Inversion of the Van der Monde matrix

Neagoe, Victor-Emil

doi:10.1109/97.489066

Cited by 57 publications

(24 citation statements)

References 1 publication

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“…The coefficients x n,k,m can be solved for by using the inverse of the Vandermonde matrix. 45 Combining the results above, we obtain the following improved gap estimator: Remarkably, this estimator is asymptotically unbiased and the limits are interchangeable (see Appendix for the analytical derivation):…”

Section: Generalized Moment Methodsmentioning

confidence: 85%

Velocity of excitations in ordered, disordered, and critical antiferromagnets

2015

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We test three different approaches, based on quantum Monte Carlo simulations, for computing the velocity c of triplet excitations in antiferromagnets. We consider the standard S = 1/2 oneand two-dimensional Heisenberg models, as well as a bilayer Heisenberg model at its critical point. Computing correlation functions in imaginary time and using their long-time behavior, we extract the lowest excitation energy versus momentum using improved fitting procedures and a generalized moment method. The velocity is then obtained from the dispersion relation. We also exploit winding numbers to define a cubic space-time geometry, where the velocity is obtained as the ratio of the spatial and temporal lengths of the system when all winding number fluctuations are equal. The two methods give consistent results for both ordered and critical systems, but the winding number estimator is more precise. For the Heisenberg chain, we accurately reproduce the exactly known velocity. For the two-dimensional Heisenberg model, our results are consistent with other recent calculations, but with an improved statistical precision; c = 1.65847(4). We also use the hydrodynamic relation c 2 = ρs/χ ⊥ (q → 0) between c, the spin stiffness ρs, and the transversal susceptibility χ ⊥ , using the smallest non-zero momentum q = 2π/L. This method also is well controlled in two dimensions, but the cubic criterion for winding numbers delivers better numerical precision. In one dimension the hydrodynamic relation is affected by logarithmic corrections which make accurate extrapolations difficult. As an application of the winding number method, for the quantum-critical bilayer model our high-precision determination of the velocity enables us to quantitatively test, at an unprecedented level, the field-theoretic low-temperature scaling forms χ = (a/c 2 )T and C = (b/c 2 )T 2 . We find agreement to within 3% with the leading 1/N results for the factors a and b in the O(N ) model, but the agreement becomes significantly worse when the first corrections in 1/N are considered.

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Section: Generalized Moment Methodsmentioning

confidence: 85%

Velocity of excitations in ordered, disordered, and critical antiferromagnets

2015

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“…That result is related to the result of a method due to D.Gottlieb, C.W. Shu, A. Solomonoff, and H. Vendeven [4] about approximation of an analytic non-periodic function, defined on a finite interval I, by algebraic polynomials with exponential decay in uniform norm.…”

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confidence: 78%

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

Vatchev

2011

Anal. Theory Appl.

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For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form L n (ψ) = n ∑ k=0 a k ψ (k) , where the constant coefficients a k ∈ R may be adapted to f . We prove that for each f ∈ C (n) (I), there is a selection of coefficients {a 1 , ··· , a n } and a corresponding linear combinationof functions ψ k (t) = e λ k t in the nullity of L which satisfies the following Jackson's type inequality:For the particular operator M n ( f ) = f + 1/(2n)! f (2n) the rate of approximation by the eigenvalues of M n for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.

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“…In its form, determinant (7) resembles the well-known Van der Monde determinant. This determinant was the subject of a very intensive analysis [15][16][17][18]. However, all of these papers were proposing only the procedures that relied on program procedures (mostly iterative ones) in determining the value of the original and inverted Van der Monde matrix with an improved efficiency.…”

Section: Problem Formulationmentioning

confidence: 99%

New method and circuit for processing of band-limited periodic signals

Petrovič

2010

SIViP

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The paper presents a reconstruction of analogue multi-harmonic signals, from a number of integrated values of input signals. Based on the value of the integral of the original input signal, with a known frequency spectrum but unknown amplitudes and phases, a reconstruction of its basic parameters is done by the means of derived analytical and summarized expressions. It is applied to signal reconstruction, spectral estimation, system identification, as well as in other important signal processing problems. The proposed method of processing can be used for precise RMS measurements (or power and energy) of periodic signal based on the presented signal reconstruction. Subsequent calculation of all relevant indicators related to the monitoring and processing of ac voltage and current signals is provided in this manner. The paper investigates the errors related to the signal parameter estimation, and there is a computer simulation that demonstrates the accuracy of these algorithms.

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Inversion of the Van der Monde matrix

Cited by 57 publications

References 1 publication

Velocity of excitations in ordered, disordered, and critical antiferromagnets

Velocity of excitations in ordered, disordered, and critical antiferromagnets

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

New method and circuit for processing of band-limited periodic signals

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