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M382 – PROJECT PROJECT 2

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M382 – PROJECT
PROJECT 2
I want you submit
1. Diary file
2. Summary file (there is an example in the attached file. It can be text file or doc
files)
3. Submit m-files and jpeg files .
When you submit your project, I want you to write all of group’s members names who
contribute it.
NOTE: THERE ARE SOME M. FILES YOU CAN USE IT. YOU CAN FIND THEM
IN THE FOLDER.
Introduction
The term “numerical quadrature” refers to the estimation of an area, or, more generally, any
integral. (You might hear the term “cubature” referring to estimating a volume in three
dimensions, but most people use the term “quadrature.”) We might want to integrate some
function or a set of tabulated data. The domain might be a finite or infinite interval, it
might be a rectangle or irregular shape, it might be a multi-dimensional volume.
We first discuss the “degree of exactness” (sometimes called the “degree of precision”) of a
quadrature rule, and its relation to the “order of accuracy.” We consider some simple tests to
measure the degree of exactness and the order of accuracy of a rule, and then describe (simple
versions of) the midpoint and trapezoidal rules. Then we consider the Newton-Cotes and
Gauss-Legendre families of rules, and discuss how to get more accurate approximations by
either increasing the order of the rule or subdividing the interval. Finally, we will consider a
way of adapting the details of the integration method to meet a desired error estimate. In the
majority of the exercises below, we will be more interested in the error (difference between
calculated and known value) than in the calculated value itself.
A word of caution. We discuss three similar-sounding concepts:
 “Degree of exactness:” the largest value of n so that all polynomials of degree n and
below are integrated exactly. (Degree of a polynomial is the highest power
of x appearing in it.)
 “Order of accuracy:” the value of n so that the error is , where measures the
subinterval size.
 “Index:” a number distinguishing one of a collection of rules from another.
These can be related to one another, but are not the same thing.
Matlab hint
As you recall, Matlab provides the capability of defining “anonymous” functions, using @
instead of writing m-files to do it. This feature is very convenient when the function to be
defined is very simple-a line of code, say-or when you have a function that requires several
arguments but you only are interested in varying one of them. You can find out about
anonymous functions on on-line reference for function_handle Suppose, for example,
you want to define a function sq(x)=x^2. You could do this by writing the following:
[email protected](x) x.^2; % define a function using @
You could then use sq(x) later, just as if you had defined it in an m-file. sq is a “function
handle” and can be used wherever a function handle is used, such as in a call from another
function. Remember, though, that another @ should not appear before the name. In the next
section you will be writing an integration function named midpoint that requires a function
handle as its first argument If you wanted to apply it to the integral , you might write
or you could write it without giving the function a name as
There is a nice way to use this form to streamline a sequence of calculations computing the
integrals of ever higher degree polynomials in order to find the degree of exactness of a
first computes using the midpoint rule, and then prints the error (=1-q because the
exact answer is 1). You would only have to change 5*x.^4 into 4*x.^3 to check the error
in , and you can make the change with judicious use of the arrow and other
keyboard keys.
Reporting Errors
Errors should be reported in scientific notation (like 1.234e-3, not .0012). You can force
Matlab to display numbers in this format using the command format short e (or
format long e for 15 decimal places). This is particularly important if you want to
visually estimate ratios of errors.
Computing ratios of errors should always be done using full precision, not the value printed
on the screen. For example, you might use code like
ratio=err20/err40
to get a ratio of errors without loss of accuracy due to reading numbers off the computer
screen.
When I compute ratios of this nature, I find it easier to compute them as “larger divided by
smaller,” yielding ratios larger than 1. It is easier to recognize that 15 is nearly (=16) than
to recognize that .0667 is nearly (=0.0625).
The Midpoint Method
In general, numerical quadrature involves breaking an interval into subintervals,
estimating or modelling the function on each subinterval and integrating it there, then adding
up the partial integrals.
Perhaps the simplest method of numerical integration is the midpoint method (presented by
Quarteroni, Sacco, and Saleri on p. 381). This method is based on interpolation of the
integrand by the constant and multiplying by the width of the interval. The
result is a form of Riemann sum that you probably saw in elementary calculus when you first
studied integration.
Break the interval into subintervals with endpoints (there
is one more endpoint than intervals, of course). Then the midpoint rule can be written as
Midpoint rule
(1)
In the exercise that follows, you will be writing a Matlab function to implement the midpoint
rule.
Exercise 1:
1. Write a function m-file called midpointquad.m with signature
5.
6. % your name and the date
where f indicates the name of a function, a and b are the lower and upper
limits of integration, and N is the number of points, not the number of intervals.
