Read the equation
The matrix combines a nearest-neighbor ring Laplacian with a diagonal cosine modulation. The ring part is circulant, but the varying diagonal generally breaks that symmetry. Consequently its eigenvalues cannot generally be obtained by simply adding a Fourier-mode ring eigenvalue to a diagonal entry.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| alpha | Ring coupling |
| kappa_intra | Diagonal-modulation amplitude |
| I_16 | Identity matrix |
| T_16 | Ring neighbor adjacency |
| D_16 | Cosine diagonal |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
Read the surrounding source passage
public:
explicit IntraStrandBlock(const CouplingParams& p = {}) : params_(p) {
build_matrix();
}
// Matrix form (Section 11.1):
// M_intra = -2α·I₁₆ + α·T₁₆ + κ_intra·D₁₆(10.6)
// where D₁₆(10.6) = diag[cos(2π·k/10.6)]
void build_matrix() {
// Initialize to zero
for (auto& row : M_) row.fill(0.0);
for (int i = 0; i < NUM_STRANDS; ++i) {
// Diagonal: -2α + κ_intra·cos(2π(i+1)/10.6)
double cos_term = std::cos(2.0 * M_PI * (i + 1) / BASE_PAIRS_PER_TURN);
M_[i][i] = -2.0 * params_.alpha + params_.kappa_intra * cos_term;
Exact original notation
M_intra = -2α·I₁₆ + α·T₁₆ + κ_intra·D₁₆(10.6)