Complexity Analysis of Algorithms in Algebraic Computation

Numerical computations with real algebraic numbers require algorithms for isolating and approximating real roots of polynomials. A classical choice for root approximation is Newton's method. For an analytic function on a Banach space, Smale introduced the concept of approximate zeros, i.e., points from which Newton's method for the function converges quadratically. To identify these approximate zeros he gave computationally verifiable convergence criteria called point estimates. However, in developing these results Smale assumed that Newton's method is computed exactly. For a system of homogeneous polynomials, Malajovich developed point estimates for a different definition of approximate zero, assuming that all operations in Newton's method are computed with fixed precision. In the first half of this dissertation, we develop point estimates for these two different definitions of approximate zeros of an analytic function on a Banach space, but assume the strong bigfloat computational model of Brent where precision of operations can vary. In this model, we derive a uniform complexity bound for approximating a root of a zero-dimensional system of integer polynomials. We also derive a non-asymptotic bound, in terms of the condition number of the system, on the precision required to implement the robust Newton method. The second part of the dissertation analyses the worst-case complexity of two algorithms for isolating real roots of a square-free polynomial with real coefficients: The Descartes method and Akritas' continued fractions algorithm. The analysis of both algorithms is based upon the Davenport-Mahler bound. For the Descartes method, we give a unified framework that encompasses both the power basis and the Bernstein basis variant of the method; we derive an O(n(L + log n)) bound on the size of the recursion tree obtained by applying the method to a square-free polynomial of degree n with integer coefficients of bit-length L, the bound is tight for L = Ω(log n); based upon this result we readily obtain the best known bit-complexity bound of Od (n4L2) for the Descartes method, where Od means we ignore logarithmic factors. Similar worst case bounds on the bit-complexity of Akritas' algorithm were not known in the literature. We provide the first such bound, Od (n12L4), for a square-free integer polynomial of degree n with coefficients of bit-length L.

[1]  Victor Y. Pan,et al.  Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations , 2005, Numerische Mathematik.

[2]  V. Y. Pan TR-2002003: Univariate Polynomial Root-Finding with a Lower Computational Precision and Higher Convergence Rates , 2002 .

[3]  Chee-Keng Yap,et al.  Robust Geometric Computation , 2016, Encyclopedia of Algorithms.

[4]  George E. Collins,et al.  Quantifier elimination and the sign variation method for real root isolation , 1989, ISSAC '89.

[5]  S. Smale,et al.  Complexity of Bézout’s theorem. I. Geometric aspects , 1993 .

[6]  Steven Fortune,et al.  Robustness Issues in Geometric Algorithms , 1996, WACG.

[7]  Chee-Keng Yap,et al.  Towards Exact Geometric Computation , 1997, Comput. Geom..

[8]  A. Sluis Upperbounds for roots of polynomials , 1970 .

[9]  Jeremy Johnson,et al.  Algorithms for polynomial real root isolation , 1992 .

[10]  Prashant Batra,et al.  Improvement of a convergence condition for the Durand-Kerner iteration , 1998 .

[11]  Steven Fortune,et al.  Polynomial root finding using iterated Eigenvalue computation , 2001, ISSAC '01.

[12]  Jeremy R. Johnson,et al.  High-performance implementations of the Descartes method , 2006, ISSAC '06.

[13]  Immo O. Kerner,et al.  Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen , 1966 .

[14]  W. Deren,et al.  The theory of Smale's point estimation and its applications , 1995 .

[15]  Kurt Mehlhorn,et al.  A Separation Bound for Real Algebraic Expressions , 2001, ESA.

[16]  C. Yap,et al.  Amortized Bound for Root Isolation via Sturm Sequences , 2007 .

[17]  Xiaoshen Wang,et al.  A Simple Proof of Descartes's Rule of Signs , 2004, Am. Math. Mon..

[18]  D. Anderson,et al.  Algorithms for minimization without derivatives , 1974 .

[19]  P. Krishnaiah,et al.  A Simple Proof of Descartes' Rule of Signs , 1963 .

[20]  R. Baker Kearfott,et al.  Interval Newton/generalized bisection when there are singularities near roots , 1991 .

[21]  Ioannis Z. Emiris,et al.  Univariate Polynomial Real Root Isolation: Continued Fractions Revisited , 2006, ESA.

