formula
Content
TFA 37 (1979-1981) 63-69
63
WOOLHOUSE’S TO PIECEWISE
FORMULA POLYNOMIAL by M.A.,
IN
RELATION FUNCTIONS
J. J. McCUTCHEON,
Ph.D.,
F.F.A.
1. INTRODUCTION Generations of actuarial students have been confronted at an early stage of their careers with Woolhouse’s formula, important for its well-known application to annuities payable several times per year. However, it is probably fair to say that most of the derivations of the formula presented to students have been in some respects unsatisfactory, since they have usually relied on “ the separation of symbols ”, manipulations of the finite difference operators D, , and (m), and have involved arguments somewhat lacking in rigour. This is not a severe criticism of the relevant textbooks, which were after all written from a practical actuarial viewpoint and which understandably enough wished to avoid the relatively difficult analysis required to derive Woolhouse’s formula in its complete generality. Woolhouse’s formula leans heavily on a more fundamental result, the so-called Euler-Maclaurin formula, which gives a relationship between n f(r) and n0f(x)dx: involving the higher derivatives (assumed
to exist) of the function f at 0 and at n and a suitable remainder term. Unfortunately, however, in general the determination of the remainder term is rather complicated and serious doubts usually arise over the convergence of the associated infinite series. The formula as normally stated is therefore mainly of theoretical interest. (We give below relevant references.) Although for a completely general function the proof of the Euler-Maclaurin formula is rather complicated and lengthy, in the special case when we restrict our attention to functions which are piecewise polynomials of a certain type (see below) a reasonably short and completely rigorous proof can be given without using advanced analysis. Moreover the resulting formula is exact and has no unwieldy remainder term. Piecewise polynomial curves (and in particular splines) are of considerable practical importance and may represent more general functions to any desired degree of accuracy, so that this restricted class of functions may still be considered most
64
Woolhouse’s
Formula
in Relation
to
useful. Accordingly we feel that the relatively simple discussion below may be of general interest and also possibly of help to future students. 2. THE BERNOUILLI NUMBERS
Before our main argument it is necessary for completeness to recall the Bernouilli numbers {Bi} (i 0). This celebrated sequence may be defined by the initial value B0 = 1 and by the recurrence relation (2.2) Clearly this last condition, combined with the initial value, defines the entire sequence. For example, putting n = 2, 3, 4, and 5, in turn we obtain 2B1+B0 = 0 3B2+3B1+B0 = 0 4B3+6B2+4B1+B0 = 0 5B4+ 10B3+ 10B2+ 5B1+ B0 = 0, from which we calculate successively (2.1)
One important numbers is
and easily established B2k+1 = 0 (k
property 1).
of the Bernouilli (2.3)
(The interested reader may refer to the references below for a simple proof of this last result and further discussion of these numbers.) We now proceed to the next step of our argument, in which use is made of the above. 3. THE BASIC INTEGRATION FORMULA
Suppose that over the interval [0, 1]f(x) is a polynomial in x of degree (2n+1) where n 1 is a given positive integer. Such a polynomial is defined by (2n+2) suitably chosen parameters. For example, the polynomial is defined by the values of f(x) and the derivatives f(2r–1) (x) (r = 1, . . . , n) at the points x = 0 and x = 1. In particular these values must determine the integral off over the interval [0, 1]. This is confirmed by the following result, which is fairly well known.
65 Lemma Suppose that n 1 and that over the interval [0, 1]f(x) is a polynomial in x of degree (2n+ 1). Then (3.1) Note that, since B2 = 1/6, when n = 1 this implies
Piecewise
Polynominal
Functions
which is easily seen to hold for cubics. The general result now follows by induction on n. (The proof depends on only the defining relationship 2.2 and the readily established equation 2.3 above. Accordingly we omit further details.) The following generalisation of equation 3.1 above is easily established. Lemma Suppose that n is a positive integer and that over the interval [a, a+h] g(x) is a polynomial in x of degree (2n+ 1). Then
(3.2) Proof: Put t = so that x = a+ht and
where f(t) = g(a+ht) is a polynomial in t of degree (2n+l). We now use 3.1 above to evaluate this last integral. Since f(r)(t) = hrg(r)(a+ht), this immediately gives the required result. 4. AN APPLICATION TO PIECEWISE CURVES POLYNOMIAL
Suppose that the function f is defined on the interval [a, a+lh] where h> 0 and l 2 is a given positive integer. We assume that f is (i) continuous and (ii) piecewise polynomial, in the sense that over each of the subintervals [a+rh, a+(r+1)h] (r = 0, . . . . l–1)f(x) is given by a polynomial in x.
