Hermite versus Simpson: the Geometry of Numerical Integration

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1 Hermite versus Simpson: te Geometry of Numerical Integration Andy Long and Cliff Long October 9, 2010 Abstract An examination of current calculus and numerical analysis texts sows tat wen composite numerical integration rules are developed, te link to parametric curve fitting (wat we call te geometry of an integration rule) is frequently ignored, or at least not exploited to its fullest. In particular, te popular Simpson s rule is not mined for its close connection wit parametric curve fitting. Wen any relationsip is developed it is often done so badly: te composite Simpson s Rule is usually derived as te integral of a continuous piecewise quadratic spline interpolant of te function values or data points, but witout even slope-continuity. We provide a better geometry to associate wit Simpson s Rule. Beyond tat, owever, we suggest tat te empasis on Simpson s Rule is outdated: we prefer anoter rule, te Corrected-Trapezoid metod (wic we call Hermite s Rule, altoug oters migt prefer Hermite Cubic quadrature [8] (p. 161)), as it as several pedogogical and logistical advantages over Simpson s Rule (especially a more interesting and useful geometry). Hermite s Rule is more general tan Simpson s Rule, as it is based on capturing derivative information (true or approximate) as well as function information. We derive one approximation to Hermite s Rule wose error term is sligtly better tan tat of Simpson s Rule, and compare te integration scemes on a number of standard calculus test functions wile focusing on te geometric aspects of eac metod. keywords: Simpson s Rule, Hermite s Rule, Hermite interpolation, finite differences, parametric curve fitting, splines. 1 Introduction Consider n+1 data values y 0, y 1,...,y n associated wit n+1 equally spaced points, x 0, x 1,...,x n. Tink of te y i as values of a function f at te points x i. How migt you approximate te definite integral I = xn x 0 f(x)dx? Te justly famous [6] Simpson s Rule may come to mind: certainly many calculus students will tink of it, because Simpson s Rule is usually te end-all of integration rules in 1

2 introductory calculus class. As for numerical analysis students, [w]ere would any book on numerical analysis be witout Mr. Simpson and is rule? [10] Mr. Simpson was Tomas Simpson ( ),...an able and self-taugt Englis matematician,... autor of several text-books,... active in perfecting trigonometry as a science. 1 Tis autor of text books would peraps be pleased to be so well-remembered by so many of tem. Simpson is credited wit [t]e first application of te Newton-Rapson 2 process to te solution of transcendental equations and was certainly an early advocate of te aritmetic mean, as we can see from te title of is paper An attempt to sow te advantage arising by taking te mean of a number of observations, in practical astronomy. Did e go beyond te mean to te weigted average known as Simpson s rule? Tere is apparently no evidence tat e developed te numerical integration rule wic bears is name. Simpson s rule, altoug tremendously popular, is not mined for its close connection wit parametric curve fitting (wic we call te geometry of an integration rule). In fact, wen te relationsip is made clear - wen te geometry is connected wit te composite Simpson s Rule - it is often done so in a bad way: te composite rule is usually derived by pasting togeter te elemental quadratic interpolants, leading tus to an integral of a continuous piecewise quadratic spline interpolant of te function values or data points, witout even slope-continuity. Te interpolant is continuous (C 0 ), but not differentiable (C 1 ). We demonstrate first of all tat one may associate a better geometry wit te composite Simpson s Rule, tat of a C 1 cubic spline interpolant. But we ave a more ambitious goal tan simply putting Simpson s Rule in closer contact wit a good parameterization: we propose tat te focus on Simpson s Rule endemic to calculus, engineering matematics, and numerical analysis texts migt be better placed on anoter rule (wic we call Hermite s Rule). Tis rule is also associated wit a C 1 cubic spline interpolant, but as several additional useful pedogogical and logistical features. Hermite s Rule itself requires derivative information. Tis may dissuade te casual reader from reading any furter. We ope, owever, to reac te imaginative reader, wo will suspend disbelief long enoug to follow us troug to te various approximations to Hermite s Rule based on finite difference approximations to te derivatives. George Polya, in How to Solve It[9], posits tat te solution to a specific problem may be best obtained by considering a more general problem. In many cases requiring numerical integration we ave only function values (or data values), so wy, ten, introduce derivatives? Tis argument is really specious: all of numerical integration is based on numerical differentiation, in some sense, it is just tat we ordinarily are not so bold as to act as if we ave te derivative values at and. Let us do so, noneteless: te advantage is tat we will derive a metod tat works if one is given te derivative information, and furtermore develop an approximation to te metod wic beats Simpson s Rule on level ground (at least for a suite of calculus test functions), even wen te true derivative is unknown. 1 Tis and oter quotes concerning Simpson are from Florian Cajori s A History of Matematics [4]. 2 aka Newton s metod, for root-finding 2

