def __radd__(self, other):
# First check if argument is a scalar
if isscalarlike(other):
new = dok_matrix(self.shape, dtype=self.dtype)
# Add this scalar to every element.
M, N = self.shape
for i in xrange(M):
for j in xrange(N):
aij = self.get((i, j), 0) + other
if aij != 0:
new[i, j] = aij
elif isinstance(other, dok_matrix):
if other.shape != self.shape:
raise ValueError("matrix dimensions are not equal")
new = dok_matrix(self.shape, dtype=self.dtype)
new.update(self)
for key in other:
new[key] += other[key]
elif isspmatrix(other):
csc = self.tocsc()
new = csc + other
elif isdense(other):
new = other + self.todense()
else:
return NotImplemented
return new
def _setdiag(self, values, k):
M, N = self.shape
if k < 0:
if values.ndim == 0:
# broadcast
max_index = min(M + k, N)
for i in xrange(max_index):
self[i - k, i] = values
else:
max_index = min(M + k, N, len(values))
if max_index <= 0:
return
for i, v in enumerate(values[:max_index]):
self[i - k, i] = v
else:
if values.ndim == 0:
# broadcast
max_index = min(M, N - k)
for i in xrange(max_index):
self[i, i + k] = values
else:
max_index = min(M, N - k, len(values))
if max_index <= 0:
return
for i, v in enumerate(values[:max_index]):
self[i, i + k] = v
def lagrange(x, w):
"""
Return a Lagrange interpolating polynomial.
Given two 1-D arrays `x` and `w,` returns the Lagrange interpolating
polynomial through the points ``(x, w)``.
Warning: This implementation is numerically unstable. Do not expect to
be able to use more than about 20 points even if they are chosen optimally.
Parameters
----------
x : array_like
`x` represents the x-coordinates of a set of datapoints.
w : array_like
`w` represents the y-coordinates of a set of datapoints, i.e. f(`x`).
Returns
-------
lagrange : numpy.poly1d instance
The Lagrange interpolating polynomial.
"""
M = len(x)
p = poly1d(0.0)
for j in xrange(M):
pt = poly1d(w[j])
for k in xrange(M):
if k == j:
continue
fac = x[j] - x[k]
pt *= poly1d([1.0, -x[k]]) / fac
p += pt
return p
def _evaluate_derivatives(self, x, der=None):
n = self.n
r = self.r
if der is None:
der = self.n
pi = np.zeros((n, len(x)))
w = np.zeros((n, len(x)))
pi[0] = 1
p = np.zeros((len(x), self.r))
p += self.c[0,np.newaxis,:]
for k in xrange(1,n):
w[k-1] = x - self.xi[k-1]
pi[k] = w[k-1]*pi[k-1]
p += pi[k,:,np.newaxis]*self.c[k]
cn = np.zeros((max(der,n+1), len(x), r), dtype=self.dtype)
cn[:n+1,:,:] += self.c[:n+1,np.newaxis,:]
cn[0] = p
for k in xrange(1,n):
for i in xrange(1,n-k+1):
pi[i] = w[k+i-1]*pi[i-1]+pi[i]
cn[k] = cn[k]+pi[i,:,np.newaxis]*cn[k+i]
cn[k] *= factorial(k)
cn[n,:,:] = 0
return cn[:der]
def _setdiag(self, values, k):
M, N = self.shape
if k < 0:
if values.ndim == 0:
# broadcast
max_index = min(M+k, N)
for i in xrange(max_index):
self[i - k, i] = values
else:
max_index = min(M+k, N, len(values))
if max_index <= 0:
return
for i,v in enumerate(values[:max_index]):
self[i - k, i] = v
else:
if values.ndim == 0:
# broadcast
max_index = min(M, N-k)
for i in xrange(max_index):
self[i, i + k] = values
else:
max_index = min(M, N-k, len(values))
if max_index <= 0:
return
for i,v in enumerate(values[:max_index]):
self[i, i + k] = v
def __init__(self, xi, yi, axis=0):
_Interpolator1DWithDerivatives.__init__(self, xi, yi, axis)
self.xi = np.asarray(xi)
self.yi = self._reshape_yi(yi)
self.n, self.r = self.yi.shape
c = np.zeros((self.n + 1, self.r), dtype=self.dtype)
c[0] = self.yi[0]
Vk = np.zeros((self.n, self.r), dtype=self.dtype)
for k in xrange(1, self.n):
s = 0
while s <= k and xi[k - s] == xi[k]:
s += 1
s -= 1
Vk[0] = self.yi[k] / float(factorial(s))
for i in xrange(k - s):
if xi[i] == xi[k]:
raise ValueError("Elements if `xi` can't be equal.")
