Test whether a matrix has tridiagonal structure

Returns TRUE when every entry \(M[i, j]\) with \(|i - j| > 1\) has absolute value below tol. Used by the dispatcher to activate the tridiagonal-series backend.

Usage

is_tridiagonal(M, tol = 1e-12)

Arguments

M

A numeric matrix.

tol

Numeric. Off-band absolute-value threshold below which an entry is treated as zero. Default 1e-12.

Value

A single logical: TRUE when only the main diagonal and the two adjacent diagonals carry mass, FALSE otherwise.

Details

The tridiagonality of \(V^\top V\) is the structural precondition for the simplified series of Fitisone & Zhou (2023, Theorem 4.1), which reduces the multi-index from \(\binom{n}{2}\) to \(n - 1\) variables. Theorem 4.1 also establishes that if \(V^\top V\) is tridiagonal, then the associated matrix \(M_n(C)\) is automatically positive definite.

References

Fitisone, A., & Zhou, Y. (2023). Solid angle measure of polyhedral cones. arXiv:2304.11102 (math.CO). https://arxiv.org/abs/2304.11102

See also

compute_dot_product_matrix for the natural input; tridiagonal_series for the downstream solver activated by a positive result; create_tridiagonal_cone for the constructor of tridiagonal cones; compute_solid_angle for the dispatcher.

Examples

is_tridiagonal(matrix(c(1, 0.5, 0,
                        0.5, 1, 0.3,
                        0, 0.3, 1), nrow = 3))                    # TRUE
#> [1] TRUE
is_tridiagonal(matrix(c(1, 0.5, 0.1,
                        0.5, 1, 0.3,
                        0.1, 0.3, 1), nrow = 3))                  # FALSE
#> [1] FALSE