def test_legvander(self) :
# check for 1d x
x = np.arange(3)
v = leg.legvander(x, 3)
assert_(v.shape == (3, 4))
for i in range(4) :
coef = [0]*i + [1]
assert_almost_equal(v[..., i], leg.legval(x, coef))
# check for 2d x
x = np.array([[1, 2], [3, 4], [5, 6]])
v = leg.legvander(x, 3)
assert_(v.shape == (3, 2, 4))
for i in range(4) :
coef = [0]*i + [1]
assert_almost_equal(v[..., i], leg.legval(x, coef))
def test_legendre_val():
"""Test Legendre polynomial (derivative) equivalence
"""
rng = np.random.RandomState(0)
# check table equiv
xs = np.linspace(-1., 1., 1000)
n_terms = 100
# True, numpy
vals_np = legendre.legvander(xs, n_terms - 1)
# Table approximation
for fun, nc in zip([_get_legen_lut_fast, _get_legen_lut_accurate],
[100, 50]):
lut, n_fact = _get_legen_table('eeg', n_coeff=nc, force_calc=True)
vals_i = fun(xs, lut)
# Need a "1:" here because we omit the first coefficient in our table!
assert_allclose(vals_np[:, 1:vals_i.shape[1] + 1], vals_i,
rtol=1e-2, atol=5e-3)
# Now let's look at our sums
ctheta = rng.rand(20, 30) * 2.0 - 1.0
beta = rng.rand(20, 30) * 0.8
lut_fun = partial(fun, lut=lut)
c1 = _comp_sum_eeg(beta.flatten(), ctheta.flatten(), lut_fun, n_fact)
c1.shape = beta.shape
# compare to numpy
n = np.arange(1, n_terms, dtype=float)[:, np.newaxis, np.newaxis]
coeffs = np.zeros((n_terms,) + beta.shape)
coeffs[1:] = (np.cumprod([beta] * (n_terms - 1), axis=0) *
(2.0 * n + 1.0) * (2.0 * n + 1.0) / n)
# can't use tensor=False here b/c it isn't in old numpy
c2 = np.empty((20, 30))
for ci1 in range(20):
for ci2 in range(30):
c2[ci1, ci2] = legendre.legval(ctheta[ci1, ci2],
coeffs[:, ci1, ci2])
assert_allclose(c1, c2, 1e-2, 1e-3) # close enough...
# compare fast and slow for MEG
ctheta = rng.rand(20 * 30) * 2.0 - 1.0
beta = rng.rand(20 * 30) * 0.8
lut, n_fact = _get_legen_table('meg', n_coeff=10, force_calc=True)
fun = partial(_get_legen_lut_fast, lut=lut)
coeffs = _comp_sums_meg(beta, ctheta, fun, n_fact, False)
lut, n_fact = _get_legen_table('meg', n_coeff=20, force_calc=True)
fun = partial(_get_legen_lut_accurate, lut=lut)
coeffs = _comp_sums_meg(beta, ctheta, fun, n_fact, False)
def test_100(self):
x, w = leg.leggauss(100)
# test orthogonality. Note that the results need to be normalized,
# otherwise the huge values that can arise from fast growing
# functions like Laguerre can be very confusing.
v = leg.legvander(x, 99)
vv = np.dot(v.T * w, v)
vd = 1/np.sqrt(vv.diagonal())
vv = vd[:,None] * vv * vd
assert_almost_equal(vv, np.eye(100))
# check that the integral of 1 is correct
tgt = 2.0
assert_almost_equal(w.sum(), tgt)
def approx(X0, Y0, X1, approx_type: ApproxType, dim):
"""
this method should perform approximation on [-1; 1] interval
:param X0: X-values (1 x N0)
:param Y0: Y-values (1 x N0)
:param X1: approximation points (1 x N1)
:param approx_type:
0 - algebraic polynomes (1, x, x^2, ...)