3 2 Simpson s Rule Simpson s Rule is undoubtably king of te numerical integration rules: open a calculus, numerical analysis, or engineering matematics textbook and ceck te index - tere it is. For an integrable function f on te closed interval [a, b], Simpson s Rule makes te approximation b f(x)dx 3 (y 0 + 4y 1 + y 2 ), (1) a were = (b a)/2; x 0 = a, x 1 = a +, and x 2 = b; and y i = f(x i ) (or is a data value corresponding to te point x i ). If f is C 4, ten te error term (defined as te exact integral minus te approximation) can be written as 5 90 f(4) (µ), wit µ (a, b). In sort, Simpson s Rule divides te interval into two equal subintervals, computes te function at te tree boundary points, and ten takes a weigted average, giving an error of te fift-order in proportional to te fourt derivative of f. Te pair of adjacent subintervals [x 0, x 1 ] and [x 1, x 2 ] constitutes wat is popularly known as a panel. Tis elemental Simpson s Rule leads to te so-called composite Simpson s rule, created by dividing te interval [a, b] into a large number of panels (tat is, an even number n of equal subintervals), invoking te elemental rule (1) on eac panel, and ten adding up te results. Te composite Simpson s rule approximation is S = 3 (y 0 + 4y 1 + 2y 2 + 4y 3 + 2y y n 4 + 4y n 3 + 2y n 2 + 4y n 1 + y n ), (2) were = (b a)/n and y i = f(a + i). If tis were a dance, we migt call it te step; Burden and Faires[3] call it...te most frequently used general-purpose quadrature algoritm. It can be sown tat te composite Simpson s Rule is of fourt-order in. More precisely, te composite Simpson s Rule as an error term of te form (b a) f (4) (µ), (3) wit µ (a, b) ([3], p. 186). Te fact tat te error is proportional to te fourt derivative implies tat Simpson s Rule gives exact results for polynomials of degree tree or less (as teir fourt derivatives are identically zero). Te order of te stepsize in te error term is one measure of a rule, wile te degree of polynomial tat it integrates exactly is anoter. We say tat te composite Simpson s Rule is a fourt-order metod aving degree of precision tree. 3 Wat s Wrong wit Simpson s? Altoug Simpson s rule is frequently presented in beginning calculus as an improvement over te rectangle and trapezoidal rules for integral approximation, we ave tree logistical or pedogical concerns about te resulting composite rule: 3

4 it only applies to an even number of subintervals; it gives te (mistaken) impression tat some interior points are twice as important as oters; and te geometry tat is frequently used to derive te composite Simpson s Rule (e.g. [3], Figure 2.7, p. 185) involves piecewise parabolic data fitting wic doesn t even ave slope continuity at te joints (Figure (1), upper left). 3.1 Odd subintervals out Tis first concern may seem rater frivolous, especially if you re a calculus professor accustomed to a formula for every function. But in today s ig-tec society, problems requiring approximate integrals often stem from automatically sampled input data rater tan from functions defined by formulas. Hence one can never assume tat an even number of subintervals will be provided or available. Te subintervals temselves are frequently equal, owever, making simple composite rules attractive. Tus we are faced wit a situation in wic we want to use a rule like te composite Simpson s rule, but would rater not be botered wit te even/odd eadace. A recent consulting experience associated wit te design of instrumentation for determining te quality of automotive windsield glass brougt tis issue to our attention. In te quality assessment process laser scanners provide glass surface information (e.g. angles of reflection) at fixed time intervals; certain integrals naturally arose, wic we wanted to calculate on an ongoing basis; te question tus became ow, and altoug we started wit Simpson s Rule, we migrated to Hermite s initially simply to avoid te problem of odd intervals. Many autors go to te trouble of talking teir readers troug te problem of applying Simpson s rule in te event of an odd number of intervals, often encouraging te use of Simpson s 3/8 rule (wic is defined on tree-subinterval panels) on one end to make up te difference. Tis means, at te very least, tat some autors believe tat Simpson s or Simpson s-like Rules are necessary and useful. We took a different approac wen we found ourselves in tis same situation: we adapted Simpson s for an odd number of subintervals as follows, and so arrived at our first example of a Hermite rule (wic we call H 3 ). If n is odd, ten te first n 1 subintervals and te last n 1 subintervals can be andled wit Simpson s rule, and te results averaged. Tis neglects, owever, te fact tat te end subintervals are added in only once, rater tan twice as are all te internal subintervals. To correct for tat, we need to add in an additional approximation for te two end subintervals before averaging. We approximated te function on te end subintervals by using a quadratic function tat interpolated te data values (2 conditions) and also matced an approximation to te slope at te internal node (given by te centered finite difference approximation). For te first interval [x 0, x 1 ], for example, tis function is q 0 (x) = y 1 + y 2 y 0 2 (x x 1) + y 2 2y 1 + y (x x 1 ) 2. 4

