“…As a first result, we present the following theorem, clarifying the spectral structure of definite pseudosymmetric matrices. It is an extension of Theorem 5 in [16], additionally clarifying the structure of the eigenvectors, and a more general variant of Theorem 3 in [46]. Our version is independent of the additional structure of Bethe-Salpeter matrices given in (9).…”
Section: Decoupling the Indefinite Eigenvalue Problem Into Two Symmet...mentioning
confidence: 92%
“…A rational function g(x) = xh(x 2 ) which maps them close to 1, i.e. approximates the scalar sign function on the interval (ℓ, 1], can be used in an iteration (16). We see from (17) that the result will be an approximation to the polar factor W , which in our setting coincides with the matrix sign function.…”
Section: Using Zolotarev Functions To Accelerate the Matrix Sign Iter...mentioning
confidence: 94%
“…In [47], the indefinite QR decomposition is presented, which improves stability by allowing 2 × 2 blocks on the diagonal of R and additional pivoting. This variant can also be computed via the (pivoted) LDL T decomposition of A T ΣA; a link which was exploited in [14] and [16]. There, the stability is improved by applying this method twice.…”
“…Consequently, employing Lemma 11 twice on a matrix A = W S with g(x) = Ẑ2r+1 (x; ℓ), we see that the eigenvalues of g(g(S)) will be in the interval [1 − 10 −15 , 1], under the condition that all eigenvalues of S are in [ℓ, 1] with ℓ ≥ 10 −16 . In this sense, G(G(A)) ≈ W has converged to the polar factor W , after two iterations of Iteration (16). Choosing a higher r, algorithms can be devised that converge in just one step.…”
Section: Using Zolotarev Functions To Accelerate the Matrix Sign Iter...mentioning
We devise a spectral divide-and-conquer scheme for matrices that are selfadjoint with respect to a given indefinite scalar product (i.e. pseudosymmetic matrices). The pseudosymmetric structure of the matrix is preserved in the spectral division, such that the method can be applied recursively to achieve full diagonalization. The method is well-suited for structured matrices that come up in computational quantum physics and chemistry. In this application context, additional definiteness properties guarantee a convergence of the matrix sign function iteration within two steps when Zolotarev functions are used. The steps are easily parallelizable. Furthermore, it is shown that the matrix decouples into symmetric definite eigenvalue problems after just one step of spectral division.
“…As a first result, we present the following theorem, clarifying the spectral structure of definite pseudosymmetric matrices. It is an extension of Theorem 5 in [16], additionally clarifying the structure of the eigenvectors, and a more general variant of Theorem 3 in [46]. Our version is independent of the additional structure of Bethe-Salpeter matrices given in (9).…”
Section: Decoupling the Indefinite Eigenvalue Problem Into Two Symmet...mentioning
confidence: 92%
“…A rational function g(x) = xh(x 2 ) which maps them close to 1, i.e. approximates the scalar sign function on the interval (ℓ, 1], can be used in an iteration (16). We see from (17) that the result will be an approximation to the polar factor W , which in our setting coincides with the matrix sign function.…”
Section: Using Zolotarev Functions To Accelerate the Matrix Sign Iter...mentioning
confidence: 94%
“…In [47], the indefinite QR decomposition is presented, which improves stability by allowing 2 × 2 blocks on the diagonal of R and additional pivoting. This variant can also be computed via the (pivoted) LDL T decomposition of A T ΣA; a link which was exploited in [14] and [16]. There, the stability is improved by applying this method twice.…”
“…Consequently, employing Lemma 11 twice on a matrix A = W S with g(x) = Ẑ2r+1 (x; ℓ), we see that the eigenvalues of g(g(S)) will be in the interval [1 − 10 −15 , 1], under the condition that all eigenvalues of S are in [ℓ, 1] with ℓ ≥ 10 −16 . In this sense, G(G(A)) ≈ W has converged to the polar factor W , after two iterations of Iteration (16). Choosing a higher r, algorithms can be devised that converge in just one step.…”
Section: Using Zolotarev Functions To Accelerate the Matrix Sign Iter...mentioning
We devise a spectral divide-and-conquer scheme for matrices that are selfadjoint with respect to a given indefinite scalar product (i.e. pseudosymmetic matrices). The pseudosymmetric structure of the matrix is preserved in the spectral division, such that the method can be applied recursively to achieve full diagonalization. The method is well-suited for structured matrices that come up in computational quantum physics and chemistry. In this application context, additional definiteness properties guarantee a convergence of the matrix sign function iteration within two steps when Zolotarev functions are used. The steps are easily parallelizable. Furthermore, it is shown that the matrix decouples into symmetric definite eigenvalue problems after just one step of spectral division.
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