Multivariate regression in Tensor Train format

Addition to the TT-Toolbox package for regression of multivariate functions. Regression algorithm is based on the Alternating Least Squares approach. Functional basis in each dimension can be specified by the user. Overall complexity of the algorithm is linear in both dimension and number of data points.

 

Rank-structured dqds algorithm

Rank-structured dqds algorithm generalizes celebrated Parlett's dqds algorithm for finding eigenvalues of real symmetric tridiagonal matrices (one of LAPACK's main eigenvalue routines — DLASQ). A collection of MATLAB functions realizes dqds iterations with O(n) complexity per step for rank-structured matrices and can be used to find roots of polynomials represented in the following bases:
  • monomials
  • real orthogonal polynomials (Chebyshev, Hermite, Legendre etc.)
  • polynomials orthogonal on the unit circle (Szegö polynomials)
Currently only real-valued version is implemented but the complex-valued version is under development and will be available soon. Download MATLAB code here.

 

MLQS 1.1: Multilevel Quasiseparable Matrices

Software Package written in C++ contains implementation of algorithms for working with quasiseparable and multilevel quasiseparable matrices. List of currently implemented algorithms:
  • Matrix operations (+/-, *, scaling).
  • Inversion and LDU factorization.
  • Matvec and system solver.
  • Compression and normalization of generators.
  • Frobenious norm.
  • Conversion to dense format.
Note that all of the above listed algorithms except the conversion to dense format are of linear complexity in the dimension of the matrix. Download the current version of MLQS here.

 

Structured Newton's Iteration Algorithm for the inversion of Toeplitz/Toeplitz-like matrices

MATLAB implementation, list of features:
  • Invert both Toeplitz and Toeplitz-like matrices.
  • Uses small number of FFTs per step.
  • Dynamically recompesses displacement generators to speed-up iterations.
  • Performs randomization of the input to enhance the convergence.
Download the current version here.