A Method for Solving Algebraic Equations Using an Automatic Computer

Muller, David E.

doi:10.2307/2001916

Cited by 486 publications

(123 citation statements)

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“…(10) using the Müller method (Muller 1956) to find the complex roots at fixed input parameters η = 0.01 and b = 0.36, and varying Alfvén-Mach numbers M A from zero to some reasonable numbers. Before starting any numerical procedure for solving the aforementioned dispersion equation, we note that for each input value of M A one can get two c k -dispersion curves one of which (for relatively small magnitudes of M A ) has a normalised phase velocity roughly equal to M A − 1 and a second dispersion curve associated with dimensionless phase velocity equal to M A + 1.…”

Section: Kink Wavesmentioning

confidence: 99%

Magnetohydrodynamic waves and their stability status in solar spicules

Zhelyazkov

2012

A&A

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Aims. We investigate conditions under which magnetohydrodynamic waves propagating along spicules become unstable because of the Kelvin-Helmholtz instability. Methods. We employ the dispersion relations of normal modes (kink and sausage waves) derived from the linearised magnetohydrodynamic equations. We assume real wave numbers and complex angular wave frequencies, namely complex wave phase velocities. The dispersion relations are solved numerically at fixed input parameters and various flow velocities. Results. It is shown that the stability of the waves depends upon three parameters, the density contrast between spicules and their environment, the ratio of the background magnetic field outside to that inside spicules, and the value of the Alfvén-Mach number (the ratio of the jet velocity to Alfvén speed inside the spicules). At certain densities and magnetic fields, an instability of the Kelvin-Helmholtz type can arise if the Alfvén-Mach number exceeds a critical value -in our case it is equal to 12.6, which means that for an Alfvén speed inside the spicules of 70 km s −1 the jet velocity should be larger than 882 km s −1 . Conclusions. It is found that only kink waves can become unstable, while the sausage ones are always unaffected by the Kelvin-Helmholtz instability.

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Section: Kink Wavesmentioning

confidence: 99%

Magnetohydrodynamic waves and their stability status in solar spicules

Zhelyazkov

2012

A&A

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“…(48) requires an initial guess, Q (1) d (z), which we will describe later. It is hoped as we iterate through i finding a minimum of (48), then the differences between the values of …”

Section: Methods Of Nonlinear Least Squaresmentioning

confidence: 99%

Extension of the continuum time-dependent Hartree-Fock method to proton states

Pardi

Stevenson

2014

Phys. Rev. E

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This paper deals with the solution of the spherically symmetric time-dependent Hartree-Fock approximation applied to nuclear giant monopole resonances in the small amplitude regime. The problem is spatially unbounded as the resonance state is in the continuum. The practical requirement to perform the calculation in a finite-sized spatial region yields an artificial boundary, which is not present physically. The question of how to ensure the boundary does not interfere with the internal solution, while keeping the overall calculation time low is studied. Here we propose an absorbing boundary condition scheme to handle the conflict. The derivation, via a Laplace transform method, and implementation is described. An inverse Laplace transform required by the absorbing boundaries is calculated using a method of nonlinear least squares. The accuracy and efficiency of the scheme is tested and results presented to support the case that they are an effective way of handling the artificial boundary.

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“…x min < ; (12) where x min is that x (k) leading to the minimum jP(x)j up to the actual iteration step and again depends on the computer accuracy. In the case of a real valued polynomial (i.e., all p are real) for every complex root also its complex conjugate is a root which can be de ated together.…”

Section: Newton's Methodsmentioning

confidence: 99%

A New and Efficient Program for Finding all Polynomial Roots

Lang¹,

Frenzel²

1994

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Finding polynomial roots rapidly and accurately is an important problem in many areas of signal processing. We present a new program which is a combination of Muller's and Newton's method. We use the former for computing a root of the de ated polynomial which is a good estimate for the root of the original polynomial. This estimate is improved by applying Newton's method to the original polynomial. Test polynomials up to the degree 10000 show the superiority of our program over the best methods to our knowledge regarding speed and accuracy, i.e., Jenkins/Traub program and the eigenvalue method.Furthermore we give a simple approach to improve the accuracy for spectral factorization in the case there are double roots on the unit circle. Finally we brie y consider the inverse problem of root nding, i.e., computing the polynomial coe cients from the roots which may lead to surprisingly large numerical errors.This research was partially supported by Alexander von Humboldt-Stiftung and by AFOSR under grant F49620-1-0006 funded by DARPA 1 Report Documentation Page Form Approved OMB No. 0704-0188Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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A Method for Solving Algebraic Equations Using an Automatic Computer

Cited by 486 publications

References 0 publications

Magnetohydrodynamic waves and their stability status in solar spicules

Magnetohydrodynamic waves and their stability status in solar spicules

Extension of the continuum time-dependent Hartree-Fock method to proton states

A New and Efficient Program for Finding all Polynomial Roots

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