The code for your m-file might look like the following:
xpts = linspace( ??? ) ;
h = ??? ; % length of subintervals
xmidpts = 0.5 * ( xpts(1:N-1) + xpts(2:N) );
fmidpts = ???
quad = h * sum ( fmidpts );
2. Test your midpointquad routine by computing . Even if you use only
one interval (i.e. N=2) you should get the exact answer because the midpoint rule
integrates linear functions exactly.
3. Use your midpoint routine to estimate the integral of our friend, the Runge
function, , over the interval . (If you do not have a copy of
the Runge function handy, you can runge.m. in the file.) The exact answer
is 2*atan(5). Fill in the following table, using scientific notation for the error
values so you can see the pattern.
4. N h Midpoint Result Error
5.
6. 11 1.0 _________________ __________________
7. 101 0.1 _________________ __________________
8. 1001 0.01 _________________ __________________
9. 10001 0.001 _________________ __________________
10. Estimate the order of accuracy (an integer power of h) by examining the behavior of
the error when h is divided by 10. (In previous labs, we have estimated such orders by
repeatedly doubling the number of subintervals. Here, we multiply by ten. The idea is
the same.)
Exactness
If a quadrature rule can compute exactly the integral of any polynomial up to some specific
degree, we will call this its degree of exactness. Thus a rule that can correctly integrate any
cubic, but not quartics, has exactness 3. Quarteroni, Sacco, and Saleri mention it on p. 429.
To determine the degree of exactness of a rule, we might look at the approximations of the
integrals
Exercise 2:
1. To study the degree of exactness of the midpoint method, use a single interval
(i.e. N = 2), and estimate the integrals of the test functions over [0,1]. The
exact answer is 1 each time.
2. func Midpoint Result Error
3.
4. 1 ___________________ ___________________
5. 2 * x ___________________ ___________________
6. 3 * x^2 ___________________ ___________________
7. 4 * x^3 ___________________ ___________________
8. What is the degree of exactness of the midpoint rule?
9. Recall that you computed the order of accuracy of the midpoint rule in
Exercise 1. For some methods, but not all, the degree of exactness is one less
than the order of accuracy. Is that the case for the midpoint rule?
The Trapezoid Method
The trapezoid rule breaks [a,b] into subintervals, approximates the integral on each
subinterval as the product of its width times the average function value, and then adds up all
the subinterval results, much like the midpoint rule. The difference is in how the function is
approximated. The trapezoid rule can be written as
Trapezoid rule
(2)
If you compare the midpoint rule (1) and the trapezoid rule (2), you will see that the midpoint
rule takes at the midpoint of the subinterval and the trapezoid takes the average of at the
endpoints. If each of the subintervals happens to have length , then the trapezoid rule
becomes
(3)
To apply the trapezoid rule, we need to generate points and evaluate the function at each
of them. Then, apply either (2) or (3) as appropriate.
Exercise 3:
1. Use your midpointquad.m m-file as a model and write a function m-file
called trapezoidquad.m to evaluate the trapezoid rule. The signature of
5.
6. % your name and the date
You may use either form of the trapezoid rule.
7. To test your routine and to study the exactness of the trapezoid rule, use a
single interval (N = 2), and estimate the integrals of the same test functions
used for the midpoint rule over [0,1]. The exact answer should be 1 each
time.
8. func Trapezoid Result Error
9.
10. 1 ___________________
___________________
11. 2 * x ___________________
___________________
12. 3 * x^2 ___________________
___________________
13. 4 * x^3 ___________________
___________________
14. What is the degree of exactness of the trapezoid rule?
15. Use the trapezoid method to estimate the integral of the Runge function
over , using the given values of N, and record the error using scientific
notation.
16. N h Trapezoid Result Error
17.
18. 11 1.0 _________________
__________________
19. 101 0.1 _________________
__________________
20. 1001 0.01 _________________
__________________
21. 10001 0.001 _________________
__________________
22. Estimate the rate at which the error decreases as decreases. (Find the power
of that best fits the error behavior.) This is the order of accuracy of the rule.
23. For some methods, but not all, the degree of exactness is one less than the
order of accuracy. Is that the case for the trapezoid rule?
Singular Integrals
The midpoint and trapezoid rules seem to have the same exactness and about the same
accuracy. There is a difference between them, though. Some integrals have perfectly welldefined values even though the integrand has some sort of mild singularity. There are some
sophisticated ways to perform these integrals, but there is a simple way that can be adequate
for the case that the singularity appears at the endpoint of an interval. Something is lost,
however.
Consider the integral
where refers to the natural logarithm. Note that the integrand “is infinite” at the left
endpoint, so you could not use the trapezoid rule to evaluate it. The midpoint rule,
conveniently, does not need the endpoint values.
