Allgower E. and Georg K. (1990), Numerical Continuous Methods, Springer.
Allgower E. and Georg K. (1993), Continuation and path following, in Acta Numerica, Vol. 2, Cambridge University Press, pp. 1–64.
Axelsson O. (1994), Iterative Solution Methods, Cambridge University Press.
Batterson S. (1994), ‘Convergence of the Francis shifted QR algorithm on normal matrices’, Linear Algebra Appl. 207, 181–195.
Batterson S. and Day D. (1992), ‘Linear convergence in the shifted QR algorithm’, Math. Comp. 59, 141–151.
Batterson S. and Smillie J. (1989), ‘The dynamics of Rayleigh quotient iteration’, SIAM J. Numer. Anal. 26, 624–636.
Batterson S. and Smillie J. (1990), ‘Rayleigh quotient iteration for nonsymmetric matrices’, Math. Comp. 55, 169–178.
Ben-Or M. (1983), Lower bounds for algebraic computation trees, in 15th Annual ACM Symposium on the Theory of Computing, pp. 80–86.
Bini D. and Pan V. (1987), ‘Sequential and parallel complexity of approximating polynomial zeros’, Computers and Mathematics (with applications) 14, 591–622.
Bini D. and Pan V. (1994), Polynomial and Matrix Computations, Birkhäuser, Basel.
Blum L., Cucker F., Shub M. and Smale S. (1996), ‘Complexity and real computation: a manifesto’, Int. J. Bifurcation and Chaos 6, 3–26. Referred to as the Manifesto.
Blum L., Cucker F., Shub M. and Smale S. (1997), Complexity and Real Computation, Springer. To appear. Referred to as BCSS (1997).
Blum L., Shub M. and Smale S. (1989), ‘On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines’, Bull. Amer. Math. Soc. 21, 1–46. Referred to as BSS (1989).
Brockett R. (1973), in Geometric Methods in Systems Theory, Proceedings of the NATO Advanced Study Institute (Mayne D. and Brockett R., eds), D. Reidel, Dordrecht.
Brownawell W. (1987), ‘Bounds for the degrees in the Nullstellensatz’, Annals of Math. 126, 577–591.
Collins G. (1975), Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, Vol. 33 of Lect. Notes in Comp. Sci., Springer, pp. 134–183.
Cuppen J. J. M. (1981), ‘A divide and conquer method for the symmetric tridiagonal eigenproblem’, Numer. Math. 36, 177–195.
Dedieu J.-P. (1997 a), Approximate solutions of numerical problems, condition number analysis and condition number theorems, in Proceedings of the Summer Seminar on ‘Mathematics of Numerical Analysis: Real Number Algorithms’, AMS Lectures in Applied Mathematics (Renegar J., Shub M. and Smale S., eds), AMS, Providence, RI. To appear.
Dedieu J.-P. (1997 b), Condition number analysis for sparse polynomial systems. Preprint.
Dedieu J.-P. (1997 c), ‘Condition operators, condition numbers and condition number theorem for the generalized eigenvalue problem’, Linear Algebra Appl. To appear.
Dedieu J.-P. (1997 d), ‘Estimations for separation number of a polynomial system’, J. Symbolic Computation. To appear.
Dégot J. and Beauzamy B. (1997), ‘Differential identities’, Trans. Amer. Math. Soc. To appear.
Dejon B. and Henrici P. (1969), Constructive Aspects of the Fundamental Theorem of Algebra, Wiley.
Demmel J. (1987), ‘On condition numbers and the distance to the nearest ill-posed problem’, Numer. Math. 51, 251–289.
Dongarra J. J. and Sorensen D. C. (1987), ‘A fully parallel algorithm for the symmetric eigenvalue problem’, SIAM J. Sci. Statist. Comput. 8, 139–154.
Du Q., Jin M., Li T. Y. and Zeng Z. (1997 a), ‘The quasi-Laguerre iteration’, Math. Comp. To appear.
Du Q., Jin M., Li T. Y. and Zeng Z. (1997 b), ‘Quasi-Laguerre iteration in solving symmetric tridiagonal eigenvalue problems’, SIAM J. Sci. Comput. To appear.
Eckart C. and Young G. (1936), ‘The approximation of one matrix by another of lower rank’, Psychometrika 1, 211–218.
Edelman A. (1988), ‘Eigenvalues and condition numbers of random matrices’, SIAM J. Matrix Anal. Appl. 9, 543–556.
Edelman A. and Kostlan E. (1995), ‘How many zeros of a random polynomial are real?’, Bull. Amer. Math. Soc. 32, 1–38.
Gauss C. F. (1973), Werke, Band X, Georg Olms, New York.
Giusti M., Heintz J., Morais J. E., Morgenstern J. and Pardo L. M. (1997), ‘Straight-line program in geometric elimination theory’, Journal of Pure and Applied Algebra. To appear.
