MA/CS 375
Fall 2002
Lecture 32
MA/CS 375 Fall 2002
1
Review
Roots of a Polynomial
• Suppose we wish to find all the roots of a
polynomial of order P
• Then there are going to be at most P roots!.
• We can use a variant of Newton’s method.
MA/CS 375 Fall 2002
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Review
Newton Scheme For Multiple
Root Finding
Initiate guesses to the roots x 1 ,x 2 ,..x P
L oop over k= 1:P
Iterate:
xk  xk 
f
 df  x k

 dx

 xk 
 i  k 1 1

 f  xk   

 i 1 x k  x i  
to find x k to a given tolerance
E nd loop
MA/CS 375 Fall 2002
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(applied to find roots of Legendre polynomials)
Multiple Root Finder
Review + Correction
MA/CS 375 Fall 2002
Should read abs(delta) > tol
4
Review
Legendre Polynomials
• Legendre polynomials are a special set
of polynomials which are orthogonal in
the L2 inner product:
1

L n  x  L m  x  dx  0 if n  m
1
MA/CS 375 Fall 2002
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Review
Legendre Polynomials
• Legendre polynomials can be calculate
using the following recursion relation:
L0  x   1
L1  x   x
 2n  1 
 n 
L n 1  x   
 x Ln  x  
 L n 1  x  n= 1,2,...
 n 1 
 n 1
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Review
Roots of the 10th Order
Legendre Polynomial
Notice how they cluster at the end points
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Numerical Quadrature
• A numerical quadrature is a set of two vectors.
• The first vector is a list of x-coordinates for
nodes where a function is to be evaluated.
• The second vector is a set of integration
weights, used to calculate the integral of a
function which is given at the nodes
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Example of Quadrature
• Say we wish to calculate an approximation to
1
the integral of f over [-1,1] :
 f  x  dx
1
• Suppose we know the value of f at a set of N
points then we would like to find a set of
weights w1,w2,..,wN so that:
1
i N
 f  x  dx   w f  x 
i
1
MA/CS 375 Fall 2002
i
i 1
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Example: Simpson’s Rule
Recall:
• The idea is to sample a function at N points.
• Then using a shifting stencil of 3 points construct
a quadratic interpolant through those 3 points.
• Then integrate the area under the interpolant in
the range bracketed by the three points.
• Sum up all the contributions from the sets of
three points.
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Example: Simpson’s Rule
1
 f  x  dx 
1
2
3( N  1)
 f  x   4 f  x   2 f  x   4 f  x  ...  f  x  
1
2
3
nodes
4
 { x1 , x 2 , x 3 , x 4 ,
N
, xN }
 n 1 
xn  1  2 

 N 1
quadrature:
w eights   w 1 , w 1 , w 1 , w 1 ,
w1 , w N 
MA/CS 375 Fall 2002
w2 , w4 , w6 ,
, w N 1 
w3 , w5 , w7 ,
, w N 2 
, w1 
2
3  N  1
8
3  N  1
4
3  N  1
11
Example: Simpson’s Rule
1
 f  x  dx 
1
2
3( N  1)
 f  x   4 f  x   2 f  x   4 f  x  ...  f  x  
1
2
3
4
N
becomes:
1
 f  x  dx  w f  x   w f  x   w f  x   .. w
1
1
2
2
3
3
N
f
 xN 
1
in summation notation:
1
nN
 f  x  dx   w f  x 
n
MA/CS 375 Fall 2002
1
n 1
n
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Newton-Cotes Formula
• The next approach we are going to use is the
well known Newton-Cotes quadrature.
• Suppose we are given a set of points
x1,x2,..,xN. Then we require that the constant
is exactly integrated:
w1 x  w 2 x 
0
1
MA/CS 375 Fall 2002
0
2
wN x
0
N
1
x 
  x dx   
 1  1
1
1
1
0
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Now we require that 1,x,x2,..,xN-1
are integrated exactly
w1 x  w 2 x 
0
1
0
2
wN x
x 
  x dx   
 1  1
1
1
N
x 
  x dx  

 2  1
1
1
0
1
w1 x  w 2 x 
1
1
1
2
wN x
2
w1 x
 w2 x
N 1
2

wN x
N 1
N

x
1
MA/CS 375 Fall 2002
1
1
1
N 1
1
1
0
N
1
N 1
1
x 
dx  

N

 1
N
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In Matrix Notation:
 1    1

1
0

x N   w1 
2
2


1    1

1
w2
xN


2



 
N 1  
xN   wN  
N
N
 1    1

N

1
 x1
 1
x
 1

 N 1
 x1
0
x
x
0
2
x
1
2
N 1
2
1











Notice anything familiar?
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It’s the transpose of the
Vandermonde matrix 
 11    1 