if s == 0:
Vk[i + 1] = (c[i] - Vk[i]) / (xi[i] - xi[k])
else:
Vk[i + 1] = (Vk[i + 1] - Vk[i]) / (xi[i] - xi[k])
c[k] = Vk[k - s]
self.c = c
def __radd__(self, other):
# First check if argument is a scalar
if isscalarlike(other):
new = dok_matrix(self.shape, dtype=self.dtype)
# Add this scalar to every element.
M, N = self.shape
for i in xrange(M):
for j in xrange(N):
aij = self.get((i, j), 0) + other
if aij != 0:
new[i, j] = aij
elif isinstance(other, dok_matrix):
if other.shape != self.shape:
raise ValueError("matrix dimensions are not equal")
new = dok_matrix(self.shape, dtype=self.dtype)
new.update(self)
for key in other:
new[key] += other[key]
elif isspmatrix(other):
csc = self.tocsc()
new = csc + other
elif isdense(other):
new = other + self.todense()
else:
return NotImplemented
return new
def _evaluate_derivatives(self, x, der=None):
n = self.n
r = self.r
if der is None:
der = self.n
pi = np.zeros((n, len(x)))
w = np.zeros((n, len(x)))
pi[0] = 1
p = np.zeros((len(x), self.r))
p += self.c[0, np.newaxis, :]
for k in xrange(1, n):
w[k - 1] = x - self.xi[k - 1]
pi[k] = w[k - 1] * pi[k - 1]
p += pi[k, :, np.newaxis] * self.c[k]
cn = np.zeros((max(der, n + 1), len(x), r), dtype=self.dtype)
cn[:n + 1, :, :] += self.c[:n + 1, np.newaxis, :]
cn[0] = p
for k in xrange(1, n):
for i in xrange(1, n - k + 1):
pi[i] = w[k + i - 1] * pi[i - 1] + pi[i]
cn[k] = cn[k] + pi[i, :, np.newaxis] * cn[k + i]
cn[k] *= factorial(k)
cn[n, :, :] = 0
return cn[:der]
def __add__(self, other):
# First check if argument is a scalar
if isscalarlike(other):
res_dtype = upcast_scalar(self.dtype, other)
new = dok_matrix(self.shape, dtype=res_dtype)
# Add this scalar to every element.
M, N = self.shape
for i in xrange(M):
for j in xrange(N):
aij = self.get((i, j), 0) + other
if aij != 0:
new[i, j] = aij
# new.dtype.char = self.dtype.char
elif isinstance(other, dok_matrix):
if other.shape != self.shape:
raise ValueError("matrix dimensions are not equal")
# We could alternatively set the dimensions to the largest of
# the two matrices to be summed. Would this be a good idea?
res_dtype = upcast(self.dtype, other.dtype)
new = dok_matrix(self.shape, dtype=res_dtype)
new.update(self)
for key in other.keys():
new[key] += other[key]
elif isspmatrix(other):
csc = self.tocsc()
new = csc + other
elif isdense(other):
new = self.todense() + other
else:
return NotImplemented
return new
def __add__(self, other):
# First check if argument is a scalar
if isscalarlike(other):
res_dtype = upcast_scalar(self.dtype, other)
new = dok_matrix(self.shape, dtype=res_dtype)
# Add this scalar to every element.
M, N = self.shape
for i in xrange(M):
for j in xrange(N):
aij = self.get((i, j), 0) + other
if aij != 0:
new[i, j] = aij
# new.dtype.char = self.dtype.char
elif isinstance(other, dok_matrix):
if other.shape != self.shape:
raise ValueError("matrix dimensions are not equal")
# We could alternatively set the dimensions to the largest of
# the two matrices to be summed. Would this be a good idea?
res_dtype = upcast(self.dtype, other.dtype)
new = dok_matrix(self.shape, dtype=res_dtype)
new.update(self)
for key in other.keys():
new[key] += other[key]
elif isspmatrix(other):
csc = self.tocsc()
new = csc + other
elif isdense(other):
new = self.todense() + other
else:
return NotImplemented
return new
def __init__(self, xi, yi, axis=0):
_Interpolator1DWithDerivatives.__init__(self, xi, yi, axis)
self.xi = np.asarray(xi)
self.yi = self._reshape_yi(yi)
self.n, self.r = self.yi.shape
c = np.zeros((self.n+1, self.r), dtype=self.dtype)
c[0] = self.yi[0]
Vk = np.zeros((self.n, self.r), dtype=self.dtype)
for k in xrange(1,self.n):
s = 0
while s <= k and xi[k-s] == xi[k]:
s += 1
s -= 1
Vk[0] = self.yi[k]/float(factorial(s))
for i in xrange(k-s):
if xi[i] == xi[k]:
raise ValueError("Elements if `xi` can't be equal.")
if s == 0:
Vk[i+1] = (c[i]-Vk[i])/(xi[i]-xi[k])
else:
Vk[i+1] = (Vk[i+1]-Vk[i])/(xi[i]-xi[k])
c[k] = Vk[k-s]
self.c = c
def lagrange(x, w):
"""
Return a Lagrange interpolating polynomial.