66
Woolhouse’s
Formula
in Relation
to is of degree
Suppose further that each of the defining polynomials at most (2n+1) (where n 1 is given) and that
(iii) for s = 1, . . . . n the left and right derivative8 of order (2s – 1) are equal at each of the point8 x = a+rh (r = 1, . . . . l–1). The last condition (iii) should be noted. What we require is that the polynomials defining f on adjacent sub-intervals should have the same value at the common boundary point of the intervals (to ensure continuity) and in addition the same value at that point of In particular the condition will each of their first n odd derivatives. be satisfied if f is a spline curve of order (2n+ 1) with the given points (In this case f is 2n-times differentiable on [a, a+lh].) as knots. However, it should be noted that the condition holds for a wider class of functions than splines. We may now apply equation 3.2 above successively to each of the Thus sub-intervals [a, a+h], [a+h, a+2h), . . ., [a+(l–1)h, a+lh]. we obtain
(In each of the above equations f(2r–1) (a+jh) must be interpreted as the common value of the left and right derivatives of order (2r–1) at the jth knot and not as a two-sided derivative, which may not exist.) By adding these last equations we obtain the integral off over the interval [a, a+lh] in the form
(4.1) which result is exact for piecewise polynomials of the type defined by conditions (i), (ii), and (iii) above. Equation 4.1 above may be regarded as a special case of the Euler-Maclaurin formula, valid when the function f is of the required
67 piecewise-polynomial form. It should be noted, however, that for such a function there is no awkward remainder term and that the result has been obtained by essentially elementary methods. Note that this last equation can be written in the form
Piecewise
Polynominal
Functions
(4.2) Woolhouse’s formula is readily inferred from equation 4.1 above. Suppose that m 2 is a positive integer and that the function f satisfies the conditions (i), (ii), and (iii) above. We split each of the sub-intervals [a+rh, a+(r+1)h] into m further equal sub-intervals, so that the interval [a, a+lh] is now divided into ml sub-intervals, with knots at the points x = The function f is clearly piecewise polynomial with respect to these ml sub-intervals, (two-sided) derivatives these new knots, where, and left derivatives are further subdivision of of degree (2n+1)
each of length h/m. Moreover
off of all orders exist at all but every mth of however, by hypothesis, the first n odd right equal. We may then apply 4.1 above to this the interval [a, a+lh]. Effectively we are
considering equation 4.1 with h replaced by h/mand l replaced by ml. We thereby obtain
(4.3) The right-hand give the value of sides of equations 4.2 and 4.3 are equal (since both f(x)dx). By equating these two expressions
68
Woolhouse’s that
Formula
in Relation
to
we see immediately
(4.4) This last equation is the precise form of Woolhouse’s formula for piecewise polynomials of the required type.
5. THE COMMON ACTUARIAL
APPROXIMATIONS
By putting n = 1 in equations 4.2 and 4.4 above we obtain exact results, valid when f is piecewise cubic on the interval [a, a+lh] over These results are sub-intervals of equal length h and differentiable.
and
(In this last equation we use
to denote the left-hand
of equation 4.4 above.) These last two equations are of course the commonly used approximations, familiar to most actuarial students. The arguments above establish by elementary means simple conditions under which these approximations contain no error. In particular, if f is a cubic spline (so that not only f‘(x) but also f"(x) exists over the entire interval) with equally spaced knots, the traditionally used approximations are in fact exact results. I am grateful to Mr. A. D. Wilkie for discussions which prompted me to write this note.
Piecewise
Polynominal REFERENCES
Functions
69
1. KNOPP, K. Theory and Application of Infinite Series (Blackie, Glasgow), 1928. 2. MILNE-THOMSON, L. M. Calculus of Finite Differences (Macmillan, London), 1960. 3. FREEMAN, H. Finite Differences for Actuarial Students (C.U.P.), 1960.
