File size: 4,553 Bytes
347a3de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | """Weighted quotient and positive-graph inverse covariance design.
These routines check the declared linear algebra; actuator realizability,
force/power conversion, and first-passage optimality are separate questions.
"""
import numpy as np
from scipy.linalg import null_space, solve_continuous_lyapunov
def shape_basis(weights):
w=np.asarray(weights,dtype=float)
if w.ndim!=1 or len(w)<2 or not np.isfinite(w).all() or np.any(w<=0):
raise ValueError('finite positive weights for at least two strands required')
q=np.sqrt(w/w.sum())
return null_space(q[None,:])
def graph_laplacian(n,edges,gains):
if not isinstance(n,(int,np.integer)) or n<2:
raise ValueError('integer n>=2 required')
g=np.asarray(gains,dtype=float)
if len(edges)!=len(g) or not np.isfinite(g).all() or np.any(g<0):
raise ValueError('finite nonnegative gain for each edge required')
L=np.zeros((n,n))
for (i,j),gain in zip(edges,g):
if i==j or min(i,j)<0 or max(i,j)>=n:
raise ValueError('edge indices must be distinct and in range')
L[i,i]+=gain;L[j,j]+=gain;L[i,j]-=gain;L[j,i]-=gain
return L
def symmetric_inverse_drift(Sigma,Q):
S=np.asarray(Sigma,dtype=float);Q=np.asarray(Q,dtype=float)
if S.ndim!=2 or S.shape[0]!=S.shape[1] or Q.shape!=S.shape:
raise ValueError('matching square matrices required')
if not np.isfinite(S).all() or not np.isfinite(Q).all():
raise ValueError('finite matrices required')
if not np.allclose(S,S.T,atol=1e-12) or not np.allclose(Q,Q.T,atol=1e-12):
raise ValueError('symmetric matrices required')
vals,O=np.linalg.eigh(S)
if vals.min()<=0 or np.linalg.eigvalsh(Q).min()<=0:
raise ValueError('positive definite Sigma and Q required')
transformed=O.T@Q@O
return O@(transformed/(vals[:,None]+vals[None,:]))@O.T
def positive_graph_target(Sigma,Q,weights,kappa,allowed_edges=None,tol=1e-10):
"""Target test with reported numerical sign/residual tolerance.
This is floating feasibility information, not an exact sign oracle.
"""
if not np.isfinite(kappa) or kappa<=0 or not np.isfinite(tol) or tol<0:
raise ValueError('positive kappa and nonnegative finite tol required')
w=np.asarray(weights,dtype=float);U=shape_basis(w)
A=symmetric_inverse_drift(Sigma,Q)
if A.shape!=(len(w)-1,len(w)-1):
raise ValueError('target shape dimension mismatch')
R=np.sqrt(w)[:,None]*U
L=R@(A-kappa*np.eye(len(w)-1))@R.T
n=len(w)
allowed={tuple(sorted(x)) for x in allowed_edges} if allowed_edges is not None else None
feasible=True;gains={};violations=[]
for i in range(n):
for j in range(i+1,n):
if L[i,j]>tol:
feasible=False;violations.append({'edge':[i,j],'positive_off_diagonal':float(L[i,j])})
if allowed is not None and (i,j) not in allowed and abs(L[i,j])>tol:
feasible=False;violations.append({'edge':[i,j],'forbidden_nonzero':float(L[i,j])})
gains[f'{i},{j}']=-float(L[i,j])
residual=np.linalg.norm(A@Sigma+Sigma@A.T-Q,ord=2)
return {'feasible_at_tolerance':feasible,'lift':L,'gains':gains,
'violations':violations,'lyapunov_residual':float(residual),'tolerance':tol}
def stationary_covariance(A,Q):
A=np.asarray(A,dtype=float);Q=np.asarray(Q,dtype=float)
if A.ndim!=2 or A.shape[0]!=A.shape[1] or Q.shape!=A.shape:
raise ValueError('matching square matrices required')
if not np.isfinite(A).all() or not np.isfinite(Q).all():
raise ValueError('finite matrices required')
if np.min(np.linalg.eigvals(A).real)<=0:
raise ValueError('positive-real-part drift required')
if not np.allclose(Q,Q.T,atol=1e-12) or np.min(np.linalg.eigvalsh(Q))<-1e-12:
raise ValueError('symmetric positive semidefinite noise required')
return solve_continuous_lyapunov(A,Q)
def weighted_action_split(velocities,weights):
v=np.asarray(velocities,dtype=float);w=np.asarray(weights,dtype=float)
shape_basis(w) # validation only
if v.ndim!=2 or v.shape[0]!=len(w) or not np.isfinite(v).all():
raise ValueError('finite [strand,coordinate] velocities required')
centroid=np.sum(w[:,None]*v,axis=0)/w.sum()
lab=np.sum(w[:,None]*v*v)
shape=np.sum(w[:,None]*(v-centroid)**2)
translation=w.sum()*float(centroid@centroid)
return {'lab_density':float(lab),'shape_density':float(shape),
'translation_density':translation,'weighted_centroid_velocity':centroid}
|