A PL homotopy for finding all the roots of a polynomial

This paper presents a constructive method which gives, for any polynomialF(Z) of the degreen, approximate values of all the roots ofF(Z).. The point of the method is on the use of a piecewise linear function (Z, t) which approximates a homotopyH(Z, t) betweenF(Z) and a polynomialG(Z) of the degreen withn known simple roots. It is shown that the set of solutions to (Z, t) = 0 includesn distinct paths,m of which converges to a root ofF(Z) if and only if the root has the multiplicitym. Starting from givenn roots ofG(Z), a complementary pivot algorithm generates thosen paths.This work was supported by grants from Management Science Development Foundation and Takeda Science Foundation.     

On the cost of approximating all roots of a complex polynomial

This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx ₀ =x, x _j =x _j–1 –f(x _j–1)/f(x _j–1),j = 1, 2,, then |x _j –z| < (1/|x ₀ –z|)|x _j–1–z|².     

On the complexity of a piecewise linear algorithm for approximating roots of complex polynomials

LetS be a triangulation of andf(z) = z ^d +a _d–1 z ^d–1++a ₀, a complex polynomial. LetF be the piecewise linear approximation off determined byS. For certainS, we establish an upper bound on the complexity of an algorithm which finds zeros ofF. This bound is a polynomial in terms ofn, max{a _i } _i , and measures of the sizes of simplices inS.     

A modification of todd's triangulationJ 3

When we apply the fixed point computing method to mappings which are affine in some variables, we show that, to generate a sequence which converges to a fixed point, the mesh size need not be decreased in these coordinates. This paper modifies the triangulationJ ₃ with continuous refinement of mesh size to a triangulation such that the mesh size of in some given coordinates is constant and the mesh size in the other coordinates shrinks to zero.     

Computation of all solutions to a system of polynomial equations

This paper proposes a homotopy continuation method for approximating all solutions to a system of polynomial equations in several complex variables. The method is based on piecewise linear approximation and complementarity theory. It utilizes a skilful artificial map and two copies of the triangulationJ ₃ with continuous refinement of grid size to increase the computational efficiency and to avoid the necessity of determining the grid size a priori. Some computational results are also reported.     

On the computational complexity of piecewise-linear homotopy algorithms

We show that piecewise-linear homotopy algorithms may take a number of steps that grows exponentially with the dimension when solving a system of linear equations whose solution lies close to the starting point. Our examples are based on an example of Murty exhibiting exponential growth for Lemke's algorithm for the linear complementarity problem.This research was supported in part by NSF grant ECS-7921279 and by a Guggenheim Fellowship.     

Some experiments with the pathological linear programs of N. Zadeh

Computational results are presented which show that N. Zadeh's pathological examples for the simplex algorithm apparently take a number of pivots approximately proportional to the number of columns in the tableau when its column order is randomized.     

Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of ordern forn 2—he has shown that to solve the problem of ordern, either of the two methods goes through 2 ⁿ pivot steps before termination.However that paper leaves it as an open question to show whether or not the same property holds if the matrix,M, in the LCP is positive definite and symmetric. The class of LCPs in whichM is positive definite and symmetric is of particular interest because of the special structure of the problems, and also because they appear in many practical applications.In this paper, we study the computational growth of each of the two methods to solve the LCP, (q, M), whenM is positive definite and symmetric and obtain similar results.This research is partially supported by Air Force Office of Scientific Research, Air Force Number AFOSR-78-3646. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon.     

An analysis of the solution set to a homotopy equation between polynomials with real coefficients

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h ₁(x, y, t)+ih ₂(x, y, t), whereh ₁ andh ₂ are polynomials from ℝ² × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh ₃ such thath ₂(x, y, t)=yh₃(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h ₁(x, 0, t)=0} and {(x+iy, t)∶h ₁(x, y, t)=0,h ₃(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.     

On the zeros of certain polynomials

We prove that certain naturally arising polynomials have all of their roots on a vertical line.     

Solving polynomials by radicals with roots of unity in minimum depth

Let be an algebraic number field. Let be a root of a polynomial which is solvable by radicals. Let be the splitting field of over . Let be a natural number divisible by the discriminant of the maximal abelian subextension of , as well as the exponent of , the Galois group of over . We show that an optimal nested radical with roots of unity for can be effectively constructed from the derived series of the solvable Galois group of over .

     

Homeomorphisms of triangulations with applications to computing fixed points

In the past decade various complementary pivoting algorithms have been developed to search for fixed points of certain functions and point to set maps. All these methods generate a sequence of simplexes which are shrinking to a point. This paper proposes a new method for shrinking the simplexes. It is shown that under certain conditions, the function whose fixed point is sought may be used to control this shrinking process. A computational method for implementing these ideas is also suggested and several examples are solved using this approach.An abstract appears in the November, 1978 issue of Notices of the American Mathematical Society.     

On the complexity of four polyhedral set containment problems

A nonempty closed convex polyhedronX can be represented either asX = {x: Ax b}, where (A, b) are given, in which caseX is called anH-cell, or in the formX = {x: x = U + V, _j = 1, 0, 0}, where (U, V) are given, in which caseX is called aW-cell. This note discusses the computational complexity of certain set containment problems. The problems of determining if , where (i)X is anH-cell andY is a closed solid ball, (ii)X is anH-cell andY is aW-cell, or (iii)X is a closed solid ball andY is aW-cell, are all shown to be NP-complete, essentially verifying a conjecture of Eaves and Freund. Furthermore, the problem of determining whether there exists an integer point in aW-cell is shown to be NP-complete, demonstrating that regardless of the representation ofX as anH-cell orW-cell, this integer containment problem is NP-complete.     

On ordered spaces of polynomials

We study quasimonotone increasing linear mappings in finite-dimensional spaces of real polynomials, ordered by the cone of nonnegative polynomials on . In particular, we prove a representation of the space of quasimonotone increasing and decreasing operators, which turns out to have dimension 3 or 4.     

On the location of critical points of polynomials

Given a polynomial of degree and with at least two distinct roots let . For a fixed root we define the quantities and . We also define and to be the corresponding minima of and as runs over . Our main results show that the ratios and are bounded above and below by constants that only depend on the degree of . In particular, we prove that , for any polynomial of degree .

     

Computing roots of polynomials on vector processing machines

The Durand and Kerner algorithm for the computation of roots of polynomials has not been og reat use up to now. The reason is the existence of more efficient methods for computers ofthe SISD type (sequential). Actually vector processing machines such as CRAY-1 or CDC CYBER 205 must bring Durand's algorithm back to honour because of its possibility of extensive vectorization. However, as the method has its maximum efficiency for a polynomial with no multiple root a criterion using Vignes' permutation-perturbation method is given to know whether the roots are all distinct or not. The optimum criterion for stopping the iterations is shown to be vectorizable and some useful properties of the method are given. Numerical examples are considered.     

On roots of random polynomials

We study the distribution of the complex roots of random polynomials of degree with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.

     

On the existence of a short pivoting sequence for a linear program

We show that given a feasible primal–dual pair of linear programs in canonical form, there exists a sequence of pivots, whose length is bounded by the minimum dimension of the constraint matrix, leading from the origin to the optimum. The sequence of pivots give a sequence of square and nonsingular submatrices of the constraint matrix. Solving two linear equations involving such a submatrix give primal–dual optimal solutions to the corresponding linear program in canonical form.