ma/cs 375 fall 20021 ma/cs 375 fall 2002 lecture 32
TRANSCRIPT
MA/CS 375 Fall 2002 1
MA/CS 375
Fall 2002
Lecture 32
MA/CS 375 Fall 2002 2
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.
Review
MA/CS 375 Fall 2002 3
Newton Scheme For Multiple Root Finding
1 2 P
1
1
Initiate guesses to the roots ,x ,..x
Loop over k=1:P
Iterate:
1
to find to a given tolerance
End loop
kk k i k
kk
i k i
k
x
f xx x
df xf x
dx x x
x
Review
MA/CS 375 Fall 2002 4
Mul
tiple
Roo
t Fin
der
(app
lied
to fi
nd
ro
ots
of L
ege
ndre
po
lyn
om
ials
)
Should read abs(delta) > tol
Review + Correction
MA/CS 375 Fall 2002 5
Legendre Polynomials
• Legendre polynomials are a special set of polynomials which are orthogonal in the L2 inner product:
1
n
1
L L 0 if mx x dx n m
Review
MA/CS 375 Fall 2002 6
Legendre Polynomials
• Legendre polynomials can be calculate using the following recursion relation:
0
1
n 1 n n 1
L 1
L
2 1L L L n=1,2,...
1 1
x
x x
n nx x x x
n n
Review
MA/CS 375 Fall 2002 7
Roots of the 10th Order Legendre Polynomial
Notice how they cluster at the end points
Review
MA/CS 375 Fall 2002 8
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
MA/CS 375 Fall 2002 9
Example of Quadrature
• Say we wish to calculate an approximation to the integral of f over [-1,1] :
• 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
1
f x dx
1
11
i N
i ii
f x dx w f x
MA/CS 375 Fall 2002 10
Example: Simpson’s RuleRecall:
• 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.
MA/CS 375 Fall 2002 11
Example: Simpson’s Rule
1
1
1 2 3 4
2 4 2 4 ...
3( 1) N
f x dx
f x f x f x f x f xN
1 2 3 4
1 1 1 1 1
1
2 4 6 1
3 5 7 2
nodes { , , , , , }
11 2
1
weights , , , , ,
2,
3 1
8, , , ,
3 1
4, , , ,
3 1
N
n
N
N
N
x x x x x
nx
N
w w w w w
w wN
w w w wN
w w w wN
quadrature:
MA/CS 375 Fall 2002 12
Example: Simpson’s Rule
1
1
1 2 3 4
2 4 2 4 ...
3( 1) N
f x dx
f x f x f x f x f xN
becomes:
1
1 1 2 2 3 3
1
.. N Nf x dx w f x w f x w f x w f x
in summation notation:
1
11
n N
n nn
f x dx w f x
MA/CS 375 Fall 2002 13
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:
11 10 0 0 0
1 1 2 2
1 11N N
xw x w x w x x dx
MA/CS 375 Fall 2002 14
11 10 0 0 0
1 1 2 2
1 1
11 21 1 1 1
1 1 2 2
1 1
111 1 1 1
1 1 2 2
1 1
1
2
N N
N N
NN N N N
N N
xw x w x w x x dx
xw x w x w x x dx
xw x w x w x x dx
N
Now we require that 1,x,x2,..,xN-1 are integrated exactly
MA/CS 375 Fall 2002 15
11
0 0 011 2 22
1 1 121 2
1 1 11 2
1 1
1
1 1
2
1 1
N
N
N N NNN NN
wx x x
wx x x
wx x x
N
In Matrix Notation:
Notice anything familiar?
MA/CS 375 Fall 2002 16
11
0 0 011 2 22
1 1 121 2
1 1 11 2
1 1
1
1 1
2
1 1
N
N
N N NNN NN
wx x x
wx x x
wx x x
N
tV w
It’s the transpose of the Vandermonde matrix
MA/CS 375 Fall 2002 17
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.
MA/CS 375 Fall 2002 18
Newton-Cotes Weights
11
1 22
2
1 1
1
1 1
2
1 1
t
N NN
w
w
w
N
1V
MA/CS 375 Fall 2002 19
Using Newton-Cotes Weights
1
11
i Nt
i ii
f x dx w f x
w f
MA/CS 375 Fall 2002 20
Using Newton-Cotes Weights(Interpretation)
1
11
1 21 21 1 1 1 1 1
1 2
i Nt
i ii
NN
f x dx w f x
N
1
w f
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.
MA/CS 375 Fall 2002 21
Matlab Function for Calculating Newton-Cotes Weights
MA/CS 375 Fall 2002 22
Demo: Matlab Function for Calculating Newton-Cotes Weights
1) set N=5 points2) build equispaced nodes3) calculate NC weights
4) evaluate F=X^3 at nodes5) evaluate integral
6) F is anti-symmetric on [-1,1] so its integral is 0
7) Answer correct
MA/CS 375 Fall 2002 23
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
MA/CS 375 Fall 2002 24
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.
MA/CS 375 Fall 2002 25
2 1
1
where:
f 1,1
where 1,1
0 where s 1,1
1,1
p
pi i
pi
p
f x If x r x s x
If x f x If
s x
r
P
P
P
P
Suppose:
MA/CS 375 Fall 2002 26
2 1
1
where:
f 1,1
where 1,1
0 where s 1,1
1,1
p
pi i
pi
p
f x If x r x s x
If x f x If
s x
r
P
P
P
P
Suppose:Remainder term, whichmust have p roots locatedat the interpolating nodes
MA/CS 375 Fall 2002 27
1 1 1
1 1 1
1
1 1
i N
i ii
f x If x r x s x
f x dx If x dx r x s x dx
w f x r x s x dx
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!.
Let’s integrate this formula for f over [-1,1]
MA/CS 375 Fall 2002 28
1 1 1
1 1 1
1 1
1 1
1
1 1
N
i N
i i Ni
f x dx If x dx s x r x dx
If x dx L x r x dx
w f x L x r x dx
• 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
MA/CS 375 Fall 2002 29
1
2 1
11
for all i N
Ni i
i
f x dx w f x f
P
i.e. the quadrature is exact for all polynomials of order up to p=(2N-1)
Hence:
MA/CS 375 Fall 2002 30
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
MA/CS 375 Fall 2002 31
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.