Golub G. and van Loan C. (1989), Matrix Computations, Johns Hopkins University Press.
Grigoriev D. (1987), in Computational complexity in polynomial algebra, Proceedings of the International Congress Math. (Berkeley, 1986), Vol. 1, 2, AMS, Providence, RI, pp. 1452–1460.
Henrici P. (1977), Applied and Computational Complex Analysis, Wiley.
Hestenes M. R. and Stiefel E. (1952), ‘Method of conjugate gradients for solving linear systems’, J. Res. Nat. Bur. Standards 49, 409–436.
Hirsch M. and Smale S. (1979), ‘On algorithms for solving f(x) = 0’, Comm. Pure Appl. Math. 32, 281–312.
Hoffman W. and Parlett B. N. (1978), ‘A new proof of global convergence for the tridiagonal QL algorithm’, SIAM J. Numer. Anal. 15, 929–937.
Isaacson E. and Keller H. (1966), Analysis of Numerical Methods, Wiley, New York.
Keller H. (1978), Global homotopic and Newton methods, in Recent Advances in Numerical Analysis, Academic Press, pp. 73–94.
Kellog R., Li T. and Yorke J. (1976), ‘A constructive proof of Brouwer fixed-point theorem and computational results’, SIAM J. Numer. Anal. 13, 473–483.
Kim M. (1988), ‘On approximate zeros and rootfinding algorithms for a complex polynomial’, Math. Comp. 51, 707–719.
Kostlan E. (1988), ‘Complexity theory of numerical linear algebra’, J. Comput. Appl. Math. 22, 219–230.
Kostlan E. (1991), ‘Statistical complexity of dominant eigenvector calculation’, J. Complexity 7, 371–379.
Kostlan E. (1993), On the distribution of the roots of random polynomials, in From Topology to Computation: Proceedings of the Smalefest (Hirsch M., Marsden J. and Shub M., eds), Springer, pp. 419–431.
Malajovich G. (1994), ‘On generalized Newton algorithms: quadratic convergence, path-following and error analysis’, Theoret. Comput. Sci. 133, 65–84.
Malajovich-Munoz G. (1993), On the complexity of path-following Newton algorithms for solving polynomial equations with integer coefficients, PhD thesis, University of California at Berkeley.
McNamee J. M. (1993), ‘A bibliography on roots of polynomials’, J. Comput. Appl. Math. 47(3), 391–394.
Milnor J. (1964), On the Betti numbers of real varieties, in Proceedings of the Amer. Math. Soc., Vol. 15, pp. 275–280.
Neff C. (1994), ‘Specified precision root isolation is in NC, J. Comput. System Sci. 48, 429–463.
Neff C. and Reif J. (1996), ‘An efficient algorithm for the complex roots problem’, J. Complexity 12, 81–115.
Oleinik O. (1951), ‘Estimates of the Betti numbers of real algebraic hypersurfaces’, Mat. Sbornik (N.S.) 28, 635–640. In Russian.
Oleinik O. and Petrovski I. (1949), ‘On the topology of real algebraic surfaces’, Izv. Akad. Nauk SSSR 13, 389–402. In Russian; English translation in Transl. Amer. Math. Soc. 1, 399–417 (1962).
Ostrowski A. (1958), ‘On the convergence of Rayleigh quotient iteration for the computation of the characteristic roots and vectors, I’, Arch. Rational Mech. Anal. 1, 233–241.
Pan V. (1997), ‘Solving a polynomial equation: some history and recent progress’, SIAM Review. To appear.
Parlett B. N. and Kahan W. (1969), ‘On the convergence of a practical QR algorithm’, Inform. Process. Lett. 68, 114–118.
Rakhmanov E. A., Saff E. B. and Zhou Y. M. (1994), ‘Minimal discrete energy on the sphere’, Mathematical Research Letters 1, 647–662.
Rakhmanov E. A., Saff E. B. and Zhou Y. M. (1995), Electrons on the sphere, in Computational Methods and Function Theory (Ali R. M., Ruscheweyh S. and Saff E. B., eds), World Scientific, pp. 111–127.
Renegar J. (1987 a), ‘On the efficiency of Newton's method in approximating all zeros of systems of complex polynomials’, Math, of Oper. Research 12, 121–148.
Renegar J. (1987 b), ‘On the worst case arithmetic complexity of approximating zeros of polynomials’, J. Complexity 3, 90–113.
Renegar J. (1996), ‘Condition numbers, the Barrier method, and the conjugate gradient method’, SIAM J. Optim. To appear.
Renegar J., Shub M. and Smale S., eds (1997), Proceedings of the Summer Seminar on ‘Mathematics of Numerical Analysis: Real Number Algorithm’, AMS Lectures in Applied Mathematics, AMS, Providence, RI.
