The Cauchy radius of a scalar polynomial is an upper bound on the magnitude of its zeros, and it ... more The Cauchy radius of a scalar polynomial is an upper bound on the magnitude of its zeros, and it is optimal among all bounds depending only on the moduli of the coefficients. It has the disadvantage of being implicit because it requires the solution of a nonlinear equation. In this note, simple and explicit upper bounds are derived that are useful approximations to the Cauchy radius. The general approach to obtain the aforementioned bounds is to embed scalar polynomials into the larger framework of their generalization to matrix polynomials and then use bounds on the eigenvalues of the latter.
Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, bas... more Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to compute the roots of the auxiliary polynomial when they exist. We derive an explicit condition for these roots to exist and, when they do, propose efficient ways to compute them. A similar auxiliary polynomial appears for the generalized Pellet theorem for matrix polynomials and it can be treated in the same way.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, a... more JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly.
This work investigates how a known eigenpair can be used to improve standard eigenvalue inclusion... more This work investigates how a known eigenpair can be used to improve standard eigenvalue inclusion sets for the remaining eigenvalues, and shows the relation of this process to eigenvalue deflation techniques. The results are applied to nonnegative and real symmetric matrices and illustrated by numerical examples.
We first generalize to complex matrix polynomials an improvement of an upper bound by Cauchy on t... more We first generalize to complex matrix polynomials an improvement of an upper bound by Cauchy on the zeros of complex scalar polynomials. The bound requires the unique positive root of a real scalar polynomial of the same degree as the given complex scalar or matrix polynomial. We then create a recursive procedure to represent a matrix polynomial by another matrix polynomial of larger size, but of lower degree. We apply this procedure to scalar polynomials, and then apply the generalized improved Cauchy bound to their matrix polynomial representation, often obtaining a better bound by solving a real scalar polynomial equation of significantly reduced degree.
We derive and then enhance a generalized matrix version of Pellet's theorem, itself based on a ge... more We derive and then enhance a generalized matrix version of Pellet's theorem, itself based on a generalized Rouché theorem for matrix-valued functions to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. The matrix version of Pellet's theorem is subsequently used to improve the original scalar theorem.
Newton's method to find the zero of a function in one variable uses the ratio of the function and... more Newton's method to find the zero of a function in one variable uses the ratio of the function and derivative values, but it does not use the information provided by these quantities separately. It is a natural question to ask what a method would look like that does take into account these values instead of just their ratio. We answer that question in the case of a polynomial with all real zeros, the result being a method that is somewhat reminiscent of Laguerre's method.
We derive, from a given matrix polynomial, a lower order matrix polynomial with the same eigenval... more We derive, from a given matrix polynomial, a lower order matrix polynomial with the same eigenvalues, which we call a companion matrix polynomial in analogy to the Frobenius companion matrix, which is a special case of our result. We derive a few nonstandard bounds on the zeros and eigenvalues of scalar and matrix polynomials, respectively, as an illustration of the usefulness of companion matrix polynomials.
In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation o... more In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation often consists of a collection of all-purpose methods that frequently do not work or are inefficient. These textbooks neglect to teach the importance of adapting a method to the given problem, and consequently also neglect to provide the tools to accomplish this. Several general techniques are described here to incorporate the specific structure or properties of a nonlinear equation into a method for solving it. This can mean the construction of a method specifically tailored to the equation, or the transformation of the equation into an equivalent one for which an existing method is well-suited. The techniques are illustrated with the help of several case studies taken from the literature.
We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rou... more We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouché theorem for matrix-valued functions, to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. Variations of the theorem are suggested to try and overcome situations where Pellet's theorem cannot be applied.
A polygon is derived that contains the numerical range of a bounded linear operator on a complex ... more A polygon is derived that contains the numerical range of a bounded linear operator on a complex Hilbert space, using only norms. In its most general form, the polygon is an octagon, symmetric with respect to the origin, and tangent to the closure of the numerical range in at least four points when the spectral norm is used.
Pellet's theorem determines when the zeros of a polynomial can be separated into two regions,... more Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to compute the roots of the auxiliary polynomial when they exist. We derive an explicit condition for these roots to exist and, when they do, propose efficient ways to compute them. A similar auxiliary polynomial appears for the generalized Pellet theorem for matrix polynomials and it can be treated in the same way.
We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general poly... more We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.
The classical Enestrom-Kakeya theorem establishes upper and lower bounds on the zeros of a polyno... more The classical Enestrom-Kakeya theorem establishes upper and lower bounds on the zeros of a polynomial with positive coefficients that are explicit functions of those coefficients. We establish a unifying framework that incorporates this theorem and several similar ones as special cases, while generating new theorems of a similar type. These establish zero inclusion and exclusion regions consisting of a single disk or the union of several disks in the complex plane. Our framework is built on two basic tools, namely a generalization of an observation by Cauchy, and a family of polynomial multipliers. Its approach is transparent and reduces algebraic manipulations to a minimum.
Newton’s method to find the zero of a function in one variable uses the ratio of the function and... more Newton’s method to find the zero of a function in one variable uses the ratio of the function and derivative values, but it does not use the information provided by these quantities separately. It is a natural question to ask what a method would look like that does take into account these values instead of just their ratio. We answer that question in the case of a polynomial with all real zeros, the result being a method that is somewhat reminiscent of Laguerre’s method.
We improve the Enestrom-Kakeya theorem, which establishes an annulus containing the zeros of a po... more We improve the Enestrom-Kakeya theorem, which establishes an annulus containing the zeros of a polynomial with positive coefficients, by defining a framework that generates such improvements in a unified way. This framework not only allows one to improve upper and lower bounds on the moduli of the zeros, but it also generates additional regions in the complex plane that contain all zeros.
