Integration
COS 323
Numerical Integration Problems
• Basic 1D numerical integration:
– Given ability to evaluate f (x) for any x, find
– Goal: best accuracy with fewest samples
• Other problems (future lectures):
–
–
–
–
Improper integration
Multi-dimensional integration
Ordinary differential equations
Partial differential equations
b
 f ( x) dx
a
Trapezoidal Rule
• Approximate function by trapezoid
f(b)
f(a)
a
b
Trapezoidal Rule
f(b)
b

a
f (a)  f (b)
f ( x) dx  (b  a)
2
f(a)
a
b
Extended Trapezoidal Rule
f(b)
b

a
f (a)  f (b)
f ( x) dx  (b  a)
2
f(a)
a
b
a
b
Divide into segments of width h:
b

a
f ( x) dx  h  12 f (a)  f ( x1 )    f ( xn 1 )  12 f (b) 
Trapezoidal Rule Error Analysis
• How accurate is this approximation?
b

a
(b  a)
 f (a)  f (b)  E
f ( x) dx 
2
• Start with Taylor series for f (x) around a
f ( x)  f (a)  ( x  a) f (a)  12 ( x  a)2 f (a)  
Trapezoidal Rule Error Analysis
• Expand LHS:
b

f ( x) dx  (b  a) f (a)  12 (b  a) 2 f (a)  16 (b  a)3 f (a)  
a
• Expand RHS
(b  a)
 f (a)  f (b)  E  12 (b  a) f (a) 
2
2
3
1
1
1

(
b

a
)
f
(
a
)

(
b

a
)
f
(
a
)

(
b

a
)
f (a)    E
2
2
4
Trapezoidal Rule Error Analysis
• So,
E   121 (b  a)3 f (a)  
• In general, error for a single segment
proportional to h3
• Error for subdividing entire ab interval
proportional to h2
– “Cubic local accuracy, quadratic global accuracy”
Determining Step Size
• Change in integral when reducing step size
is a reasonable guess for accuracy
• For trapezoidal rule, easy to go from h  h/2
without wasting previous samples
a
b
Simpson’s Rule
• Approximate integral by
parabola through
three points
f(b)
f(a)
a
b

a
h
f ( x) dx   f (a)  4 f (a  h)  f (b)   O(h5 )
3
• Better accuracy for same # of evaluations
b
Richardson Extrapolation
• Better way of getting higher accuracy for a
given # of samples
• Suppose we’ve evaluated integral for step size
h and step size h/2:
Fh  F  h 2  h 4  
Fh / 2  F    h2     h2   
2
4
• Then
4
3
Fh / 2  13 Fh  F  O(h 4 )
Richardson Extrapolation
• This treats the approximation as a function of h
and “extrapolates” the result to h=0
• Can repeat:
Fh
–1/3
–1/15
4/3
Fh / 2
16/15
–1/63
64/63
Fh / 4
Fh / 8
O(h 2 )
O(h 4 )
O(h6 )
O ( h8 )
Open Methods
• Trapezoidal rule won’t work if function
undefined at one of the points where evaluating
– Most often: function infinite at one endpoint
1
dx
0 x 2
• Open methods only evaluate function on the
open interval (i.e., not at endpoints)
Midpoint Rule
• Approximate function by rectangle evaluated at
midpoint
f ( a 2 b )
a
b
Extended Midpoint Rule
b

f ( x) dx  (b  a) f ( a 2 b )
a
a
b
a
b
Divide into segments of width h:
b

a
f ( x) dx  h  f (a  h2 )  f (a  32h )    f (b  h2 ) 
Midpoint Rule Error Analysis
• Following similar analysis to trapezoidal rule,
find that local accuracy is cubic,
quadratic global accuracy
• Formula suitable for adaptive method,
Richardson extrapolation,
but can’t halve intervals without
wasting samples
Discontinuities
• All the above error analyses assumed nice
(continuous, differentiable) functions
• In the presence of a discontinuity, all methods
revert to accuracy proportional to h
• Locally-adaptive methods: do not subdivide
all intervals equally, focus on those with large error
(estimated from change with a single subdivision)