Simpson's Rule is Exact for Quintics (Preliminary Report)

Louis A. Talman

Department of Mathematical & Computer Sciences
Metropolitan State College of Denver

Presented at the annual meeting of the Rocky Mountain Section of the Mathematical Association of America, in Colorado Springs, CO; April 16, 2004.

Background

It is well known that if is a bound for on the interval , then the error in replacing by its Trapezoidal Rule approximation satisfies the inequality

 , (1)

where is the number of subdivisions used in applying the Trapezoidal Rule. It is also well known that the error in replacing by its Simpson's Rule approximation with subdivisions satisfies the inequality

 , (2)

the number now being a bound for on .  (Here and elsewhere in this talk, the error in a numerical approximation means the true value of the integral minus the value given by the approximation.)  The result appears in most elementary calculus books, but the proof does not.  Instead, the reader is referred to a text on numerical analysis, where the proof is usually accomplished by means of Lagrange interpolation in the context of general Newton-Cotes quadratures--and so is inaccessible to freshmen.

Results

In "An Elementary Proof of Error Estimates for the Trapezoidal Rule," D. Cruz-Uribe and C. J. Neugebauer [Mathematics Magazine, 76(2003), pp 303-306] gave an elementary argument establishing inequality (1).  They also discussed error estimates for functions that do not have bounded second derivatives.  However, they were unable to extend their methods to Simpson's Rule.  Here we show how use elementary techniques to prove the following:

Theorem 1:

If is a function on for which exists throughout , then there is a number such that the error that arises in replacing with its -subdivision Trapezoidal Rule approximation is given by

 . (3)

Theorem 2:

If is a function on for which exists throughout , then there is a number such that the error that arises in replacing with its -subdivision Simpson's Rule approximation is given by

 . (4)

Remark:

Our techniques can be used to establish error estimates for the other numerical integration schemes of elementary calculus (the Right-Hand Rule, the Left-Hand Rule, and the Mid-Point Rule); they can also be used to establish error estimates for Simpson's Rule when the fourth derivative of the integrand isn't bounded--and even when the integrand doesn't have higher order derivatives.

Preliminary Facts

It is known, though one hesitates to say "well" known, that derivatives have the Intermediate Value Property:

Proposition

Suppose that is a function differentiable on , and that for certain , .  There is a number between and such that .

Proof:

Proof:  Assume, WLOG, that .  Consider the function defined on by .  Then .  Thus   Thus, neither nor can yield a minimum value for on .  However, must have a minimum in that interval, which must therefore occur at some interior point .  But then , whence .■

This fact has an important corollary that is not as well known as it should be:

Corollary

Suppose that is a function differentiable on .  Let be a positive integer, , for each .  If for each , is a positive real number, then there is such that

 .■ (5)

The standard Mean Value Theorem takes on a special form for quadratic functions:

Let be a quadratic function.  Then

Proof:

Let .  Then

 (6)
 (7)
 (8)
 (9)

We will also make use of the well-known Taylor Expansion with Integral Remainder:

Theorem

Let be a function on an interval centered at , and let be in that interval.  Then

 (10)

Proof:

By the Fundamental Theorem of Calculus, .  Expand the integral using integration by parts, taking ; ; ; .  Repeat inductively as many times as necessary.■

Error In Simpson's Rule

We restate Theorem 2:

Theorem 2:

If is a function on an interval , if is defined throughout , and if is a positive integer, then there is a such that

 (11)

where .

Proof:

We first consider the error in a single one of the summands, and we simplify matters by assuming that the interval corresponding to that summand is centered at the origin.  We define an error function by

 . (12)

Let us consider also the first three derivatives of .

In[1]:=

Out[1]=

In[2]:=

Out[2]=

In[3]:=

Out[3]=

In[4]:=

Out[4]=

Note that :

In[5]:=

Out[5]=

In[6]:=

Out[6]=

In[7]:=

Out[7]=

Consequently, applying Taylor with Integral Remainder, we may write:

 . (13)

Let us now define a function on by

 (14)

Note that as .  Because is a function, is continuous on , and we may write:

 (15)

By the First Mean Value Theorem for Integrals, we now can find such that

 , (16)

or,

 (17)

because

In[8]:=

Out[8]=

Applying the Mean Value Theorem to this latter expression for , we find such that

 . (18)

This expression gives the error in a single summand, corresponding to an interval of width , in Simpson's Rule.

Returning now to the interval , we find , , so that the error contributed by the -th term in the sum is .  Then we find so that .  We now have

 .■ (19)

Corollary:

We can obtain the exact value of from Simpson's Rule when is a polynomial of degree not exceeding 5.

Proof:

This is well known when is a polynomial of degree not exceeding 3, so we need only prove it for polynomials of degree 4 and of degree 5.

Let be a polynomial of degree 4.  Denoting the Simpson's Rule approximation to by , where is the number of subdivisions, we now need only note that for a certain we have to have

 . (20)

Because is a polynomial of degree 4, is constant, and so the right side of the equation gives the exact value of the integral on the left when we replace with 24 times the leading coefficient of the polynomial .

Now let be a polynomial of degree 5.  Recall equation (17):

 (21)

Now is quadratic, so .  Translating this from the symmetric interval to , we find that for an arbitrary polynomial of degree 5, , we therefore have

 (22)

Created by Mathematica  (April 10, 2004)