Methods for the simultaneous approximate derivation of the roots of algebraic, trigonometric and exponential equations

Angelova, Evgenia; Semerdzhiev, Khr.

doi:10.1016/0041-5553(82)90179-3

Cited by 17 publications

(21 citation statements)

References 1 publication

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“…Firstly, the continuation method of [3] is sketched which gives method (MN) using the Taylor method of N-th order for solving a certain initial value problem. Secondly, it is proved that (M2) is method (M) in the unified approach of [7] which is Newton-Raphson's method for a certain system of nonlinear equations.…”

Section: Continuation Process Unified Method Euler Methodsmentioning

confidence: 99%

“…(1)). [] In [3], [9] and [7] only (M2) is considered and convergence is proved only for particular cases. We stress that Theorem 2 gives convergence for all methods simultaneously and also for all methods of higher order N.…”

Section: Z~v) (T) 2~i 0v~ (1 -T)q(z) + Tf(z)/mentioning

confidence: 99%

“…4 that (M2) is method (M) from [7]. Moreover, (Mr) is a natural generalization of the continuation process in [3] of order N which was firstly considered for algebraic polynomials in [-5]. For n = 1, the so-called Eulermethods are obtained.…”

Section: (K)(xj) Q(r)(xj) K!q'(xy)' Q*:= K!q'(xj)mentioning

confidence: 99%

“…Following the continuation process from [3], assume that there exists a holomorphic function Q: D--* IE having exactly the simple zeros x~ ..... x, in D. In the particular cases of algebraic, trigonometric and exponential polynomials Q is easily obtained explicitly. In general, Q can be defined as the remainder of the interpolation of f using a n-dimensional Chebyshev space U and the knodes x~ ..... x..…”

Section: Introductionmentioning

confidence: 99%

“…Using the argument principle higher order methods for simultaneous computation of all zeros of generalized polynomials (like algebraic, trigonometric and exponential polynomials or exponential sums) are derived. The methods can also be derived following the continuation principle from [3]. Thereby, the unified approach of I-7] is enlarged to arbitrary order N. The local convergence as well as a-priori and a-posteriori error estimates for these methods are treated on a general level.…”

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confidence: 99%

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On a class of higher order methods for simultaneous rootfinding of generalized polynomials

Carstensen

Reinders

1993

Numer. Math.

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Summary. Using the argument principle higher order methods for simultaneous computation of all zeros of generalized polynomials (like algebraic, trigonometric and exponential polynomials or exponential sums) are derived. The methods can also be derived following the continuation principle from [3]. Thereby, the unified approach of I-7] is enlarged to arbitrary order N. The local convergence as well as a-priori and a-posteriori error estimates for these methods are treated on a general level. Numerical examples are included Subject Classification (1991): 65H05, 65H10 Mathematics

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Section: Continuation Process Unified Method Euler Methodsmentioning

confidence: 99%

Section: Z~v) (T) 2~i 0v~ (1 -T)q(z) + Tf(z)/mentioning

confidence: 99%

Section: (K)(xj) Q(r)(xj) K!q'(xy)' Q*:= K!q'(xj)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On a class of higher order methods for simultaneous rootfinding of generalized polynomials

Carstensen

Reinders

1993

Numer. Math.

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A unified approach to methods for the simultaneous computation of all zeros of generalized polynomials

Frommer

1988

Numer. Math.

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Summary. Applying Newton's method to a particular system of nonlinear equations we derive methods for the simultaneous computation of all zeros of generalized polynomials. These generalized polynomials are from a function space satisfying a condition similar to Haar's condition. By this approach we bring together recent methods for trigonometric and exponential polynomials and a well-known method for ordinary polynomials. The quadratic convergence of these methods is an immediate consequence of our approach and needs not to be proved explicitly. Moreover, our approach yields new interesting methods for ordinary, trigonometric and exponential polynomials and methods for other functions occuring in approximation theory.

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The Durand-Kerner polynomials roots-finding method in case of multiple roots

Fraigniaud

1991

BIT

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Abstract.The convergence of the Durand-Kerner algorithm is quadratic in case of simple roots but only linear in case of multiple roots. This paper shows that, at each step, the mean of the components converging to the same root approaches it with a n error proportional to the square of the error at the previous step. Since it is also shown that it is possible to estimate the multiplicity order of the roots during the algorithm, a modification of the Durand-Kerner iteration is proposed to preserve a quadratic-like convergence even in case of multiple zeros.Subject Classifications: AMS(MOS):65H05; CR:5.15.

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Methods for the simultaneous approximate derivation of the roots of algebraic, trigonometric and exponential equations

Cited by 17 publications

References 1 publication

On a class of higher order methods for simultaneous rootfinding of generalized polynomials

On a class of higher order methods for simultaneous rootfinding of generalized polynomials

A unified approach to methods for the simultaneous computation of all zeros of generalized polynomials

The Durand-Kerner polynomials roots-finding method in case of multiple roots

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