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Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations

Erik Parkinson, Kate Wall, Jane Slagle, Daniel Treuhaft, Xander de la Bruere, Samuel Goldrup, Timothy Keith, Peter Call, Tyler J. Jarvis

Abstract

We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has R-quadratic convergence locally near simple zeros of the system. We also analyze the temporal complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to three that the method is both fast and accurate on a wide range of problems. The algorithm should also work well in higher dimensions. Our tests show that the algorithm outperforms other standard methods on this problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available on GitHub.

Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations

Abstract

We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in . It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has R-quadratic convergence locally near simple zeros of the system. We also analyze the temporal complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to three that the method is both fast and accurate on a wide range of problems. The algorithm should also work well in higher dimensions. Our tests show that the algorithm outperforms other standard methods on this problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available on GitHub.
Paper Structure (49 sections, 16 theorems, 40 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 49 sections, 16 theorems, 40 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Overview and Relation to the Chebyshev Proxy Method
  3. Broad Picture of the Algorithm
  4. Why Chebyshev?
  5. Outline
  6. Detailed Description of the Algorithm
  7. Chebyshev Proxy
  8. Approximation Degrees
  9. Bounding Approximation Error
  10. Functions with Large Dynamic Range
  11. Chebyshev Solver
  12. Elimination Checks
  13. Interval Reduction
  14. First Reduction Method
  15. Second Reduction Method
  16. ...and 34 more sections

Key Result

Theorem 3.2

Let $J_0$ be an interval containing only one zero ${{{\mathbf{z}}} }$ of the Chebyshev proxy system ${p}_1,\dots, {p}_n$, and assume also that ${{{\mathbf{z}}} }$ is a simple zero of the proxy system. If $J_0$ is sufficiently small, the second reduction method, using all of the linear terms (see Sec let $\gamma_{k,m} = b_{k,m} - a_{k,m}$ be the width of $J_k$ in coordinate $m$, and let ${\hat{\eps

Figures (8)

  • Figure 1: The computed values of $\tau_{0.5,{\beta}}$ for large degree $n$ along with the values from Conjecture \ref{['Conj:Actual Tau']}, which the approximations seem to converge to.
  • Figure 2: Histogram (in blue) of the errors (distance to the true value) for all of the zeros computed by our algorithm for the first $1000$ Chebyshev monomials $T_d(x)$. The $x$-axis is the size of the error, and the $y$-axis is the density of the zeros with that error. Plotted in black is the density of the errors of the best-possible numerical solutions (the nearest floating-point number to the true zero).
  • Figure 3: Average time for the Chebyshev polynomial solver to solve ten systems of Chebyshev-basis polynomials of varying total degree ${d}$ with coefficients drawn form the standard normal distribution in dimensions one through five. The legend gives the observed arithmetic complexity of each, estimated from the slope of the plotted almost-straight lines. As functions of degree ${d}$ the smaller dimensional problems appear to be better even than the conjectured arithmetic complexity of $O(d^{n+1})$ (see Section \ref{['sec:numerical-temporal-complexity']}).
  • Figure 4: The results of running the random-polynomial tests in two (left panel) and three (right panel) dimensions, with degree on the $x$-axis and the timing in log scale on the $y$-axis. Each of the different methods is plotted as a different colored line. Chebfun2 does not appear in the right panel because it is not yet implemented in dimensions greater than two. We expect that implementing YRoots in a faster language like Julia or C would make it competitive with MTV in low degree.
  • Figure 5: The maximum (worst) residuals for the solutions found by each of the the different solvers on the 2-dimensional (left panel) and 3-dimensional (right panel) random-polynomial tests, with degree on the $x$-axis and the residual in log scale on the $y$-axis. Each of the different methods plotted as a different colored line. The lines for Reduce and NSolveValues are only plotted for low degrees because in all the missing higher degrees they failed to terminate, or when they terminated they failed to find all of the roots. Note that for all of these methods, if the final results are sufficiently close to the correct answers, then the computed roots could be Newton polished to arbitrary precision. In such cases, the the practical impact of the differences in residual is minimal. For more on this, see the discussion in Subsection \ref{['sec:random-polynomial-results']}.
  • ...and 3 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • ...and 29 more