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This study presents fast methods for solving least squares problems with matrices of small displacement rank. Efficient Cholesky factorization is achieved for specific matrix structures, enabling rapid solutions.

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Area of Science:

  • Numerical Analysis
  • Linear Algebra
  • Matrix Computations

Background:

  • Least squares problems are fundamental in data analysis and model fitting.
  • Matrices with small displacement rank, such as Toeplitz and Hankel matrices, appear in various applications.
  • Efficiently solving linear systems involving these matrices is computationally challenging.

Purpose of the Study:

  • To develop efficient algorithms for solving least squares problems involving matrices with small displacement rank.
  • To investigate the Cholesky factorization of M(H)M for matrices with specific properties.
  • To accelerate solutions for least squares, total least squares, and regularized total least squares problems.

Main Methods:

  • Developing formulas for generators of M(H)M based on generators of M.
  • Analyzing conditions for fast Cholesky factorization: Z1 near unitary, Z2 triangular and nilpotent.
  • Applying methods to classes of matrices including Toeplitz, block Toeplitz, Hankel, and block Hankel.

Main Results:

  • Formulas for matrix generators of M(H)M are derived.
  • Fast Cholesky factorization is demonstrated under specific conditions for Z1 and Z2.
  • The methods are applicable to structured matrices like Toeplitz and Hankel types.

Conclusions:

  • Fast Cholesky factorization of M(H)M is achievable for matrices with small displacement rank under given conditions.
  • This enables significantly faster solutions for various least squares problems.
  • The findings are relevant for efficiently processing structured matrices in scientific computing.