0 x8 x
f (x)
a b
0 1 1 2 1 2 7 8
0 1 2 7 8
7 1
0 8 0
1 1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2
( ) 2 ( ) 2 ( ) 2 ( ) ( )2
( ) 2 ( ) ( ) ( ) 2 ( ) ( ) ;2 2
n
i i nni i
h h h hf x f x f x f x f x f x f x f x
hf x f x f x f x f x
h h b af x f x f x f x f x f x h
n
Area =
x4 x1 x2 x3 x5 x6 x7
N=1 linear interpolation within each interval:
more intervals to improve accuracy
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Lecture 11 8
Composite Trapezoidal Rule
Notice that this composite formula can be written in the generic form:
where the weights are given by
The truncation error of the composite trapezoidal rule is of order
Error per interval times number of intervals
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Lecture 11 9
Simpson’s 1/3 rule
x
f (x)
x
f (x)
Basic idea: interpolate between 3 points using a parabola (N=2
polynomial) rather than a straight line (as in trapezoid rule)
x
f (x)
red
parabola
blue
parabola
x
f (x)
trapezoid Simpson
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Lecture 11 10
Simpson’s 1/3 rule: derivation
x
f (x)
x0 x2 x1
Interpolation: find equation for a
parabola (N=2 polynomial) passing
through 3 points
1
2
0 0
0
0
( )
x x
x x
f x x x
at
at
at
0
2
1 1
0
0
( )
x x
x x
f x x x
at
at
at
0
1
2 2
0
0
( )
x x
x x
f x x x
at
at
at
( )p x
p(x) is a parabola (quadratic in x)
p(x) goes through 3 desired points
Use Lagrange form for convenience:
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Lecture 11 11
Simpson’s 1/3 rule: derivation
for 2 intervals, 3 points
x
f (x)
x0 x2 x1
( )p x
2
0
( )x
xp x dx Area
2
0
(4) 5
0 1 2
1( ) ( ) 4 ( ) ( ) ; ( )
3 90
x
tx
hp x dx f x f x f x E f h Area
2 1 1 0h x x x x
Remember from calculus:
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Lecture 11 12
Composite Simpson’s 1/3 rule:
(n intervals, n+1 points, n even)
compare with trapezoidal rule
The formula is thus:
The error is:
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Lecture 11 13
Closed Newton-Cotes Formulas
We have just derived the first two closed Newton-Cotes formulas for equally spaced points that include the end points of the interval [a,b]
Basic truncation errors • Trapezoidal (1 interval, 2 points): O(h3)
• Simpson’s 1/3 (2 intervals, 3 points): O(h5)
Composite formula truncation errors (n intervals, n+1 points) • Trapezoidal (“first-order accurate”): O(h2(b-a)) = O(1/n2)
• Simpson’s 1/3 (“third-order accurate”): O(h4(b-a)) = O(1/n4)
Higher order formulas can evidently be developed by using local polynomials with N>2 to interpolate with a larger number of points
It is also easy to adapt these formulas to composite formulas with unequally spaced intervals, h1 h2 h3 … if the data is unevenly spaced
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Lecture 11 14
Example: Simpson’s 1/3 Rule
Let’s try out the composite Simpson’s 1/3 rule on the integral:
Exact result is 2[sin(1)-cos(1)] =0.602337… See SimpsonL11.m
n error
4 2.20 x 10-3
8 1.35 x 10-4
16 8.34 x 10-6
Errors
consistent with
~1/n4 scaling
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Lecture 11 15
Open Newton-Cotes formulas
Same idea as closed formulas, but end points x0 = a, xn = b are not included
These are useful if
• Data is not supplied at the edges of the interval [a,b]
• The integrand has a singularity at the edges (see later)
A simple and useful example – the composite midpoint rule:
where x1/2 is located at the midpoint
between x0 and x1, etc. Same accuracy as
composite TR
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Lecture 11 16
Options for higher accuracy
If we want highly accurate results for an integral, we have various options:
• Use more points n in Simpson’s rule (since error ~ 1/n4) –
computationally costly
• Derive a higher-order Newton-Cotes formula that fits more points to a polynomial with N>2 – analytically cumbersome for small gains
• Use repeated Richardson extrapolation to develop higher order approximations Romberg integration
• Switch to a method based on global approximants:
• Clenshaw-Curtis Quadrature
• Gauss Quadrature
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Lecture 11 17
Clenshaw-Curtis Quadrature
Clenshaw-Curtis quadrature is based on integrating a global Chebyshev polynomial interpolant through all N+1 points
The abscissas are fixed at the Chebyshev points:
If we integrate p(x):
Finally, the weights are obtained by substituting the solution of the collocation equations [T] {c} = {f} into this expression for the ci values
The M-file clencurt.m from L.N. Trefethen, “Spectral Methods in MatLab” (SIAM, 2000) encapsulates these weight calculations
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Lecture 11 18
Example: Clenshaw-Curtis
See ClenCurtL11.m
Compares the error vs. N for integrals of 4 functions:
Spectral accuracy
for the two smooth
functions:
Error below
machine precision
for N > 10!
