Use of Newton-Cotes Formulas and eventual shortcomings

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Use of Newton-Cotes Formulas and eventual
shortcomings Matlab code to increase number of
segments with trapezoid, 1/3 and 3/8 rule
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Methods to achieve better accuracy at lower
effort have been developed Romberg integration -
uses Richardson extrapolation Idea behind
Richardson extrapolation - improve the estimate
at iteration j by using information from
iteration j-1
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True integral value can be written
This is true for any iteration
Using
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So
and
which leads to
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Plugging back into
If
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Combine two O(h2) estimates to get an O(h4)
estimate Can also combine two O(h4) estimates to
get an O(h6) estimate
Can combine two O(h6) estimates to get an O(h8)
estimate
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General pattern is called Romberg Integration

• j - level of accuracy - j1 is more accurate
  (more segments)
• k - level of integration - k1 is original
  trapezoid estimate (O(h2)), k2 is improved
  (O(h4)), etc.

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Excel example
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Gauss quadrature Idea is that if we evaluate the
function at certain points, and sum with certain
weights, we will get accurate integral
Evaluation points and weights are tabulated
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Development of Gauss-Legendre quadrature Assume
a and b are limits of integration
Trapezoid rule should give exact results for
constant function or straight line
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Trapezoid rule always works in these cases
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Now instead of trapezoid, which has fixed end
points (a,b), let them float 4 unknowns -
x0,x1,c0,c1 4 equations - constant, linear (had
before), quadratic, cubic
From -1 to 1 to simplify math
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Can solve these equations to get
and two point Gauss-Legendre formula
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Used cubic so this is third order accurate
To go to -1 to 1 from other limits - use linear
transformation If lower limit is a
If upper limit is b
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Solve and get
So that
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Example Evaluate
using two-point Gauss quadrature Exact value is
0.512076
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First transform
Substituting
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Evaluate that equation at
and
So the integral is 0.630444 Et is 23 - better
than single application of Simpsons 1/3
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Can develop higher order Gauss-Legendre forms
using
Values for cs and xs are tabulated Use the same
transformation
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Example do the same integral using 6-point Gauss
Legendre quadrature
Evaluate at these x and multiply by c
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A final example Determine mass of concrete slab
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Mass is density times volume
Volume is thickness times area
Say thickness is 1 ft Determine area
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30 ft
27 ft
Take measurements at chosen points use symmetry
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Symmetry of slab around x-axis Find area of one
half and X 2
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Use Simpsons 1/3 rule Area for 1/2 is 265.22 ft2
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