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Polynomial approximations for the matrix logarithm with computation graphs

Elias Jarlebring, Jorge Sastre, J. Javier Ibáñez González

TL;DR

This work tackles the computational burden of the matrix logarithm by introducing a graph-based polynomial evaluation framework within the inverse scaling and squaring method. It blends Taylor-based polynomials for small iteration counts with min–max, computation-graph optimized polynomials for larger counts, leveraging the transformation $-\,\log(I-A)=\sum_{i=1}^m \frac{A^i}{i}$ and a degree-$2^k$ structure to minimize matrix multiplications. A thorough stability and backward error analysis is provided to guarantee reliability under finite-precision arithmetic, and the approach is validated through extensive numerical experiments showing competitive runtime and accuracy relative to Padé and prior polynomial methods. The work also offers practical MATLAB implementations and clear guidelines for selecting polynomial schemes based on cost and accuracy, highlighting potential for further optimization.

Abstract

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. The main computational effort lies in matrix-matrix multiplications and left matrix division. In this work we illustrate that the number of such operations can be substantially reduced, by using a graph based representation of an efficient polynomial evaluation scheme. A technique to analyze the rounding error is proposed, and backward error analysis is adapted. We provide substantial simulations illustrating competitiveness both in terms of computation time and rounding errors.

Polynomial approximations for the matrix logarithm with computation graphs

TL;DR

This work tackles the computational burden of the matrix logarithm by introducing a graph-based polynomial evaluation framework within the inverse scaling and squaring method. It blends Taylor-based polynomials for small iteration counts with min–max, computation-graph optimized polynomials for larger counts, leveraging the transformation and a degree- structure to minimize matrix multiplications. A thorough stability and backward error analysis is provided to guarantee reliability under finite-precision arithmetic, and the approach is validated through extensive numerical experiments showing competitive runtime and accuracy relative to Padé and prior polynomial methods. The work also offers practical MATLAB implementations and clear guidelines for selecting polynomial schemes based on cost and accuracy, highlighting potential for further optimization.

Abstract

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. The main computational effort lies in matrix-matrix multiplications and left matrix division. In this work we illustrate that the number of such operations can be substantially reduced, by using a graph based representation of an efficient polynomial evaluation scheme. A technique to analyze the rounding error is proposed, and backward error analysis is adapted. We provide substantial simulations illustrating competitiveness both in terms of computation time and rounding errors.
Paper Structure (10 sections, 30 equations, 2 figures, 6 tables, 1 algorithm)

This paper contains 10 sections, 30 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction and notation
  2. Related methods
  3. Methods for the matrix logarithm
  4. Efficient polynomial evaluation
  5. Polynomial approximations of the matrix logarithm
  6. Stability of the polynomial evaluation methods
  7. Backward error analysis
  8. Algorithm for computing $\log(A)$
  9. Numerical experiments
  10. Conclusions

Figures (2)

  • Figure 1: Comparison of the degree optimal polynomial obtain using the optimization scheme \ref{['eq:five_mult_degopt']}. For comparison, also the Taylor approximation obtained with a Paterson--Stockmeyer evaluation with the same number of matrix-matrix multiplications is also provided.
  • Figure 2: Experimental rounding error results for test set 3.

Theorems & Definitions (1)

  • proof