Given two 1-D arrays `x` and `w,` returns the Lagrange interpolating
polynomial through the points ``(x, w)``.
Warning: This implementation is numerically unstable. Do not expect to
be able to use more than about 20 points even if they are chosen optimally.
Parameters
----------
x : array_like
`x` represents the x-coordinates of a set of datapoints.
w : array_like
`w` represents the y-coordinates of a set of datapoints, i.e. f(`x`).
Returns
-------
lagrange : numpy.poly1d instance
The Lagrange interpolating polynomial.
"""
M = len(x)
p = poly1d(0.0)
for j in xrange(M):
pt = poly1d(w[j])
for k in xrange(M):
if k == j:
continue
fac = x[j]-x[k]
pt *= poly1d([1.0, -x[k]])/fac
p += pt
return p
def test_cascade(self):
for J in xrange(1, 7):
for i in xrange(1, 5):
lpcoef = wavelets.daub(i)
k = len(lpcoef)
x, phi, psi = wavelets.cascade(lpcoef, J)
assert_(len(x) == len(phi) == len(psi))
assert_equal(len(x), (k - 1) * 2 ** J)
def test_cascade(self):
for J in xrange(1, 7):
for i in xrange(1, 5):
lpcoef = wavelets.daub(i)
k = len(lpcoef)
x, phi, psi = wavelets.cascade(lpcoef, J)
assert_(len(x) == len(phi) == len(psi))
assert_equal(len(x), (k - 1) * 2 ** J)
def test_improvement(self):
import time
start = time.time()
for i in xrange(100):
quad(self.lib.sin, 0, 100)
fast = time.time() - start
start = time.time()
for i in xrange(100):
quad(math.sin, 0, 100)
slow = time.time() - start
assert_(fast < 0.5 * slow, (fast, slow))
def test_correspond_2_and_up(self):
# Tests correspond(Z, y) on linkage and CDMs over observation sets of
# different sizes.
for i in xrange(2, 4):
y = np.random.rand(i * (i - 1) // 2)
Z = linkage(y)
self.assertTrue(correspond(Z, y))
for i in xrange(4, 15, 3):
y = np.random.rand(i * (i - 1) // 2)
Z = linkage(y)
self.assertTrue(correspond(Z, y))
def test_improvement(self):
import time
start = time.time()
for i in xrange(100):
quad(self.lib.sin, 0, 100)
fast = time.time() - start
start = time.time()
for i in xrange(100):
quad(math.sin, 0, 100)
slow = time.time() - start
assert_(fast < 0.5*slow, (fast, slow))
def test_correspond_2_and_up(self):
# Tests correspond(Z, y) on linkage and CDMs over observation sets of
# different sizes.
for i in xrange(2, 4):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
self.assertTrue(correspond(Z, y))
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
self.assertTrue(correspond(Z, y))
def piecefuncgen(num):
Mk = order // 2 - num
if (Mk < 0):
return 0 # final function is 0
coeffs = [(1 - 2 * (k % 2)) * float(comb(order + 1, k, exact=1)) / fval
for k in xrange(Mk + 1)]
shifts = [-bound - k for k in xrange(Mk + 1)]
def thefunc(x):
res = 0.0
for k in range(Mk + 1):
res += coeffs[k] * (x + shifts[k]) ** order
return res
return thefunc
def test_nd_simplex(self):
# simple smoke test: triangulate a n-dimensional simplex
for nd in xrange(2, 8):
points = np.zeros((nd+1, nd))
for j in xrange(nd):
points[j,j] = 1.0
points[-1,:] = 1.0
tri = qhull.Delaunay(points)
tri.vertices.sort()
assert_equal(tri.vertices, np.arange(nd+1, dtype=np.int)[None,:])
assert_equal(tri.neighbors, -1 + np.zeros((nd+1), dtype=np.int)[None,:])
def test_nd_simplex(self):
# simple smoke test: triangulate a n-dimensional simplex
for nd in xrange(2, 8):
points = np.zeros((nd+1, nd))
for j in xrange(nd):
points[j,j] = 1.0
points[-1,:] = 1.0
tri = qhull.Delaunay(points)
tri.vertices.sort()
assert_equal(tri.vertices, np.arange(nd+1, dtype=np.int)[None,:])
assert_equal(tri.neighbors, -1 + np.zeros((nd+1), dtype=np.int)[None,:])
def test_improvement(self):
def myfunc(x): # Euler's constant integrand
return -exp(-x)*log(x)
import time
start = time.time()
for i in xrange(20):
quad(self.lib._multivariate_indefinite, 0, 100)
fast = time.time() - start
start = time.time()
for i in xrange(20):
quad(myfunc, 0, 100)
slow = time.time() - start
# 2+ times faster speeds generated by nontrivial ctypes
# function (single variable)
assert_(fast < 0.5*slow, (fast, slow))
def test_num_obs_linkage_multi_matrix(self):