Exercise 4: Apply the midpoint rule to the above integral, and fill in the following
table.
n h Midpoint Result Error
11 0.1 _________________ __________________
101 0.01 _________________ __________________
1001 0.001 _________________ __________________
10001 0.0001 _________________ __________________
Estimate the rate of convergence (power of ) as . You should see that the
singularity causes a loss in the rate of convergence.
.
Newton-Cotes Rules
Look at the trapezoid rule for a minute. One way of interpreting that rule is to say that if the
function is roughly linear over the subinterval , then the integral of is the
integral of the linear function that agrees with (i.e., interpolates ) at the endpoints of the
interval. What about trying higher order methods? It turns out that Simpson’s rule can be
derived by picking triples of points, interpolating the integrand by a quadratic polynomial,
and integrating the quadratic. The trapezoid rule and Simpson’s rule are Newton-Cotes rules
of index one and index two, respectively. In general, a Newton-Cotes formula uses the idea
that if you approximate a function by a polynomial interpolant on uniformly-spaced points in
each subinterval, then you can approximate the integral of that function with the integral of
the polynomial interpolant. This idea does not always work for derivatives but usually does
for integrals. The polynomial interpolant in this case being taken on a uniformly distributed
set of points, including the end points. The number of points used in a Newton-Cotes rule is a
fundamental parameter, and can be used to characterize the rule. The “index” of a NewtonCotes rule is commonly defined as one fewer than the number of points it uses, although this
common usage is not universal.
We applied the trapezoid rule to an interval by breaking it into subintervals and repeatedly
applying a simple formula for the integral on a single subinterval. Similarly, we will be
constructing higher-order rules by repeatedly applying Newton-Cotes rules over subintervals.
But Newton-Cotes formulæ are not so simple as the trapezoid rule, so we will first write a
helper function to apply the rule on a single subinterval.
Over a single interval, all (closed) Newton-Cotes formulæ can be written as
where is a function and are evenly-spaced points between and . The
weights can be computed from the Lagrange interpolation polynomials as
(The Lagrange interpolation polynomials arise because we are doing a polynomial
interpolation.) The weights do not depend on , and depend on and in a simple manner,
so they are often tabulated for the unit interval. In the exercise below, I will provide them to
you in the form of a function.
Remark: There are also open Newton-Cotes formulæ that do not require values at endpoints,
which we will not consider.
Remark: There are also open Newton-Cotes formulæ that do not require values at endpoints,
but there is not time to consider them in this lab.
Exercise 5:
2. Write a routine called nc_single.m with the signature
3. function quad = nc_single ( func, a, b, N )
4. % quad = nc_single ( func, a, b, N )
6.
7. % your name and the date
There are no subintervals in this case. The coding might look like something
like this:
xvec = linspace ( a, b, N );
wvec = nc_weight ( N );
fvec = ???
quad = (b-a) * sum(wvec .* fvec);
8. Test your function by showing its exactness is at least 1 for
N=2: exactly.
9. Fill in the following table by computing the integrals over [0,1] of the
indicated integrands using nc_single. (Quarteroni, Sacco, and Saleri,
Theorem 9.2) indicates that the degree of exactness is equal to the (N-1) when
n is even and the degree of exactness is N when N is odd . Your results should
agree, further confirming that your function is correct. (Hint: You can use
anonymous functions to simplify your work.)
10. func Error Error Error
11. N=4 N=5 N=6
12. 4 * x^3 __________ __________ ___________
13. 5 * x^4 __________ __________ ___________
14. 6 * x^5 __________ __________ ___________
15. 7 * x^6 __________ __________ ___________
16. Degree ___ ___ ___
The objective of numerical quadrature rules is to accurately approximate integrals. We have
already seen that polynomial interpolation on uniformly spaced points does not always
converge, so it should be no surprise that increasing the order of Newton-Cotes integration
Exercise 6: Attempt to get accurate estimates of the integral of the Runge function
over the interval [-5,5]. Recall that the exact answer is 2*atan(5). Fill in the
following table
n nc_single Result Error
3 _________________ __________________
7 _________________ __________________
11 _________________ __________________
15 _________________ __________________
The results of Exercise 6 should have convinced you that you raising in a Newton-Cotes
rule is not the way to get increasing accuracy. One alternative to raising is breaking the
interval into subintervals and using a Newton-Cotes rule on each subinterval. This is the idea
of a “composite” rule. In the following exercise you will use nc_single as a helper
function for a composite Newton-Cotes routine. You will also be using the “partly quadratic”
function from Lab 9:

whose Matlab implementation is
% input x (possibly a vector or matrix)
% output y, where
% for x<=0, y=0
% for x>0, y=x(1-x)
y=(heaviside(x)-heaviside(x-1)).*x.*(1-x);
Clearly,
Exercise 7:
1. Write a function m-file called nc_quad.m to perform a composite NewtonCotes integration. Use the following signature.
numSubintervals)
5.