[22]  Pengyuan Chen Approximate zeros of quadratically convergent algorithms , 1994 .

[23]  D. P. Mitchell Robust ray intersection with interval arithmetic , 1990 .

[24]  Arnold Schönhage,et al.  The fundamental theorem of algebra in terms of computational complexity - preliminary report , 1982 .

[25]  Kurt Mehlhorn,et al.  A Generalized and improved constructive separation bound for real algebraic expressions , 2000 .

[26]  Chee-Keng Yap,et al.  Fundamental problems of algorithmic algebra , 1999 .

[27]  Myong-Hi Kim,et al.  Polynomial Root-Finding Algorithms and Branched Covers , 1994, SIAM J. Comput..

[28]  I. Emiris,et al.  Real Algebraic Numbers: Complexity Analysis and Experimentations , 2008 .

[29]  Freeman J. Dyson,et al.  The approximation to algebraic numbers by rationals , 1947 .

[30]  Ansi Ieee,et al.  IEEE Standard for Binary Floating Point Arithmetic , 1985 .

[31]  Peter Duren,et al.  Coefficients of univalent functions , 1977 .

[32]  P. Zimmermann,et al.  Efficient isolation of polynomial's real roots , 2004 .

[33]  M. Mignotte,et al.  Polynomials: An Algorithmic Approach , 1999 .

[34]  W. Boehm,et al.  Bezier and B-Spline Techniques , 2002 .

[35]  M. SIAMJ.,et al.  NEWTON’S METHOD IN FLOATING POINT ARITHMETIC AND ITERATIVE REFINEMENT OF GENERALIZED EIGENVALUE PROBLEMS∗ , 1999 .

[36]  Rida T. Farouki,et al.  On the numerical condition of polynomials in Bernstein form , 1987, Comput. Aided Geom. Des..

[37]  K. Mahler An inequality for the discriminant of a polynomial. , 1964 .

[38]  S. Smale,et al.  On the complexity of path-following newton algorithms for solving systems of polynomial equations with integer coefficients , 1993 .

[39]  Victor Y. Pan,et al.  Solving a Polynomial Equation: Some History and Recent Progress , 1997, SIAM Rev..

[40]  E. Kreyszig Introductory Functional Analysis With Applications , 1978 .

[41]  Arnold Schönhage Storage Modification Machines , 1980, SIAM J. Comput..

[42]  Rida T. Farouki,et al.  Algorithms for polynomials in Bernstein form , 1988, Comput. Aided Geom. Des..

[43]  Oliver Aberth,et al.  Iteration methods for finding all zeros of a polynomial simultaneously , 1973 .

[44]  Christoph M. Hoffmann,et al.  The problems of accuracy and robustness in geometric computation , 1989, Computer.

[45]  Joachim von zur Gathen,et al.  Fast algorithms for Taylor shifts and certain difference equations , 1997, ISSAC.

[46]  Ľ.,et al.  Polynomial Zero Finders Based on Szeg } O Polynomials , 2022 .

[47]  Fabrice Rouillier,et al.  Bernstein's basis and real root isolation , 2004 .

[48]  Sylvain Pion,et al.  Constructive root bound method for k-aray rational input numbers , 2003 .

[49]  Alkiviadis G. Akritas,et al.  Elements of Computer Algebra with Applications , 1989 .

[50]  Myong-Hi Kim On approximate zeros and rootfinding algorithms for a complex polynomial , 1988 .

[51]  Chee Yap,et al.  Towards robust geometric computation (invited white paper) , 2004 .

[52]  Alkiviadis G. Akritas,et al.  There is no “Uspensky's method.” , 1986, SYMSAC '86.

[53]  B. Mourrain,et al.  The Bernstein Basis and Real Root Isolation , 2007 .

[54]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[55]  Wilhelm Werner,et al.  On the simultaneous determination of polynomial roots , 1982 .

[56]  W. Burnside,et al.  Theory of equations , 1886 .

[57]  S. Smale,et al.  Complexity of Bezout's theorem IV: probability of success; extensions , 1996 .

[58]  A. Ostrowski Solution of equations in Euclidean and Banach spaces , 1973 .

[59]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[60]  Doru Stefanescu,et al.  New Bounds for Positive Roots of Polynomials , 2005, J. Univers. Comput. Sci..