1 - legendre polynomes
2 - harmonic
:param dim: dimension
:return Y1: approximated Y-values (1 x N1)
:return a: vector (1 x dim) of approximation coefficients
:return P: (for approx_type 0 and 1) coefficients of approximation polynome P (1 x dim)
"""
if approx_type is ApproxType.algebraic:
Q = np.vander(X0, dim, increasing=True)
mat = Q.T @ Q
b = Q.T @ Y0
P = np.linalg.solve(mat, b)
y = polyval(X1, P)
return y, [], P
if approx_type is ApproxType.legendre:
Q = legvander(X0, dim - 1)
mat = Q.T @ Q
b = Q.T @ Y0
P = np.linalg.solve(mat, b)
y = legval(X1, P)
D = [[0 for i in range(dim)] for j in range(dim)]
D[0][0] = 1
D[0][1] = 0
D[1][0] = 0
D[1][1] = 1
i = 2
while i < (dim):
D[i][0] = -(i - 1) / (i) * D[i - 2][0]
j = 0
while j <= i:
D[i][j] = (2 * i -
1) / i * D[i - 1][j - 1] - (i - 1) / i * D[i - 2][j]
j += 1
i += 1
D = np.array(D)
G = P @ D
return y, P, G
raise Exception(f'approximation of type {approx_type} not supported yet')
def test_legendre_val():
"""Test Legendre polynomial (derivative) equivalence
"""
# check table equiv
xs = np.linspace(-1., 1., 1000)
n_terms = 100
# True, numpy
vals_np = legendre.legvander(xs, n_terms - 1)
# Table approximation
for fun, nc in zip([_get_legen_lut_fast, _get_legen_lut_accurate],
[100, 50]):
lut, n_fact = _get_legen_table('eeg', n_coeff=nc, force_calc=True)
vals_i = fun(xs, lut)
# Need a "1:" here because we omit the first coefficient in our table!
assert_allclose(vals_np[:, 1:vals_i.shape[1] + 1], vals_i,
rtol=1e-2, atol=5e-3)
# Now let's look at our sums
ctheta = np.random.rand(20, 30) * 2.0 - 1.0
beta = np.random.rand(20, 30) * 0.8
lut_fun = partial(fun, lut=lut)
c1 = _comp_sum_eeg(beta.flatten(), ctheta.flatten(), lut_fun, n_fact)
c1.shape = beta.shape
# compare to numpy
n = np.arange(1, n_terms, dtype=float)[:, np.newaxis, np.newaxis]
coeffs = np.zeros((n_terms,) + beta.shape)
coeffs[1:] = (np.cumprod([beta] * (n_terms - 1), axis=0) *
(2.0 * n + 1.0) * (2.0 * n + 1.0) / n)
# can't use tensor=False here b/c it isn't in old numpy
c2 = np.empty((20, 30))
for ci1 in range(20):
for ci2 in range(30):
c2[ci1, ci2] = legendre.legval(ctheta[ci1, ci2],
coeffs[:, ci1, ci2])
assert_allclose(c1, c2, 1e-2, 1e-3) # close enough...
# compare fast and slow for MEG
ctheta = np.random.rand(20 * 30) * 2.0 - 1.0
beta = np.random.rand(20 * 30) * 0.8
lut, n_fact = _get_legen_table('meg', n_coeff=10, force_calc=True)
fun = partial(_get_legen_lut_fast, lut=lut)
coeffs = _comp_sums_meg(beta, ctheta, fun, n_fact, False)
lut, n_fact = _get_legen_table('meg', n_coeff=20, force_calc=True)
fun = partial(_get_legen_lut_accurate, lut=lut)
coeffs = _comp_sums_meg(beta, ctheta, fun, n_fact, False)
def leginterpol(x_coords, y_coords):
deg = len(x_coords)
A = np.ndarray((deg, deg))
A_ = np.ndarray((deg, deg))
B = y_coords
# print(A)
for i in range(deg):
A[i] = leg.legval(x_coords, np.ones((i + 1, 1)))
A = A.T
A_ = leg.legvander(x_coords, deg - 1)
# print('A=', A)
# print('A_=', A_)
# A*coefs = B
coefs = np.linalg.solve(A_, y_coords)
return coefs
def fitlegendre(wavepts, fluxpts, sigpts, order):
from numpy.linalg import svd
vander = L.legvander(wavepts, order)
design = np.zeros(vander.shape)
for i in range(len(wavepts)):