6. % your name and the date
This function will perform these steps: (1) break the interval
into numSubintervals subintervals; (2) use nc_single to integrate over
each subinterval; and, (3) add them up.
7. The most elementary test to make when you write this kind of routine is to
check that you get the same answer when numSubintervals=1 as you
would have obtained using nc_single. Choose at least one line from the
table in Exercise 6 and make sure you get the same result using nc_quad.
8. Test your routine by computing using at
least N=3 and numSubintervals=2. Explain why your result should have
an error of zero or roundoff-sized.
9. Test your routine by computing using at
least N=3 and numSubintervals=3. Explain why your result
should not have an error of zero or roundoff-sized.
10. Test your routine by checking the following value
11. nc_quad(@runge, -5, 5, 4, 10) = 2.74533025
12. Fill in the following table using the Runge function on [-5,5].
14. tervals N Error Err ratio
15.
16. 10 2 _____________ __________
17. 20 2 _____________ __________
18. 40 2 _____________ __________
19. 80 2 _____________ __________
20. 160 2 _____________ __________
21. 320 2 _____________
22.
23. 10 3 _____________ __________
24. 20 3 _____________ __________
25. 40 3 _____________ __________
26. 80 3 _____________ __________
27. 160 3 _____________ __________
28. 320 3 _____________
29.
30. 10 4 _____________ __________
31. 20 4 _____________ __________
32. 40 4 _____________ __________
33. 80 4 _____________ __________
34. 160 4 _____________ __________
35. 320 4 _____________
36. For each index, estimate the order of convergence by taking the sequence of
ratios of the error for num subintervals divided by the error
for (2*num) subintervals and guessing the power of two that best
approximates the limit of the sequence.
In the previous exercise, the table served to illustrate the behavior of the integration routine.
Suppose, on the other hand, that you had an integration routine and you wanted to be sure it
had no errors. It is not good enough to just see that you can get “good” answers. In addition, it
must converge at the correct rate. Tables such as the previous one are one of the most
powerful debugging and verification tools a researcher has.
polynomial and integrates the polynomial. Instead of uniformly spaced points, GaussLegendre uses optimally-spaced points. Furthermore, Gauss-Legendre converges as degree
gets large, unlike Newton-Cotes, as we saw above. Of course, in real applications, one does
not use higher and higher degrees of quadrature; instead, one uses more and more
subintervals, each with some fixed degree of quadrature.
The disadvantage of Gauss-Legendre quadrature is that there is no easy way to compute the
node points and weights. Tables of values are generally available. We will be using a Matlab
function to serve as a table of node points and weights.
Normally, Gauss-Legendre quadrature is characterized by the number of integration points.
For example, we speak of “three-point” Gauss.
The following two exercises involve writing m-files analogous
Exercise 8:
1. Download the file gl_weight.m. This file returns both the node points and
weights for Gauss-Legendre quadrature for points.
2. Write a routine called gl_single.m with the signature
3. function quad = gl_single ( func, a, b, N )
4. % quad = gl_single ( func, a, b, N )
6.
7. % your name and the date
As with nc_single there are no subintervals in this case. Your coding might
look like something like this:
[xvec, wvec] = gl_weight ( a, b, N );
fvec = ???
quad = sum( wvec .* fvec );
8. Test your function by showing its exactness is at least 1 for N=1 and one
interval: exactly. If the exactness is not at least 1, fix your code
now.
9. Fill in the following table by computing the integrals over [0,1] of the
indicated integrands using gl_single. It is known that the degree of
exactness of the method is , and your results should agree, further
confirming that your function is correct. (Hint: You can use anonymous
10. f Error Error
11. N=2 N=3
12. 3 * x^2 __________ ___________
13. 4 * x^3 __________ ___________
14. 5 * x^4 __________ ___________
15. 6 * x^5 __________ ___________
16. 7 * x^6 __________ ___________
17. Degree ___ ___
18. Get accuracy estimates of the integral of the Runge function over the interval [-
5,5]. Recall that the exact answer is 2*atan(5). Fill in the following table
19. N gl_single Result Error
20.
21. 3 _________________ __________________
22. 7 _________________ __________________
23. 11 _________________ __________________
24. 15 _________________ __________________
You might be surprised at how much better Gauss-Legendre integration is than NewtonCotes, using a single interval. There is a similar advantage for composite integration, but it is
hard to see for small N. When Gauss-Legendre integration is used in a computer program, it is
generally in the form of a composite formulation because it is difficult to compute the weights
and integration points accurately for high order Gauss-Legendre integration. The efficiency of
Gauss-Legendre integration is compounded in multiple dimensions, and essentially all
computer programs that use the finite element method use composite Gauss-Legendre
integration rules to compute the coefficient matrices.