Module structured 

Structured matrices support (Toeplitz, circulant) for efficient representations

This module provides implementations of various structured matrices that have special properties allowing for efficient storage and operations.

Overview

Structured matrices are matrices with special patterns that allow them to be represented using far fewer parameters than general matrices. This provides significant memory savings and computational advantages for large matrices.

Types of Structured Matrices

• Toeplitz matrices: Matrices where each descending diagonal from left to right is constant
• Circulant matrices: Special Toeplitz matrices where each row is a cyclic shift of the first row
• Hankel matrices: Matrices where each ascending diagonal from left to right is constant

Examples

use scirs2_core::ndarray::{Array1, array};
use scirs2_linalg::structured::{ToeplitzMatrix, StructuredMatrix};

// Create a Toeplitz matrix from its first row and column
let first_row = array![1.0, 2.0, 3.0];
let first_col = array![1.0, 4.0, 5.0];

let toeplitz = ToeplitzMatrix::new(first_row.view(), first_col.view()).unwrap();

// The matrix represented is:
// [1.0, 2.0, 3.0]
// [4.0, 1.0, 2.0]
// [5.0, 4.0, 1.0]

// Apply to a vector efficiently without forming the full matrix
let x = array![1.0, 2.0, 3.0];
let y = toeplitz.matvec(&x.view()).unwrap();

Structs

CirculantMatrix

Circulant matrix implementation

HankelMatrix

Hankel matrix implementation

ToeplitzMatrix

Toeplitz matrix implementation

Traits

StructuredMatrix

A trait for structured matrices that can be represented efficiently

Functions

circulant_determinant

Compute determinant of a circulant matrix

circulant_eigenvalues

Compute eigenvalues of a circulant matrix

circulant_inverse_fft

FFT-based circulant matrix inverse

circulant_matvec_direct

Direct circulant matrix-vector multiplication

circulant_matvec_fft

FFT-based circulant matrix-vector multiplication

dftmatrix_multiply

Discrete Fourier Transform matrix multiplication

fast_toeplitz_inverse

Fast Toeplitz matrix inversion using the Gohberg-Semencul formula

gohberg_semencul_inverse

Gohberg-Semencul formula for efficient Toeplitz matrix inversion

hadamard_transform

Fast Walsh-Hadamard transform

hankel_determinant

Compute determinant of a Hankel matrix

hankel_matvec

Hankel matrix-vector multiplication

hankel_matvec_fft

FFT-based Hankel matrix-vector multiplication

hankel_svd

Compute SVD of a Hankel matrix

levinson_durbin

Levinson-Durbin algorithm for solving Toeplitz systems

solve_circulant

Solve a Circulant system

solve_circulant_fft

FFT-based circulant system solver

solve_toeplitz

Solve a Toeplitz system using the Levinson algorithm

solve_tridiagonal_lu

Solve tridiagonal system using LU decomposition

solve_tridiagonal_thomas

Solve tridiagonal system using Thomas algorithm

structured_to_operator

Convert a structured matrix to a matrix-free operator

tridiagonal_determinant

Compute determinant of a tridiagonal matrix

tridiagonal_eigenvalues

Compute eigenvalues of a tridiagonal matrix

tridiagonal_eigenvectors

Compute eigenvectors of a tridiagonal matrix

tridiagonal_matvec

Tridiagonal matrix-vector multiplication

yule_walker

Yule-Walker equations solver