## Description

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

q=midpointquad(sq,0,1,11)

or you could write it without giving the function a name as

q=midpointquad(@(x) x.^2,0,1,11)

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

quadrature rule. The following statement

q=midpointquad(@(x) 5*x.^4,0,1,11);1-q

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

err20=midpointquad(@runge,-5,5,20)-2*atan(5);

err40=midpointquad(@runge,-5,5,40)-2*atan(5);

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

2. function quad = midpointquad( func, a, b, N)

3. % quad = midpointquad( func, a, b, N)

4. % comments

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

your m-file should be

2. function quad = trapezoidquad( func, a, b, N )

3. % quad = trapezoidquad( func, a, b, N )

4. % more comments

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:

1. Download nc_weight.m.

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 )

5. % more comments

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

might not produce accurate quadratures.

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

function y=partly_quadratic(x)

% y=partly_quadratic(x)

% 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.

2. function quad = nc_quad( func, a, b, N,

numSubintervals)

3. % quad = nc_quad( func, a, b, N, numSubintervals)

4. % comments

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].

13. Subin- nc_quad

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.

Gauss Quadrature

Like Newton-Cotes quadrature, Gauss-Legendre quadrature interpolates the integrand by a

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

to nc_single.m and nc_quad.m.

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 )

5. % comments

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

functions to simplify your work.)

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.