# Tests num_obs_linkage with observation matrices of multiple sizes.
for n in xrange(2, 10):
X = np.random.rand(n, 4)
Y = pdist(X)
Z = linkage(Y)
self.assertTrue(num_obs_linkage(Z) == n)
def test_call(self):
poly = []
for n in xrange(5):
poly.extend([
x.strip() for x in ("""
orth.jacobi(%(n)d,0.3,0.9)
orth.sh_jacobi(%(n)d,0.3,0.9)
orth.genlaguerre(%(n)d,0.3)
orth.laguerre(%(n)d)
orth.hermite(%(n)d)
orth.hermitenorm(%(n)d)
orth.gegenbauer(%(n)d,0.3)
orth.chebyt(%(n)d)
orth.chebyu(%(n)d)
orth.chebyc(%(n)d)
orth.chebys(%(n)d)
orth.sh_chebyt(%(n)d)
orth.sh_chebyu(%(n)d)
orth.legendre(%(n)d)
orth.sh_legendre(%(n)d)
""" % dict(n=n)).split()
])
olderr = np.seterr(all='ignore')
try:
for pstr in poly:
p = eval(pstr)
assert_almost_equal(p(0.315),
np.poly1d(p)(0.315),
err_msg=pstr)
finally:
np.seterr(**olderr)
def test_num_obs_linkage_4_and_up(self):
# Tests num_obs_linkage(Z) on linkage on observation sets between sizes
# 4 and 15 (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
self.assertTrue(num_obs_linkage(Z) == i)
def setUp(self):
self.tck = splrep([0,1,2,3,4,5], [0,10,-1,3,7,2], s=0)
self.test_xs = np.linspace(-1,6,100)
self.spline_ys = splev(self.test_xs, self.tck)
self.spline_yps = splev(self.test_xs, self.tck, der=1)
self.xi = np.unique(self.tck[0])
self.yi = [[splev(x, self.tck, der=j) for j in xrange(3)] for x in self.xi]
def test_is_valid_linkage_4_and_up(self):
# Tests is_valid_linkage(Z) on linkage on observation sets between
# sizes 4 and 15 (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_(is_valid_linkage(Z) == True)
def test_derivatives(self):
P = PiecewisePolynomial(self.xi,self.yi,3)
m = 4
r = P.derivatives(self.test_xs,m)
#print r.shape, r
for i in xrange(m):
assert_almost_equal(P.derivative(self.test_xs,i),r[i])
def piecefuncgen(num):
Mk = order // 2 - num
if (Mk < 0):
return 0 # final function is 0
coeffs = [(1 - 2 * (k % 2)) * float(comb(order + 1, k, exact=1)) / fval
for k in xrange(Mk + 1)]
shifts = [-bound - k for k in xrange(Mk + 1)]
#print "Adding piece number %d with coeffs %s and shifts %s" \
# % (num, str(coeffs), str(shifts))
def thefunc(x):
res = 0.0
for k in range(Mk + 1):
res += coeffs[k] * (x + shifts[k]) ** order
return res
return thefunc
def test_num_obs_linkage_multi_matrix(self):