design[i] = vander[i] / sigpts[i]
U, s, v = svd(design, full_matrices=False, compute_uv=True)
V = v.transpose()
solvec = np.zeros(order + 1)
for i in range(order + 1):
solvec += np.dot(U[:, i], fluxpts / sigpts) / s[i] * V[:, i]
### Build covariance matrix
covmtx = np.zeros([order + 1, order + 1])
for j in range(order + 1):
for k in range(order + 1):
covmtx[j, k] = sum(V[:, j] * V[:, k] / s)
return solvec, covmtx
def poles(self):
orders = 4
xi = self.xi
r = self.r
mu = self.mu
Nr, Nmu = self.xi.shape
v = legvander(mu, orders)
poles = numpy.empty((len(r), orders+1))
sym = (mu >= 0).all()
for i in range(orders + 1):
norm = simps(v[:, i] ** 2, mu)
if sym and i % 2 == 1:
poles[:, i] = 0
else:
poles[:, i] = simps(xi * v[:, i][None, :], mu) / norm
return poles
def pose_postprocess(self, pose_3d):
#
focus_time = None
time_length = len(pose_3d)
t_axis = np.linspace(0, time_length, time_length + 1)
#
nsamples = self.end - self.start
# The independent variable sampled at regular intervals
xs = np.linspace(-1, 1, nsamples)
# The Legendre Vandermonde matrix
V = leg.legvander(xs, self.approx_deg)
# Generate some data to fit
ys = xs * np.pi
# Do the fit for all 17 poses
resampledpose = {}
for ii in range(self.num_joints):
respsoexyz = []
for ij in range(3):
jnt_coord = [pose_3d[i][ij, ii] for i in range(time_length)]
#
coeffs = np.linalg.lstsq(V,
jnt_coord[self.start:self.end],
rcond=None)[0]
# Evaluate the fit for plotting purposes
g = leg.legval(xs, coeffs)
focus_val, _time = signal.resample(g,
len(g) * self.resamp,
t_axis[self.start:self.end])
focus_val = list(focus_val)
if focus_time == None:
focus_time = list(_time)
for ti in range(self.start):
focus_time.insert(0, ti)
# adding the inactive times
for ti in range(self.start):
focus_val.insert(0, focus_val[0])
respsoexyz.append(focus_val)
#
resampledpose.update({'joint_' + str(ii): np.array(respsoexyz)})
return resampledpose, focus_time
def detrend_poly(self, npol=20, istart=None, iend=None):
flux = self.flux[istart:iend]
covs = self.covariates[istart:iend, [2, 4, 5, 6]].copy()
covs -= covs.mean(0)
covs /= covs.std(0)
x = (self.time - self.time[0]) / diff(self.time[[0, -1]]) * 2 - 1
pol = legvander(x, npol)
pol[:, 1:] /= pol[:, 1:].ptp(0)
covs = concatenate([covs, pol], 1)
c, _, _, _ = lstsq(covs, flux, rcond=None)
cbl = c.copy()
cbl[:4] = 0.
css = c.copy()
css[4:] = 0.
fbl = dot(covs, cbl)
fsys = dot(covs, css)
fall = dot(covs, c)
return self.time[istart:iend], flux - fall + 1, fsys, fbl
def culledLegForwardTransform(orders, locations, functionVals, threshold=None):
# inspired by : A Simple Regularization of the Polynomial Interpolation For the Runge Phenomemenon
if len(locations.shape)==1:
vandermonde=leg.legvander(locations, orders[0])
elif locations.shape[1]==2:
vandermonde=leg.legvander2d(locations[:,0], locations[:,1], orders)
elif locations.shape[1]==3:
vandermonde=leg.legvander3d(locations[:,0],locations[:,1],locations[:,2], orders)
elif locations.shape[1]==4:
vandermonde=legvander4d(locations,orders)
elif locations.shape[1]==5:
vandermonde=legvander5d(locations,orders)
else:
raise NotImplementedError # there's a bad startup joke about this being good enough for the paper.
