Integration

1
Integration

• COS 323

2
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

3
Trapezoidal Rule

• Approximate function by trapezoid

4
Trapezoidal Rule
5
Extended Trapezoidal Rule
Divide into segments of width h
6
Trapezoidal Rule Error Analysis

• How accurate is this approximation?
• Start with Taylor series for f (x) around a

7
Trapezoidal Rule Error Analysis

• Expand LHS
• Expand RHS

8
Trapezoidal Rule Error Analysis

• So,
• 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

9
Determining Step Size

• Change in integral when reducing step sizeis a
  reasonable guess for accuracy
• For trapezoidal rule, easy to go from h ?
  h/2without wasting previous samples

a
b
10
Simpsons Rule
f(b)

• Approximate integral byparabola throughthree
  points
• Better accuracy for same of evaluations

f(a)
a
b
11
Richardson Extrapolation

• Better way of getting higher accuracy for agiven
  of samples
• Suppose weve evaluated integral for step sizeh
  and step size h/2
• Then

12
Richardson Extrapolation

• This treats the approximation as a function of h
  and extrapolates the result to h0
• Can repeat

1/3
1/15
4/3
16/15
1/63
64/63
13
Open Methods

• Trapezoidal rule wont work if function undefined
  at one of the points where evaluating
• Most often function infinite at one endpoint
• Open methods only evaluate function on the open
  interval (i.e., not at endpoints)

14
Midpoint Rule

• Approximate function by rectangle evaluated at
  midpoint

15
Extended Midpoint Rule
Divide into segments of width h
16
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 cant halve intervals
  withoutwasting samples

17
Discontinuities

• All the above error analyses assumed nice
  (continuous, differentiable) functions
• In the presence of a discontinuity, all
  methodsrevert to accuracy proportional to h
• Locally-adaptive methods do not subdivideall
  intervals equally, focus on those with large
  error(estimated from change with a single
  subdivision)