Numerical Integration ProblemsNumerical Integration Problems
• Basic 1D numerical integration:Basic 1D numerical integration:– GivenGiven ability to evaluate ability to evaluate ff ((xx)) for any for any xx, , findfind
– Goal:Goal: best accuracy with fewest samples best accuracy with fewest samples
• Other problems (future lectures):Other problems (future lectures):– Improper integration Improper integration
– Multi-dimensional integrationMulti-dimensional integration
– Ordinary differential equationsOrdinary differential equations
– Partial differential equationsPartial differential equations
b
a
dxxf )(b
a
dxxf )(
Trapezoidal RuleTrapezoidal Rule
• Approximate function by trapezoidApproximate function by trapezoid
aa bb
f(a)f(a)
f(b)f(b)
Trapezoidal RuleTrapezoidal Rule
aa bb
f(a)f(a)
f(b)f(b)
2
)()()()(
bfafabdxxf
b
a
2
)()()()(
bfafabdxxf
b
a
Extended Trapezoidal RuleExtended Trapezoidal Rule
aa bb
f(a)f(a)
f(b)f(b)
2
)()()()(
bfafabdxxf
b
a
2
)()()()(
bfafabdxxf
b
a
)()()()()( 21
1121 bfxfxfafhdxxf n
b
a
)()()()()( 21
1121 bfxfxfafhdxxf n
b
a
Divide into segments of width Divide into segments of width hh::
aa bb
Trapezoidal Rule Error AnalysisTrapezoidal Rule Error Analysis
• How accurate is this approximation?How accurate is this approximation?
• Start with Taylor series for Start with Taylor series for ff ((xx)) around around aa
)()()()()()( 221 afaxafaxafxf )()()()()()( 221 afaxafaxafxf
E
)()(2
)()( bfaf
abdxxf
b
a
E
)()(2
)()( bfaf
abdxxf
b
a
Trapezoidal Rule Error AnalysisTrapezoidal Rule Error Analysis
• Expand LHS:Expand LHS:
• Expand RHSExpand RHS
)()()()()()()( 3612
21 afabafabafabdxxf
b
a
)()()()()()()( 3612
21 afabafabafabdxxf
b
a
E
E
)()()()()()(
)()()()(2
)(
3412
21
21
21
afabafabafab
afabbfafab
E
E
)()()()()()(
)()()()(2
)(
3412
21
21
21
afabafabafab
afabbfafab
Trapezoidal Rule Error AnalysisTrapezoidal Rule Error Analysis
• So,So,
• In general, error for a In general, error for a singlesingle segment segment proportional to proportional to hh33
• Error for subdividing entire aError for subdividing entire ab interval b interval proportional to proportional to hh22
– ““Cubic local accuracy, quadratic global Cubic local accuracy, quadratic global accuracy”accuracy”
)()( 3121 afabE )()( 3
121 afabE
Determining Step SizeDetermining Step Size
• Change in integral when reducing step Change in integral when reducing step sizesizeis a reasonable guess for accuracyis a reasonable guess for accuracy
• For trapezoidal rule, easy to go from h For trapezoidal rule, easy to go from h h/2h/2without wasting previous sampleswithout wasting previous samples
aa bb
• Approximate integral byApproximate integral byparabola throughparabola throughthree pointsthree points
• Better accuracy for same # of Better accuracy for same # of evaluationsevaluations
Simpson’s RuleSimpson’s Rule
aa bb
f(a)f(a)
f(b)f(b)
)()()(4)(3
)( 5hObfhafafh
dxxfb
a
)()()(4)(3
)( 5hObfhafafh
dxxfb
a
Richardson ExtrapolationRichardson Extrapolation
• Better way of getting higher accuracy for aBetter way of getting higher accuracy for agiven # of samplesgiven # of samples
• Suppose we’ve evaluated integral for step Suppose we’ve evaluated integral for step sizesizeh and step size h/2:h and step size h/2:
• ThenThen
4
2
2
22/
42
hhh
h
FF
hhFF
4
2
2
22/
42
hhh
h
FF
hhFF
)( 431
2/34 hOFFF hh )( 4
31
2/34 hOFFF hh
Richardson ExtrapolationRichardson Extrapolation
• This treats the approximation as a This treats the approximation as a function of h and “extrapolates” the function of h and “extrapolates” the result to h=0result to h=0
• Can repeat:Can repeat:
)()()()( 8642
8/
4/
2/
hOhOhOhO
F
F
F
F
h
h
h
h
)()()()( 8642
8/
4/
2/
hOhOhOhO
F
F
F
F
h
h
h
h ––1/31/3
4/34/3 ––1/151/15
16/1516/15 ––1/631/63
64/6364/63
Open MethodsOpen Methods
• Trapezoidal rule won’t work if function Trapezoidal rule won’t work if function undefined at one of the points where undefined at one of the points where evaluatingevaluating– Most often: function infinite at one endpointMost often: function infinite at one endpoint
• Open methods only evaluate function Open methods only evaluate function on the on the openopen interval (i.e., not at interval (i.e., not at endpoints)endpoints)
1
02x
dx1
02x
dx
Midpoint RuleMidpoint Rule
• Approximate function by rectangle Approximate function by rectangle evaluated at midpointevaluated at midpoint
aa bb
)( 2baf )( 2baf
Extended Midpoint RuleExtended Midpoint Rule
)()()( 2ba
b
a
fabdxxf )()()( 2ba
b
a
fabdxxf
)()()()( 223
2hhh
b
a
bfafafhdxxf )()()()( 223
2hhh
b
a
bfafafhdxxf
Divide into segments of width Divide into segments of width hh::
aa bb
aa bb
Midpoint Rule Error AnalysisMidpoint Rule Error Analysis
• Following similar analysis to trapezoidal Following similar analysis to trapezoidal rule,rule,find that local accuracy is cubic,find that local accuracy is cubic,quadratic global accuracyquadratic global accuracy
• Formula suitable for adaptive method,Formula suitable for adaptive method,Richardson extrapolation,Richardson extrapolation,but can’t halve intervals withoutbut can’t halve intervals withoutwasting sampleswasting samples
DiscontinuitiesDiscontinuities
• All the above error analyses assumed nice All the above error analyses assumed nice (continuous, differentiable) functions(continuous, differentiable) functions
• In the presence of a discontinuity, all methodsIn the presence of a discontinuity, all methodsrevert to accuracy proportional to hrevert to accuracy proportional to h
• Locally-adaptive methods: do not subdivideLocally-adaptive methods: do not subdivideall intervals equally, focus on those with large all intervals equally, focus on those with large errorerror(estimated from change with a single (estimated from change with a single subdivision)subdivision)