# preconditioner = np.diag((0.94) ** (2* (np.arange(vandermonde.shape[0]))))
# vandermonde=np.dot(preconditioner, vandermonde)
U,S,Vh=np.linalg.svd(vandermonde)
numTake=0
filtS=S
if threshold is None:
Eps= np.finfo(functionVals.dtype).eps
Neps = np.prod(cmn.numpyze(orders)) * Eps * S[0] #truncation due to ill-conditioning
Nt = max(np.argmax(Vh, axis=0)) #"Automatic" determination of threshold due to Runge's phenomenon
threshold=min(Neps, Nt)
while numTake<=0:
filter=S>threshold
numTake=filter.sum()
if numTake>0:
filtU=U[:,:numTake]; filtS=S[:numTake]; filtVh=Vh[:numTake, :]
else:
if threshold>1e-13:
threshold=threshold/2
warnings.warn('cutting threshold for eigenvalues to '+str(threshold))
else:
warnings('seems all eigenvalues are zero (<1e-13), setting to zero and breaking')
filtS=np.zeros_like(S)
truncVander=np.dot(filtU,np.dot(np.diag(filtS),filtVh))
ret, _, _, _=npl.lstsq(truncVander, functionVals, rcond=None)
return np.reshape(ret, np.array(orders).flatten()+1)
def __init__(self,n,k,a,m):
self.k2 = 2*k
# Uniform grid points
x = np.linspace(-1,1,n)
# Legendre Polynomials on grid
self.V = legvander(x,m-1)
# Do QR factorization of Vandermonde for least squares
self.Q,self.R = qr(self.V,mode='economic')
I = np.eye(m)
D = np.zeros((m,m))
D[:-self.k2,:] = legder(I,self.k2)
# Legendre modal approximation of differential operator
self.A = I-a*D
# Store LU factors for repeated solves
self.PLU = lu_factor(self.A[:-self.k2,self.k2:])
def __init__(self, n, k, a, m):
self.k2 = 2 * k
# Uniform grid points
x = np.linspace(-1, 1, n)
# Legendre Polynomials on grid
self.V = legvander(x, m - 1)
# Do QR factorization of Vandermonde for least squares
self.Q, self.R = qr(self.V, mode='economic')
I = np.eye(m)
D = np.zeros((m, m))
D[:-self.k2, :] = legder(I, self.k2)
# Legendre modal approximation of differential operator
self.A = I - a * D
# Store LU factors for repeated solves
self.PLU = lu_factor(self.A[:-self.k2, self.k2:])
def __init__(self,
name: str,
dfile: Path,
zero_epoch: float,
period: float,
nsamples: int = 10,
trdur: float = 0.125,
bldur: float = 0.3,
nlegendre: int = 2,
ctx=None,
queue=None):
tb = Table.read(dfile)
self.bjdrefi = tb.meta['BJDREFI']
zero_epoch = zero_epoch - self.bjdrefi
df = tb.to_pandas().dropna(subset=['TIME', 'SAP_FLUX', 'PDCSAP_FLUX'])
self.lc = lc = KeplerLC(df.TIME.values, df.PDCSAP_FLUX.values,
zeros(df.shape[0]), zero_epoch, period, trdur,
bldur)
self.nlegendre = nlegendre
super().__init__(name,
zero_epoch,
period, ['TESS'],
times=lc.time_per_transit,
fluxes=lc.normalized_flux_per_transit,
pbids=lc.nt * [0],
nsamples=nsamples,
exptimes=[0.00139],
cl_ctx=ctx,
cl_queue=queue)
self.mtimes = [t - t.mean() for t in self.times]
self.windows = window = concatenate(self.mtimes).ptp()
self.mtimes = [t / window for t in self.mtimes]
self.legs = [legvander(t, self.nlegendre) for t in self.mtimes]
self.ofluxa = self.ofluxa.astype('d')
self.lnlikelihood = self.lnlikelihood_nb
def getVandermondeLegendre(self):
from numpy.polynomial.legendre import legvander
return legvander(self.Nodes, self.Order)
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
import numpy.polynomial.legendre as leg
n = 16
def f(x, y):
return np.sin(2 * np.pi * x) * np.sin(2 * np.pi * y)
def curve(a, b, c, x):
return a + b * x + c * x**2
Qx, Qw = leg.leggauss(n)
nx, ny = np.meshgrid(Qx, Qx)
Le = leg.legvander(Qx, n - 1)
#La = sp.interpolate.lagrange(n)
Q, R = np.linalg.qr(Le)
def vandermonde(self, x):
return leg.legvander(x, 1)
def _get_legen(x, n_coeff=100):
"""Get Legendre polynomials expanded about x"""
return legendre.legvander(x, n_coeff - 1)
def legvander(x, deg) :
from numpy.polynomial.legendre import legvander
return legvander(x, deg)
def _get_legen(x, n_coeff=100):
"""Get Legendre polynomials expanded about x"""
return legendre.legvander(x, n_coeff - 1)
def ortholegvander(x, deg):
"""See numpy.polynomial.legendre.legvander(). Uses an orthonormal basis."""
