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Any O(N^2) method for finding the eigenvalues of J-Hermitian (complex Hamiltonian) matrix?
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I came across the problem of finding the eigenvalue spectrum of a general J-Hermitian (complex Hamiltonian) matrix. Is there any method (available algorithms) that would work faster than the "standard" methods for finding the eigenvalues of non-structured (no symmetry) complex-valued matrices (the number of flops is O(N^3), with N being the dimension of the matrix). In the picture below $gamma_k$ are the complex Fourier-expansion coefficients of some given complex-valued functionHere is the part of the paper text where that matrix eigenproblem is defined with more details regarding its elements and structure
matrices eigenvalues-eigenvectors
asked Aug 1 at 20:08
Yary Prilly
61
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up vote
1
down vote
favorite
I came across the problem of finding the eigenvalue spectrum of a general J-Hermitian (complex Hamiltonian) matrix. Is there any method (available algorithms) that would work faster than the "standard" methods for finding the eigenvalues of non-structured (no symmetry) complex-valued matrices (the number of flops is O(N^3), with N being the dimension of the matrix). In the picture below $gamma_k$ are the complex Fourier-expansion coefficients of some given complex-valued functionHere is the part of the paper text where that matrix eigenproblem is defined with more details regarding its elements and structure
matrices eigenvalues-eigenvectors
asked Aug 1 at 20:08
Yary Prilly
61
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I came across the problem of finding the eigenvalue spectrum of a general J-Hermitian (complex Hamiltonian) matrix. Is there any method (available algorithms) that would work faster than the "standard" methods for finding the eigenvalues of non-structured (no symmetry) complex-valued matrices (the number of flops is O(N^3), with N being the dimension of the matrix). In the picture below $gamma_k$ are the complex Fourier-expansion coefficients of some given complex-valued functionHere is the part of the paper text where that matrix eigenproblem is defined with more details regarding its elements and structure
matrices eigenvalues-eigenvectors
asked Aug 1 at 20:08
Yary Prilly
61
I came across the problem of finding the eigenvalue spectrum of a general J-Hermitian (complex Hamiltonian) matrix. Is there any method (available algorithms) that would work faster than the "standard" methods for finding the eigenvalues of non-structured (no symmetry) complex-valued matrices (the number of flops is O(N^3), with N being the dimension of the matrix). In the picture below $gamma_k$ are the complex Fourier-expansion coefficients of some given complex-valued functionHere is the part of the paper text where that matrix eigenproblem is defined with more details regarding its elements and structure
matrices eigenvalues-eigenvectors
asked Aug 1 at 20:08
Yary Prilly
61
asked Aug 1 at 20:08
Yary Prilly
61
asked Aug 1 at 20:08
Yary Prilly
61
asked Aug 1 at 20:08
Yary Prilly
61
61
add a comment |Â
add a comment |Â
1 Answer
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The complexity of the eigenvalue problem for complex Hamiltonian matrices has been studied in A Note on the Numerical Solution of Complex Hamiltonian and Skew-Hamiltonian Eigenvalue Problems. There is no improvement over the usual $N^3$ complexity, however, what you can achieve is an algorithm that respects the structure of the matrix, so that if $lambda$ is an eigenvalue also $-barlambda$ is one. In particular, the routine returns eigenvalues with real part exactly equal to zero.
answered Aug 1 at 20:59
Carlo Beenakker
42448
Thank you. I have also seen some works related to the structure preserving methods, and I though that maybe if we forget about the structure preservation, we can have something faster
– Yary Prilly
Aug 3 at 12:14
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The complexity of the eigenvalue problem for complex Hamiltonian matrices has been studied in A Note on the Numerical Solution of Complex Hamiltonian and Skew-Hamiltonian Eigenvalue Problems. There is no improvement over the usual $N^3$ complexity, however, what you can achieve is an algorithm that respects the structure of the matrix, so that if $lambda$ is an eigenvalue also $-barlambda$ is one. In particular, the routine returns eigenvalues with real part exactly equal to zero.
answered Aug 1 at 20:59
Carlo Beenakker
42448
Thank you. I have also seen some works related to the structure preserving methods, and I though that maybe if we forget about the structure preservation, we can have something faster
– Yary Prilly
Aug 3 at 12:14
add a comment |Â
up vote
0
down vote
The complexity of the eigenvalue problem for complex Hamiltonian matrices has been studied in A Note on the Numerical Solution of Complex Hamiltonian and Skew-Hamiltonian Eigenvalue Problems. There is no improvement over the usual $N^3$ complexity, however, what you can achieve is an algorithm that respects the structure of the matrix, so that if $lambda$ is an eigenvalue also $-barlambda$ is one. In particular, the routine returns eigenvalues with real part exactly equal to zero.
answered Aug 1 at 20:59
Carlo Beenakker
42448
Thank you. I have also seen some works related to the structure preserving methods, and I though that maybe if we forget about the structure preservation, we can have something faster
– Yary Prilly
Aug 3 at 12:14
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The complexity of the eigenvalue problem for complex Hamiltonian matrices has been studied in A Note on the Numerical Solution of Complex Hamiltonian and Skew-Hamiltonian Eigenvalue Problems. There is no improvement over the usual $N^3$ complexity, however, what you can achieve is an algorithm that respects the structure of the matrix, so that if $lambda$ is an eigenvalue also $-barlambda$ is one. In particular, the routine returns eigenvalues with real part exactly equal to zero.
answered Aug 1 at 20:59
Carlo Beenakker
42448
The complexity of the eigenvalue problem for complex Hamiltonian matrices has been studied in A Note on the Numerical Solution of Complex Hamiltonian and Skew-Hamiltonian Eigenvalue Problems. There is no improvement over the usual $N^3$ complexity, however, what you can achieve is an algorithm that respects the structure of the matrix, so that if $lambda$ is an eigenvalue also $-barlambda$ is one. In particular, the routine returns eigenvalues with real part exactly equal to zero.
answered Aug 1 at 20:59
Carlo Beenakker
42448
answered Aug 1 at 20:59
Carlo Beenakker
42448
answered Aug 1 at 20:59
Carlo Beenakker
42448
answered Aug 1 at 20:59
Carlo Beenakker
42448
42448
Thank you. I have also seen some works related to the structure preserving methods, and I though that maybe if we forget about the structure preservation, we can have something faster
– Yary Prilly
Aug 3 at 12:14
add a comment |Â
Thank you. I have also seen some works related to the structure preserving methods, and I though that maybe if we forget about the structure preservation, we can have something faster
– Yary Prilly
Aug 3 at 12:14
Thank you. I have also seen some works related to the structure preserving methods, and I though that maybe if we forget about the structure preservation, we can have something faster
– Yary Prilly
Aug 3 at 12:14
Thank you. I have also seen some works related to the structure preserving methods, and I though that maybe if we forget about the structure preservation, we can have something faster
– Yary Prilly
Aug 3 at 12:14
add a comment |Â
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