The metric for the case of two comparable masses cannot be solved in closed form and therefore one has to resort to approximation techniques such as the post-Newtonian approximation or numerical approximations.
42.
An example of this is the use of numerical approximations to the Navier Stokes equations to describe aerodynamic flow over an aircraft, or the use of Miner's rule to calculate fatigue damage.
43.
Exact solutions generally only exist for fairly simple systems, so often the eigenfunctions are found either via numerical approximation or by approximating complex systems in terms of simpler systems that are analytically solvable.
44.
In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, Variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior.
45.
The iterative numerical approximation methods for solving equations, which use a calculated estimate for the solution, for use in calculating the next, improved, solution-estimate, differ only by how their calculated solution-estimates are made.
46.
Of his extensive investigations into the solutions ( especially by numerical approximation ) of certain classes of physical differential equations, only a few items have been published, at intervals, in the " Philosophical Magazine ".
47.
One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection ( YBC 7289 ), which gives a sexagesimal numerical approximation of \ sqrt { 2 }, the length of the diagonal in a unit square.
48.
The exact circular frequency \ omega increases in the numerical approximation by a factor of 1 + \ tfrac1 { 24 } \ omega ^ 2 \ Delta t ^ 2 + O ( \ Delta t ^ 4 ).
49.
Lyness and Moler showed in 1966 that using undetermined coefficients for the polynomials in Neville's algorithm, one can compute the Maclaurin expansion of the final interpolating polynomial, which yields numerical approximations for the derivatives of the function at the origin.
50.
But, there is considerable uncertainty in the measurement of some of the relevant key physical constants, and the QCD calculations are so difficult that solutions to these equations are almost always numerical approximations ( reached by several very different methodologies ).
