def update_polynomial_coefficients(list_x):
matrix_A = sm.matrix(
[[x**k for k in range(len(list_x)-1)] + [ 0 if j==0 else (-1)**j ] \
for j,x in enumerate(list_x)])
vector_b = sm.matrix([sm.sin(x) for x in list_x])
u = sm.lu_solve(matrix_A, vector_b)
# a[0],...,a[n], d
return u[:-1], u[u.rows - 1]
def update_polynomial_coefficients(list_x):
matrix_A = sm.matrix(
[[x**k for k in range(len(list_x)-1)] + [ 0 if j==0 else (-1)**j ] \
for j,x in enumerate(list_x)])
vector_b = sm.matrix([sm.sin(x) for x in list_x])
u = sm.lu_solve(matrix_A, vector_b)
# a[0],...,a[n], d
return u[:-1], u[u.rows-1]
def test_msvd():
output("""\
msvd_:{[b;x]
$[3<>count usv:.qml.msvd x;::;
not (.qml.mdim[u:usv 0]~2#d 0) and (.qml.mdim[v:usv 2]~2#d 1) and
.qml.mdim[s:usv 1]~d:.qml.mdim x;::;
not mortho[u] and mortho[v] and
all[0<=f:.qml.mdiag s] and mzero s _'til d 0;::;
not mzero x-.qml.mm[u] .qml.mm[s] flip v;::;
b;1b;(p*/:m#/:u;m#m#/:s;
(p:(f>.qml.eps*f[0]*max d)*1-2*0>m#u 0)*/:(m:min d)#/:v)]};""")
for A in subjects:
U, S, V = map(mp.matrix, mp.svd(mp.matrix(A)))
m = min(A.rows, A.cols)
p = mp.diag([mp.sign(s * u) for s, u in zip(mp.chop(S), U[0, :])])
U *= p
S = mp.diag(S)
V = V.T * p
test("msvd_[0b", A, (U, S, V))
reps(250)
for Aq in large_subjects:
test("msvd_[1b", qstr(Aq), qstr("1b"))
reps(10000)
def test_nstr():
m = matrix([[0.75, 0.190940654, -0.0299195971],
[0.190940654, 0.65625, 0.205663228],
[-0.0299195971, 0.205663228, 0.64453125e-20]])
assert nstr(m, 4, min_fixed=-inf) == \
'''[ 0.75 0.1909 -0.02992]
[ 0.1909 0.6563 0.2057]
[-0.02992 0.2057 0.000000000000000000006445]'''
assert nstr(m, 4) == \
'''[ 0.75 0.1909 -0.02992]
def test_nstr():
m = matrix([[0.75, 0.190940654, -0.0299195971],
[0.190940654, 0.65625, 0.205663228],
[-0.0299195971, 0.205663228, 0.64453125e-20]])
assert nstr(m, 4, min_fixed=-inf) == \
'''[ 0.75 0.1909 -0.02992]
[ 0.1909 0.6563 0.2057]
[-0.02992 0.2057 0.000000000000000000006445]'''
assert nstr(m, 4) == \
'''[ 0.75 0.1909 -0.02992]
def mc_compute_stationary(P, precision=None, tol=None):
n = P.shape[0]
if precision is None:
# Compute eigenvalues and eigenvectors
eigvals, eigvecs = la.eig(P, left=True, right=False)
# Find the index for unit eigenvalues
index = np.where(abs(eigvals - 1.) < 1e-12)[0]
# Pull out the eigenvectors that correspond to unit eig-vals
uniteigvecs = eigvecs[:, index]
stationary_dists = uniteigvecs / np.sum(uniteigvecs, axis=0)
else:
# Create a list to store eigvals
stationary_dists_list = []
if tol is None:
# If tolerance isn't specified then use 2*precision
tol = mp.mpf(2 * 10**(-precision + 1))
with mp.workdps(precision):
eigvals, eigvecs = mp.eig(mp.matrix(P), left=True, right=False)
for ind, el in enumerate(eigvals):
if el >= (mp.mpf(1) - mp.mpf(tol)) and el <= (mp.mpf(1) +
mp.mpf(tol)):
stationary_dists_list.append(eigvecs[ind, :])
stationary_dists = np.asarray(stationary_dists_list).T
stationary_dists = (stationary_dists /
sum(stationary_dists)).astype(np.float)
# Check to make sure all of the elements of invar_dist are positive
if np.any(stationary_dists < -1e-16):
warn("Elements of your invariant distribution were negative; " +
"Re-trying with additional precision")
if precision is None:
stationary_dists = mc_compute_stationary(P, precision=18, tol=tol)
elif precision is not None:
raise ValueError("Elements of your stationary distribution were" +
"negative. Try computing with higher precision")
# Since we will be accessing the columns of this matrix, we
# might consider adding .astype(np.float, order='F') to make it
# column major at beginning
return stationary_dists.squeeze()
def mc_compute_stationary(P, precision=None, tol=None):
n = P.shape[0]
if precision is None:
# Compute eigenvalues and eigenvectors
eigvals, eigvecs = la.eig(P, left=True, right=False)
# Find the index for unit eigenvalues
index = np.where(abs(eigvals - 1.) < 1e-12)[0]
# Pull out the eigenvectors that correspond to unit eig-vals
uniteigvecs = eigvecs[:, index]
stationary_dists = uniteigvecs/np.sum(uniteigvecs, axis=0)
else:
# Create a list to store eigvals
stationary_dists_list = []
if tol is None:
# If tolerance isn't specified then use 2*precision
tol = mp.mpf(2 * 10**(-precision + 1))
with mp.workdps(precision):
eigvals, eigvecs = mp.eig(mp.matrix(P), left=True, right=False)
for ind, el in enumerate(eigvals):
if el>=(mp.mpf(1)-mp.mpf(tol)) and el<=(mp.mpf(1)+mp.mpf(tol)):
stationary_dists_list.append(eigvecs[ind, :])
stationary_dists = np.asarray(stationary_dists_list).T
stationary_dists = (stationary_dists/sum(stationary_dists)).astype(np.float)
# Check to make sure all of the elements of invar_dist are positive
if np.any(stationary_dists < -1e-16):
warn("Elements of your invariant distribution were negative; " +
"Re-trying with additional precision")
if precision is None:
stationary_dists = mc_compute_stationary(P, precision=18, tol=tol)
elif precision is not None:
raise ValueError("Elements of your stationary distribution were" +
"negative. Try computing with higher precision")
# Since we will be accessing the columns of this matrix, we
# might consider adding .astype(np.float, order='F') to make it
# column major at beginning
return stationary_dists.squeeze()
def test_mev():
output("""\
reim:{$[0>type x;1 0*x;2=count x;x;'`]};
mc:{((x[0]*y 0)-x[1]*y 1;(x[0]*y 1)+x[1]*y 0)};
mmc:{((.qml.mm[x 0]y 0)-.qml.mm[x 1]y 1;(.qml.mm[x 0]y 1)+.qml.mm[x 1]y 0)};
mev_:{[b;x]
if[2<>count wv:.qml.mev x;'`length];
if[not all over prec>=abs
mmc[flip vc;flip(flip')(reim'')flip x]-
flip(w:reim'[wv 0])mc'vc:(flip')(reim'')(v:wv 1);'`check];
/ Normalize sign; LAPACK already normalized to real
v*:1-2*0>{x a?max a:abs x}each vc[;0];
(?'[prec>=abs w[;1];w[;0];w];?'[b;v;0n])};""")
for A in eigenvalue_subjects:
if A.rows <= 3:
V = []
for w, n, r in A.eigenvects():
w = sp.simplify(sp.expand_complex(w))
if len(r) == 1:
r = r[0]
r = sp.simplify(sp.expand_complex(r))
r = r.normalized() / sp.sign(max(r, key=abs))
r = sp.simplify(sp.expand_complex(r))
else:
r = None
V.extend([(w, r)]*n)
V.sort(key=lambda (x, _): (-abs(x), -sp.im(x)))
else:
Am = mp.matrix(A)
# extra precision for complex pairs to be equal in sort
with mp.extradps(mp.mp.dps):
W, R = mp.eig(Am)
V = []
for w, r in zip(W, (R.column(i) for i in range(R.cols))):
w = mp.chop(w)
with mp.extradps(mp.mp.dps):
_, S, _ = mp.svd(Am - w*mp.eye(A.rows))
if sum(x == 0 for x in mp.chop(S)) == 1:
# nullity 1, so normalized eigenvector is unique
r /= mp.norm(r) * mp.sign(max(r, key=abs))
r = mp.chop(r)
else:
r = None
V.append((w, r))
V.sort(key=lambda (x, _): (-abs(x), -x.imag))
W, R = zip(*V)
test("mev_[%sb" % "".join("0" if r is None else "1" for r in R), A,
(W, [r if r is None else list(r) for r in R]), complex_pair=True)
def diag(diagonal):
mat = mp.matrix(np.eye(len(diagonal)))
for i, value in enumerate(diagonal):
mat[i, i] = value
return mat
from scipy.linalg import toeplitz
import sympy.mpmath as mp
import numpy as np
n = 25
A = toeplitz(np.r_[2, 1, np.zeros(n)])
# Numpy ---
lmbda, U = np.linalg.eig(A)
Lmbda = np.diag(lmbda)
print np.linalg.norm(A - U.dot(Lmbda.dot(U.T)))
# Mpmath
def diag(diagonal):
mat = mp.matrix(np.eye(len(diagonal)))
for i, value in enumerate(diagonal):
mat[i, i] = value
return mat
A = mp.matrix(A.tolist())
E, Q = mp.eigsy(A)
E = diag(E)
# Sort for comparison
Lmbda = np.diag(np.sort(np.diagonal(Lmbda)))
print np.linalg.norm(Lmbda - np.array(E.tolist(), dtype=float))