c.f. Simpson’s rule error of 10-5 at N=16
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Lecture 11 19
Gauss Quadrature:
Global like Clenshaw-Curtis, but also
optimally choose abscissas
-1 0 1 -1 0 1 1
1( )I f x dx
x1 x2
poor
approximation much better approximation
obtained by adjusting x1 & x2
values and w1 and w2 values
2 point
case:
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Lecture 11 20
Finding optimal choice of
parameters
-1 0 1 x1 x2
Note that there are 4 unknowns to be
determined.
Gauss procedure: we choose the 4 unknowns so that
the integral gives an exact answer whenever f (x) is a
polynomial of order 3 or less
2 3( ) 1, ( ) , ( ) , ( ) .f x f x x f x x f x x
it would have to give the exact answer for the following four cases:
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Lecture 11 21
Determining the unknowns: Gauss-
Legendre Quadrature (N=2 case)
•More generally, for an N point formula, the abscissas are the N roots of the
Legendre Polynomial PN(x). The weights can be obtained by solving a
linear system with a tridiagonal matrix.
•Notice that Gauss-Legendre is an open formula, unlike Clenshaw-Curtis
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Lecture 11 22
Example: Gauss-Legendre
See gauss.m (for abscissas and weights) and GaussL11.m
Compares the error vs. N for integrals of same 4 functions:
Spectral accuracy
for the two smooth
functions:
Error below
machine precision
for N > 6!
Faster
convergence than
even Clenshaw-
Curtis for smooth
functions c.f. Simpson’s rule error of 10-5 at N=16
Gauss exactly integrates a
polynomial of degree 2N-1!
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Lecture 11 23
Adjusting the limits of
integration
The spectral accuracy of the Gauss-Legendre and Clenshaw-Curtis methods can be traced to the fact that they employ global polynomial interpolation and cluster their abscissas at the edges of the interval
Remember that the Clenshaw-Curtis and Gauss-Legendre formulas apply only for integrals with limits of -1 and +1!
The limits need to be rescaled for other integrals:
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Lecture 11 24
Integrals with Singularities
Occasionally you will encounter integrals with weak (integrable) singularities, e.g.
In this case there is an inverse square root singularity at the endpoint x=0. We obviously cannot apply a closed integration formula in this case!
Alternatives are: • Apply an open formula, like composite midpoint rule or Gauss-Legendre
quadrature
• Use an open formula that explicitly includes the x-1/2 factor (better option)
• Rescale x variable as x = t2, so the singularity is removed (best option, if available):
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Lecture 11 25
Indefinite (improper) integrals
Often you will encounter integrals with infinite limits, e.g.
There are several alternatives: • Apply a Newton-Cotes formula to a similar integral, but with 1 replaced
with a large number R
• Rescale x variable as x = - ln t, (assuming resulting integral not singular):
A better approach is to use a Gaussian quadrature formula appropriate for the interval [0,1), such as the Gauss-Laguerre formula:
Don’t forget to factor the
xexp(-x) out of your function!