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 )(
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
Simpson’s Rule: Suppose f(x)=x
22
2
1 abdxx
b
a
bfab
fafab2
46
1
Let a = 2 and b = 8
½(64-4) = 30
1/6(6)(2 + 4*5 +8) = 30
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: 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.
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.
.
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
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