The lectures in this volume are the proceedings from the 1995 AMS-SIAM Summer Seminar in Applied Mathematics held in Park City, UT. The mathematical theory of real number algorithms was the subject of the conference, with emphasis on geometrical, algebraic, analytic, and foundational perspectives. Investigations on efficiency played a special role. The goal of the conference was to give the topic of numerical analysis greater coherence by focusing on the mathematical side. Particular attention was aimed at strengthening the unity of mathematics and numerical analysis and narrowing the gap between pure and applied mathematics.The conference was international in character, with strong representation from the most mathematically developed parts of numerical analysis. Seminars in the following areas were held: linear algebra, nonlinear systems-path following, differential equations, linear programming, interval arithmetic, algebraic questions, foundations, information based complexity, lower bounds, and approximation theory.

Table of Contents:
Panel discussion: Does numerical analysis need a model of computation? by M. Shub On numerical solving of nonlinear Polaron equations by P. G. Akishin, I. V. Puzynin, and Y. S. Smirnov Symmetry reductions for the numerical solution of boundary value problems by E. L. Allgower and P. J. Aston The combinatorics of real algebraic splines over a simplicial complex by C. L. Bajaj QMR and TFQMR method for sparse nonsymmetric problems on massively parallel systems by A. Basermann On multigrid techniques for thin plate spline interpolation in two dimensions by R. K. Beatson, G. Goodsell, and M. D. Powell Sparse matrix reordering schemes for browsing hypertext by M. W. Berry, B. Hendrickson, and P. Raghavan Algebraic settings for the problem ""P $\neq$ NP?"" by L. Blum, F. Cucker, M. Shub, and S. Smale A new algorithm for computing the spectral matrix for higher-order differential equations and the location of discrete eigenvalues by B. M. Brown, M. P. Eastham, and D. R. McCormack An asymptotically optimal non-adaptive algorithm for minimization of Brownian motion by J. M. Calvin On two iterative methods for approximating the roots of a polynomial by J.-P. Cardinal Algebraic approach of residues and applications by J. P. Cardinal and B. Mourrain Nash trees and Nash complexity by F. Cucker and T. Lickteig Operator equations, multiscale concepts and complexity by W. Dahmen, A. Kunoth, and R. Schneider Approximate solutions of numerical problems, condition number analysis and condition number theorem by J.-P. Dedieu Computing the distance from a point to an algebraic hypersurface by J. P. Dedieu, X. Gourdon, and J. C. Yakoubsohn Local analysis of a Newton-type method based on partial linearization by A. L. Dontchev Approximations and complexity for computing algebraic curves by B. C. Eaves and U. G. Rothblum Numerical univariate polynomial GCD by I. Z. Emiris, A. Galligo, and H. Lombardi A parallel preconditioned GMRES algorithm for sparse matrices by J. Erhel An optimal algorithm for the local solution of integral equations by K. Frank Descriptive complexity theory over the real numbers by E. Gradel and K. Meer Complexity theory of Monte Carlo algorithms by S. Heinrich Qualitative numerical analysis of ordinary differential equations by A. Iserles and A. Zanna Tapia indicators and finite termination of infeasible-interior-point methods for degenerate LCP by J. Ji and F. A. Potra Quasi-Monte Carlo methods in computer graphics: The global illumination problem by A. Keller Numerical algorithms with automatic result verification by U. Kulisch Random product homotopy with minimal BKK bound by T. Y. Li, T. Wang, and X. Wang Computational complexity over the 2-adic numbers by M. Maller and J. Whitehead Optimal reconstruction of stochastic evolutions by P. Mathe Lagrangian globalization: Solving nonlinear equations via constrained optimization by J. L. Nazareth Polynomial time methods in convex programming by A. Nemirovski Effective parallel computations with Toeplitz and Toeplitz-like matrices filled with integers by V. Y. Pan An efficient discretization for solving ill-posed problems by S. V. Pereverzev and S. G. Solodky Survey of computational complexity with noisy information by L. Plaskota Lazy analysis and elementary numbers by D. Richardson On the average case complexity of solving Poisson equations by K. Ritter and G. W. Wasilkowski On the average number of real roots of certain random sparse polynomial systems by J. M. Rojas Computations in real algebraic geometry by M.-F. Roy Path following for large nonlinear equations by implicit block elimination based on recursive projections by H. Schwetlick, G. Timmermann, and R. Losche Computability with neural networks by H. T. Siegelmann Numerical algebraic geometry by A. J. Sommese and C. W. Wampler Wavelets from filter banks by G. Strang A pragmatic overview of fast multipole methods by J. H. Strickland and R. S. Baty Topological complexity of root-finding algorithms by V. A. Vassiliev On the relationship between layered least squares and affine scaling steps by S. A. Vavasis and Y. Ye Enclosure methods for capricious solutions of ordinary differential equations by W. Walter QR-like algorithms-An overview of convergence theory and practice by D. S. Watkins The complexity of the Poisson problem for spaces of bounded mixed derivatives by A. G. Werschulz Overview of information-based complexity by H. Wozniakowski.