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README.md

ESS

Extended SemiSeparable solver, which scales as O(N). The algorithm relies on embedding the semi-separable structure in an extended sparse matrix.

To run the solver key in the following:

make -f makefile_ESS.mk

Then run the executable with the number of unknowns and rank of the semi-separability

./ESS N m c

where N is the number of unknowns, m is the rank of semi-separability and c is a character to denote, if we want to compare with usual method or not. The semi-separable matrix obtained is a random one. To change the code, to get your desired semi-separable matrix, edit the file testESS.cpp. The lines you might need to change include the definition of the matrices U, V to obtain the semi-separability, vector of diagonal entries d and right hand side vector rhs.

Dan provides the python bindings, which also works for complex matrices (alpha and beta complex), and has kinldy made it available here.

Examples

To solve a 5000 by 5000 dense system with semi-separable rank being 4 and to compare with the usual dense solver, key in

./ESS 5000 4 y

To not compare with the dense solver, key in

./ESS 500000 4 n

Typically, larger the value of N (say beyond 10000), it is not advisable to compare with the usual solver, since the usual solver will take excruciatingly long time to solve.



GRP

The Generalized Rybicki Press algorithm can be interpreted as a special case of ESS. However, a naive ESS would lead to numerical issues (such as ill-conditioning, overflow/underflow) due to the presence of the exponential terms in the extended sparse matrix. The GRP circumvents this issue by a judicious analytic preconditioning (rescale the appropriate variables). To test that the GRP code is working, do the following:

make -f makefile_GRP.mk

Then run the executable with the number of unknowns and rank of the semi-separability

./GRP N m c

where N is the number of unknowns, m is the rank of semi-separability, i.e., the number of exponential sums in the covariance matrix and c is a character to denote, if we want to compare with usual method or not. The covariance matrix is of the form

K(i,j) = sum_k a_kexp(-b_k|t_i-t_j|)	when i != j
K(i,j) = d						 		when i = j

The timestamps t_i, a_k, b_k, d are chosen at random. To change the code, to get your desired covariance matrix, edit the file testGRP.cpp. The lines you might need to change include the definition of the vectors alpha (i.e., a_k's), beta (i.e., b_k's), t (i.e., the time stamps), the diagonal d and the right hand side rhs.

Examples

To solve a 5000 by 5000 dense system with semi-separable rank being 4 and to compare with the usual dense solver, key in

./GRP 5000 4 y

To not compare with the dense solver, key in

./GRP 500000 4 n

Typically, larger the value of N (say beyond 10000), it is not advisable to compare with the usual solver, since the usual solver will take excruciatingly long time to solve.

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Extended SemiSeparable solver

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