result = leg.legvander(x, deg)
factors = np.array([np.sqrt((2 * n + 1) / 2) for n in range(0, 1 + deg)])
return result * factors
def __init__(self,
target: str,
photometry: list,
tid: int,
cids: list,
filters: tuple,
aperture_lims: tuple = (0, inf),
use_opencl: bool = False,
n_legendre: int = 0,
use_toi_info=True,
with_transit=True,
with_contamination=False,
radius_ratio: str = 'achromatic'):
assert radius_ratio in ('chromatic', 'achromatic')
self.use_opencl = use_opencl
self.planet = None
self.photometry_frozen = False
self.with_transit = with_transit
self.with_contamination = with_contamination
self.chromatic_transit = radius_ratio == 'chromatic'
self.radius_ratio = radius_ratio
self.n_legendre = n_legendre
self.toi = None
if self.with_transit and 'toi' in target.lower() and use_toi_info:
self.toi = get_toi(float(target.lower().strip('toi')))
# Set photometry
# --------------
self.phs = photometry
self.nph = len(photometry)
# Set the aperture ranges
# -----------------------
self.min_apt = amin = min(max(aperture_lims[0], 0),
photometry[0].flux.aperture.size)
self.max_apt = amax = max(
min(aperture_lims[1], photometry[0].flux.aperture.size), 0)
self.napt = amax - amin
# Target and comparison star IDs
# ------------------------------
self.tid = atleast_1d(tid)
if self.tid.size == 1:
self.tid = tile(self.tid, self.nph)
self.cids = atleast_2d(cids)
if self.cids.shape[0] == 1:
self.cids = tile(self.cids, (self.nph, 1))
assert self.tid.size == self.nph
assert self.cids.shape[0] == self.nph
self.covnames = 'intercept sky airmass xshift yshift entropy'.split()
times = [array(ph.bjd) for ph in photometry]
fluxes = [
array(ph.flux[:, tid, 1]) for tid, ph in zip(self.tid, photometry)
]
fluxes = [f / nanmedian(f) for f in fluxes]
self.apertures = ones(len(times)).astype('int')
self.t0 = floor(times[0].min())
times = [t - self.t0 for t in times]
self._tmin = times[0].min()
self._tmax = times[0].max()
covariates = []
for ph in photometry:
covs = concatenate(
[ones([ph._fmask.sum(), 1]),
array(ph.aux)[:, [1, 3, 4, 5]]], 1)
covariates.append(covs)
self.airmasses = [array(ph.aux[:, 2]) for ph in photometry]
wns = [ones(ph.nframes) for ph in photometry]
if use_opencl:
import pyopencl as cl
ctx = cl.create_some_context()
queue = cl.CommandQueue(ctx)
tm = QuadraticModelCL(klims=(0.005, 0.25),
nk=512,
nz=512,
cl_ctx=ctx,
cl_queue=queue)
else:
tm = QuadraticModel(interpolate=True,
klims=(0.005, 0.25),
nk=512,
nz=512)
super().__init__(target,
filters,
times,
fluxes,
wns,
arange(len(photometry)),
covariates,
arange(len(photometry)),
tm=tm)
self.legendre = [
legvander((t - t.min()) / (0.5 * t.ptp()) - 1,
self.n_legendre)[:, 1:] for t in self.times
]
# Create the target and reference star flux arrays
# ------------------------------------------------
self.ofluxes = [
array(ph.flux[:, self.tid[i], amin:amax + 1] /
ph.flux[:, self.tid[i], amin:amax + 1].median('mjd'))
for i, ph in enumerate(photometry)
]
self.refs = []
for ip, ph in enumerate(photometry):
self.refs.append([
pad(array(ph.flux[:, cid, amin:amax + 1]), ((0, 0), (1, 0)),
mode='constant') for cid in self.cids[ip]
])
self.set_orbit_priors()
x = Symbol("x")
u = (1.0 - x**2)**2 * cos(
np.pi * 4 * x) * (x - 0.25)**3 + b * (1 + x) / 2. + a * (1 - x) / 2.
kx = np.sqrt(5)
f = -u.diff(x, 2) + kx**2 * u
fl = lambdify(x, f, "numpy")
ul = lambdify(x, u, "numpy")
n = 32
domain = sys.argv[-1] if len(sys.argv) == 2 else "C1"
# Chebyshev-Gauss nodes and weights
points, w = leg.leggauss(n + 1)
# Chebyshev Vandermonde matrix
V = leg.legvander(points, n)
scl = 1.0
# First derivative matrix zero padded to (n+1)x(n+1)
D1 = np.zeros((n + 1, n + 1))
D1[:-1, :] = leg.legder(np.eye(n + 1), 1, scl=scl)
Vx = np.dot(V, D1)
# Matrix of trial functions
P = np.zeros((n + 1, n + 1))
P[:, 0] = (V[:, 0] - V[:, 1]) / 2
P[:, 1:-1] = V[:, :-2] - V[:, 2:]
P[:, -1] = (V[:, 0] + V[:, 1]) / 2
def generate_Vandermonde_matrixu(x_coords):
""" Returns the pseudo-Vandermonde matrix of degree len(x_coords) and sample points x_coords"""
return legndr.legvander(x_coords, len(x_coords) - 1)
def named_target_matrix(name, size):
"""
Parameter:
name: name of the target matrix
Return:
target_matrix: (n, n) numpy array for real matrices or (n, n, 2) for complex matrices.