5 Figure 1: Te top two functions are geometries associated wit te composite Simpson s Rule on tree panels (C 0 quadratic patces, and a C 1 cubic spline wit te left-slope given by te five-point rule (eq. (9)), te same slope used by H 5 ). Te bottom two functions are geometries associated wit approximations H 3 (left) and H 5 (rigt) to Hermite s Rule. 5

6 Tese two functions, integrated over teir respective subintervals, give contributions of 12 (5y 0 + 8y 1 y 2 ) and 12 ( y n 2 + 8y n 1 + 5y n ). Averaging tese various elements via te matrix product yields te weigts 1 [ ] [ ] ; (4) 24 te inner product of tese weigts wit te vector of points [ y 0 y 1... y n ] gives te approximation wic we ave since named Hermite s Rule H 3 (we will sow wy in a moment); and H 3 ultimately inspired us to write tis paper. 3.2 Wo says odd points are better tan even points? It is clear wy students may get te impression tat Simpson s rule values te odd numbered points twice as muc as te even points: 4 is, after all, twice 2. Tis impression can be rectified by rewriting (2) as n S = y i + i=0 3 ( 2y 0 + y 1 y 2 + y 3 y y n 1 + y n 2y n ). (5) But tis form, too, provokes troubling questions: ow was te pase of tis wave cosen? Wy does it multiply even-numbered points by -1 and odd-numbered points by 1? Does tis sinusoidal wave, superimposed on te data, solicite contrast information across te domain of te function? Te autors of Numerical Recipes[10] ave a teory all teir own: Many people believe tat te wobbling alternation someow contains information about te integral of teir function wic is not apparent to mortal eyes. One way to explain away tis wave is to claim tat it is merely an artifact of te metod of construction (pasting togeter elemental units to generate a composite rule). However, as we are about to sow, te composite Simpson s rule can be derived witout pasting anyting at all. 3.3 Te geometry of Simpson s Rule All te usual numerical integration rules are associated wit some type of approximating function. Te rectangle rules are so named because te data are treated as toug tey 6