[61]  V. Pan Optimal and nearly optimal algorithms for approximating polynomial zeros , 1996 .

[62]  Christopher J. Van Wyk,et al.  Efficient exact arithmetic for computational geometry , 1993, SCG '93.

[63]  L. Kantorovich,et al.  Functional analysis and applied mathematics , 1963 .

[64]  Stefan Schirra,et al.  Robustness and Precision Issues in Geometric Computation , 2000, Handbook of Computational Geometry.

[65]  Giuseppe Fiorentino,et al.  Numerical Computation of Polynomial Roots Using MPSolve Version 2 . 2 , 2001 .

[66]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[67]  Tetsuro Yamamoto,et al.  A unified derivation of several error bounds for Newton's process , 1985 .

[68]  S. Smale,et al.  Computational complexity: on the geometry of polynomials and a theory of cost. I , 1985 .

[69]  C. Jacobi Observatiunculae ad theoriam aequationum pertinentes. , 1835 .

[70]  Kurt Mehlhorn,et al.  A Separation Bound for Real Algebraic Expressions , 2001, Algorithmica.

[71]  Hoon Hong,et al.  Bounds for Absolute Positiveness of Multivariate Polynomials , 1998, J. Symb. Comput..

[72]  Chee-Keng Yap,et al.  A new constructive root bound for algebraic expressions , 2001, SODA '01.

[73]  Chee-Keng Yap,et al.  Almost tight recursion tree bounds for the Descartes method , 2006, ISSAC '06.

[74]  L. Zoretti Sur la résolution des équations numériques , 1909 .

[75]  Richard E. Ewing,et al.  "The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics" , 1986 .

[76]  Stephen Smale,et al.  Computational Complexity: On the Geometry of Polynomials and a Theory of Cost: II , 1986, SIAM J. Comput..

[77]  Melvin R. Spencer Polynomial real root finding in Bernstein form , 1994 .

[78]  Michael N. Vrahatis,et al.  On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree , 2002, J. Complex..

[79]  Bud Mishra,et al.  Counting Real Zeros , 1991 .

[80]  Carsten Carstensen,et al.  Weierstrass formula and zero-finding methods , 1995 .

[81]  A. Edelman,et al.  Polynomial roots from companion matrix eigenvalues , 1995 .

[82]  S. Smale Newton’s Method Estimates from Data at One Point , 1986 .

[83]  James H. Curry,et al.  On zero finding methods of higher order from data at one point , 1989, J. Complex..

[84]  J. McNamee A bibliography on roots of polynomials , 1993 .

[85]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[86]  C. A. Neff,et al.  An O(n^1+epsilon log b) Algorithm for the Complex Roots Problem , 1994, FOCS 1994.

[87]  Chee-Keng Yap,et al.  A core library for robust numeric and geometric computation , 1999, SCG '99.

[88]  Li Chen,et al.  CORE Library Tutorial: A Library for Robust Geometric Computation , 1999 .

[89]  R. Riesenfeld,et al.  Bounds on a polynomial , 1981 .

[90]  Lothar Reichel,et al.  Polynomial zerofinders based on Szego polynomials , 2001 .

[91]  V. Pan,et al.  Improved initialization of the accelerated and robust QR-like polynomial root-finding. , 2004 .

[92]  関川 浩 Using Interval Computation with the Mahler Measure for Zero Determination of Algebraic Numbers , 1998 .

[93]  Alkiviadis G. Akritas,et al.  Polynomial real root isolation using Descarte's rule of signs , 1976, SYMSAC '76.

[94]  J B Kiostelikis,et al.  Bounds for positive roots of polynomials , 1986 .

[95]  Victor Y. Pan,et al.  Root-Finding with Eigen-Solving , 2007 .

[96]  K. Hensel Journal für die reine und angewandte Mathematik , 1892 .

[97]  Maurice Mignotte,et al.  On the distance between the roots of a polynomial , 1995, Applicable Algebra in Engineering, Communication and Computing.

[98]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[99]  Daniel Reischert Asymptotically fast computation of subresultants , 1997, ISSAC.

[100]  Alkiviadis G. Akritas A Correction on a Theorem By Uspensky , 1978 .

[101]  Miodrag S. Petkovic,et al.  Safe convergence of simultaneous methods for polynomial zeros , 2004, Numerical Algorithms.