"""
if name == 'dft':
return LA.dft(size, scale='sqrtn')[:, :, None].view('float64')
elif name == 'idft':
return np.ascontiguousarray(LA.dft(size, scale='sqrtn').conj().T)[:, :, None].view('float64')
elif name == 'dft2':
size_sr = int(math.sqrt(size))
matrix = np.fft.fft2(np.eye(size_sr**2).reshape(-1, size_sr, size_sr), norm='ortho').reshape(-1, size_sr**2)
# matrix1d = LA.dft(size_sr, scale='sqrtn')
# assert np.allclose(np.kron(m1d, m1d), matrix)
# return matrix[:, :, None].view('float64')
from butterfly.utils import bitreversal_permutation
br_perm = bitreversal_permutation(size_sr)
br_perm2 = np.arange(size_sr**2).reshape(size_sr, size_sr)[br_perm][:, br_perm].reshape(-1)
matrix = np.ascontiguousarray(matrix[:, br_perm2])
return matrix[:, :, None].view('float64')
elif name == 'dct':
# Need to transpose as dct acts on rows of matrix np.eye, not columns
# return dct(np.eye(size), norm='ortho').T
return dct(np.eye(size)).T / math.sqrt(size)
elif name == 'dst':
return dst(np.eye(size)).T / math.sqrt(size)
elif name == 'hadamard':
return LA.hadamard(size) / math.sqrt(size)
elif name == 'hadamard2':
size_sr = int(math.sqrt(size))
matrix1d = LA.hadamard(size_sr) / math.sqrt(size_sr)
return np.kron(matrix1d, matrix1d)
elif name == 'b2':
size_sr = int(math.sqrt(size))
from butterfly import Block2x2DiagProduct
b = Block2x2DiagProduct(size_sr)
matrix1d = b(torch.eye(size_sr)).t().detach().numpy()
return np.kron(matrix1d, matrix1d)
elif name == 'convolution':
np.random.seed(0)
x = np.random.randn(size)
return LA.circulant(x) / math.sqrt(size)
elif name == 'hartley':
return hartley_matrix(size) / math.sqrt(size)
elif name == 'haar':
return haar_matrix(size, normalized=True) / math.sqrt(size)
elif name == 'legendre':
grid = np.linspace(-1, 1, size + 2)[1:-1]
return legendre.legvander(grid, size - 1).T / math.sqrt(size)
elif name == 'hilbert':
H = hilbert_matrix(size)
return H / np.linalg.norm(H, 2)
elif name == 'randn':
np.random.seed(0)
return np.random.randn(size, size) / math.sqrt(size)
elif name == 'permutation':
np.random.seed(0)
perm = np.random.permutation(size)
P = np.eye(size)[perm]
return P
elif name.startswith('rank-unnorm'):
r = int(name[11:])
np.random.seed(0)
G = np.random.randn(size, r)
H = np.random.randn(size, r)
M = G @ H.T
# M /= math.sqrt(size*r)
return M
elif name.startswith('rank'):
r = int(name[4:])
np.random.seed(0)
G = np.random.randn(size, r)
H = np.random.randn(size, r)
M = G @ H.T
M /= math.sqrt(size*r)
return M
elif name.startswith('sparse'):
s = int(name[6:])
# 2rn parameters
np.random.seed(0)
mask = sparse.random(size, size, density=s/size, data_rvs=np.ones)
M = np.random.randn(size, size) * (mask.toarray())
M /= math.sqrt(s)
return M
elif name.startswith('toeplitz'):
r = int(name[8:])
G = np.random.randn(size, r) / math.sqrt(size*r)
H = np.random.randn(size, r) / math.sqrt(size*r)
M = toeplitz_like(G, H)
return M
elif name == 'fastfood':
n = size
S = np.random.randn(n)
G = np.random.randn(n)
B = np.random.randn(n)
# P = np.arange(n)
P = np.random.permutation(n)
H = hadamard(n)
# SHGPHB
# print(H)
# print((H*B)[P,:])
# print((H @ (G[:,np.newaxis] * (H * B)[P,:])))
F = S[:,np.newaxis] * (H @ (G[:,np.newaxis] * (H * B)[P,:])) / n
return F
# x = np.random.randn(batch_size,n)
# HB = hadamard_transform(B)
# PHBx = HBx[:, P]
# HGPHBx = hadamard_transform(G*PHBx)
# return S*HGPHBx
elif name == 'butterfly':
# n (log n+1) params in the hierarchy
b = Butterfly(in_size=size, out_size=size, bias=False, tied_weight=False, param='odo', nblocks=0)
M = b(torch.eye(size))
return M.cpu().detach().numpy()
else:
assert False, 'Target matrix name not recognized or implemented'
# Some exact solution
x = Symbol("x")
u = (1.0-x**2)**2*cos(np.pi*4*x)*(x-0.25)**3 + b*(1 + x)/2. + a*(1 - x)/2.