7 are a sample of a step-function, and te integral of te step function (a sum of areas of rectangles) serves as an approximation to te integral of te function itself. Te trapezoidal rule T = n 1 i=0 2 (y i+1 + y i ) = n y i i=0 2 (y 0 + y n ), (6) (te simplest Newton-Cotes rule) is derived by creating a linear spline between data points: tat is, on te i t subinterval a linear function (te linear Lagrange interpolant 3 ) joins te points (x i, y i ) and (x i+1, y i+1 ), and te composite trapezoidal rule is derived by joining tese linear segments togeter to form a C 0 spline, ten integrating te result (a sum of areas of trapezoids). Simpson s Rule is also a member of te Newton-Cotes family of intergration formulae, eac of wic is derived (in elemental form) by integrating Lagrange interpolating polynomials. Tis is te positive aspect of te relationsip between parametric curve fitting and Newton-Cotes numerical integration rules. Unfortunately, tis sensible treatment does not usually extend to te development of te composite integration rules. Te elemental Simpson s Rule is derived by integrating te Lagrange quadratic interpolant on a panel, rater tan on a subinterval. Tat is, we consider te panel composed of te tree points {(x 0, y 0 ), (x 1, y 1 ), (x 2, y 2 )}, for wic te Lagrange quadratic interpolant is given by q(x) = (x x 1)(x x 2 ) (x 0 x 1 )(x 0 x 2 ) y 0 + (x x 0)(x x 2 ) (x 1 x 0 )(x 1 x 2 ) y 1 + (x x 0)(x x 1 ) (x 2 x 0 )(x 2 x 1 ) y 2. Integrating q from x 0 to x 2 gives te elemental Simpson s Rule (eq. (1)). Moving on to te composite rule, werein we divide te interval [a, b] into n panels, te inclination is to simply calculate te Lagrange quadratic for eac panel and paste tem togeter to give rise to te geometry of te composite rule. Tis results again in a C 0 spline, but is unsatisfactory due to te slope discontinuities (as sown, for example, in Figure (1)). To interpolate data points or function values geometrically wit piecewise curves wic are not even slope continuous sends a strange message to our students (especially to any wit interests in engineering or computer-aided design, for example). Note tat our concern about geometry is somewat pedogogical (some migt even say pedantic!): first of all, te geometry does not affect te numerical answer; and secondly, Simpson s Rule is not tied to a single geometry, altoug some autors tend to act as toug it were. For example, one textbook guses tat [Simpson s rule] yields exact results for cubic polynomials even toug it is derived from a parabola! [5] Our cief concern is tat te temptation is great to carelessly adopt te quadratic geometry of te elemental Simpson s Rule (a geometry wic we do appreciate) to te composite rule, rater tan to retink te issue. 4 We now establis tat te geometry of Simpson s Rule is not unique, and tat we can, witout muc ado, associate instead a C 1 cubic-spline interpolant to te composite Simpson s Rule. Since Simpson s error of approximation is proportional to te fourt-derivative, we can add any cubic c to te quadratic q suc tat c(x i ) = 0 at te panel values x 0, x 1, and x 2. 3 Tese will be defined sortly. Te impatient will want to refer to eq. (8). 4 We will demonstrate ow to tink te issue below, wit te derivation of Hermite s Rule. 7

8 Tis clearly leaves te approximation of Simpson s Rule uncanged, x2 x 0 (q(x) + c(x))dx = 3 [q(x 0) + c(x 0 ) + 4(q(x 1 ) + c(x 1 )) + q(x 2 ) + c(x 2 )] = 3 [q(x 0) + 4q(x 1 ) + q(x 2 )], and since Simpson s Rule is exact for cubics, te exact integral of tis cubic q(x) + c(x) is given by Simpson s Rule. Tus we may turn te tables on te original derivation of Simpson s Rule as te integral of a quadratic and agree tat Simpson s Rule will encefort be associated wit any cubic interpolant of te tree points on te panel (including te degenerate cubic wic is te Lagrange interpolating quadratic). Examine te top two plots in Figure (1): tese two functions one a quadratic C 0 spline, te oter a cubic C 1 spline ave te same integral value on te intervals sown! In te case of a (non-degenerate) cubic, interpolation of te tree data values leaves one degree of freedom to be used as we wis (a cubic is determined by four constraints). Wy not, terefore, use te extra degree of freedom wen constructing te composite rule? For example, we could prescribe te slope at eac left endpoint of a panel. Tis gives us just enoug freedom to create a C 1 spline, as follows: te coice of te slope at te left endpoint of te first cubic (along wit te tree constraints of interpolation) dictates te slope at te rigt endpoint; we ten coose te left slope of te second panel s cubic to matc te first s rigt slope, wic in turn dictates te second s rigt slope; and so on. Continuing in tis manner we see tat te coice of te first cubic s slope at te left endpoint completely determines all subsequent cubics, resulting in a C 1 cubic-spline interpolant. Tese cubic splines on panels are in te same family as quadratic splines wit slope continuity on subintervals (wic are unpopular due to our inerent inability to control slope beavior at bot ends of te spline). For tese splines te tail wags te dog : te slope at one side of te interpolant wags te oter. Imagine grasping te sipwrigt s spline 5, tis C 1 interpolant, wic is pegged to all te data points, at one end and varying te slope: you would ave no control over te slope at te oter end. It would wiggle as it pleases. Quartics defined on tree-subinterval panels, quintics on fours, etc. all can be made C 1, but will wag in tis manner: from a design standpoint, one is at te mercy (rater tan in command) of te curve. 4 Hermite s Rule Te Hermite of Hermite s Rule is Carles Hermite ( ), about wom Poincaré (is student) said Talk wit M. Hermite. He never evokes a concrete image, yet you soon perceive tat te most abstract entities are to im like living creatures. 6 In 1873 e proved te transcendence of e (meaning tat e cannot be expressed as te root of a polynomial equation wit integral coefficients), and aving done so wrote to a friend tat I sall risk noting on an attempt to prove te transcendence of te number π. If oters undertake tis 5 A spline was originally a flexible rod serving as a design tool for sip builders. 6 Tis and following quotes are from [12] and [1]. 8

