MATH2070: LAB 9: Quadrature

Introduction	Exercise 1
Matlab hint	Exercise 2
The Midpoint Method	Exercise 3
Exactness	Exercise 4
The Trapezoid Method	Exercise 5
Singular Integrals	Exercise 6
Newton-Cotes Rules	Exercise 7
Gauss Quadrature	Exercise 8
Adaptive quadrature	Exercise 9
Integration by Monte Carlo methods (Extra)	Exercise 10
	Exercise 11
	Exercise 12
	Exercise 13
	Exercise 14
	Exercise 15
	Extra Credit

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 $f(x)$ 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 $O(h^n)$, where $h$ 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.

This lab will take four sessions. If you print this lab, you may prefer to use the pdf version.

Matlab hint

Matlab provides the capability of defining functions ``in line'' instead of writing m-files to do it. This feature is most 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. One way to do this is to use the inline function. (The full help description is available from the command line using help inline, from the help facility on your PC, or from the Mathworks on-line reference.) (More recent versions of Matlab include a more general facility called anonymous functions, defined using the @ symbol. You may use this facility if you wish, but it does not work on the computers in GSCC.) Suppose, for example, you want to define a function sq(x)=x^2. You could do this by writing

  sq=inline('x.^2')

(Letters such as x are understood to mean independent variables.) or, less ambiguously, by writing

  sq=inline('x.^2','x')

You could then use sq(x) later, just as if you had defined it in an m-file. Alternatively, you could use it where a function name is needed. In the next section you will be writing an integration function named midpoint that requires a function name as its first argument. If you wanted to apply it to the integral $\int_0^1x^2dx$, you might write

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

There is a nice way to use inline 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(inline('5*x.^4','x'),0,1,11);1-q

first computes $\int_0^1 5x^4dx$ 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 $\int_0^1 4x^3dx$, and you can make the change with judicious use of the arrow keys.

A second remark is that 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. In this lab, you should use full-exactness values when you compute ratios of errros. 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.

The Midpoint Method

In general, numerical quadrature involves breaking an interval $[a,b]$ 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 $f(x)$ by the constant $f(\frac{a+b}2)$ 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 $[a,b]$ into $N-1$ subintervals with endpoints $x_1,x_2,\ldots,x_{N-1},x_N$ (there is one more endpoint than intervals, of course). Then the midpoint rule can be written as

$$=\sum_{k=1}^{N-1} (x_{k+1}-x_k)f(\frac{x_k+x_{k+1}}2).$$ (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

   function quad = midpointquad( f, a, b, n)
   % quad = midpointquad( f, a, b, n)
   % comments
   
   % 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 $\int_0^1 2xdx=1$. 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, $f(x)=1/(1+x^2)$, over the interval $[-5,5]$. (If you do not have a copy of the Runge function handy, you can download my version of runge.m.) 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.

        n  h      Midpoint Result     Error
   
       11  1.0   _________________   __________________
      101  0.1   _________________   __________________
     1001  0.01  _________________   __________________
    10001  0.001 _________________   __________________
   

4. 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

$$\int_0^1 1dx = [x]_0^1$$

$$\int_0^1 2xdx = [x^2]_0^1$$

$$\int_0^1 3x^2dx = [x^3]_0^1$$

$$\vdots$$

$$\int_0^1 (k+1)x^kdx = [x^{k+1}]_0^1$$

The exact value of each one of these integrals is 1.

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.

        f    Midpoint Result        Error
   
     1       ___________________    ___________________
     2 * x   ___________________    ___________________
     3 * x^2 ___________________    ___________________
     4 * x^3 ___________________    ___________________
   

2. What is the degree of exactness of the midpoint rule?
3. 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

$$=\sum_{k=1}^{N-1} (x_{k+1}-x_k)\frac{f(x_k)+f(x_{k+1})}2$$ (2)

If you compare the midpoint rule (1) and the trapezoid rule, you will see that the midpoint rule takes $f$ at the midpoint of the subinterval and the trapezoid takes the average of $f$ at the endpoints. If each of the subintervals happens to have length $h$, then the trapezoid rule becomes

$$\frac{h}{2}f(x_1) + \frac{h}{2}f(x_{N}) + h\sum_{k=2}^{N-1}f(x_k).$$ (3)

