Population Flow

If a system's transition matrix is fixed at compile-time, its steady-state behavior never needs to be recomputed at runtime. Take the example from Using Eigenvectors to Find Steady State Population Flows and apply it using consteig. The transition matrix is fixed at compile-time, so the steady-state fractions can be stored as static constexpr values. At runtime, computing the distribution for any total population is then a single multiply. There is no need to iterate the transition matrix until it converges.

The population.cpp example demonstrates this:

Problem

Consider a simple two-city model where each year: - 95% of Seattle's population stays, 5% moves to Portland - 20% of Portland's population moves to Seattle, 80% stays

This gives a transition matrix:

A = [0.95  0.20]
    [0.05  0.80]

The steady-state distribution is the eigenvector corresponding to eigenvalue 1.

Solution with consteig

#include <consteig/consteig.hpp>

static constexpr consteig::Matrix<double, 2, 2> A{{
    {0.95, 0.20},
    {0.05, 0.80}
}};

// Eigenvalues computed at compile time
static constexpr auto eigs = consteig::eigenvalues(A);
// eigs(0,0).real ~ 1.0   (steady state)
// eigs(1,0).real ~ 0.75  (transient mode)

// Verify at compile time
static_assert(consteig::checkEigenValues(A, eigs, 1e-9), "Bad eigenvalues");

Output

Population Transition Matrix (A)
0.95 0.2
0.05 0.8

Initial Population Vector (u0)
500000
500000

Eigenvalues (lambda):
1
0.75

To find the steady-state population distribution, we need to find the eigenvector 'v'
that corresponds to the eigenvalue of 1 by solving (A - lambda*I)v = 0.

Steady-State Population Flow (Total: 1000000):
Seattle:  800000 (80.00%)
Portland: 200000 (20.00%)

Steady-State Population Flow (Total: 2500000):
Seattle:  2000000 (80.00%)
Portland: 500000 (20.00%)

The same compile-time eigenvectors drive both outputs; no matrix iteration happens at runtime.