kx = np.sqrt(5)
f = -u.diff(x, 2) + kx**2*u
fl = lambdify(x, f, "numpy")
ul = lambdify(x, u, "numpy")
n = 32
domain = sys.argv[-1] if len(sys.argv) == 2 else "C1"
# Chebyshev-Gauss nodes and weights
points, w = leg.leggauss(n+1)
# Chebyshev Vandermonde matrix
V = leg.legvander(points, n)
scl=1.0
# First derivative matrix zero padded to (n+1)x(n+1)
D1 = np.zeros((n+1,n+1))
D1[:-1,:] = leg.legder(np.eye(n+1), 1, scl=scl)
Vx = np.dot(V, D1)
# Matrix of trial functions
P = np.zeros((n+1,n+1))
P[:,0] = (V[:,0] - V[:,1])/2
P[:,1:-1] = V[:,:-2] - V[:,2:]
P[:,-1] = (V[:,0] + V[:,1])/2
def setX(self, x):
self.x = numpy.copy(x)
self.x = xshift(self.x)
self.vander = legendre.legvander(self.x, self.order)
print "[INFO] Yp_test.shape (E-DMD) " + repr(Yp_test.shape);
print "[INFO] Yf_test.shape (E-DMD) " + repr(Yf_test.shape);
Yp_final_train = Yp_train; # straight up quadratics training data 0,...,n-1
Yf_final_train = Yf_train; # straight up quadratics training data 1 ,... n 1 step forward propagation of Yp_train
Yp_final_test = Yp_test; #straight up quadratics test data -
Yf_final_test = Yf_test; # straight up quadratics test data - 1-step forward propagation of Yp_test
# # # - - - - - - Legendre polynomial dictionary generation - - - - - # # #
if use_legendre:
#print Yp.shape
deg_polynomial = poly_deg
Yp_leg = legvander(Yp,deg_polynomial)
n_points,n_channels,n_deg_polyp1= Yp_leg.shape
Yp_leg = np.reshape(Yp_leg,(n_deg_polyp1,n_points,n_channels));
#print Yp_leg.shape
Yf_leg = legvander(Yf,deg_polynomial)
n_points,n_channels,n_deg_polyp1= Yf_leg.shape
Yf_leg = np.reshape(Yf_leg,(n_deg_polyp1,n_points,n_channels));
#print Yf_leg.shape
Yp_leg_stack = np.concatenate((Yp_leg[:,:,0],Yp_leg[:,:,1])) ;
Yf_leg_stack = np.concatenate((Yf_leg[:,:,0],Yf_leg[:,:,1])) ;
for i in range(2,n_channels):
if i%1000==0:
#print i
#print Yp_leg_stack.shape
print Yf_leg_stack.shape
def named_target_matrix(name, size):
"""
Parameter:
name: name of the target matrix
Return:
target_matrix: (n, n) numpy array for real matrices or (n, n, 2) for complex matrices.