To apply the trapezoid rule, we need to generate $N$ 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

   function quad = trapezoidquad( f, a, b, n )
   % quad = trapezoidquad( f, a, b, n )
   % more comments
   
   % your name and the date
   

   You may use either form of the trapezoid rule.
2. 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.

        f    Trapezoid Result       Error
   
     1       ___________________    ___________________
     2 * x   ___________________    ___________________
     3 * x^2 ___________________    ___________________
     4 * x^3 ___________________    ___________________
   

3. What is the degree of exactness of the trapezoid rule?
4. Use the trapezoid method to estimate the integral of the Runge function over $[-5,5]$, using the given values of n, and record the error using scientific notation.

        n  h     Trapezoid Result    Error
   
       11  1.0   _________________   __________________
      101  0.1   _________________   __________________
     1001  0.01  _________________   __________________
    10001  0.001 _________________   __________________
   

5. Estimate the rate at which the error decreases as $h$ decreases. (Find the power of $h$ that best fits the error behavior.) This is the order of accuracy of the rule.
6. 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 well-defined 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

$$I=\int_0^1 \log(x) dx = -1,$$

where $\log$ 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 $h$) as $h\rightarrow0$. 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 $f$ is roughly linear over the subinterval $[x_{k},x_{k+1}]$, then the integral of $f$ is the integral of the linear function that agrees with $f$ (i.e., interpolates $f$) 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 $f$ 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 work very well for derivatives. 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 Newton-Cotes 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

$$\int_a^b f(x) dx \approx Q_n(f)=\sum_{k=0}^n w_{k,n}f(x_k)$$

where $f$ is a function and $x_k$ are $n+1$ evenly-spaced points between $a$ and $b$. The weights $w_{k,n}$ can be computed from the Lagrange interpolation polynomials $\ell_{k,n}$ as

$$w_{k,n}=(b-a)\int_0^1\ell_{k,n}(\xi)d\xi.$$

(The Lagrange interpolation polynomials arise because we are doing a polynomial interpolation. See Quarteroni, Sacco, and Saleri, p. 387 ). The weights do not depend on $f$, and depend on $a$ and $b$ 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.

Exercise 5:

1. Download nc_weight.m.
2. Write a routine called nc_single.m with the signature

   function quad = nc_single ( f, a, b, n )
   % quad = nc_single ( f, a, b, n )
   % more comments
   
   % 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);
   

3. Test your function by showing its exactness is at least 1 for n=2: $\int_0^1 2xdx=1$ exactly.
4. 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) indicate 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 inline to simplify your work.)

        f     Error         Error        Error
               n=4           n=5          n=6
     4 * x^3  __________    __________   ___________
     5 * x^4  __________    __________   ___________
     6 * x^5  __________    __________   ___________
     7 * x^6  __________    __________   ___________
     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

  index  nc_single Result    Error

      3  _________________   __________________
      7  _________________   __________________
     11  _________________   __________________
     15  _________________   __________________

The results of the previous exercise should have convinced you that you need a composite Newton-Cotes rule in order to accurately do numerical integration. In the following exercise you will use nc_single as a helper function for a composite Newton-Cotes routine.

Exercise 7:

1. Write a function m-file called nc_quad.m to perform a composite Newton-Cotes integration. Use the following signature.

   function quad = nc_quad( f, a, b, n, numSubintervals)
   % quad = nc_quad( f, a, b, n, numSubintervals)
   % comments
   
   % 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.
2. 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 5 and make sure you get the same result using nc_quad.
3. Test your routine by checking the following value

     nc_quad('runge', -5, 5, 4, 10) = 2.74533025
   

4. Fill in the following table using the Runge function on [-5,5].

   Subin-         nc_quad
   tervals    n   Error          Err ratio 
   
       10     2   _____________  __________
       20     2   _____________  __________
       40     2   _____________  __________
       80     2   _____________  __________
      160     2   _____________  __________
      320     2   _____________            
   
       10     3   _____________  __________
       20     3   _____________  __________
       40     3   _____________  __________
       80     3   _____________  __________
      160     3   _____________  __________
      320     3   _____________            
   
       10     4   _____________  __________
       20     4   _____________  __________
       40     4   _____________  __________
       80     4   _____________  __________
      160     4   _____________  __________
      320     4   _____________
   

5. For each index, estimate the order of convergence by taking the sequence of ratios of the error for N subintervals divided by the error for (2*N) 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, Gauss-Legendre 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-one uses more and more subintervals, each with some given degree of quadrature.