[102]  R. Tapia,et al.  Optimal Error Bounds for the Newton–Kantorovich Theorem , 1974 .

[103]  A. Ostrowski Note on Vincent's Theorem , 1950 .

[104]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[105]  Michel Coste,et al.  Thom's Lemma, the Coding of Real Algebraic Numbers and the Computation of the Topology of Semi-Algebraic Sets , 1988, J. Symb. Comput..

[106]  Chee-Keng Yap,et al.  Robust Approximate Zeros , 2005, ESA.

[107]  John J. Sopka,et al.  Introductory Functional Analysis with Applications (Erwin Kreyszig) , 1979 .

[108]  J. Hubbard,et al.  How to find all roots of complex polynomials by Newton’s method , 2001 .

[109]  James Hardy Wilkinson,et al.  Rounding errors in algebraic processes , 1964, IFIP Congress.

[110]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

[111]  S. Smale On the efficiency of algorithms of analysis , 1985 .

[112]  S. Smale The fundamental theorem of algebra and complexity theory , 1981 .

[113]  Jeremy R. Johnson,et al.  Architecture-aware classical Taylor shift by 1 , 2005, ISSAC.

[114]  T. A. Brown,et al.  Theory of Equations. , 1950, The Mathematical Gazette.

[115]  Herbert S. Wilf A Global Bisection Algorithm for Computing the Zeros of Polynomials in the Complex Plane , 1978, JACM.

[116]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[117]  A. Ostrowski Solution of equations and systems of equations , 1967 .

[118]  Michael Ben-Or,et al.  Simple algorithms for approximating all roots of a polynomial with real roots , 1990, J. Complex..

[119]  J. Liouville Mémoire sur les transcendantes elliptiques de 1$^{re}$ et de 2$^{me}$ espèce, considérées comme fonctions de leur module. , 1840 .

[120]  R. B. Kearfott,et al.  Abstract generalized bisection and a cost bound , 1987 .

[121]  Alkiviadis G. Akritas Reflections on a Pair of Theorems by Budan and Fourier , 1982 .

[122]  C. A. Neff,et al.  An O(nls'log b) Algorithm for the Complex Roots Problem , 1994 .

[123]  Tetsuro Yamamoto,et al.  Error Bounds for Newton’s Method Under the Kantorovich Assumptions , 1986 .

[124]  Richard P. Brent,et al.  Fast Multiple-Precision Evaluation of Elementary Functions , 1976, JACM.

[125]  K. F. Roth,et al.  Rational approximations to algebraic numbers , 1955 .

[126]  Thomas Lickteig,et al.  Sylvester-Habicht Sequences and Fast Cauchy Index Computation , 2001, J. Symb. Comput..

[127]  Florian Cajori,et al.  Historical Note on the Newton-Raphson Method of Approximation , 1911 .

[128]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[129]  Stephen Smale,et al.  Complexity of Bezout's Theorem: III. Condition Number and Packing , 1993, J. Complex..

[130]  Kurt Mehlhorn,et al.  New bounds for the Descartes method , 2005, SIGS.

[131]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[132]  Mukarram Ahmad,et al.  Continued fractions , 2019, Quadratic Number Theory.

[133]  J Erˆome,et al.  A CONDITION NUMBER THEOREM FOR UNDERDETERMINED POLYNOMIAL SYSTEMS , 2000 .

[134]  Rémi Vaillancourt,et al.  A composite polynomial zerofinding matrix algorithm , 1995 .

[135]  Axel Thue Über Annäherungswerte algebraischer Zahlen. , 1909 .

[136]  Bahman Kalantari,et al.  An infinite family of bounds on zeros of analytic functions and relationship to Smale's bound , 2004, Math. Comput..

[137]  Tjalling J. Ypma,et al.  Historical Development of the Newton-Raphson Method , 1995, SIAM Rev..

[138]  Maurice Mignotte,et al.  Some inequalities about univariate polynomials , 1981, SYMSAC '81.

[139]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[140]  Paul Turán,et al.  On a new method of analysis and its applications , 1984 .

[141]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[142]  Susanne Schmitt,et al.  The Diamond Operator - Implementation of Exact Real Algebraic Numbers , 2005, CASC.

[143]  S. Basu,et al.  Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics) , 2006 .

[144]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.