Algorithm 419: zeros of a complex polynomial [C2]

Jenkins, M. A.; Traub, Joseph F.

doi:10.1145/361254.361262

Cited by 73 publications

(29 citation statements)

References 3 publications

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“…A change of 0.5% in the second coefficient yields a polynomial of x 4 -1O.05x 3 + 35x 2 -50x + 24, which has roots of (x -0.992)(x -2.340 -jO.2269)(x -2.340 + j0.2269)(x -4.378). The choice of a complex rooting routine by Jenkins and Traub and double precision calculations are the tools used to reduce errors in this section of the algorithm [45]. While the three stage rooting algorithm has performed well, the sensitivity of the roots of a polynomial to the coefficient values indicates the choice of algorithms in step one may be the major contributing factor to the accuracy of the final model.…”

Section: Prony's Methodsmentioning

confidence: 99%

A prony algorithm for shallow water waveguide analysis

Diemer¹

1987

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Low frequency acoustic propagation in shallow water is examined from a normal mode context. By modelling the far field pressure field as a modal sum, propagating mode characteristics of wavenumber, initial phase, attennation and amplitude may be estimated using a high resolution parameter modeling technique. The advantages of such an algorithm are the resolution of closely spaced modes in a range independent environment and the ability to analyze range dependent waveguides.This thesis presents the application of a Prony algorithm to the shallow water environment. The algorithm operates directly on the signal matrix. Synthetically generated, range independent pressure fields are used to analyze the technique'S performance and to observe its sensitivity to variations in model specifications. Noise is added to determine the threshold of acceptable performance. As a consequence of field data tests, further enhancements to the algorithm are suggested.Range dependent performance is evaluated on a coastal wedge example and geoacoustic parameter shift example.

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Section: Prony's Methodsmentioning

confidence: 99%

A prony algorithm for shallow water waveguide analysis

Diemer¹

1987

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“…(9), to zero and solving the resulting equations. Here, this is done numerically, using the method described by Jenkins and Traub (1972), and examining the negative real root values of z -1 . For the case of T=1, K 1 =5, K 2 =20 and various values of α from 0 to 1, the results for various values of N in Table 1 [ Fig.…”

Section: Analysis Of a 2-component Responsementioning

confidence: 99%

Root selection methods in flood analysis

Parmentier

Dooge

Bruen

2003

Hydrol. Earth Syst. Sci.

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In the 1970s, de Laine developed a root-matching procedure for estimating unit hydrograph ordinates from estimates of the fast component of the total runoff from multiple storms. Later, Turner produced a root selection method which required only data from one storm event and was based on recognising a pattern typical of unit hydrograph roots. Both methods required direct runoff data, i.e. prior separation of the slow response. This paper introduces a further refinement, called root separation, which allows the estimation of both the unit hydrograph ordinates and the effective precipitation from the full discharge hydrograph. It is based on recognising and separating the quicker component of the response from the much slower components due to interflow and/or baseflow. The method analyses the z-transform roots of carefully selected segments of the full hydrograph. The root patterns of these separate segments tend to be dominated by either the fast response or the slow response. This paper shows how their respective time-scales can be distinguished with an accuracy sufficient for practical purposes. As an illustration, theoretical equations are derived for a conceptual rainfall-runoff system with the input split between fast and slow reservoirs in parallel. These are solved analytically to identify the reservoir constants and the input splitting parameter. The proposed method, called "root separation", avoids the subjective selection of rainfall roots in the Turner method as well as the subjective matching of roots in the original de Laine method.

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“…(4)]. It is important to consider two subcases: If all coefficients are numeric, there are subroutines (see [7]) which compute the roots and inform the user about the existence of multiple or complex roots. When the polynomial involves literal coefficients, the situation becomes more complex.…”

Section: Stepmentioning

confidence: 99%

“…The current capabilities of the program which performs this step are summarized in Table I. It should be remarked that, when dealing with homogeneous equations whose coefficients are numeric, it is possible to use strictly numeric routines such as [5], [7], [12] to replace the time-consuming symbolic computations of steps 2, 4, and 5.…”

Section: Stepmentioning

confidence: 99%

Symbolic Solution of Finite-Difference Equations

Cohen

Katcoff

1977

ACM Trans. Math. Softw.

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An interactive computer program which has some capability for solving systems of finite difference equations is described. Although this capability is limited to linear systems, a knowledgeable user can, with the help of the program, solve a wider class of equations. Over 100 examples, covering a variety of cases, have been solved by using the program. Some representative examples are presented. Additional features which would improve the versatility of the program are also discussed.

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Algorithm 419: zeros of a complex polynomial [C2]

Cited by 73 publications

References 3 publications

A prony algorithm for shallow water waveguide analysis

A prony algorithm for shallow water waveguide analysis

Root selection methods in flood analysis

Symbolic Solution of Finite-Difference Equations

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