The disadvantage of Gauss-Legendre quadrature is that there is no simple formula for the node points and weights. See Quarteroni, Sacco, and Saleri, Section 10.2 and their program zplege.m for further information. 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. We speak of three-point Gauss, etc.

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 index=$n$ points.
2. Write a routine called gl_single.m with the signature

   function quad = gl_single ( f, a, b, n )
   % quad = gl_single ( f, a, b, n )
   % comments
   
   % 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 );
   

3. Test your function by showing its exactness is at least 1 for n=1 and one interval: $\int_0^1 2xdx=1$ exactly. If the exactness is not at least 1, fix your code now.
4. Fill in the following table by computing the integrals over [0,1] of the indicated integrands using gl_single. Quarteroni, Sacco, and Saleri, Corollary 10.2, show that the degree of exactness of the method is $2n-1$, and your results should agree, further confirming that your function is correct. (Hint: You can use inline to simplify your work.)

        f       Error        Error
                 n=2          n=3
     3 * x^2  __________   ___________
     4 * x^3  __________   ___________
     5 * x^4  __________   ___________
     6 * x^5  __________   ___________
     7 * x^6  __________   ___________
     Degree      ___          ___
   

5. 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

     index  gl_single Result    Error
   
         3  _________________   __________________
         7  _________________   __________________
        11  _________________   __________________
        15  _________________   __________________
   

You might be surprised at how much better Gauss-Legendre integration is than Newton-Cotes, 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.

Exercise 9:

1. Write a function m-file called gl_quad.m to perform a composite Gauss-Legendre integration. Use the following signature.

   function quad = gl_quad( f, a, b, n, numSubintervals)
   % quad = gl_quad( f, a, b, n, numSubintervals)
   % comments
   
   % your name and the date
   

   This function will perform two steps: (1) break the interval into numSubintervals subintervals; (2) use gl_single to integrate over each subinterval; and, (3) add them up.
2. 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 gl_single. Choose at least one line from the table in the previous exercise (8) and make sure you get the same result using gl_quad.
3. Test your routine by checking the following value

     gl_quad('runge', -5, 5, 4, 10) = 2.7468113
   

4. Fill in the following table using the Runge function on [-5,5].

   Subin-        gl_quad
   tervals index Error          Err ratio 
   
       10    1   ____________  __________
       20    1   ____________  __________
       40    1   ____________  __________
       80    1   ____________  __________
      160    1   ____________  __________
      320    1   ____________            
   
       10    2   ____________  __________
       20    2   ____________  __________
       40    2   ____________  __________
       80    2   ____________  __________
      160    2   ____________  __________
      320    2   ____________            
   
       45    3   ____________  __________
       90    3   ____________
   
       46    3   ____________  __________
       92    3   ____________
   
       47    3   ____________  __________
       94    3   ____________
   
       48    3   ____________  __________
       96    3   ____________
   
       49    3   ____________  __________
       98    3   ____________
   

5. For indices 1 and 2, estimate the order of convergence by taking the sequence of ratios of the error for N subintervals divided by the error for (2*N) subintervals and guessing the power of two that best approximates the limit of the sequence.
6. Estimate the order of accuracy for index 3 by taking the five ratios of errors $r_k=e_k/e_{2k}$ for $k=45,\dots,49$, take their geometric mean $r=(\prod_{k=45}^{49}r_k)^{1/5}$, and guess the power of two that best approximates $r$. The reason that this case is different from the others is that the errors become near roundoff and averaging is necessary to smooth out the resulting values. Geometric averaging is appropriate because the theoretical error curve is a straight line on a log-log plot and geometrical averaging is the same as arithmetic averaging of logs.

The proofs you have seen about convergence of Gauss quadrature rely on bounds on higher derivatives of the function. When bounds are not available, higher-order convergence might not be observed.

