Multiple quadrature with central differences on one line

Abstract

The coefficients A 2 m n A_{2m}^n in the n-fold quadrature formulas for the stepwise integration of (1) y ( n ) = f ( x , y , y ′ , ⋯ , y ( n − 1 ) ) {y^{(n)}} = f(x,y,y’, \cdots , {y^{(n - 1)}}) , at intervals of h, namely, for n even, (2) δ n y 0 = h n ∑ m = 1 10 ( 1 + A 2 m n δ 2 m ) f 0 + ⋯ {\delta ^n}{y_0} = {h^n}\sum \nolimits _{m = 1}^{10} {(1 + A_{2m}^n{\delta ^{2m}}){f_0} + \cdots } , for n odd, (3) μ δ n y 0 = h n ∑ m = 1 10 ( 1 + A 2 m n δ 2 m ) f 0 + ⋯ \mu {\delta ^n}{y_0} = {h^n}\sum \nolimits _{m = 1}^{10} {(1 + A_{2m}^n{\delta ^{2m}}){f_0} + \cdots } , are tabulated exactly for n = 1(1)6, m = 1(1)10. They were calculated from the well-known symbolic formulas (4) δ n y = ( δ / D ) n f {{\delta }^{n}}y = {{({\delta }/{D})}^{n}}f , (5) ( δ / D ) n = ( δ h / 2 sinh − 1 ( δ / 2 ) ) n {{({\delta }/{D})}^{n}} = {{({{{\delta h}/{2\,\sinh }}^{-1}}({\delta }/{2}))}^{n}} and (6) μ = ( 1 + δ 2 / 4 ) 1 / 2 = 1 + δ 2 8 − δ 4 128 + δ 6 1024 − 5 δ 8 32768 + ⋯ \mu = {{(1 + {{{\delta }^{2}}}/{4})}^{{1}/{2}}} = 1 + \frac {{{\delta }^{2}}}{8}\,-\,\frac {{{\delta }^{4}}}{128} + \frac {{{\delta }^{6}}}{1024}\,-\,\frac {5{{\delta }^{8}}}{32768} + \cdots . For calculating y ( r ) {y^{(r)}} , replace n by n - r in (2) and (3). Use of (2) and (3) avoids the solution of (1) by simultaneous lower-order systems for n > 1 n > 1 , as well as mid-interval tabular arguments, requires only even-order differences, on a single line, and provides great accuracy due to rapid decrease of A 2 m n A_{2m}^n as m increases. However, the integration may be slowed down by the need to estimate and refine iteratively the later values of y , y ′ , ⋯ , y ( n − 1 ) y,y’, \cdots , {y^{(n - 1)}} required in δ 2 m f 0 {\delta ^{2m}}{f_0} . Reference to earlier collected formulas of Legendre, Oppolzer, Thiele, Lindow, Salzer, Milne and Buckingham, reveals that Thiele and Buckingham come closest to (2), (3), as their works contain schemes that involve just tabular arguments throughout. For n odd, they give formulas that are based upon the series in δ 2 m {\delta ^{2m}} for ( 1 / μ ) ( δ / D ) n ({1}/{\mu }){{({\delta }/{D})}^{n}} instead of μ ( δ / D ) n \mu {{({\delta }/{D})}^{n}} as in the present arrangement.