1

Numerical Integration

1. General considerations

2. Trapezoid rule

3. Simpson’s rule

4. Error estimates

5. Gaussian quadrature

Consider the integral

The general strategy for estimating I is to

evaluate f(x) at a number of points and

approximate I by a sum

where the wi are weight factors.

b

a

dxxfI )(

b

a

N

iii

iiNN

wfdxxfI

xffbxxxxa

1

121

)(

)(

a b x

f(x)

Criteria for the choice of a method are the desired accuracy, the efficiency of the method and particular properties of the integrand (for example singularities).

2

Trapezoid rule

• N evenly spaced points xi

• n=N-1 intervals of size h

• in each interval, approximate the curve f(x) by a straight line (Taylor expansion to first order) a=x1 b=x5

x

f(x)

x2 x3 x4

h

T1 T2 T3 T4

N=5, n=4

N

N

iiiT

N

iiNN

N

N

N

NNNNnT

iiiiiii

i

ffffh

hhhh

wfwfwI

fhffh

f

w

hf

whf

whf

w

h

ffh

ffh

ffh

ffh

TTTII

ffh

ffxxT

NihiaxN

abh

,,, and2

,,,,,2

with

)(222

)(2

)(2

)(2

)(2

trapezoid theof area)(2

)(2

1)(

,,1,)1(1

211

1

211

1

2

2

1

1

112322121

111

3

Simpson’s rule

• N evenly spaced points xi

• n=N-1 intervals of size h

• in each pair of intervals, approximate the curve f(x) by a parabola (Taylor expansion to second order)

h

a=x1 b=x5x

f(x)

x2 x3 x4

N=5

)2(1

)])()([)]()(([1

))2

()2

((1

)(but

)(3

1)(2

3

1)(

2

1

2

1)()()(

2

1)()()(

112

2

3

321

1

2

iii

iiiiiii

ii

h

hi

h

hi

h

hi

i

i

h

h

iii

fffh

hxfxfxfhxfh

hxf

hxf

hxf

xfhxhf

yxfyxfyxfx

xyxfyxfxfdydxxf

hyhxxx

y

xxxfyxxxfxfxf iiiiiii

11

2

2 for)()(2

1)()()()(

4

N

N

iiiS

NNNS

b

a

iiiiii

x

x

i

ffffhhhhhhh

wfwfwI

hfhfhfhfhf

hf

hfhfhfhfIdxxfI

ban

hfhfhffffh

hhfdxxfi

i

,,, and3

,3

4,,

3

2,

3

4,

3

2,

3

4,

3with

3

1

3

4

3

1

3

1

3

4

32

3

1

3

1

3

4

3

1)(

to from integral for the valueobtain the tointervals of pairs 2

theof onscontributi theadd

3

1

3

4

3

1)2(

1

3

12)(

211

1254

3

3321

1112

31

1

• The Simpson rule approximation IS to the integral I is called Sn in Cooper, A Matlab Companion … , pp 172-173

• The number of intervals (n = N-1) has to be even!

• The sum over the weights provides a useful check of the derivation

N

ii abnhhNw

1

)()1(

5

Error estimates

5)4(

5

4)4(34)4(3

11

2

4)4(32

)(60

1

52

)(!4

1

0

)(!3

1)(

!4

1)(

!3

1

:intervals ofpair single aover integral in the error the

estimatingby start We.sorder termfourth and third the todue valueintegral in the

error theestimate now usLet .order toup termsconsidered have werule, sSimpson'For

)(!4

1)(

!3

1)(

2

1)()()(

writebefore, as ; about )( ofexpansion Taylor heConsider t

hxf

h

dyyxfdyyxfyxfyxfdy

hyhxxx

IE

y

yxfyxfyxfyxfxfxf

xxyxxf

i

h

h

i

h

h

iii

h

h

S

iiS

SS

iiiii

ii

Note: the value of the prefactor for S is incorrect because we have also approximated the second derivative of f (x) when we derived Simpson’s rule. A more careful analysis yields:

ban

abfhabfE

nEhf

n

ab

n

abh

xxhf

S

SSS

iiS

such that somefor )(

)(180

1))((

180

1

2for and)(

)(

90

1 find we

)( Using

such that somefor )(90

1

4

5)4(4)4(

4)4(

115)4(

6

• Even though we have used second order polynomials to approximate the function over the pairs of intervals, the Simpson rule yields exact results for polynomials up to third order, since the error is proportional to a fourth-order derivative.

• ES 1/n4 doubling the number of intervals reduces the error by a factor 16 .

• The error estimates for the Trapezoid rule are derived in a similar way and given by

4factor aby error thereduces intervals ofnumber thedoubling

such that somefor )(

)(12

1)()(

12

1

such that somefor )(12

2

32

1

3

ban

abfhfabE

xxfh

T

iiT

Matlab offers a numerical integration routine quad that uses an adaptive Simpson quadrature. Quadruature is the general term for numerical integration methods (from counting squares under a curve). An adaptive integration method uses different interval sizes depending on the behavior of the function and the desired tolerance.

Assume we want to evaluate the integral between two points u and v with au<vb: 1. Split [u,v] into two intervals of size h and use Simpson’s rule, call the result S(2) 2. Split [u,v] into four intervals of size h/2 and use Simpson’s rule, call the result S(4) 3. Calculate the error estimate (2) | S(4)- S(2)|4. If the error is smaller than the tolerance, accept the result and go on to the next interval [v,w]. Otherwise, split [u,v] into 8 intervals of size h/4 , etc.

7

Gaussian quadrature

Please read the handouts from: A. L. Garcia, Numerical Methods for Physics, pp 325-328, and Landau and Paez, Computational Physics, pp 55-57

Remember the general form of numerical integration:

)(andwith)( 1211

iiNN

b

a

N

iii xffbxxxxawfdxxfI

Gaussian quadratures use evaluation points xi (not evenly spaced) and weights wi that yield an optimum approximation to the integral for specific types of functions.

Gaussian quadrature is the preferred method if you can choose a type (Legendre, Laguerre, Chebyshev, etc. ) appropriate for your problem. In that case, you can achieve much higher accuracy with many fewer evaluation points compared to the other integration methods.

However, in general, it is difficult to estimate the error and there is no guarantee that the result is a good approximation to the integral.

Matlab offers a routine quadl that employs a variant of Gaussian quadrature. It may work very well, but be careful when using it and check your results with a different method from time to time.