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They are robust towards noise and can be calculated efficiently using the fast Fourier transform (FFT) method.
The FBF coefficients can be calculated efficiently using the Levinson-Durbin recursion.
We note in passing that, as usual, all the inner products involving complex sinusoids of different frequencies can be calculated efficiently using FFTs.
Hence, a local optimum of a GP problem is also a global optimum, which can always be calculated efficiently using interior-point methods [29].
As before, M g (A i ) can be calculated efficiently using the Subset Summation Tree.
ACS and shustrings can be calculated efficiently using suffix trees (Weiner, 1973).
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The steepest descent direction can be calculated efficiently by using the symmetry of numerical Green's function derived from a mixed finite element method proposed by Nédélec for Maxwell's equation, without calculating the Jacobian matrix explicitly.
By using the Poisson summation formula with Ewald's trick the effect of image vortices can be calculated efficiently.
Both of the expectations can be calculated efficiently (see Supplementary Material).
The MVN densities can be calculated efficiently with the diagonalization matrices given in step 1.
By applying a model predictive control (MPC) method, we formulate the problem of finding the optimal joint train regulation and passenger flow control strategy as the problem of solving a set of quadratic programming (QP) problems, under which an optimal control law can be numerically calculated efficiently using a quadratic programming algorithm.
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