Finally, one can construct a polynomial with assigned zeros by the repeated use of the following result. A polynomial with and as its isolated zeros with multiplicities and , respectively, and a sphere of zeros with multiplicity can be constructed through the chain. An alternative syntax for the function addresses the problem of constructing a polynomial in fact it constructs a chain once one knows the nature and multiplicity of its roots.
We reconsider here Example 6 of [ 4 ]. An example of a polynomial that has as a zero of multiplicity three, as a zero of multiplicity two and as a sphere of zeros with multiplicity two is. We confirm this using the function with the new syntax. Recall that a real root is always an isolated root, and two roots in the same congruence class cannot be isolated. This article has discussed implementation issues related to the manipulation, evaluation and factorization of quaternionic polynomials.
We recommend that interested readers download the support file to get complete access to all the implemented functions. The increasing interest in the use of quaternions in areas such as number theory, robotics, virtual reality and image processing [ 18 ] makes us believe that developing a computational tool for operating in the quaternions framework will be useful for other researchers, especially taking into account the power of Mathematica as a symbolic language.
In the ring of quaternionic polynomials, new problems arise mainly because the structure of zero sets, as we have described, is very different from the complex case. In this article, we did not discuss the problem of computing the roots or the factor terms of a polynomial; all the results we have presented assumed that either the zeros or the factor terms of a given polynomial are known. Methods for computing the roots or factor terms of a quaternionic polynomial are considered in Part II. Her research interests are numerical analysis, hypercomplex analysis and scientific software.
Fernando Miranda is an assistant professor in the Department of Mathematics and Applications of the University of Minho. His research interests are differential equations, quaternions and related algebras and scientific software. Ricardo Severino is an assistant professor in the Department of Mathematics and Applications of the University of Minho. His research interests are dynamical systems, quaternions and related algebras and scientific software.
Her research interests are numerical analysis, wavelets mainly in applications to economics, and quaternions and related algebras. In analogy with the complex case , we define:. Real part of ,. Vector part of ,. Here is some arithmetic in. With a polynomial as in equation 1 and a quaternion, define:. Theorem 1 Let and be two polynomials in and consider the polynomial and. If , then , where. If is a real polynomial , then. Theorem 2 — Euclidean division If and are polynomials in with , then there exist unique , , and such that.
Since , is a right divisor of and. On the other hand, does not right-divide but it is a left divisor. The greatest common right or left divisor polynomial of two polynomials can now be computed using the Euclidean algorithm by a basic procedure similar to the one used in the complex setting. The function implements this procedure for the case of the greatest common right divisor.
Here is defined as follows. We say that a quaternion is congruent or similar to a quaternion and write if there exists a nonzero quaternion such that. A zero of is called an isolated zero of if contains no other zeros of. Otherwise, is called a spherical zero of and is referred to as a sphere of zeros. Theorem 3 [ 9 ] Let and. The following conditions are equivalent: There is an such that. The characteristic polynomial of , , is a divisor of the companion polynomial of.
Theorem 4 [ 9 ], [ 10 ] A nonreal zero is a spherical zero of if and only if any of the following equivalent conditions hold: The characteristic polynomial of is a right divisor of ; that is, there exists a polynomial such that. Theorem 5 — Factor theorem [ 11 ], [ 12 ] Let and. Theorem 6 — Fundamental theorem of algebra [ 13 ] Any nonconstant polynomial in always has a zero in.
Theorem 7 — Factorization into linear terms Any monic polynomial of degree in factors into linear factors; that is, there exist such that. In a factorization of of the form 5 , the quaternions are called factor terms of and the -tuple is called a factor terms chain associated with or simply a chain of. Theorem 8 Let be a chain of a polynomial. Theorem 9 [ 12 — 14 ] Let be a chain of the polynomial. Theorem 10 — Zeros from factors [ 12 ] Consider a chain of the polynomial. If the similarity classes are distinct, then has exactly zeros , which are given by: Theorem 11 — Factors from zeros [ 9 ] If are quaternions such that the similarity classes are distinct, then there is a unique polynomial of degree with zeros that can be constructed from the chain , where.
Theorem 12 [ 9 — 15 ] Let be a quaternionic polynomial of degree. Moreover, if a chain associated with a polynomial has property 7 , is a polynomial of degree such that is its unique zero and , then the polynomial of degree has only two zeros, namely and. The multiplicity of a zero of is defined as the maximum degree of the right factors of with as their unique zero and is denoted by.
