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This study simplifies and unifies fast matrix-vector multiplication for structured dense matrices. It identifies matrix classes enabling sub-quadratic computation, crucial for efficient algorithms.

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Area of Science:

  • Numerical Analysis
  • Computational Linear Algebra
  • Computer Science Theory

Background:

  • Matrix-vector multiplication is a fundamental computation, typically requiring quadratic time complexity.
  • Structured matrices can allow for faster multiplication, but identifying and utilizing these structures is challenging.
  • Efficient computation of matrix inverses for structured matrices remains an open problem.

Purpose of the Study:

  • To identify classes of structured dense matrices representable with linear parameters.
  • To enable sub-quadratic time matrix-vector multiplication and potentially other operations like solvers.
  • To unify and generalize existing results on fast structured matrix multiplication.

Main Methods:

  • Exploration of orthogonal polynomial transforms and low displacement rank matrices.
  • Development of a unified framework for structured matrix representations.
  • Reduction of applications like multipoint polynomial evaluation to low recurrence width matrices.

Main Results:

  • A unified and simplified approach to structured matrix-vector multiplication.
  • Identification of new classes of structured matrices with efficient computational properties.
  • Demonstration of applicability to problems like multipoint evaluations of multivariate polynomials.

Conclusions:

  • Significant progress in unifying and simplifying fast structured matrix multiplication.
  • Broadened the scope of matrices amenable to efficient linear algebraic computations.
  • Established connections between structured matrices and applications in polynomial computations.