A fast algorithm for particle simulations
Journal of Computational Physics
Variable order panel clustering
Computing
H2-matrix approximation of integral operators by interpolation
Applied Numerical Mathematics
Multiscale Bases for the Sparse Representation of Boundary Integral Operators on Complex Geometry
SIAM Journal on Scientific Computing
Approximation of Integral Operators by Variable-Order Interpolation
Numerische Mathematik
Hybrid cross approximation of integral operators
Numerische Mathematik
A Fast Direct Solver for a Class of Elliptic Partial Differential Equations
Journal of Scientific Computing
Proceedings of the 48th Design Automation Conference
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices
SIAM Journal on Matrix Analysis and Applications
A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix
SIAM Journal on Matrix Analysis and Applications
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For hierarchical matrices, approximations of the matrix-matrix sum and product can be computed in almost linear complexity, and using these matrix operations it is possible to construct the matrix inverse, efficient preconditioners based on approximate factorizations or solutions of certain matrix equations.**-matrices are a variant of hierarchical matrices which allow us to perform certain operations, like the matrix-vector product, in ``true'' linear complexity, but until now it was not clear whether matrix arithmetic operations could also reach this, in some sense optimal, complexity.We present algorithms that compute the best-approximation of the sum and product of two **-matrices in a prescribed **-matrix format, and we prove that these computations can be accomplished in linear complexity. Numerical experiments demonstrate that the new algorithms are more efficient than the well-known methods for hierarchical matrices.