The multiplicity of a sphere of zeros of , denoted by , is the largest for which divides. This book will be invaluable to anyone doing research in polynomial roots, or teaching a graduate course on that topic. First comprehensive treatment of Root-Finding in several decades Gives description of high-grade software and where it can be down-loaded Very up-to-date in mid; long chapter on matrix methods Includes Parallel methods, errors where appropriate Invaluable for research or graduate course This book along with volume 2 covers most of the traditional methods for polynomial root-finding such as Newton's, as well as numerous variations on them invented in the last few decades.
Additional Details Series Volume Number. Evaluation, Convergence, Bounds 2.
Sturm Sequences and Greatest Common Divisors 3. Real Roots by Continued Fractions 4. Newton's and Related Methods 6. A smoother density can be obtained by increasing the option. We consider again the polynomial of Example 3 , whose roots are the isolated roots , and. The following code produces the plots corresponding to the choice of in one of the following regions:. As was already pointed out, Theorem 2 can be applied only in ; this is why both methods produce the same plots in this case.
Here is the behavior of the -Newton methods in. The plots produced by give information on the number of iterations required by each of the quaternionic Newton methods to converge within a certain precision to any of the roots of the polynomial under consideration.
Computational Aspects of Quaternionic Polynomials « The Mathematica Journal
However, those plots do not give any information about the root and how the convergence occurs. This issue can be easily overcome by plotting the basins of attraction of the roots with respect to the iterative function. More precisely, we introduce a new input parameter in the function with the information of the root for which we want to compute the basin of attraction. A new function takes into account the existence of spheres of zeros. The functions and give the number of iterations needed to observe convergence to an isolated root or a spherical one, respectively.
These functions return when the corresponding convergence test fails. The functions that plot the basin of attraction of an isolated root or a spherical root have an input parameter associated to that root. The color coding used is the following: For a sphere of zeros , all the points that converge to a point in have the color assigned to. Dark shades of a color mean fast convergence, while lighter-colored points lead to slower convergence.
As before, white regions mean that the method does not converge. We consider once more the polynomial of Example 4, now from the perspective of the basins of attraction of each of the roots , and. We associate with these roots the colors red, blue and green, respectively, and consider the domains , and , described in Example 4.
The corresponding plots can be obtained as follows it can take some time to produce the figures.
Сведения о продавце
Here are the basins of attraction in left. Here are the basins of attraction in left and right. This example concerns the polynomial studied in Example 2 , which has an isolated root 0 red and a sphere of zeros blue. The corresponding plots can be obtained as follows. Here are the basins of attraction in left ; as expected, the behavior is similar to that in , since. The Weierstrass method is one of the most popular iterative methods for obtaining simultaneously approximations to all the roots of a polynomial with complex coefficients.
Let be a complex monic polynomial of degree with roots and let be distinct numbers. The classical Weierstrass method for approximating the roots is defined by the iterative scheme:.
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The iteration procedure 12 computes one approximation at a time based on the already computed approximations. For this reason, it is usually referred to as the total-step or parallel mode. The convergence of the method can be accelerated by using a variant — the so-called single-step, serial or sequential mode — that makes use of the most recent updated approximations to the roots as soon as they are available:.
In a recent article [ 11 ], we adapted the Weierstrass method to the quaternion algebra setting. We refer to [ 2 ] and references therein to recall the main concepts and properties of the ring of unilateral quaternionic polynomials. In particular, we recall the factorization of polynomials in into linear terms and the relation between zeros and factors of. Theorem 3 — Factorization into linear terms.
Following the idea of the Weierstrass method in its sequential version 13 , the next results show how to obtain sequences converging, at a quadratic rate, to the factor terms in 14 of a given polynomial. Moreover, by making use of Theorem 4 , it is possible to construct sequences converging quadratically to the roots of. The functions , and are implemented as the functions , and , respectively. The support file associated with [ 2 ] needs to be loaded.
Computational Mathematics Volume V
The iterative functions associated with 17 and 21 are built into the function. The quaternionic Weierstrass iterative method is implemented in the function. The usual convergence test has been replaced in by. Since we also include a test on the value of , there is no risk of misidentifying an isolated root. We consider now the application of the Weierstrass method to the computation of the roots of the polynomial of Example 3 , which we recall are , and. All of the initial approximations , and have to lie in distinct congruence classes.
Our next test example is a polynomial that also fulfills the assumptions of Theorem 5 and has simple zeros see [ 11 ], Example 1. First, we check that the polynomial. The polynomial has an isolated root and a sphere of zeros.