Module linalg 

Dense complex linear algebra for the small matrices of MIMO processing.

The matrices of a MIMO link are a few antennas wide, so the routines here favour simplicity and accuracy over asymptotic speed: modified Gram–Schmidt with re-orthogonalisation for the QR decomposition, Cholesky for Hermitian positive definite systems, and the one-sided Jacobi method (Hestenes) for the singular value decomposition, which is accurate to the relative precision of the singular values. Matrices are ndarray::Array2<C64> (CMat).

Model

• qr: $A = Q R$ with $Q^{H} Q = I$ and $R$ upper triangular with a real, non-negative diagonal ($A$ is $m \times n$, $m \ge n$);
• cholesky: $A = L L^{H}$ with $L$ lower triangular ($A$ Hermitian positive definite), and the systems of solve_hpd, or of solve_cholesky with the factor computed once;
• svd: $A = U \Sigma V^{H}$ with the singular values in decreasing order;
• solve_tridiagonal: a tridiagonal system by the Thomas algorithm.

Examples

use comnum_math::linalg::{matmul, hermitian, svd};
use comnum_math::C64;
use ndarray::array;
let a = array![[C64::new(1.0, 1.0), C64::new(0.5, 0.0)], [C64::new(0.0, -2.0), C64::new(1.0, 0.0)]];
let (u, s, v) = svd(&a.view());
// U Σ V^H rebuilds A
let mut us = u.clone();
for (j, sj) in s.iter().enumerate() {
    us.column_mut(j).mapv_inplace(|x| x * *sj);
}
let b = matmul(&us.view(), &hermitian(&v.view()).view());
assert!(a.iter().zip(b.iter()).all(|(x, y)| (x - y).norm() < 1e-12));

References

1. G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013 (Sec. 5.2, 4.2, 8.6).

Functions

cholesky

The Cholesky factor $L$ of a Hermitian positive definite matrix, $A = L L^{H}$.

gram

The Gram matrix $A^{H} A$.

hermitian

The conjugate transpose $A^{H}$.

identity

The identity of size n.

inverse_hpd

The inverse of a Hermitian positive definite matrix.

log2_det_hpd

$\log_{2} \det A$ of a Hermitian positive definite matrix.

matmul

The product $A B$.

matvec

The product $A x$ of a matrix and a vector.

qr

The thin QR decomposition $A = Q R$ of an $m \times n$ matrix, $m \ge n$.

solve_cholesky

Solve $L L^{H} X = B$ from the Cholesky factor $L$ of cholesky (forward, then backward substitution): solve_hpd with the factor computed once, for systems that share the matrix.

solve_hpd

Solve $A X = B$ for a Hermitian positive definite $A$ (by Cholesky).

solve_tridiagonal

The solution of the tridiagonal system lower[i] u[i-1] + diag[i] u[i] + upper[i] u[i+1] = rhs[i] (lower[0] and the last upper unused), by the Thomas algorithm without pivoting: for diagonally dominant systems, as those of cubic splines.

svd

The thin singular value decomposition $A = U \Sigma V^{H}$ by one-sided Jacobi.

Type Aliases

CMat

A dense complex matrix.