Exercise 10: Consider the integral from Exercise 4

$$I=\int_0^1 \log(x) dx = -1.$$

1. Use gl_quad to fill in the following table.

                    log(x)
   Subin-        gl_quad
   tervals index Error          Err ratio 
   
       10    1   ____________  __________
       20    1   ____________  __________
       40    1   ____________  __________
       80    1   ____________
   

   What is the order of accuracy of the method using index=1?
2. Use gl_quad to fill in the following table.

                    log(x)
   Subin-        gl_quad
   tervals index Error          Err ratio 
   
       10    2   ____________  __________
       20    2   ____________  __________
       40    2   ____________  __________
       80    2   ____________
   

   What is the order of accuracy of the method using index=2?
3. Use gl_quad to fill in the following table.

                    log(x)
   Subin-        gl_quad
   tervals index Error          Err ratio 
   
       10    3   ____________  __________
       20    3   ____________  __________
       40    3   ____________  __________
       80    3   ____________
   

   What is the order of accuracy of the method using index=3?

Adaptive quadrature

Our final task will consider ``adaptive quadrature.'' Adaptive quadrature employs non-uniform division of the interval of integration into subintervals of non-equal length. It uses smaller subintervals where the integrand is changing rapidly and larger subintervals where it is flatter. The advantage of this approach is that it minimizes the work necessary to compute a given integral. Adaptive quadrature is discussed by Quarteroni, Sacco, and Saleri on page 402. In this section, you will investigate a recursive algorithm for adaptively computing quadratures.

Numerical integration is used often with integrands that are very complicated and take a long time to compute. Although this section will use simple integrands for illustrative purposes, you should think of each evaluation of the integrand (``function call'') to take a long time. Thus, the objective is to reach a given accuracy with a minimum number of function calls. A good strategy for achieving a specified accuracy efficiently is to attempt to uniformly distribute the error over each subinterval. If no interval is particularly bad, there is no obvious place to improve the estimate and if no interval is particularly good, no work has been wasted.

Recall that, if an integration method has degree of exactness $p$, then the local error on an integration interval of length $h$ can be estimated by dividing the interval into two subintervals of length $h/2$ each. Denote by $Q_h$ the integral over the interval of length $h$ and $Q^L_{h/2}$ and $Q^R_{h/2}$ the two estimates of the integral on the left and right subintervals. Then

$$Q_h =Q+Ch^{p+2}f^{(p+1)}(\xi)+O(h^{p+3})$$

$$Q^L_{h/2}+Q^R_{h/2} =Q+C(h/2)^{p+2}(f^{(p+1)}(\xi^L)+f^{(p+1)}(\xi^R)) +O(h^{p+3})$$

where $C$ is a constant, and $\xi$, $\xi^L$, and $\xi^R$ are appropriately chosen. Assuming that $f^{(p+1)}$ is roughly constant, $f^{(p+1)}(\xi)=f^{(p+1)}(\xi^L)= f^{(p+1)}(\xi^R)=f^{(p+1)}$, and that the higher order terms can be neglected yields

$$Q_h =Q+Ch^{p+2}f^{(p+1)}$$ (4)

$$Q^L_{h/2}+Q^R_{h/2} =Q+C(h/2)^{p+2}(2f^{(p+1)}).$$ (5)

Solving Equation (4) for $Q$, plugging it into Equation (5), and defining the error as $Q^L_{h/2}+Q^R_{h/2}-Q$ yields the expression

$$=\frac{\vert Q^L_{h/2}+Q^R_{h/2}-Q_h\vert}{2^{p+1}-1}$$ (6)

The basic structure of one simple adaptive algorithm depends on using the error estimate (6) over each integration subinterval. If the error is acceptably small over each interval, the process stops, and, if not, continues recursively. In the following exercise, you will write a recursive function to implement this procedure.