Reznick B. (1992), Sums of Even Powers of Real Linear Forms, Vol. 463 of Memoirs of the American Mathematical Society, AMS, Providence, RI.
Rice J. R. (1966), ‘A theory of condition’, SIAM J. Numer. Anal. 3, 287–310.
Santal L.ó (1976), Integral Geometry and Geometric Probability, Addison-Wesley, Reading, MA.
Schönhage A. (1982), The fundamental theorem of algebra in terms of computational complexity, Technical report, Math. Institut der Universitat Tubingen.
Schönhage A. (1987), Equation solving in terms of computational complexity, in Proceedings of the International Congress of Mathematicans, AMS, Providence, RI.
Shub M. (1993), On the work of Steve Smale on the theory of computation, in From Topology to Computation: Proceedings of the Smalefest (Hirsch M., Marsden J. and Shub M., eds), Springer, pp. 443–455.
Shub M. and Smale S. (1985), ‘Computational complexity: on the geometry of polynomials and a theory of cost I’, Ann. Sci. École Norm. Sup. 18, 107–142.
Shub M. and Smale S. (1986), ‘Computational complexity: on the geometry of polynomials and a theory of cost II’, SIAM J. Comput. 15, 145–161.
Shub M. and Smale S. (1993 a), ‘Complexity of Bézout's theorem I: geometric aspect’, J. Amer. Math. Soc. 6, 459–501. Referred to as Bez I.
Shub M. and Smale S. (1993 b), Complexity of Bézout's theorem II: volumes and probabilities, in Computational Algebraic Geometry (Eyssette F. and Galligo A., eds), Vol. 109 of Progress in Mathematics, pp. 267–285. Referred to as Bez II.
Shub M. and Smale S. (1993 c), ‘Complexity of Bézout's theorem III: condition number and packing’, J. Complexity 9, 4–14. Referred to as Bez III.
Shub M. and Smale S. (1994), ‘Complexity of Bézout's theorem V: polynomial time’, Theoret. Comput. Sci. 133, 141–164. Referred to as Bez V.
Shub M. and Smale S. (1996), ‘Complexity of Bézout's theorem IV: probability of success; extensions’, SIAM J. Numer. Anal. 33, 128–148. Referred to as Bez IV.
Smale S. (1976), ‘A convergent process of price adjustment and global Newton method’, J. Math. Economy 3, 107–120.
Smale S. (1981), ‘The fundamental theorem of algebra and complexity theory’, Bull. Amer. Math. Soc. 4, 1–36.
Smale S. (1985), ‘On the efficiency of algorithms of analysis’, Bull. Amer. Math. Soc. 13, 87–121.
Smale S. (1986), Newton's method estimates from data at one point, in The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Ewing R., Gross K. and Martin C., eds), Springer, pp. 185–196.
Smale S. (1987 a), Algorithms for solving equations, in Proceedings of the International Congress of Mathematicians, AMS, Providence, RI, pp. 172–195.
Smale S. (1987b), ‘On the topology of algorithms I’, J. Complexity 3, 81–89.
Smale S. (1990), ‘Some remarks on the foundations of numerical analysis’, SIAM Review 32, 211–220.
Steele J. and Yao A. (1982), ‘Lower bounds for algebraic decision trees’, Journal of Algorithms 3, 1–8.
Stein E. and Weiss G. (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press.
Thom R. (1965), Sur l'homologie des variétés algébriques réelles, in Differential and Combinatorial Topology (Cairns S., ed.), Princeton University Press.
Traub J. and Wozniakowski H. (1979), ‘Convergence and complexity of Newton iteration for operator equations’, J. Assoc. Comput. Mach. 29, 250–258.
Traub J., Wasilkowski G. and Wozniakowski H. (1988), Information-Based Complexity, Academic Press.
Trefethen L. N. (preprint), Why Gaussian elimination is stable for almost all matrices.
Vassiliev V. A. (1992), Complements of Discriminants of Smooth Maps: Topology and Applications, Vol. 98 of Transl. of Math. Monographs, AMS, Providence, RI. Revised 1994.
Wang X. (1993), Some results relevant to Smale's reports, in From Topology to Computation: Proceedings of the Smalefest (Hirsch M., Marsden J. and Shub M., eds), Springer, pp. 456–465.
Weyl H. (1932), The Theory of Groups and Quantum Mechanics, Dover.
Wilkinson J. (1963), Rounding Errors in Algebraic Processes, Prentice-Hall.
Wilkinson J. (1968), ‘Global convergence of tridiagonal QR algorithm with origin shifts’, Linear Algebra Appl. I, 409–420.
Wozniakowski H. (1977), ‘Numerical stability for solving non-linear equations’, Nu-mer. Math. 27, 373–390.