The Cauchy radius of a scalar polynomial is an upper bound on the magnitude of its zeros, and it ... more The Cauchy radius of a scalar polynomial is an upper bound on the magnitude of its zeros, and it is optimal among all bounds depending only on the moduli of the coefficients. It has the disadvantage of being implicit because it requires the solution of a nonlinear equation. In this note, simple and explicit upper bounds are derived that are useful approximations to the Cauchy radius. The general approach to obtain the aforementioned bounds is to embed scalar polynomials into the larger framework of their generalization to matrix polynomials and then use bounds on the eigenvalues of the latter.
Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, bas... more Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to compute the roots of the auxiliary polynomial when they exist. We derive an explicit condition for these roots to exist and, when they do, propose efficient ways to compute them. A similar auxiliary polynomial appears for the generalized Pellet theorem for matrix polynomials and it can be treated in the same way.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, a... more JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly.
This work investigates how a known eigenpair can be used to improve standard eigenvalue inclusion... more This work investigates how a known eigenpair can be used to improve standard eigenvalue inclusion sets for the remaining eigenvalues, and shows the relation of this process to eigenvalue deflation techniques. The results are applied to nonnegative and real symmetric matrices and illustrated by numerical examples.
We first generalize to complex matrix polynomials an improvement of an upper bound by Cauchy on t... more We first generalize to complex matrix polynomials an improvement of an upper bound by Cauchy on the zeros of complex scalar polynomials. The bound requires the unique positive root of a real scalar polynomial of the same degree as the given complex scalar or matrix polynomial. We then create a recursive procedure to represent a matrix polynomial by another matrix polynomial of larger size, but of lower degree. We apply this procedure to scalar polynomials, and then apply the generalized improved Cauchy bound to their matrix polynomial representation, often obtaining a better bound by solving a real scalar polynomial equation of significantly reduced degree.
We derive and then enhance a generalized matrix version of Pellet's theorem, itself based on a ge... more We derive and then enhance a generalized matrix version of Pellet's theorem, itself based on a generalized Rouché theorem for matrix-valued functions to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. The matrix version of Pellet's theorem is subsequently used to improve the original scalar theorem.
Newton's method to find the zero of a function in one variable uses the ratio of the function and... more Newton's method to find the zero of a function in one variable uses the ratio of the function and derivative values, but it does not use the information provided by these quantities separately. It is a natural question to ask what a method would look like that does take into account these values instead of just their ratio. We answer that question in the case of a polynomial with all real zeros, the result being a method that is somewhat reminiscent of Laguerre's method.
We derive, from a given matrix polynomial, a lower order matrix polynomial with the same eigenval... more We derive, from a given matrix polynomial, a lower order matrix polynomial with the same eigenvalues, which we call a companion matrix polynomial in analogy to the Frobenius companion matrix, which is a special case of our result. We derive a few nonstandard bounds on the zeros and eigenvalues of scalar and matrix polynomials, respectively, as an illustration of the usefulness of companion matrix polynomials.
In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation o... more In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation often consists of a collection of all-purpose methods that frequently do not work or are inefficient. These textbooks neglect to teach the importance of adapting a method to the given problem, and consequently also neglect to provide the tools to accomplish this. Several general techniques are described here to incorporate the specific structure or properties of a nonlinear equation into a method for solving it. This can mean the construction of a method specifically tailored to the equation, or the transformation of the equation into an equivalent one for which an existing method is well-suited. The techniques are illustrated with the help of several case studies taken from the literature.
We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rou... more We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouché theorem for matrix-valued functions, to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. Variations of the theorem are suggested to try and overcome situations where Pellet's theorem cannot be applied.
A polygon is derived that contains the numerical range of a bounded linear operator on a complex ... more A polygon is derived that contains the numerical range of a bounded linear operator on a complex Hilbert space, using only norms. In its most general form, the polygon is an octagon, symmetric with respect to the origin, and tangent to the closure of the numerical range in at least four points when the spectral norm is used.
Pellet's theorem determines when the zeros of a polynomial can be separated into two regions,... more Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to compute the roots of the auxiliary polynomial when they exist. We derive an explicit condition for these roots to exist and, when they do, propose efficient ways to compute them. A similar auxiliary polynomial appears for the generalized Pellet theorem for matrix polynomials and it can be treated in the same way.
We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general poly... more We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.
The classical Enestrom-Kakeya theorem establishes upper and lower bounds on the zeros of a polyno... more The classical Enestrom-Kakeya theorem establishes upper and lower bounds on the zeros of a polynomial with positive coefficients that are explicit functions of those coefficients. We establish a unifying framework that incorporates this theorem and several similar ones as special cases, while generating new theorems of a similar type. These establish zero inclusion and exclusion regions consisting of a single disk or the union of several disks in the complex plane. Our framework is built on two basic tools, namely a generalization of an observation by Cauchy, and a family of polynomial multipliers. Its approach is transparent and reduces algebraic manipulations to a minimum.
Newton’s method to find the zero of a function in one variable uses the ratio of the function and... more Newton’s method to find the zero of a function in one variable uses the ratio of the function and derivative values, but it does not use the information provided by these quantities separately. It is a natural question to ask what a method would look like that does take into account these values instead of just their ratio. We answer that question in the case of a polynomial with all real zeros, the result being a method that is somewhat reminiscent of Laguerre’s method.
We improve the Enestrom-Kakeya theorem, which establishes an annulus containing the zeros of a po... more We improve the Enestrom-Kakeya theorem, which establishes an annulus containing the zeros of a polynomial with positive coefficients, by defining a framework that generates such improvements in a unified way. This framework not only allows one to improve upper and lower bounds on the moduli of the zeros, but it also generates additional regions in the complex plane that contain all zeros.
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