Exercise 11:

1. Create a function m-file named adaptquad.m with the following code, and fill in the places marked ``???''.

   function [Q,errEst,x,recursions]= ...
            adaptquad(f,x0,x1,tol,recursions)
   % [Q,errEst,x,recursions]=
   %        adaptquad(f,x0,x1,tol,recursions)
   % adaptive quadrature
   %     input parameters
   % f          = function to integrate
   % x0         = left end point
   % x1         = right end point
   % tol        = desired accuracy
   % recursions = number of allowable recursions left
   %
   %     output parameters
   % Q          = estimate of the value of the integral
   % errEst     = estimate of error in Q
   % x          = all intermediate integration points
   % recursions = minimum number of recursions remaining
   %              after convergence
   
   % Add a mid-point and re-estimate integral
   xmid=(x0+x1)/2;
   
   % Qleft and Qright are integrals over two halves
   N=3;
   Qboth=gl_single(f,x0,x1,N);
   Qleft=gl_single(f,x0,xmid,N);
   Qright=gl_single( ??? );
   
   % p=degree of exactness of Gauss-Legendre
   p=2*N-1;  
   errEst= ??? ;
   
   if errEst<tol | recursions<=0  %vertical bar means "or"
     % either ran out of recursions or converged
     Q= ??? ;
     x=[x0 xmid x1];
   else
     % not converged -- do it again
     [Qleft,estLeft,xleft,recursLeft]=adaptquad(f, ...
                    x0,xmid,tol/2,recursions-1);
     [Qright,estRight,xright,recursRight]=adaptquad(f, ...
                     ??? );
     % recursive work is all done, return answers
     % don't want xmid to appear twice in x
     x=[xleft xright(2:length(xright))];  
     Q= ??? ;
     errEst= ??? ;
     recursions=min(recursLeft,recursRight);
   end
   

   Note: The input and output parameter recursions is not theoretically necessary, but is used to guard against infinite recursion. The output vector x is not necessary, either, but will be used to show the effect of the adaptation.

2. Test adaptquad by applying it to the polynomial $f_5(x)=6x^5$ on the interval $[0,1]$, using tol=1.e-5 and recursions=50. Because Gauss-Legendre integration of index 3 is exact for this polynomial, the integral should equal 1 (i.e. error should be zero or roundoff), and recursions=50, and the values of x should be three equally-spaced points in the interval.

3. Test adaptquad by applying it to the polynomial $f_6(x)=7x^6$ on the interval $[0,1]$, using tol=1.e-5 and recursions=50. Because Gauss-Legendre integration of index 3 is not exact for this polynomial, the integral should be close to 1. It turns out that a single set of refinements is performed, so recursions=49, and the values of x should be five equally-spaced points in the interval. The estimated and true errors should be extremely close. Please include both the estimated and true errors in your summary.

4. Test adaptquad by applying it to the Runge function on the interval [-5,5]. Use recursions=50. Recall that the exact answer is 2*atan(5). Fill in the following table

     adaptquad for Runge
    tol    est. error  exact error
    1.e-3  __________  __________
    1.e-6  __________  __________
    1.e-9  __________  __________
   

   You should find that the estimated and exact errors are close in size, and smaller than tol. For the two more accurate cases, the estimated error is slightly larger than the exact error. As you can see, the estimated error is not so good for the case that tol=1.e-3.

Exercise 12: Consider the following situation.

• A quadrature is being attempted with the call

  [Q,estErr,x,recursions]=adaptquad('func',0,1,tol,50);
  

• The estimated error is larger than tol at first, so new calls with tol/2 are made for the intervals $I_{\mbox{left}}=[0,0.5]$ and $I_{\mbox{right}}=[0.5,1]$.
• Assume that the call for the interval $I_{\mbox{left}}$ satisfies the convergence criterion.
• Assume that the call for the interval $I_{\mbox{right}}$ does not satisfy the convergence criterion, thus requiring two more calls to adaptquad. Assume that each of these calls satisfies the convergence criterion.

What are the final values of x and recursions after the adaptquad function has completed its work?

The next few exercises will help you look a little more closely at the results of this recursive adaptive algorithm. Some of the points that will be made are listed below.

• You will see the advantage of adaptive algorithms. They save work over fixed algorithms such as gl_quad.
• You will see the pattern of the integration points take. They can be distributed in a very nonuniform fashion.
• You will see what happens when you try ``harder'' integrands.
• You will see some of the weaknesses of the algorithm.