1
0

x N   w1 
2
2


1    1

1
w2
xN


2



 
N 1  
xN   wN  
N
N
 1    1

N

1
 x1
 1
x1
t

V w 

 N 1
 x1
0
MA/CS 375 Fall 2002
x
0
2
x
1
2
N 1
x2











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Integration by Interpolation
• In essence this approach uses the unique
(N-1)’th order interpolating polynomial If and
integrates the area under the If instead of
the area under f
• Clearly, we can estimate the approximation
error using the estimates for the error in the
interpolation we used before.
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Newton-Cotes Weights
 11    1 

1

 12    1  2


2


 N
N
 1    1

N

1
 w1

w2



 wN
MA/CS 375 Fall 2002


1
t
  V 














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Using Newton-Cotes Weights
1

1
MA/CS 375 Fall 2002
f
 x  dx 
i N

wi f
 xi  w f
t
i 1
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Using Newton-Cotes Weights
(Interpretation)
i N
1
 f  x  dx   w f  x   w
i
f
i 1
1
 11    1 


1

i
t
1
1   1
2
2
2
1   1
N
N
N

1
V
f



i.e. we calculate the coefficients of the interpolating polynomial
expansion using the Vandermonde, then since we know the
integral of each term we can sum up the integral of each term
to get the total.
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Matlab Function for Calculating
Newton-Cotes Weights
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Demo: Matlab Function for
Calculating Newton-Cotes Weights
1) set N=5 points
2) build equispaced nodes
3) calculate NC weights
4) evaluate F=X^3 at nodes
5) evaluate integral
6) F is anti-symmetric on
[-1,1] so its integral is 0
7) Answer correct 
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Individual Exercise
• Download the contents of:
http://www.math.unm.edu/~timwar/MA375F02/Integration
• make sure your matlab path points to your copy of
this directory
• using a script figure out what order polynomial the
weights produced with newtoncotes can exactly
integrate for a given set of N points (say
N=3,4,5,6,7,8) created with linspace
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Gauss Quadrature
• The construction of the Newton-Cotes
weights does not utilize the ability to choose
the distribution of nodes for greater accuracy.
• We can in fact choose the set of nodes to
increase the order of polynomial that can be
integrated exactly using just N points.
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Suppose:
f
x 
x  rxsx
If
w here:
f P
If
2 p 1
 xi  
   1,1 
f
 xi 
w here If  P
s  x i   0 w here s  P
rP
MA/CS 375 Fall 2002
p
p 1
p
   1,1  
   1,1  
   1,1 
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Remainder term, which
must have p roots located
at the interpolating nodes
Suppose:
f
x 
x  rxsx
If
w here:
f P
If
2 p 1
 xi  
   1,1 
f
 xi 
w here If  P
s  x i   0 w here s  P
rP
MA/CS 375 Fall 2002
p
p 1
p
   1,1  
   1,1  
   1,1 
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Let’s integrate this formula for f over [-1,1]
f
x 
1
x  rxsx
If
1
 f  x  dx  
1
1
If
1
 x dx   r  x  s  x  dx
1
i N

1
 w f  x    r  x  s  x  dx
i
i 1
i
1
At this point we can choose the nodes {xi}.
If we choose them so that they are the p+1 roots of
the (p+1)’th order Legendre function then s(x) is in
fact the N=(p+1)’th order Legendre function itself!.
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1
1
 f  x  dx  
1
1
If
1
 x d x   s  x  r  x  d x
1
1


1
If
1
 x d x  
1
i N

LN  x  r  x  dx
1
 w f  x  
i
i 1
i
LN  x  r  x  dx
1
• But we also know that if r is a lower order polynomial than
(p+1)’th order, it can be expressed as a linear combination
of Legendre polynomials {L1, L2, L3 , … , LN }.
• By the orthogonality of the Legendre polynomials we know
that the s is in fact orthogonal to Lp+1
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Hence:
1
i N
 f  x  dx   w f  x 
i
1
i
for all f  P
2 N 1
i 1
i.e. the quadrature is exact for all polynomials
of order up to p=(2N-1)
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Summary of Gauss Quadrature
• We can use the multiple root finder to locate the
roots of the N’th order Legendre polynomial.
• We can then use the Newton-Cotes formula
with the roots of the N’th order Legendre
polynomial to calculate a set of N weights.
• We now have a quadrature !!! which will
integrate polynomials of order 2N-1 with N
points
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Team Exercise
• Use the root finder (gaussNR) and Newton-Cotes routines
(newtoncotes) to build a quadrature for N points (N arbitrary).
• Use it to integrate exp(x) over the interval [-1,1]
• Use it to integrate 1./(1+25*x.^2) over the interval [-1,1]
• For N=2,3,4,5,6,7,8,9 plot the integration error for both
functions on the same graph.
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