# Tests num_obs_linkage with observation matrices of multiple sizes.
for n in xrange(2, 10):
X = np.random.rand(n, 4)
Y = pdist(X)
Z = linkage(Y)
self.assertTrue(num_obs_linkage(Z) == n)
def test_concurrent_ok(self):
f = lambda t, y: 1.0
for k in xrange(3):
for sol in ('vode', 'zvode', 'lsoda', 'dopri5', 'dop853'):
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_allclose(r.y, 0.1)
assert_allclose(r2.y, 0.2)
for sol in ('dopri5', 'dop853'):
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_allclose(r.y, 0.3)
assert_allclose(r2.y, 0.2)
def test_num_obs_linkage_4_and_up(self):
# Tests num_obs_linkage(Z) on linkage on observation sets between sizes
# 4 and 15 (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i * (i - 1) // 2)
Z = linkage(y)
self.assertTrue(num_obs_linkage(Z) == i)
def test_is_valid_linkage_4_and_up(self):
# Tests is_valid_linkage(Z) on linkage on observation sets between
# sizes 4 and 15 (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i * (i - 1) // 2)
Z = linkage(y)
assert_(is_valid_linkage(Z) == True)
def test_exponential(self):
degree = 5
p = approximate_taylor_polynomial(np.exp, 0, degree, 1, 15)
for i in xrange(degree + 1):
assert_almost_equal(p(0), 1)
p = p.deriv()
assert_almost_equal(p(0), 0)
def _get_slice(self, i, start, stop, stride, shape):
"""Returns a copy of the elements
[i, start:stop:string] for row-oriented matrices
[start:stop:string, i] for column-oriented matrices
"""
if stride != 1:
raise ValueError("slicing with step != 1 not supported")
if stop <= start:
raise ValueError("slice width must be >= 1")
# TODO make [i,:] faster
# TODO implement [i,x:y:z]
indices = []
for ind in xrange(self.indptr[i], self.indptr[i + 1]):
if self.indices[ind] >= start and self.indices[ind] < stop:
indices.append(ind)
index = self.indices[indices] - start
data = self.data[indices]
indptr = np.array([0, len(indices)])
return self.__class__((data, index, indptr),
shape=shape,
dtype=self.dtype)
def piecefuncgen(num):
Mk = order // 2 - num
if (Mk < 0):
return 0 # final function is 0
coeffs = [(1 - 2 * (k % 2)) * float(comb(order + 1, k, exact=1)) / fval
for k in xrange(Mk + 1)]
shifts = [-bound - k for k in xrange(Mk + 1)]
#print "Adding piece number %d with coeffs %s and shifts %s" \
# % (num, str(coeffs), str(shifts))
def thefunc(x):
res = 0.0
for k in range(Mk + 1):
res += coeffs[k] * (x + shifts[k]) ** order
return res
return thefunc
def test_exponential(self):
degree = 5
p = approximate_taylor_polynomial(np.exp, 0, degree, 1, 15)
for i in xrange(degree+1):
assert_almost_equal(p(0),1)
p = p.deriv()
assert_almost_equal(p(0),0)
def setUp(self):
self.tck = splrep([0,1,2,3,4,5], [0,10,-1,3,7,2], s=0)
self.test_xs = np.linspace(-1,6,100)
self.spline_ys = splev(self.test_xs, self.tck)
self.spline_yps = splev(self.test_xs, self.tck, der=1)
self.xi = np.unique(self.tck[0])
self.yi = [[splev(x, self.tck, der=j) for j in xrange(3)] for x in self.xi]
def test_improvement(self):
def myfunc(x): # Euler's constant integrand
return -exp(-x) * log(x)
import time
start = time.time()
for i in xrange(20):
quad(self.lib._multivariate_indefinite, 0, 100)
fast = time.time() - start
start = time.time()
for i in xrange(20):
quad(myfunc, 0, 100)
slow = time.time() - start
# 2+ times faster speeds generated by nontrivial ctypes
# function (single variable)
assert_(fast < 0.5 * slow, (fast, slow))
def test_call(self):
poly = []
for n in xrange(5):
poly.extend([x.strip() for x in
("""
orth.jacobi(%(n)d,0.3,0.9)
orth.sh_jacobi(%(n)d,0.3,0.9)
orth.genlaguerre(%(n)d,0.3)
orth.laguerre(%(n)d)
orth.hermite(%(n)d)
orth.hermitenorm(%(n)d)
orth.gegenbauer(%(n)d,0.3)
orth.chebyt(%(n)d)
orth.chebyu(%(n)d)
orth.chebyc(%(n)d)
orth.chebys(%(n)d)
orth.sh_chebyt(%(n)d)
orth.sh_chebyu(%(n)d)
orth.legendre(%(n)d)
orth.sh_legendre(%(n)d)
""" % dict(n=n)).split()
])
olderr = np.seterr(all='ignore')
try:
for pstr in poly:
p = eval(pstr)
assert_almost_equal(p(0.315), np.poly1d(p)(0.315), err_msg=pstr)
finally:
np.seterr(**olderr)
def test_derivatives(self):
P = PiecewisePolynomial(self.xi,self.yi,3)
m = 4
r = P.derivatives(self.test_xs,m)
#print r.shape, r
for i in xrange(m):
assert_almost_equal(P.derivative(self.test_xs,i),r[i])
def test_concurrent_ok(self):
f = lambda t, y: 1.0
for k in xrange(3):
for sol in ('vode', 'zvode', 'lsoda', 'dopri5', 'dop853'):
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_allclose(r.y, 0.1)
assert_allclose(r2.y, 0.2)
for sol in ('dopri5', 'dop853'):
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_allclose(r.y, 0.3)
assert_allclose(r2.y, 0.2)
def setdiag(self, values, k=0):
"""
Set diagonal or off-diagonal elements of the array.
Parameters
----------
values : array_like
New values of the diagonal elements.
Values may have any length. If the diagonal is longer than values,
then the remaining diagonal entries will not be set. If values if
longer than the diagonal, then the remaining values are ignored.