In the following exercises, you will examine two functions that are more difficult to integrate. The first is a scaled version of the Runge function, $1/(a^2+x^2)$, where $a=10^{-3}$ The value of the integral on [-1,1] of the scaled Runge function is

$$\int_{-1}^1 \frac{1}{a^2+x^2}dx= \frac{2}{a}\tan^{-1}\frac{1}{a}.$$

The scaled Runge function has a peak value of $1/a^2=10^6$, so is much more strongly peaked than the unscaled Runge function.

The second function is $\sqrt{\vert x\vert}$, and the value of its integral over the interval [-1,1] is

$$\int_{-1}^1 \sqrt{\vert x\vert}dx=\frac{4}{3}.$$

This function has a singularity in its derivatives at $x=0$, thus invalidating the proof that the error estimator is reliable.

A third function that is difficult to integrate is $x^{-0.99}$. This function has an integrable singularity at $x=0$ that is close to being nonintegrable.

Exercise 13:

1. Write a function m-file named srunge.m for the scaled Runge function $f(x)=1/(a^2+x^2)$ with $a=10^{-3}$.
2. Evaluate the integral of srunge on the interval [-1,1] using the following call.

   [Q,estErr,x,recursions]=adaptquad('srunge',-1,1,1.e-10,50);
   

   What are the estimated and true errors? Is recursions larger than zero?
3. Use gl_quad with index=3 on the scaled Runge function. Use trial-and-error to find the value of numSubintervals that yields an error roughly comparable to estErr from adaptquad. How does this compare with length(x)-1, the number of subintervals that adaptquad used?
4. Plot the sizes of the subintervals that adaptquad used with the following command.

   xave=(x(2:end)+x(1:end-1))/2;
   dx= x(2:end)-x(1:end-1);
   semilogy(xave,dx,'*')
   

   A semilog plot is appropriate here because of the wide range of interval sizes. Please include this plot with your summary.
5. What are the lengths of the largest and smallest intervals? Explain (one sentence) where you might expect to find the smallest intervals for an arbitrary function.

Exercise 14:

1. Approximate the integral of the function $f(x)=\sqrt{\vert x\vert}$ over the interval [-1,1] to a tolerance of 1.e-10. The exact value of this integral is 4/3. What are the estimated and true errors?
2. What is the returned value of recursions? It should be positive, indicating that the subinterval convergence criterion was always reached successfully.
3. Plot the subinterval sizes using the following command

   xave=(x(2:end)+x(1:end-1))/2;
   dx= x(2:end)-x(1:end-1);
   semilogy(xave,dx,'*')
   

   where x is replaced by the variable name you used. A semilog plot is appropriate here because of the wide range of interval sizes. Please include this plot with your summary.

In the following exercise you will apply adaptquad to a function that is almost not integrable. You will see the benefit of the recursions variable, whose reduction to 0 indicates that convergence was not achieved.

Exercise 15:

1. Use adaptquad to approximate the integral
   $$\int_0^1 x^{-0.99}dx=100$$
   to a tolerance of 1.e-10. Use recursions=50. Notice that the integration interval is [0,1]. What is the computed value of the integral? What are the estimated error and the true error? What is the value returned for recursions?
2. Do the same approximation starting with recursions=100. What are the computed value of the integral, the estimated error, the true error, and the returned value of recursions? (You may have to use the command set(0,'RecursionLimit',200) in order that Matlab will allow sufficient recursions.) Warning: This may take more than a minute on the computers in GSCC.
3. If the variable recursions were not used, this recursive function would ``never'' terminate because the convergence test would never be satisfied. (Actually, it would abort because there is a practical limit on the recursion depth.) The reason is that in the presence of the singularity, halving the interval does result in a reduction of error, but the reduction is half or less (c.f. Exercise 10), so it never passes the convergence test.

Extra credit: Integration by Monte Carlo methods (8 points)

There is another approach to approximating integrals, one that can be used even when the integrand is not smooth or piecewise smooth, when the integral is taken over a region $\Omega\in\mathbb{R}^n$ that is not easily characterized or when its dimension, $n$ is high. This versatility comes at a cost: the method is probabalistic and also slowly convergent. It is called the ``Monte Carlo'' method.

The basic idea behind the Monte Carlo method is that the formula for computing the average value of a function over a region

$$\langle f\rangle = \frac{1}{\vert\Omega\vert} \int_\Omega f$$

can be used to compute the integral on the right side if the average on the left side is known.