9 enterprise, no one will be appier tan I at teir success, but believe me...it will not fail to cost tem some efforts. We may be excused for imagining tat e actually said tis in Frenc. Hermite s name is well-known in quantum mecanics (Hermite functions, Hermite polynomials), in algebra (Hermitian forms), and in numerical analysis (Hermite interpolation). As so often appens, te pysicists must be grateful to a man wo was doing number teory - noting really practical at all. Hermite corresponded voluminously wit matematicians all over Europe, and was known for is kindly, encouraging manner. At te time of is deat, e was loved te world over. One tinks, peraps, of Erdos in our own time. We now proceed to te derivation of Hermite s Rule, so named because of its relationsip to Hermite interpolation. Hermite s Rule and its associated approximations work for even or odd numbers of subintervals; use derivative information (if available, or approximations if not); are associated wit a very satisfying geometry; and ave error terms comparable to (and, in some cases, superior to) Simpson s Rule. We will derive Hermite s Rule based on te geometry of te C 1 cubic-spline interpolant. Consider n + 1 data values y 0, y 1,..., y n associated wit n + 1 equally spaced points, x 0, x 1,...,x n. We will suppose tat te slopes m i at te points x i for i {0,...,n} are given. Ten between any two data points a unique cubic is completely defined by te four constraints of matcing te data and slopes for tat subinterval. Tis is te Hermite cubic interpolator[13], and it provides us wit anoter elemental metod for numerical integration. Te cubic approximation to te function on te interval [x i, x i+1 ] can be expressed as c i (x) = y i + m i (x x i ) + 1 2[3(y i+1 y i ) (m i+1 + 2m i )](x x i ) [ 2(y i+1 y i ) + (m i+1 + m i )](x x i ) 3, as one can verify by calculation (c i (x i ) = y i, c i (x i+1 ) = y i+1, c i (x i) = m i, and c i (x i+1) = m i+1 ). Tis cubic as an especially nice feature as a bonus: its derivative serves as a piecewise quadratic interpolator of te derivative function of f at te locations x i and x i+1. We integrate c i on te interval [x i, x i+1 ] to give te elemental integration rule 12 [6(y i + y i+1 ) (m i+1 m i )], wic, wen applied to an interval divided into many subintervals, yields te elegantly simple composite rule n H y i 2 (y 0 + y n ) 2 12 (m n m 0 ). (7) i=0 9

10 Cancellation of te internal derivative terms leads to tis beautiful result. We ave written te approximation in tis form to illuminate its similarity to te trapezoidal rule (eq. (6)): H = T 2 12 (m n m 0 ). One may well marvel at te simplicity of tis formula: it is te trapezoidal rule wit a sligt slope adjustment (wic explains wy tis rule is more commonly known as te Corrected trapezoid(al) rule ) 7. Note tat te values of te internal slopes, wile fundamental to te geometry, ave become irrelevant from te point of view of te estimate: tey disappear from te formula (see Kaaner, Moler and Nas[8] (p. 162) for more on te case of te disappearing internal slopes ). Higer-order (and more complicated) metods can be created by considering Hermite interpolators for N > 2 points (matcing data and slopes over panels, rater tan subintervals). Te Hermite polynomial is given in general by N H(x) = li 2 (x){[1 2l i (x i)(x x i )]f i + (x x i )f i }, i=1 were te l i are Lagrange polynomials, wic can be written as l i (x) = N j=1,j i (x x j ) (x i x j ) (8) (see Bucanan and Turner[2]). Te error of approximation of tese Hermite polynomials is were f(x) H(x) = f(2n) (ξ) [L N (x)] 2, (2N)! N L N (x) = (x x j ). j=1 and were ξ is an element of te smallest interval containing x and all te x i 8. For our purposes we consider only two-point (N=2) Hermite interpolation - tat is, Hermite cubics. Hermite s Rule tus as local error on eac subinterval of f (4) (ξ) Wen we build a composite rule wit tis elemental rule, we integrate over te subintervals of te interval [a, b] to produce an error of [ ] 1 (b a) f (4) (ξ), 7 Additional details concerning tis formula are given in te appendix. 8 Greek letters like ξ will usually designate unknown elements of tese sorts of intervals (unspecified) in te following. 10