If a scalar value is given, all of the diagonal is set to it.
k : int, optional
Which off-diagonal to set, corresponding to elements a[i,i+k].
Default: 0 (the main diagonal).
"""
M, N = self.shape
if (k > 0 and k >= N) or (k < 0 and -k >= M):
raise ValueError("k exceeds matrix dimensions")
if k < 0:
if np.asarray(values).ndim == 0:
# broadcast
max_index = min(M+k, N)
for i in xrange(max_index):
self[i - k, i] = values
else:
max_index = min(M+k, N, len(values))
if max_index <= 0:
return
for i,v in enumerate(values[:max_index]):
self[i - k, i] = v
else:
if np.asarray(values).ndim == 0:
# broadcast
max_index = min(M, N-k)
for i in xrange(max_index):
self[i, i + k] = values
else:
max_index = min(M, N-k, len(values))
if max_index <= 0:
return
for i,v in enumerate(values[:max_index]):
self[i, i + k] = v
def setdiag(self, values, k=0):
"""
Set diagonal or off-diagonal elements of the array.
Parameters
----------
values : array_like
New values of the diagonal elements.
Values may have any length. If the diagonal is longer than values,
then the remaining diagonal entries will not be set. If values if
longer than the diagonal, then the remaining values are ignored.
If a scalar value is given, all of the diagonal is set to it.
k : int, optional
Which off-diagonal to set, corresponding to elements a[i,i+k].
Default: 0 (the main diagonal).
"""
M, N = self.shape
if (k > 0 and k >= N) or (k < 0 and -k >= M):
raise ValueError("k exceeds matrix dimensions")
if k < 0:
if np.asarray(values).ndim == 0:
# broadcast
max_index = min(M + k, N)
for i in xrange(max_index):
self[i - k, i] = values
else:
max_index = min(M + k, N, len(values))
if max_index <= 0:
return
for i, v in enumerate(values[:max_index]):
self[i - k, i] = v
else:
if np.asarray(values).ndim == 0:
# broadcast
max_index = min(M, N - k)
for i in xrange(max_index):
self[i, i + k] = values
else:
max_index = min(M, N - k, len(values))
if max_index <= 0:
return
for i, v in enumerate(values[:max_index]):
self[i, i + k] = v
def comb(N,k,exact=False,repetition=False):
"""
The number of combinations of N things taken k at a time.
This is often expressed as "N choose k".
Parameters
----------
N : int, ndarray
Number of things.
k : int, ndarray
Number of elements taken.
exact : bool, optional
If `exact` is False, then floating point precision is used, otherwise
exact long integer is computed.
repetition : bool, optional
If `repetition` is True, then the number of combinations with
repetition is computed.
Returns
-------
val : int, ndarray
The total number of combinations.
Notes
-----
- Array arguments accepted only for exact=False case.
- If k > N, N < 0, or k < 0, then a 0 is returned.
Examples
--------
>>> k = np.array([3, 4])
>>> n = np.array([10, 10])
>>> sc.comb(n, k, exact=False)
array([ 120., 210.])
>>> sc.comb(10, 3, exact=True)
120L
>>> sc.comb(10, 3, exact=True, repetition=True)
220L
"""
if repetition:
return comb(N + k - 1, k, exact)
if exact:
if (k > N) or (N < 0) or (k < 0):
return 0
val = 1
for j in xrange(min(k, N-k)):
val = (val*(N-j))//(j+1)
return val
else:
k,N = asarray(k), asarray(N)
cond = (k <= N) & (N >= 0) & (k >= 0)
vals = binom(N, k)
if isinstance(vals, np.ndarray):
vals[~cond] = 0
elif not cond:
vals = np.float64(0)
return vals
def _check_dot(self, jac_cls, complex=False, tol=1e-6, **kw):
np.random.seed(123)
N = 7
def rand(*a):
q = np.random.rand(*a)
if complex:
q = q + 1j * np.random.rand(*a)
return q
def assert_close(a, b, msg):
d = abs(a - b).max()
f = tol + abs(b).max() * tol
if d > f:
raise AssertionError('%s: err %g' % (msg, d))
self.A = rand(N, N)
# initialize
x0 = np.random.rand(N)
jac = jac_cls(**kw)
jac.setup(x0, self._func(x0), self._func)
# check consistency
for k in xrange(2 * N):
v = rand(N)
if hasattr(jac, '__array__'):
Jd = np.array(jac)
if hasattr(jac, 'solve'):
Gv = jac.solve(v)
