Package jline.solvers.fluid.moments

Class FluidLyapunov

java.lang.Object

jline.solvers.fluid.moments.FluidLyapunov

public final class FluidLyapunov
extends Object

Stationary covariance of the linear noise approximation. Java twin of the MATLAB fluid_lyapunov.

Around a fixed point x* of the fluid drift, the fluctuation process Z = (X - x*) obeys the linear stochastic differential equation dZ = A*Z*dt + sqrt(Qdiff)*dW, whose stationary covariance solves the Lyapunov equation

   A*Sigma + Sigma*A' + Qdiff = 0,   Qdiff = D*diag(r(x*))*D'
 

A is singular whenever the model conserves population: every closed class contributes a left null vector, so the equation has no unique solution on the full state space. It does have one on the reachable subspace, which is exactly range(D): the state can only move along jump directions, so the fluctuation lives there and nowhere else. Both A = D*diag(rateBase)*G and Qdiff map into range(D) as well, so restricting to an orthonormal basis V of range(D) is an exact reduction, not an approximation, and the reduced Lyapunov equation is nonsingular whenever the fixed point is stable.

See Also:

• FluidMomentTerms

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static final class	FluidLyapunov.Result	The stationary covariance together with the diagnostics of the reduced solve.

Method Summary

Modifier and Type	Method	Description
static Matrix	orth(Matrix A)	Orthonormal basis of the column space, mirroring MATLAB orth: the left singular vectors whose singular value exceeds max(size(A))*eps*max(s).
static FluidLyapunov.Result	solve(Matrix A, Matrix Qdiff, Matrix D, double tol)	Solves the Lyapunov equation on the reachable subspace.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

solve

public static FluidLyapunov.Result solve(Matrix A,
 Matrix Qdiff,
 Matrix D,
 double tol)

Solves the Lyapunov equation on the reachable subspace.

Parameters:

A - drift Jacobian at the fixed point (n x n)

Qdiff - diffusion matrix D*diag(r)*D' (n x n)

D - jump matrix (n x nevents), spanning the reachable subspace

tol - stability margin; eigenvalues of the reduced A with real part above -tol are reported as non-hyperbolic

Returns:

the stationary covariance, supported on range(D)

orth

public static Matrix orth(Matrix A)

Orthonormal basis of the column space, mirroring MATLAB orth: the left singular vectors whose singular value exceeds max(size(A))*eps*max(s).

Parameters:

A - any matrix

Returns:

an (rows x rank) matrix with orthonormal columns