11 were ξ (a, b). Tis fourt-order sceme wit degree of precision tree as approximately one-fourt te error of Simpson s Rule. Wile Hermite s Rule tus beats Simpson s Rule in terms of error, and as a more pleasing and sensible geometry, te obvious criticism of tis rule is tat it requires derivative information (at least at te boundary points x 0 and x n ). Take tat away and wat ave we got? In te absence of te actual derivative information, we simply approximate it: we replace te true slopes wit approximations using forward and backward difference formulae. If we make our goal a metod of fourt-order and degree of precision tree (for comparison wit Simpson s Rule), ten we will not succeed by coosing te most obvious difference metods. Te usual calculus class approximations, e.g. f (x) = f(x + ) f(x) + 2 f (ξ), will acieve neiter fourt order, nor precision tree. To do bot, we must use at least te tree-point forward and backward difference formulae and f (x 0 ) = 1 2 [3y 0 4y 1 + y 2 ] f(3) (ξ 0 )) f (x n ) = 1 2 [y n 2 4y n 1 + 3y n ] f(3) (ξ n )). Using tese approximation we arrive at te formula H 3 = T 24 (3y 0 4y 1 + y 2 + y n 2 4y n 1 + 3y n ), were te subscript (3) indicates tat we ave used te tree-point difference sceme to approximate H. Tis is exactly te same weigted sum as tat seen earlier (eq. (4)), derived by adapting Simpson s Rule to odd numbers of subintervals. One oter surprising link to Simpson s rule can be made: if one applies te H 3 rule to a single panel, it yields exactly Simpson s Rule: H 3 = 2 (y 0 + 2y 1 + y 2 ) 24 (3y 0 4y 1 + y 2 + y 0 4y 1 + 3y 2 ) = 3 (y 0 + 4y 1 + y 2 ). Once we pus beyond a single panel, owever, tis connection to Simpson s rule is lost. Even so, it suggests tat we could associate a quadratic geometry wit endpoint subintervals (remember tat we first derived H 3 by tacking on integrals of quadratics on te ends). Rater tan tink quadratically, owever, we continue to associate te Hermite cubicspline geometry wit tese Hermite rules created via derivative approximations. Rater tan te exact slopes, we use te best approximations available. Because te internal slopes disappear, we can use any approximation watsoever to tem: an often reasonable estimate is provided by centered finite differences, suc tat at eac of te internal points (i {1,..., n 1}) we estimate te slope at (x i, y i ) to be te symmetric difference y i+1 y i

12 Wit tese slopes given, along wit te interpolation conditions, te internal cubics are completely defined. We are left to determine te cubics associated wit te endpoints. At te endpoints temselves (i = 0 and i = n) we use te tree-point forward and backward difference approximation to te derivative for te H 3 Hermite rule. Putting it all togeter we get te approximate Hermite cubic-spline interpolator, wic is still C 1 and is, from a practical standpoint, an interpolator wose derivative continues to provide a reasonable approximation to te derivative function of f. See te bottom left plot in Figure (1) for tis interpolator to our sample data. Te order of error of H 3 can be derived by combining te error of Hermite s Rule wit te error of te derivative approximations: I H 3 = 1 4 = 1 4 [ ] (b a) f (4) (ξ) 4 36 (f(3) (ξ n ) f (3) (ξ 0 )) [ ] (b a) f (4) (ξ) 4 36 f(4) (ξ m )(ξ n ξ 0 ) (wic is a consequence of treating te last term by te fundamental teorem of calculus). Ten I H 3 = 1 [ ] (b a) f (4) (ξ) 4 36 f(4) (ξ m )ρ(b a) (b a) = [ρf 4 (4) (ξ m ) 1 ] f(4) (ξ), were ρ 1 as 0. Tus, in te limit, tis metod does not ave as good an error profile as Simpson s Rule (by a factor of about five). A disappointing start, but a journey of a tousand miles begins wit a single step. Let s take anoter step (well, two steps): instead of H 3, we consider H 5, using te five-point difference scemes applied to te endpoints. Tat is, we approximate te endpoint derivatives by and f (x 0 ) = 1 12 [ 25y y 1 36y y 3 3y 4 ] f(5) (ξ 0 ) (9) f (x n ) = 1 12 [3y n 4 16y n y n 2 48y n y n ] f(5) (ξ n ). Tese approximations lead to te rule H 5 = T As for te error, 144 [25(y 0 + y n ) 48(y 1 + y n 1 ) + 36(y 2 + y n 2 ) 16(y 3 + y n 3 ) + 3(y 4 + y n 4 )]. wic we can write as I H 5 = (b a) f (4) (ξ) (f(5) (ξ n ) f (5) (ξ 0 )), = (b a) [f (4) (ξ) ρf (6) (ξ m )] 12

