def test_legvander3d(self) :
# also tests polyval3d for non-square coefficient array
x1, x2, x3 = self.x
c = np.random.random((2, 3, 4))
van = leg.legvander3d(x1, x2, x3, [1, 2, 3])
tgt = leg.legval3d(x1, x2, x3, c)
res = np.dot(van, c.flat)
assert_almost_equal(res, tgt)
# check shape
van = leg.legvander3d([x1], [x2], [x3], [1, 2, 3])
assert_(van.shape == (1, 5, 24))
def test_legvander3d(self) :
# also tests polyval3d for non-square coefficient array
x1, x2, x3 = self.x
c = np.random.random((2, 3, 4))
van = leg.legvander3d(x1, x2, x3, [1, 2, 3])
tgt = leg.legval3d(x1, x2, x3, c)
res = np.dot(van, c.flat)
assert_almost_equal(res, tgt)
# check shape
van = leg.legvander3d([x1], [x2], [x3], [1, 2, 3])
assert_(van.shape == (1, 5, 24))
def legForwardTransform(orders, locations, functionVals):
if len(locations.shape)==1:
return np.array(leg.legfit(locations, functionVals, orders[0]))
else:
if locations.shape[1]==2:
V=leg.legvander2d(locations[:,0], locations[:,1], orders)
elif locations.shape[1]==3:
V=leg.legvander3d(locations[:,0],locations[:,1],locations[:,2], orders)
elif locations.shape[1]==4:
V=legvander4d(locations,orders)
elif locations.shape[1]==5:
V=legvander5d(locations,orders)
else:
raise NotImplementedError # there's a bad startup joke about this being good enough for the paper.
ret, _, _, _=npl.lstsq(V, functionVals, rcond=None)
return np.reshape(ret, (np.array(orders)+1).flatten())
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)