"""
if name == 'dft':
return LA.dft(size, scale='sqrtn')[:, :, None].view('float64')
elif name == 'idft':
return np.ascontiguousarray(LA.dft(
size, scale='sqrtn').conj().T)[:, :, None].view('float64')
elif name == 'dft2':
size_sr = int(math.sqrt(size))
matrix = np.fft.fft2(np.eye(size_sr**2).reshape(-1, size_sr, size_sr),
norm='ortho').reshape(-1, size_sr**2)
# matrix1d = LA.dft(size_sr, scale='sqrtn')
# assert np.allclose(np.kron(m1d, m1d), matrix)
# return matrix[:, :, None].view('float64')
from butterfly.utils import bitreversal_permutation
br_perm = bitreversal_permutation(size_sr)
br_perm2 = np.arange(size_sr**2).reshape(
size_sr, size_sr)[br_perm][:, br_perm].reshape(-1)
matrix = np.ascontiguousarray(matrix[:, br_perm2])
return matrix[:, :, None].view('float64')
elif name == 'dct':
# Need to transpose as dct acts on rows of matrix np.eye, not columns
# return dct(np.eye(size), norm='ortho').T
return dct(np.eye(size)).T / math.sqrt(size)
elif name == 'dst':
return dst(np.eye(size)).T / math.sqrt(size)
elif name == 'hadamard':
return LA.hadamard(size) / math.sqrt(size)
elif name == 'hadamard2':
size_sr = int(math.sqrt(size))
matrix1d = LA.hadamard(size_sr) / math.sqrt(size_sr)
return np.kron(matrix1d, matrix1d)
elif name == 'b2':
size_sr = int(math.sqrt(size))
import torch
from butterfly import Block2x2DiagProduct
b = Block2x2DiagProduct(size_sr)
matrix1d = b(torch.eye(size_sr)).t().detach().numpy()
return np.kron(matrix1d, matrix1d)
elif name == 'convolution':
np.random.seed(0)
x = np.random.randn(size)
return LA.circulant(x) / math.sqrt(size)
elif name == 'hartley':
return hartley_matrix(size) / math.sqrt(size)
elif name == 'haar':
return haar_matrix(size, normalized=True) / math.sqrt(size)
elif name == 'legendre':
grid = np.linspace(-1, 1, size + 2)[1:-1]
return legendre.legvander(grid, size - 1).T / math.sqrt(size)
elif name == 'hilbert':
H = hilbert_matrix(size)
return H / np.linalg.norm(H, 2)
elif name == 'randn':
np.random.seed(0)
return np.random.randn(size, size) / math.sqrt(size)
else:
assert False, 'Target matrix name not recognized or implemented'
def _get_exp_coeffs(self, y: np.ndarray, x: np.ndarray,
initial_guess: Optional[np.ndarray] = None
) -> Tuple[np.ndarray, np.ndarray]:
r"""Calculate the exponential coefficients
As a first step to fitting, calculate the exponential "rate" factors
:math:`\lambda_k`.
Parameters
----------
y
Function values corresponding to `x`.
x
Independent variable values
Returns
-------
exp_coeff
List of exponential coefficients
ode_coeff
Optimal coefficients of the ODE involved in calculating the
exponential coefficients.
Other parameters
----------------
initial_guess
An initial guess for determining the parameters of the ODE (if you
don't know what this is about, don't bother). The array is 1D and
has `n_exp` + 1 entries. If `None`, use ``numpy.ones(n_exp + 1)``.
"""
if initial_guess is None:
initial_guess = np.ones(self.n_exp + 1)
s = _ODESolver(self.n_exp, self.poly_order)
# Map x to [-1, 1]
x_min = np.min(x)
x_range = np.ptp(x)
x_mapped = 2 * (x - x_min) / x_range - 1
# Weights (trapezoidal)
d_x = np.diff(x)
w = np.zeros(len(x))
w[0] = d_x[0]
w[1:-1] = d_x[1:] + d_x[:-1]
w[-1] = d_x[-1]
w /= x_range
leg = legvander(x_mapped, self.n_exp - 1)
def residual(coeffs):
s.coefficients = coeffs
# The following is the residual condition
# \hat{y}_k = (k + 1 / 2) \sum_{i=1}^n w_i z_i P(t_i)
rc = leg.T @ (w * y)
rc *= np.arange(0.5, self.n_exp)
# Solve the ODE in Legendre space
# \sum_{k=0}^p a_k D^k \hat{y} = e_1
rhs = np.zeros(self.poly_order)
rhs[0] = 1
sol_hat = s.solve(rc, rhs)
# transform to real space
sol = legval(x_mapped, sol_hat)
return y - sol
def jacobian(coeffs):
s.coefficients = coeffs
jac = np.zeros((len(x), self.n_exp + 1))
tan = s.tangent()
for i in range(self.n_exp + 1):
jac[:, i] = -legval(x_mapped, tan[:, i])
return jac
ode_coeff = leastsq(residual, initial_guess, Dfun=jacobian)[0]
exp_coeff = np.roots(ode_coeff[::-1]) * 2 / x_range
return exp_coeff, ode_coeff
def legvander(x, deg):
from numpy.polynomial.legendre import legvander
return legvander(x, deg)
def frompoles(self, poles):
v = legvander(self.mu, poles.shape[1] - 1)
self.xi[...] = numpy.einsum('jl,il->ij', v, poles)
if hasattr(self, 'poles'):
del self.poles