13 provided f is six times differentiable, and were ρ 1 as 0. Tis, too, is a fourt-order metod wit degree of precision tree. As 0 te error of tis metod tends to te error of Hermite s Rule, as we see ere and as we sall see again wen we test te scemes on some standard functions from calculus wile forcing to get very small. Once again we stick wit te geometry wic as served us so well, approximating slopes at internal points by centered differences and at te endpoints by te five-point forward and backward difference approximations (see te bottom rigt plot of Figure (1)). 5 Test Results For purposes of comparing te integration rules, it is necessary to revert to functions wit known integrals. Burden and Faires[3] provide a set of test functions wic we used to compare te metods, but we also added several more of our own, as well as a number cosen from te integral tables in te back of a favorite 9 calculus text[11]. We wanted to give a rater complete set of functions, suc as one is likely to encounter in calculus class. As one can see from te following tables, Simpson s Rule does better tan Hermite s Rule H 3 by a factor of 4 or so in general, wile Hermite s Rule H 5 beats Simpson s by a similar factor. In te limit as 0 Hermite s Rule itself sows tat it is about 4 times better tan Simpson s Rule, as expected. We ave included a couple of functions wic are not differentiable, or not twice differentiable on te domain: results for tese functions are arder to classify. It seems fairly clear, toug, tat for a reasonable number of subintervals (e.g. 80) te results of H 5 are superior to tose of Simpson s Rule on almost all functions. And wat if te data come from an automated process? Certainly muc industrial-strengt integral estimation from data involves at least 81 points; tus, provided tat te date come from relatively smoot processes, te better rule to use is H 5. 6 Conclusions Putting te rigt geometry wit an integration rule is just good matematics. A careless connection (or, worse yet, no connection) of integration tecniques to function approximation and interpolation is a wasted opportunity. One migt argue tat te geometry really doesn t matter (especially if it isn t even unique); tat wat matters is te numerical answer. But wat matters in our minds is tat students make te strong connection between subjects wic really sould go and-in-and. We present Hermite s Rule as a useful replacement for te overworked and at times unwieldy Simpson s Rule. Hermite s Rule avoids te tree concerns of Simpson s rule mentioned earlier, namely te problem of even/odd numbers of subintervals, te curious empasis on every oter point, and te C 0 geometry commonly associated wit te composite Simpson s Rule. Hermite s Rule as te distinct advantage of empasizing modern data sampling problems wile introducing piecewise differentiable cubic interpolating curves wit known slopes Rule! 9 Favorite in spite of te fact tat no geometry is associated wit te derivation of te composite Simpson s 13

14 at te nodes, wic play suc a significant role in computer-aided geometric design. It is clear tat te H 5 approximation to Hermite s Rule as a better error tan Simpson s rule on most of te standard test functions, yet does not require any explicit derivative information, as a formula as simple to apply as Simpson s, and is associated wit a useful geometry wose derivative also serves as an approximation to te derivative of f. So next time you find yourself using or teacing numerical integration tecniques, consider Hermite s Rule, and consider teacing your rules in conjunction wit piecewise interpolation of data points. Integration tecniques provide one more opportunity to broac tis important subject, and to demonstrate to our students te interdependence of important topics in matematics and modern life. And peraps te next time tey look out an automobile windsield tey won t just see telepone poles and lines - tey ll also see integration scemes and interpolators. Table 1: Te first six functions in tis table were used by Burden and Faires [3] to compare integration scemes, and integrated from 0 to 2. Additional functions were added to tese to offer a more complete sample of typical functions from calculus. We used 4 panels (i.e. 8 subintervals) to obtain tese results. It appears tat te error in Hermite s Rule H 3 is often around 3 to 4 times te error in Simpson s Rule, wile Hermite s Rule H 5 beats Simpson s by a similar factor. Results ave not settled down for te majority of tese rules (n is still rater small). Function True ɛ H3 /ɛ S ɛ S /ɛ H5 ɛ S /ɛ H x NaN NaN NaN x /(x + 1) x sin(x) e x ln(x + 1) /(x 2 + 1) / x cos(2x) cos(5x) cos(10x) x NaN NaN NaN 5x x x x (5/2) x 3/ signum(x pi/4) (x pi/4) x pi/

