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The largest 5th pivot may be the root of a 61st degree polynomial

James Chen, Alan Edelman, John Urschel

Abstract

This paper introduces a number of new techniques in the study of the famous question from numerical linear algebra: what is the largest possible growth factor when performing Gaussian elimination with complete pivoting? This question is highly complex, due to a complicated set of polynomial inequalities that need to be simultaneously satisfied. This paper introduces the JuMP + Groebner basis + discriminant polynomial approach as well as the use of interval arithmetic computations. Thus, we are introducing a marriage of numerical and exact mathematical computations. In 1988, Day and Peterson performed numerical optimization on $n=5$ with NPSOL and obtained a largest seen value of $4.1325...$. This same best value was reproduced by Gould with LANCELOT in 1991. We ran extensive comparable experiments with the modern software tool JuMP and also saw the same value $4.1325...$. While the combinatorial explosion of possibilities prevents us from knowing there may not be a larger maximum, we succeed in obtaining the exact mathematical value: the number $4.1325...$ is exactly the root of a 61st degree polynomial provided in this work, and is a maximum given the equality constraints seen by JuMP. In light of the numerics, we pose the conjecture that this lower bound is indeed the maximum. We also apply this technique to $n = 6$, $7$, and $8$. Furthermore, in 1969, an upper bound of $4\frac{17}{18}\approx 4.94$ was produced for the maximum possible growth for $n = 5$. We slightly lower this upper bound to $4.84$.

The largest 5th pivot may be the root of a 61st degree polynomial

Abstract

This paper introduces a number of new techniques in the study of the famous question from numerical linear algebra: what is the largest possible growth factor when performing Gaussian elimination with complete pivoting? This question is highly complex, due to a complicated set of polynomial inequalities that need to be simultaneously satisfied. This paper introduces the JuMP + Groebner basis + discriminant polynomial approach as well as the use of interval arithmetic computations. Thus, we are introducing a marriage of numerical and exact mathematical computations. In 1988, Day and Peterson performed numerical optimization on with NPSOL and obtained a largest seen value of . This same best value was reproduced by Gould with LANCELOT in 1991. We ran extensive comparable experiments with the modern software tool JuMP and also saw the same value . While the combinatorial explosion of possibilities prevents us from knowing there may not be a larger maximum, we succeed in obtaining the exact mathematical value: the number is exactly the root of a 61st degree polynomial provided in this work, and is a maximum given the equality constraints seen by JuMP. In light of the numerics, we pose the conjecture that this lower bound is indeed the maximum. We also apply this technique to , , and . Furthermore, in 1969, an upper bound of was produced for the maximum possible growth for . We slightly lower this upper bound to .
Paper Structure (14 sections, 6 theorems, 27 equations, 3 figures, 1 table)

This paper contains 14 sections, 6 theorems, 27 equations, 3 figures, 1 table.

Table of Contents

  1. The maximum growth factor sequence for $n=1,\ldots,5$
  2. Gaussian elimination with complete pivoting
  3. The algebraic complexity of the growth factor
  4. A JuMP + Groebner Basis + Discriminant Polynomial Approach
  5. Our structure for $n=5$
  6. The role of the Groebner basis
  7. The role of the discriminant polynomial
  8. The growth value 4.1325... is optimal for Sequence (\ref{['eq:suits']})
  9. What about $n =6$, $7$, and $8$?
  10. Upper bounds on the growth of $5 \times 5$ matrices with complete pivoting
  11. High level technique
  12. A test for $3 \times 3$ matrices
  13. Construction of covers
  14. Infeasibility of Inequality \ref{['eq:low_rank_5x5_pivot']}

Key Result

Proposition 1.1

The largest possible growth factor for a $n \times n$ real matrix under complete pivoting is an algebraic number of degree at most $(2n)^{n^2-1}$.

Figures (3)

  • Figure 1: The 61st degree polynomial with only 12 Mathematica input cells
  • Figure 2: 2D Projections of $S_{2.2}$ and $S_{2.15}$ onto entries $A_{21}$ and $A_{12}$ (left), entries $A_{31}$ and $A_{13}$ (middle), and entries $A_{32}$ and $A_{23}$ (right). Note that $S_{2.2} \subset S_{2.15}$ and thus every box plotted that is part of $S_{2.2}$ is also part of $S_{2.15}$.
  • Figure 3: Under the assumption that $p_3 p_3' > 4.84$, the gray section is ruled out trivially. Computations involving $S_{2.2}$ and $S_{2.15}$ rule out the red and blue regions respectively.

Theorems & Definitions (14)

  • Proposition 1.1
  • proof
  • Conjecture 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3: Pruning boxes
  • proof
  • ...and 4 more