Gregory formula

for the approximate integration of a function $ f $

The formula

$$ { \frac{1}{h} } \int\limits _ { a } ^ { {a } + nh } f ( x) dx = \ { \frac{1}{2} } y _ {0} + y _ {1} + \dots + y _ {n - 1 } + { \frac{1}{2} } y _ {n} + $$

$$ + A _ {2} ( \Delta ^ {1} y _ {n - 1 } - \Delta ^ {1} y _ {0} ) - A _ {3} ( \Delta ^ {2} y _ {n - 2 } - \Delta ^ {2} y _ {0} ) + $$

$$ + A _ {4} ( \Delta ^ {3} y _ {n - 3 } - \Delta ^ {3} y _ {0} ) - A _ {5} ( \Delta ^ {4} y _ {n - 4 } + \Delta ^ {4} y _ {0} ) + \dots + R, $$

where $ y _ {j} = f ( a + jh) $, and $ \Delta ^ {l} y _ {j} $ are the differences of the function $ y $ of order $ l = 1, 2 \dots $ at the points $ a + jh $, $ j = 0 \dots n, $ and

$$ A _ {k} = \ \int\limits _ { 0 } ^ { 1 } \frac{t ( t - 1) \dots ( t - k + 1) }{( k!) } dt,\ \ k = 2, 3 ,\dots . $$

In particular,

$$ A _ {2} = - { \frac{1}{12} } ,\ \ A _ {3} = { \frac{1}{24} } , $$

$$ A _ {4} = - { \frac{19}{720} } ,\ A _ {5} = { \frac{3}{160} } ,\dots . $$

Gregory's formula is obtained by the integration of the interpolation polynomial with nodes at $ a, a + h \dots a + nh $. Gregory's formula with differences of orders up to $ n $ inclusive can be obtained from the Newton–Cotes quadrature formula of closed type (cf. Cotes formulas), and for this reason the residual terms of the two formulas are identical. The simplest variant of Gregory's formula was proposed by J. Gregory in 1668.

References

[1]	I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)

Comments

In [1] above, the Gregory formulas are derived by means of the Fraser diagram and by integrating the resulting polynomial interpolation formula in each interval $ [ a + jh , a + jh + h ] $. In this reference it is also observed that they can be derived from the Euler–MacLaurin formula by suitable discretization of the "end corrections" . It turns out that this approach is rather straightforward and yields explicit expressions in terms of Stirling numbers and Bernoulli numbers for the coefficients $ A _ {k} $( cf. [a2]). Yet another approach for deriving the Gregory formulas integrates the "initial-value problem" $ y ^ \prime ( x) = f ( x) $, $ y ( a) = 0 $ from $ x = a $ until $ x = a + n h $ by a numerical integration method with step size $ h $. Let $ Y ( n h ) $ denote the numerical approximation obtained for $ y ( n h ) $. Since $ y ( n h ) $ equals the integration of the function $ f $ over the interval $ [ a , a + n h ] $, one may consider $ Y ( n h ) $ as an approximation integration of (or quadrature formula for) the function $ f $ over the interval $ [ a , a + n h ] $. If the numerical integration method used for integrating $ y ^ \prime ( x) = f ( x) $, $ y ( a) = 0 $ is a linear multi-step method generated by polynomials $ \rho $ and $ \sigma $, then the quadrature formula determined by $ Y ( n h ) $ is called a $ ( \rho , \sigma ) $- reducible quadrature formula. If the linear multi-step method $ ( \rho , \sigma ) $ is an Adams–Moulton method (cf. Adams method) with starting values derived from interpolatory quadrature formulas, then $ Y ( n h ) $ is identical with a Gregory formula. The notion of $ ( \rho , \sigma ) $- reducible quadrature formulas was introduced by J. Matthys [a4]. The general theory of these formulas is due to P.H.M. Wolkenfelt [a6]. The particular case of the Gregory formulas is discussed in [a5] and [a2], and, especially, in [a1]. These references give expressions for the residual term (or quadrature error) of Gregory formulas with differences of arbitrary orders. Gregory formulas play an important role in the numerical solution of Volterra equations (cf. [a5], [a3] and [a2]).

References

[a1]	H. Brass, "Quadraturverfahren" , Vandenhoeck & Ruprecht (1977)
[a2]	H. Brunner, P.J. van der Houwen, "The numerical solution of Volterra equations" , North-Holland (1986)
[a3]	H. Brunner, J.D. Lambert, "Stability of numerical methods for Volterra integro-differential equations" Computing , 12 (1974) pp. 75–89
[a4]	J. Matthys, "A stable linear multistep method for Volterra integro-differential equations" Numer. Math. , 27 (1976) pp. 85–94
[a5]	J. Steinberg, "Numerical solution of Volterra integral equations" Numer. Math. , 19 (1972) pp. 212–217
[a6]	P.H.M. Wolkenfelt, "The construction of reducible quadrature rules for Volterra integral and integro-differential equations" IMA J. Numer. Anal. , 2 (1982) pp. 131–152