63
WOOLHOUSE’S TO PIECEWISE
FORMULA POLYNOMIAL by M.A.,
IN
RELATION FUNCTIONS
J. J. McCUTCHEON,
Ph.D.,
F.F.A.
1. INTRODUCTION Generations of actuarial students have been confronted at an early stage of their careers with Woolhouse’s formula, important for its well-known application to annuities payable several times per year. However, it is probably fair to say that most of the derivations of the formula presented to students have been in some respects unsatisfactory, since they have usually relied on “ the separation of symbols ”, manipulations of the finite difference operators D, , and (m), and have involved arguments somewhat lacking in rigour. This is not a severe criticism of the relevant textbooks, which were after all written from a practical actuarial viewpoint and which understandably enough wished to avoid the relatively difficult analysis required to derive Woolhouse’s formula in its complete generality. Woolhouse’s formula leans heavily on a more fundamental result, the so-called Euler-Maclaurin formula, which gives a relationship between n f(r) and n0f(x)dx: involving the higher derivatives (assumed
to exist) of the function f at 0 and at n and a suitable remainder term. Unfortunately, however, in general the determination of the remainder term is rather complicated and serious doubts usually arise over the convergence of the associated infinite series. The formula as normally stated is therefore mainly of theoretical interest. (We give below relevant references.) Although for a completely general function the proof of the Euler-Maclaurin formula is rather complicated and lengthy, in the special case when we restrict our attention to functions which are piecewise polynomials of a certain type (see below) a reasonably short and completely rigorous proof can be given without using advanced analysis. Moreover the resulting formula is exact and has no unwieldy remainder term. Piecewise polynomial curves (and in particular splines) are of considerable practical importance and may represent more general functions to any desired degree of accuracy, so that this restricted class of functions may still be considered most
64
Woolhouse’s
Formula
in Relation
to
useful. Accordingly we feel that the relatively simple discussion below may be of general interest and also possibly of help to future students. 2. THE BERNOUILLI NUMBERS
Before our main argument it is necessary for completeness to recall the Bernouilli numbers {Bi} (i 0). This celebrated sequence may be defined by the initial value B0 = 1 and by the recurrence relation (2.2) Clearly this last condition, combined with the initial value, defines the entire sequence. For example, putting n = 2, 3, 4, and 5, in turn we obtain 2B1+B0 = 0 3B2+3B1+B0 = 0 4B3+6B2+4B1+B0 = 0 5B4+ 10B3+ 10B2+ 5B1+ B0 = 0, from which we calculate successively (2.1)
One important numbers is
and easily established B2k+1 = 0 (k
property 1).
of the Bernouilli (2.3)
(The interested reader may refer to the references below for a simple proof of this last result and further discussion of these numbers.) We now proceed to the next step of our argument, in which use is made of the above. 3. THE BASIC INTEGRATION FORMULA
Suppose that over the interval [0, 1]f(x) is a polynomial in x of degree (2n+1) where n 1 is a given positive integer. Such a polynomial is defined by (2n+2) suitably chosen parameters. For example, the polynomial is defined by the values of f(x) and the derivatives f(2r–1) (x) (r = 1, . . . , n) at the points x = 0 and x = 1. In particular these values must determine the integral off over the interval [0, 1]. This is confirmed by the following result, which is fairly well known.
65 Lemma Suppose that n 1 and that over the interval [0, 1]f(x) is a polynomial in x of degree (2n+ 1). Then (3.1) Note that, since B2 = 1/6, when n = 1 this implies
Piecewise
Polynominal
Functions
which is easily seen to hold for cubics. The general result now follows by induction on n. (The proof depends on only the defining relationship 2.2 and the readily established equation 2.3 above. Accordingly we omit further details.) The following generalisation of equation 3.1 above is easily established. Lemma Suppose that n is a positive integer and that over the interval [a, a+h] g(x) is a polynomial in x of degree (2n+ 1). Then
(3.2) Proof: Put t = so that x = a+ht and
where f(t) = g(a+ht) is a polynomial in t of degree (2n+l). We now use 3.1 above to evaluate this last integral. Since f(r)(t) = hrg(r)(a+ht), this immediately gives the required result. 4. AN APPLICATION TO PIECEWISE CURVES POLYNOMIAL
Suppose that the function f is defined on the interval [a, a+lh] where h> 0 and l 2 is a given positive integer. We assume that f is (i) continuous and (ii) piecewise polynomial, in the sense that over each of the subintervals [a+rh, a+(r+1)h] (r = 0, . . . . l–1)f(x) is given by a polynomial in x.