The Monte Carlo method is discussed briefly by Quarteroni, Sacco, and Saleri on page 416-417, and in many places on the web. You can find a very clear description, with considerable detail at
http://farside.ph.utexas.edu/teaching/329/lectures/node109.html.

Eric Weisstein has an article in MathWorld,
http://mathworld.wolfram.com/MonteCarloIntegration.html that states, for $\Omega\in\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$

$$\int_\Omega f(\mathbf{x}) dx_1\ldots dx_n \approx \vert\Omega\ver... ...gle\pm \vert\Omega\vert\sqrt{\frac{\langle f^2\rangle - \langle f\rangle^2}{N}}$$ (7)

where the angle brackets denote an average taken over randomly-chosen points $\{\mathbf{x}_k\}_{k=1}^N\Omega$,

$$\begin{array}{r@{}l} \langle f\rangle &= \frac1N\sum_{k=1}^Nf... ... f^2\rangle &= \frac1N\sum_{k=1}^Nf(\mathbf{x}_k)^2 \end{array}$$ (8)

and $\vert\Omega\vert$ denotes the volume of $\Omega$.

As an example, consider the problem of computing the area of the unit circle in $\mathbb{R}^2$. In this case, $f$ is simply the characteristic function of the circle

$$f(x_1,x_2)=\left\{\begin{array}{ll}1 & x_1^2+x_2^2 \leq 1 [3pt] 0 & \text{otherwise} \end{array}\right.$$

The following code, using x and y to denote $x_1$ and $x_2$ will estimate the area of the circle

CHUNK=10000;  % chosen for efficiency
NUM_CHUNKS=100;

VOLUME=4;

totalPoints=0;
insidePoints=0;
for k=1:NUM_CHUNKS
  x=(2*rand(CHUNK,1)-1);
  y=(2*rand(CHUNK,1)-1);
  z=(x.^2+y.^2<=1);   % z=1 for "true" and z=0 for "false"
  insidePoints=insidePoints+sum(z);
  totalPoints=totalPoints+CHUNK;
end
average=insidePoints/totalPoints;
a=VOLUME*average;
disp(strcat('approx area=',num2str(a), ...
            ' with true error=',num2str(pi-a), ...
            ' and estimated error=', ... 
     num2str(VOLUME*sqrt((average-average^2)/totalPoints))));

Remark: Note that the points are divided into chunks of 10,000 points each. Then the function evaluation is done using vectors whose lengths are the chunk size. Working with these long vectors, Matlab will do the calculations efficiently. You don't, however, want to do all 1,000,000 points at once, because such large vectors can take a while just for the memory to be allocated. That is why the problem is first divided into chunks and then repeated a number of times.

Exercise 16:

1. Copy the above code to a script m-file and execute it several times. You should observe that the estimated area changes slightly each time and that the estimated error is usually, but not always, larger than the actual error. (These are probabalistic qunatities, after all.) Explain why both $\langle f \rangle$ and $\langle f^2 \rangle$ are given the value average in computing the error.
2. Use Monte Carlo integration to estimate the following integral
   $$\int_0^2 \sin^2\pi x dx$$
   along with both the true and estimated errors. Use enough trials to achieve an estimated accuracy of $\pm0.005$ or smaller. This function is easily integrated exactly. What is the true error of your estimate? It should be smaller, or at least not much bigger, than your error estimate.
3. Computing the volume of the intersection of two unit cylinders with orthogonal axes in $\mathbb{R}^3$ is an exercise often given to calculus students because it is difficult, but not impossible, to compute using elementary calculus. This volume is called a Steinmetz Solid and more information can be found on the web at http://mathworld.wolfram.com/SteinmetzSolid.html. This volume is known to be 16/3. Use Monte Carlo integration to estimate this volume along with both the true and estimated errors. Use enough trials to achieve an estimated accuracy of $\pm0.005$ or smaller.
4. Compute the integral of $2e^{x_1x_2x_3}$ over the volume of the intersection of two unit cylinders along with the estimated error. Use enough trials to achieve an estimated accuracy of $\pm0.005$ or smaller.

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