where [C.sub.p+1] is called as an error constant
of the method.
It is interesting to observe that only a single coefficient [A.sub.0] in [[phi].sub.4] ([x.sub.n], [y.sub.n]) contributes to its role in the construction of the desired asymptotic error constant
as can be seen in Theroem 1.
With reference to the definition in Lambert , Henrici , and Butcher , the order and error constant
of the (m+ 1)th-step block method follow Definition 6.
Other than that, as v [right arrow] 0, PFAFTDRK4(6) will have the same error constant
The term [[??].sub.p+2] is called the error constant
and the local truncation error is given by
(10), and [rho], [tau], [lambda] is the error constant
Therefore, the order of three-step Adam's method is five; (p = 5) with error constant
[[-11/1440 -1/10080 0 -1/126 -3/160 -9/560].sup.T].
therefore, [C.sub.s+1] is the error constant
and [C.sub.s+1] [h.sup.(s+1)] [y.sup.(s+1)([t.sub.n])] the principal local truncation error at the point [t.sub.n].
and proved that if [alpha] [member of] [0.21,1), then all methods of class (5) are linearly convergent with asymptotic error constant
[absolute value of 2[alpha] - 1].
It is worth pointing out that the y-intercept equals b = log K, where K is the absolute value of the error constant
The coefficients a(i), b(i) i = 1,..., 4 are numerically determined to minimize an error constant
that measures Hamiltonian (total energy) truncation errors.
n m Integral Relative Error Error Constant
1 2 0.2485091E+01 0.2665E-01 0.3101E-01 2 4 0.2550594E+01 0.9960E-03 0.1217E-03 3 6 0.2553083E+01 0.2123E-04 0.2113E-06 4 8 0.2553135E+01 0.6903E-06 0.2062E-09 5 10 0.2553138E+01 0.1502E-06 0.1289E-12 Similar results were obtained for other values of [Omega] [is greater than or equal to] 0.5, including integer and near-integer values, as well as for m = n and m = 2 (even for m = 0, since the poles are not very close to the interval [0,1]).