66
Woolhouse’s
Formula
in Relation
to is of degree
Suppose further that each of the defining polynomials at most (2n+1) (where n 1 is given) and that
(iii) for s = 1, . . . . n the left and right derivative8 of order (2s – 1) are equal at each of the point8 x = a+rh (r = 1, . . . . l–1). The last condition (iii) should be noted. What we require is that the polynomials defining f on adjacent sub-intervals should have the same value at the common boundary point of the intervals (to ensure continuity) and in addition the same value at that point of In particular the condition will each of their first n odd derivatives. be satisfied if f is a spline curve of order (2n+ 1) with the given points (In this case f is 2n-times differentiable on [a, a+lh].) as knots. However, it should be noted that the condition holds for a wider class of functions than splines. We may now apply equation 3.2 above successively to each of the Thus sub-intervals [a, a+h], [a+h, a+2h), . . ., [a+(l–1)h, a+lh]. we obtain
(In each of the above equations f(2r–1) (a+jh) must be interpreted as the common value of the left and right derivatives of order (2r–1) at the jth knot and not as a two-sided derivative, which may not exist.) By adding these last equations we obtain the integral off over the interval [a, a+lh] in the form
(4.1) which result is exact for piecewise polynomials of the type defined by conditions (i), (ii), and (iii) above. Equation 4.1 above may be regarded as a special case of the Euler-Maclaurin formula, valid when the function f is of the required
67 piecewise-polynomial form. It should be noted, however, that for such a function there is no awkward remainder term and that the result has been obtained by essentially elementary methods. Note that this last equation can be written in the form
Piecewise
Polynominal
Functions
(4.2) Woolhouse’s formula is readily inferred from equation 4.1 above. Suppose that m 2 is a positive integer and that the function f satisfies the conditions (i), (ii), and (iii) above. We split each of the sub-intervals [a+rh, a+(r+1)h] into m further equal sub-intervals, so that the interval [a, a+lh] is now divided into ml sub-intervals, with knots at the points x = The function f is clearly piecewise polynomial with respect to these ml sub-intervals, (two-sided) derivatives these new knots, where, and left derivatives are further subdivision of of degree (2n+1)
each of length h/m. Moreover
off of all orders exist at all but every mth of however, by hypothesis, the first n odd right equal. We may then apply 4.1 above to this the interval [a, a+lh]. Effectively we are
considering equation 4.1 with h replaced by h/mand l replaced by ml. We thereby obtain
(4.3) The right-hand give the value of sides of equations 4.2 and 4.3 are equal (since both f(x)dx). By equating these two expressions
68
Woolhouse’s that
Formula
in Relation
to
we see immediately
(4.4) This last equation is the precise form of Woolhouse’s formula for piecewise polynomials of the required type.
5. THE COMMON ACTUARIAL
APPROXIMATIONS
By putting n = 1 in equations 4.2 and 4.4 above we obtain exact results, valid when f is piecewise cubic on the interval [a, a+lh] over These results are sub-intervals of equal length h and differentiable.
and
(In this last equation we use
to denote the left-hand
of equation 4.4 above.) These last two equations are of course the commonly used approximations, familiar to most actuarial students. The arguments above establish by elementary means simple conditions under which these approximations contain no error. In particular, if f is a cubic spline (so that not only f‘(x) but also f"(x) exists over the entire interval) with equally spaced knots, the traditionally used approximations are in fact exact results. I am grateful to Mr. A. D. Wilkie for discussions which prompted me to write this note.
Piecewise
Polynominal REFERENCES
Functions
69
1. KNOPP, K. Theory and Application of Infinite Series (Blackie, Glasgow), 1928. 2. MILNE-THOMSON, L. M. Calculus of Finite Differences (Macmillan, London), 1960. 3. FREEMAN, H. Finite Differences for Actuarial Students (C.U.P.), 1960.
