Newton Cotes method

8/14/2019 Newton Cotes method

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Newton Cotes Method


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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 Newtons method.

Review


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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

k

k ki k

k

k

i k i

k

x

f xx x

df xf x

dx x xx

Review


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MultipleRo

otFinder

(appliedto

findrootsofL

egendrepolyn

omials)

Should read abs(delta) > tol

Review + Correct ion


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Legendre Polynomials

Legendre polynomials are a special set

of polynomials which are orthogonal in

the L2 inner product:

1

n

1

L L 0 ifmx x dx n m

Review


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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 xn n

Review


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Roots of the 10thOrder

Legendre Polynomial

Notice how they cluster at the end points

Review


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Numerical Quadrature

A numerical quadrature is a set of two vectors.

The first vector is a list of x-coordinates fornodes where a function is to be evaluated.

The second vector is a set of integrationweights, 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 tothe 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,..,wNso that:

1

1

f x dx

1

11

i N

i i

i

f x dx w f x


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Example: Simpsons 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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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 21

weights , , , , ,

2,

3 18

, , , ,3 1

4, , , ,

3 1

N

n

N

N

N

x x x x x

nxN

w w w w w

w w

N

w w w wN

w w w wN

quadrature:


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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 n

n

f x dx w f x


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Newton-Cotes Formula

The next approach we are going to use is thewell known Newton-Cotes quadrature.

Suppose we are given a set of pointsx1,x2,..,xN. Then we require that the constant

is exactly integrated:

11 10 0 0 0

1 1 2 2

1 11

N N

xw x w x w x x dx


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11 10 0 0 0

1 1 2 2

1 1

1

1 21 1 1 1

1 1 2 2

1 1

11

1 1 1 1

1 1 2 2

1 1

1

2

N N

N N

N

N 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


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11

0 0 011 2 22

1 1 1

21 2

1 1 1

1 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?


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11

0 0 0

11 2 221 1 1

21 2

1 1 11 2

1 1

1

1 1

2

1 1

N

N

N N N NN NN

wx x xwx x x

wx x x

N

tV w

Its the transpose of the

Vandermonde matrix


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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 ofthe area under f

Clearly, we can estimate the approximationerror using the estimates for the error in the

interpolation we used before.


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Newton-Cotes Weights

11

1 22

2

1 1

1

1 1

2

1 1

t

N NN

ww

w

N

1

V


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Using Newton-Cotes Weights

1

11

i Nt

i i

i

f x dx w f x

w f


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Using Newton-Cotes Weights

(Interpretation)

1

11

1 21 21 1 1 1 1 1

1 2

i Nt

i i

i

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.


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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 ofthis directory

using a script figure out what order polynomial the

weights produced with newtoncotes can exactlyintegrate for a given set of N points (say

N=3,4,5,6,7,8) created with l inspace


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Gauss Quadrature

The construction of the Newton-Cotes

weights does no tutilize 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 beintegrated exactly using just N points.


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2 1

1

where:

f 1,1

where 1,1

0 where s 1,1

1,1

p

p

i i

p

i

p

f x If x r x s x

If x f x If

s x

r

P

P

P

P

Suppose:

Remainder term which


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2 1

1

where:

f 1,1

where 1,1

0 where s 1,1

1,1

p

p

i i

p

i

p

f x If x r x s x

If x f x If

s x

r

P

P

P

P

Suppose:Remainder term, which

must have p roots located

at the interpolating nodes


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1 1 1

1 1 1

1

1 1

i N

i i

i

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!.

Lets integrate this formula for f over [-1,1]


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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


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1

2 1

11

for alli N

N

i 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:


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Summary of Gauss Quadrature

We can use the multiple root finder to locate theroots of the Nth order Legendre polynomial.

We can then use the Newton-Cotes formulawith the roots of the Nth order Legendrepolynomial to calculate a set of N weights.

We now have a quadrature !!! which willintegrate polynomials of order 2N-1 with Npoints


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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 bothfunctions on the same graph.

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Newton Cotes method

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