Gv2 = np.linalg.solve(Jd, v)
assert_close(Gv, Gv2, 'solve vs array')
if hasattr(jac, 'rsolve'):
Gv = jac.rsolve(v)
Gv2 = np.linalg.solve(Jd.T.conj(), v)
assert_close(Gv, Gv2, 'rsolve vs array')
if hasattr(jac, 'matvec'):
Jv = jac.matvec(v)
Jv2 = np.dot(Jd, v)
assert_close(Jv, Jv2, 'dot vs array')
if hasattr(jac, 'rmatvec'):
Jv = jac.rmatvec(v)
Jv2 = np.dot(Jd.T.conj(), v)
assert_close(Jv, Jv2, 'rmatvec vs array')
if hasattr(jac, 'matvec') and hasattr(jac, 'solve'):
Jv = jac.matvec(v)
Jv2 = jac.solve(jac.matvec(Jv))
assert_close(Jv, Jv2, 'dot vs solve')
if hasattr(jac, 'rmatvec') and hasattr(jac, 'rsolve'):
Jv = jac.rmatvec(v)
Jv2 = jac.rmatvec(jac.rsolve(Jv))
assert_close(Jv, Jv2, 'rmatvec vs rsolve')
x = rand(N)
jac.update(x, self._func(x))
def _check_dot(self, jac_cls, complex=False, tol=1e-6, **kw):
np.random.seed(123)
N = 7
def rand(*a):
q = np.random.rand(*a)
if complex:
q = q + 1j*np.random.rand(*a)
return q
def assert_close(a, b, msg):
d = abs(a - b).max()
f = tol + abs(b).max()*tol
if d > f:
raise AssertionError('%s: err %g' % (msg, d))
self.A = rand(N, N)
# initialize
x0 = np.random.rand(N)
jac = jac_cls(**kw)
jac.setup(x0, self._func(x0), self._func)
# check consistency
for k in xrange(2*N):
v = rand(N)
if hasattr(jac, '__array__'):
Jd = np.array(jac)
if hasattr(jac, 'solve'):
Gv = jac.solve(v)
Gv2 = np.linalg.solve(Jd, v)
assert_close(Gv, Gv2, 'solve vs array')
if hasattr(jac, 'rsolve'):
Gv = jac.rsolve(v)
Gv2 = np.linalg.solve(Jd.T.conj(), v)
assert_close(Gv, Gv2, 'rsolve vs array')
if hasattr(jac, 'matvec'):
Jv = jac.matvec(v)
Jv2 = np.dot(Jd, v)
assert_close(Jv, Jv2, 'dot vs array')
if hasattr(jac, 'rmatvec'):
Jv = jac.rmatvec(v)
Jv2 = np.dot(Jd.T.conj(), v)
assert_close(Jv, Jv2, 'rmatvec vs array')
if hasattr(jac, 'matvec') and hasattr(jac, 'solve'):
Jv = jac.matvec(v)
Jv2 = jac.solve(jac.matvec(Jv))
assert_close(Jv, Jv2, 'dot vs solve')
if hasattr(jac, 'rmatvec') and hasattr(jac, 'rsolve'):
Jv = jac.rmatvec(v)
Jv2 = jac.rmatvec(jac.rsolve(Jv))
assert_close(Jv, Jv2, 'rmatvec vs rsolve')
x = rand(N)
jac.update(x, self._func(x))
def matrixmultiply(a, b):
if len(b.shape) == 1:
b_is_vector = True
b = b[:,newaxis]
else:
b_is_vector = False
assert_equal(a.shape[1], b.shape[0])
c = zeros((a.shape[0], b.shape[1]), common_type(a, b))
for i in xrange(a.shape[0]):
for j in xrange(b.shape[1]):
s = 0
for k in xrange(a.shape[1]):
s += a[i,k] * b[k, j]
c[i,j] = s
if b_is_vector:
c = c.reshape((a.shape[0],))
return c
def test_vector(self):
xs = [0, 1, 2]
ys = np.array([[0,1],[1,0],[2,1]])
P = BarycentricInterpolator(xs,ys)
Pi = [BarycentricInterpolator(xs,ys[:,i]) for i in xrange(ys.shape[1])]
test_xs = np.linspace(-1,3,100)
assert_almost_equal(P(test_xs),
np.rollaxis(np.asarray([p(test_xs) for p in Pi]),-1))
def test_incremental(self):
with warnings.catch_warnings():
warnings.filterwarnings('ignore', category=DeprecationWarning)
P = PiecewisePolynomial([self.xi[0]], [self.yi[0]], 3)
for i in xrange(1, len(self.xi)):
P.append(self.xi[i], self.yi[i], 3)
assert_almost_equal(P(self.test_xs), self.spline_ys)
def test_inverse(self):
for n in xrange(1, 10):
a = hilbert(n)
b = invhilbert(n)
# The Hilbert matrix is increasingly badly conditioned,
# so take that into account in the test
c = cond(a)
assert_allclose(a.dot(b), eye(n), atol=1e-15*c, rtol=1e-15*c)
def test_is_valid_im_4_and_up(self):
# Tests is_valid_im(R) on im on observation sets between sizes 4 and 15
# (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i * (i - 1) // 2)
Z = linkage(y)
R = inconsistent(Z)
assert_(is_valid_im(R) == True)
def matrixmultiply(a, b):
if len(b.shape) == 1:
b_is_vector = True
b = b[:, newaxis]
else:
b_is_vector = False
assert_equal(a.shape[1], b.shape[0])
c = zeros((a.shape[0], b.shape[1]), common_type(a, b))
for i in xrange(a.shape[0]):
for j in xrange(b.shape[1]):
