Early Cowles Commission work on simultaneous equations estimation centered on Koopman and Hood s ( 1953 ) algorithms from the economics of transportation and optimal routing, with maximum likelihood estimation, and closed form algebraic calculations, as iterative solution search techniques were limited in the days before computers.
32.
The voltage of the source V _ S is a known given value, but V _ D is on both sides of the equation, which forces an iterative solution : a starting value for V _ D is guessed and put into the right side of the equation.
33.
The third approach is 4D-Var, which in the implementation described uses the initial conditions and surface forcing as control variables to be modified in order to be consistent with the observations as well as a numerical representation of the equations of motion through iterative solution of a giant optimization problem.
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If you solve the equations of motion ( Buzz Aldrin wrote up a paper on exact iterative solutions to Kepler's laws, which should be " good enough " ), you can forward-project the actual orbits following the TLI burn, and see if the trajectory would be anywhere near the L1 when it crosses the Earth-Moon radial line.
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S . Kotz suggest that for " not too small " shape parameter estimates \ hat { \ alpha }, \ hat { \ beta }, the logarithmic approximation to the digamma function \ psi ( \ hat { \ alpha } ) \ approx \ ln ( \ hat { \ alpha }-\ tfrac { 1 } { 2 } ) may be used to obtain initial values for an iterative solution, since the equations resulting from this approximation can be solved exactly:
36.
There are no mathematical techniques for calculating the exact solution of high order non-linear difference equations . ( The Scarf Algorithm cannot be used in this context because the number of markets is infinite; i . e ., there is no assumed end of the world . ) Auerbach and Kotlikoff devised an iterative solution method that entails guessing how the economy's ratio of capital to labor will evolve and then updating the guesses based on deviations of annual capital and labor supplies from their respective annual demands and continuing in this manner until the economy's capital-labor transition path converges to a fixed-point path ( more precisely, until the guessed ratio of the annual demands for capital relative to labor equal the annual supplies of capital relative to labor ).
