Mathematics of Computation
Computational frameworks for the fast Fourier transform
Computational frameworks for the fast Fourier transform
Fast Gaussian elimination with partial pivoting for matrices with displacement structure
Mathematics of Computation
Displacement structure: theory and applications
SIAM Review
On Robbins boundary conditions, elliptic equations, and FTT methods
Journal of Computational Physics
SIAM Review
A Fast Direct Solution of Poisson's Equation Using Fourier Analysis
Journal of the ACM (JACM)
A multigrid tutorial: second edition
A multigrid tutorial: second edition
Fast reliable algorithms for matrices with structure
Fast reliable algorithms for matrices with structure
Matrix decomposition algorithms for elliptic boundary value problems: a survey
Numerical Algorithms
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We present a fast direct method for solving the two-dimensional Helmholtz equation: ∂2φ/∂x2+∂2φ/∂y2+λφ = f(x,y), on a rectangular grid [0,a1] × [0,a2] with Robbins boundary conditions ( ∂φ/∂x-p0φ) (0y) = α0(y), ( ∂φ/∂x-p1φ) (a1,y) = α1(y), ( ∂φ/∂y-q0φ) (x,0) = β0(x), ( ∂φ/∂y-q1φ) (x,a2) = β1(x), where λ, P0, P1, q0 and q1 are constants. Because we can solve the Helmholtz equation with Neumann boundary conditions in a fast way using the discrete cosine transform, we can split the problem above into two smaller problems. One of these problems can be solved using the same techniques as in the Neumann-boundary case. The second, and the hardest problem of the two, can be solved using low displacement rank techniques. When dividing [0, a1] into n1 and [0,a2] into n2 equal parts, the total complexity of the overall algorithm is 10n1n2 log n2 + O(n12+n1n2), which gives us a fast algorithm.