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Full Description
Faster Algorithms via Approximation Theory illustrates how classical and modern techniques from approximation theory play a crucial role in obtaining results that are relevant to the emerging theory of fast algorithms. The key lies in the fact that such results imply faster ways to approximate primitives such as products of matrix functions with vectors and, to compute matrix eigenvalues and eigenvectors, which are fundamental to many spectral algorithms.
The first half of the book is devoted to the ideas and results from approximation theory that are central, elegant, and may have wider applicability in theoretical computer science. These include not only techniques relating to polynomial approximations but also those relating to approximations by rational functions and beyond. The remaining half illustrates a variety of ways that these results can be used to design fast algorithms.
Faster Algorithms via Approximation Theory is self-contained and should be of interest to researchers and students in theoretical computer science, numerical linear algebra, and related areas.
Contents
Introduction. I APPROXIMATION THEORY: 1. Uniform Approximations 2. Chebyshev Polynomials 3. Approximating Monomials 4. Approximating the Exponential 5. Lower Bounds for Polynomial Approximations 6. Approximating the Exponential using Rational Functions 7. Rational Approximations to the Exponential with Negative Poles II APPLICATIONS: 8. Simulating Random Walks 9. Solving Linear Equations via the Conjugate Gradient Method 10. Computing Eigenvalues via the Lanczos Method 11. Computing the Matrix Exponential 12. Matrix Inversion via Exponentiation. References.
