Prashant Batra

On necessary conditions for real robust Schur-stability

A coefficient diameter of real Schur-stable interval polynomials is bounded using meromorphic functions. The resulting bound is the best possible. A known bound for the ratio of the two leading coefficients is improved for large diameters.

Simultaneous Point Estimates for Newton's Method

Beside the classical Kantorovich theory there exist convergence criteria for the Newton iteration which only involve data at one point, i.e. point estimates. Given a polynomial P, these conditions imply the point evaluation of n = deg(P) functions (from a certain Taylor expansion). Such sufficient conditions ensure quadratic convergence to a single zero and have been used by several authors in the design and analysis of robust, fast and efficient root-finding methods for polynomials.In this paper a sufficient condition for the simultaneous convergence of the one-dimensional Newton iteration for polynomials will be given. The new condition involves only n point evaluations of the Newton correction and the minimum mutual distance of approximations to ensure “simultaneous” quadratic convergence to the pairwise distinct n roots.

Newton's method and the Computational Complexity of the Fundamental Theorem of Algebra

Several different uses of Newtons method in connection with the Fundamental Theorem of Algebra are pointed out. Theoretical subdivision schemes have been combined with the numerical Newton iteration to yield fast root-approximation methods together with a constructive proof of the fundamental theorem of algebra. The existence of the inverse near a simple zero may be used globally to convert topological methods like path-following via Newtons method to numerical schemes with probabilistic convergence. Finally, fast factoring methods which yield root-approximations are constructed using some algebraic Newton iteration for initial factor approximations.

On the Quality of Some Root-Bounds

Bounds for the maximum modulus of all positive or all complex roots of a polynomial are a fundamental building block of algorithms involving algebraic equations. We apply known results to show which are the salient features of the Lagrange real root-bound as well as the related bound by Fujiwara. For a polynomial of degree n, we construct a bound of relative overestimation at most 1.72n which overestimates the Cauchy root by a factor of two at most. This can be carried over to the bounds by Kioustelidis and Hong. Giving a very short variant of a recent proof presented by Collins, we sketch a way to further definite, measurable improvement.

Near optimal subdivision algorithms for real root isolation

Abstract Isolating real roots of a square-free polynomial in a given interval is a fundamental problem in computational algebra. Subdivision based algorithms are a standard approach to solve this problem. For instance, Sturms method, or various algorithms based on the Descartess rule of signs. For the benchmark problem of isolating all the real roots of a polynomial of degree n and root separation σ , the size of the subdivision tree of most of these algorithms is bounded by O ( log ⁡ 1 / σ ) (assume σ 1 ). Moreover, it is known that this is optimal for subdivision algorithms that perform uniform subdivision, i.e., the width of the interval decreases by some constant. Recently Sagraloff (2012) and Sagraloff–Mehlhorn (2016) have developed algorithms for real root isolation that combine subdivision with Newton iteration to reduce the size of the subdivision tree to O ( n ( log ⁡ ( n log ⁡ 1 / σ ) ) ) . We describe a subroutine that reduces the size of the subdivision tree of any subdivision algorithm for real root isolation. The subdivision tree size of our algorithm using predicates based on either the Descartess rule of signs or Sturm sequences is bounded by O ( n log ⁡ n ) , which is close to the optimal value of O ( n ) . The corresponding bound for the algorithm EVAL , which uses certain interval-arithmetic based predicates, is O ( n 2 log ⁡ n ) . Our analysis differs in two key aspects from earlier approaches. First, we use the general technique of continuous amortization from Burr–Krahmer–Yap (2009) , and extend it to handle non-uniform subdivisions; second, we use the geometry of clusters of roots instead of root bounds. The latter aspect enables us to derive a bound on the subdivision tree that is independent of the root separation σ . The number of Newton iterations is bounded by O ( n log ⁡ log ⁡ ( 1 / σ ) ) .

Bound for all coefficient diameters of real Schur-stable interval polynomials

A sharp diameter bound for all coefficient intervals of real robustly Schur-stable polynomials is established. The results may be used in a preprocessing rejection scheme when testing for robust Schur-stability.

On Coefficient Diameters of Real Schur-Stable Interval Polynomials

Abstract A new necessary, sharp condition for Schur-stability of real interval polynomials is established here. Moreover, a way to study perturbation effects on the coefficient range is outlined. The results may be used in a preprocessing rejection scheme when testing for robust Schur-stability.

Improvements of Lagrange's bound for polynomial roots

Abstract An upper bound for the roots of X d + a 1 X d − 1 + ⋯ + a d − 1 X + a d is given by the sum of the largest two of the terms | a i | 1 / i . This bound by Lagrange has gained attention from different sides recently, while a succinct proof seems to be missing. We present a short, original proof of Lagranges bound. Our approach leads to some definite improvements. To benefit computationally from these improvements, we construct a modified Lagrange bound which at the same asymptotic computational complexity is at most 11 per cent from optimal for degrees d ≥ 16 .

Globally Convergent, Iterative Path-Following for Algebraic Equations

AbstractHomotopy methods are of great importance for the solution of systems of equations. It is a major problem to ensure well-defined iterations along the homotopy path. Many investigations have considered the complexity of path-following methods depending on the unknown distance of some given path to the variety of ill-posed problems. It is shown here that there exists a construction method for safe paths for a single algebraic equation. A safe path may be effectively determined with bounded effort. Special perturbation estimates for the zeros together with convergence conditions for Newton’s method in simultaneous mode allow our method to proceed on the safe path. This yields the first globally convergent, never-failing, uniformly iterative path-following algorithm. The maximum number of homotopy steps in our algorithm reaches a theoretical bound forecast by Shub and Smale i.e., the number of steps is at most quadratic in the condition number. A constructive proof of the fundamental theorem of algebra meeting demands by Gauß, Kronecker and Weierstraß is a consequence of our algorithm.

On Gol’dberg’s Constant A2

Gol’dberg considered the class of functions analytic in the unit disc with unequal positive numbers of zeros and ones there. The maximum modulus of zero- and one-places in this class is non-trivially bounded from below by the universal constant A2. This constant determines a fundamental limit of controller design in engineering, and has applications when estimating covering regions for composites of fixed point free functions with schlicht functions. The lower bound for A2 is improved in this note by considering simultaneously the extremal functions f and 1 — f together with their reciprocals.