Numerical IntegrationUC Berkeley

Fall 2004, E77http://jagger.me.berkeley.edu/~pack/e77

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IntegrationFor a function f,

The “integral of f from a to b” is the area under the graph of the function.

If f is continuous, then the area is well defined, as the common limit of upper and lower sums. The integral is denoted

a b x

graph of f(x)

b

a

dxxf )(

Integration: Fundamental theorem of Calculus

If a function g is the antiderivative of f, namely

for all x, then the fundamental theorem of calculus gives that the integral of f can be computed by evaluating g

But finding antiderivatives can be hard…

In many cases of engineering and scientific (and economics, etc) interest

–the functions do not have known antiderivatives, so…–the integral (area) must be approximated by a finite number of

function evaluations.

)()()( agbgdxxfb

a

xfxg

Numerical Integration: Trapezoid

a b x

graph of f(x)

a b x

graph of f(x)

approximate this region

with this trapezoid

afbfab

2

AREA

Composite Trapezoid

a b x

graph of f(x)

Apply “trapezoid” approximation to k subdivisions of [a b]

a b x

graph of f(x)

“trapezoid” approximation with 3 subdivisions of [a b]

Composite Trapezoid

a b x

graph of f(x)

“trapezoid” approximation with 3 subdivisions of [a b]

h

Let h=(b-a)/3. The sum of all the approximations is

)()2(2)(2)(2

)()2(2

)2()(2

)()(2

bfhafhafafh

bfhafh

hafhafh

hafafh

I

Derivation of Simpson’s Rule: Suppose f(x)=x3

443

4

1 abdxx

b

a

3223

4

1abaabbab

332233

3

2

3

1

3

1

3

2

4

1aabaabbbab

333

3

2

3

1

3

2

4

1aabbab

33

3

3

2

23

8

3

2

4

1a

abbab

33

3

24

6

1a

abbab

bfab

fafab2

46

1

Simpson’s Rule: Suppose f(x)=x2

332

3

1 abdxx

b

a

bfab

fafab2

46

1

Simpson’s Rule: Suppose f(x)=x

22

2

1 abdxx

b

a

bfab

fafab2

46

1

Simpson’s Rule: Suppose f(x)=1

abdxb

a

1

bfab

fafab2

46

1

Simpson’s Rule: Derivation

bfab

fafabdxxfb

a 24

6

1)(

Put all of those together, along with

Hence: If f is any cubic polynomial, then

This is the basis for Simpson’s rule.

b

a

b

a

b

a

dxxgdxxfdxxgxf )()()()(

b

a

b

a

dxxfdxxf )()(

Simpson’s Rule

bfab

fafabI2

46

1Simpson

For any function f, the Simpson’s approximation to

is

b

a

dxxf )(

Evaluate the function at the endpoints

and in the middle

Composite Simpson’s Rule

a b x

graph of f(x)

Simpson on 3 subdivisions of [a b]

h

h=(b-a)/3

hafafafh

I hS 21 4

6

hafafhafh

I hS 24

62

32

bfafhafh

I hS 2

53 426

Add them up. Total of 7 function evaluations.

Composite Simpson’s Rule

a b x

graph of f(x)

Simpson on 2 subdivisions of [a b]h=(b-a)/2

hafafafh

I hS 21 4

6

bfafhafh

I hS 2

32 46 5 function

evaluations

bfafhafafafh

hh 23

2 42)(4)(6

giving the approximation as IS=I1S+I2

S

Numerical Integration: Ad-Hoc stopping criteria

Pick a method (trapezoid, or Simpson’s).

Set a stopping tolerance TOL.

Pick k, an initial number of subdivisions

Iterate as below–Apply composite method using k divisions–Apply composite method using 2k divisions–If answers are within TOL, stop, and return the 2k division

answer– If answers are not within TOL, increase k and repeat.

With only this much analysis, the computed answer is not specifically known to be accurate to any specific precision.

Adaptive Stepsize

Only use small h (the “stepsize”) where the convergence demands it. Recursive implementation is straightforward.

function I = adr(fh,a,b,tol)

Compute I1 using 1 subdivision

Compute I2 using 2 subdivisions

If the answers are within tol, I = I2;

Else

m = (a+b)/2;

ILeft = adr(fh,a,m,tol/2);

IRight = adr(fh,m,b,tol/2);

I = ILeft + IRight;

end

Error Analysis

More error analysis can be done making assumptions about the magnitude of the derivatives of f

–A bound on the magnitude of 3rd derivative of f yields accuracy bounds for the Trapezoid method

–A bound on the magnitude of 5th derivative of f yields accuracy bounds for the Simpson’s method

Learn more about these in Math 128A

Matlab functions

trapz –fixed trapezoidal approximation

–data is vectors x and y (y represents f(x))

–1 division between each data pair

quad–Adaptive Simpson’s method–Data is function handle, interval endpoints, stopping tolerance