15 Table 2: Te same table, using 40 panels (i.e. 80 subintervals). We now see good convergence of te scemes toward te expected values of about 5, for te ratio of error in H 3 to Simpson s, to 4 for te oter two ratios. Again, te non-differentiable functions aren t giving consistent results. Function True ɛ H3 /ɛ S ɛ S /ɛ H5 ɛ S /ɛ H x NaN NaN NaN x /(x + 1) x sin(x) e x ln(x + 1) /(x 2 + 1) / x cos(2x) cos(5x) cos(10x) x NaN NaN NaN 5x x x x (5/2) x 3/ signum(x pi/4) (x pi/4) x pi/ Appendix Anoter derivation of Hermite s Rule is provided by consideration of te asymptotic error in te trapezoidal rule as 0 (te following is based on [2]): ɛ T (f) lim n 2 wic is a Riemann sum; tus = lim n [ 12 n 1 k=0 f (η k ) ] = 1 12 lim n n 1 k=0 f (η k ), lim ɛ T(f) = 2 b n 12 lim f (x)dx = 2 n a 12 [f (b) f (a)], te asymptotic error. Tis process can be continued, leading to te IMT formula[7], or Euler-Maclaurin summation formula[10]: b a f(x)dx = T m r=1 2r B 2r (2r)! [f(2r 1) (b) f (2r 1) (a)] + R m, 15

16 Table 3: Using 100 panels. Function True ɛ H3 /ɛ S ɛ S /ɛ H5 ɛ S /ɛ H x NaN NaN NaN x /(x + 1) x sin(x) e x ln(x + 1) /(x 2 + 1) / x cos(2x) cos(5x) cos(10x) x NaN NaN NaN 5x x x x (5/2) x 3/ signum(x pi/4) (x pi/4) x pi/ were [ R m = 2m+1 1 n 1 ] B 2m (t) f (2m) (a + k + t) dt, (2m)! 0 k=0 and were B n (t) are te Bernoulli polynomials of degree n and te B n are Bernoulli numbers (B 0 = 1, B 2 = 1/6, B 4 = 1/30, B 6 = 1/42, B 8 = 1/30, and so on), generated by t e t 1 = n=0 B n t n n!. According to [10], tis is...an asymptotic expansion wose error wen truncated at any point is always less tan twice te magnitude of te first neglected term. References [1] E. T. Bell. Men of Matematics. Simon and Scuster, New York, [2] J. L. Bucanan and P. R. Turner. Numerical Metods and Analysis. McGraw Hill, New York, [3] R. L. Burden and J. D. Faires. Numerical Analysis. PWS Publising Company, Boston, fift edition,

17 [4] F. Cajori. A History of Matematics. Te MacMillan Company, London, second edition, [5] S. C. Capra and R. P. Canale. Numerical Metods for Engineers. McGraw-Hill Book Company, New York, second edition, [6] S. D. Conte and C. de Boor. Elementary Numerical Analysis. McGraw Hill, New York, tird edition, [7] K. Itô, editor. Encyclopedic Dictionary of Matematics. Te MIT Press, Cambridge, MA, second edition, [8] D. Kaaner, C. Moler, and S. Nas. Numerical Metods and Software. Prentice Hall, Englewood Cliffs, N.J., [9] G. Polya. How to Solve It. Princeton University Press, second edition, [10] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes: te Art of Scientific Computing. Cambridge University Press, New York, [11] S. L. Salas and E. I. Hille. Calculus: One and Several Variables. Wiley, New York, tird edition, [12] G. F. Simmons. Differential Equations wit Applications and Historical Notes. McGraw- Hill, Inc., New York, second edition, [13] G. Strang. Introduction to Applied Matematics. Wellesley-Cambridge Press, Wellesley, MA, first edition,

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