s = 0
for k in xrange(a.shape[1]):
s += a[i, k] * b[k, j]
c[i, j] = s
if b_is_vector:
c = c.reshape((a.shape[0], ))
return c
def test_vector(self):
xs = [0, 1, 2]
ys = np.array([[0,1],[1,0],[2,1]])
P = BarycentricInterpolator(xs,ys)
Pi = [BarycentricInterpolator(xs,ys[:,i]) for i in xrange(ys.shape[1])]
test_xs = np.linspace(-1,3,100)
assert_almost_equal(P(test_xs),
np.rollaxis(np.asarray([p(test_xs) for p in Pi]),-1))
def test_inverse(self):
for n in xrange(1, 10):
a = hilbert(n)
b = invhilbert(n)
# The Hilbert matrix is increasingly badly conditioned,
# so take that into account in the test
c = cond(a)
assert_allclose(a.dot(b), eye(n), atol=1e-15 * c, rtol=1e-15 * c)
def test_is_valid_im_4_and_up(self):
# Tests is_valid_im(R) on im on observation sets between sizes 4 and 15
# (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
R = inconsistent(Z)
assert_(is_valid_im(R) == True)
def test_incremental(self):
with warnings.catch_warnings():
warnings.filterwarnings('ignore', category=DeprecationWarning)
P = PiecewisePolynomial([self.xi[0]], [self.yi[0]], 3)
for i in xrange(1,len(self.xi)):
P.append(self.xi[i],self.yi[i],3)
assert_almost_equal(P(self.test_xs),self.spline_ys)
def _boolrelextrema(data, comparator,
axis=0, order=1, mode='clip'):
"""
Calculate the relative extrema of `data`.
Relative extrema are calculated by finding locations where
``comparator(data[n], data[n+1:n+order+1])`` is True.
Parameters
----------
data : ndarray
Array in which to find the relative extrema.
comparator : callable
Function to use to compare two data points.
Should take 2 numbers as arguments.
axis : int, optional
Axis over which to select from `data`. Default is 0.
order : int, optional
How many points on each side to use for the comparison
to consider ``comparator(n,n+x)`` to be True.
mode : str, optional
How the edges of the vector are treated. 'wrap' (wrap around) or
'clip' (treat overflow as the same as the last (or first) element).
Default 'clip'. See numpy.take
Returns
-------
extrema : ndarray
Boolean array of the same shape as `data` that is True at an extrema,
False otherwise.
See also
--------
argrelmax, argrelmin
Examples
--------
>>> testdata = np.array([1,2,3,2,1])
>>> _boolrelextrema(testdata, np.greater, axis=0)
array([False, False, True, False, False], dtype=bool)
"""
if((int(order) != order) or (order < 1)):
raise ValueError('Order must be an int >= 1')
datalen = data.shape[axis]
locs = np.arange(0, datalen)
results = np.ones(data.shape, dtype=bool)
main = data.take(locs, axis=axis, mode=mode)
for shift in xrange(1, order + 1):
plus = data.take(locs + shift, axis=axis, mode=mode)
minus = data.take(locs - shift, axis=axis, mode=mode)
results &= comparator(main, plus)
results &= comparator(main, minus)
if(~results.any()):
return results
return results
def _boolrelextrema(data, comparator,
axis=0, order=1, mode='clip'):
"""
Calculate the relative extrema of `data`.
Relative extrema are calculated by finding locations where
``comparator(data[n], data[n+1:n+order+1])`` is True.
Parameters
----------
data : ndarray
Array in which to find the relative extrema.
comparator : callable
Function to use to compare two data points.
Should take 2 numbers as arguments.
axis : int, optional
Axis over which to select from `data`. Default is 0.
order : int, optional
How many points on each side to use for the comparison
to consider ``comparator(n,n+x)`` to be True.
mode : str, optional
How the edges of the vector are treated. 'wrap' (wrap around) or
'clip' (treat overflow as the same as the last (or first) element).
Default 'clip'. See numpy.take
Returns
-------
extrema : ndarray
Boolean array of the same shape as `data` that is True at an extrema,
False otherwise.
See also
--------
argrelmax, argrelmin
Examples
--------
>>> testdata = np.array([1,2,3,2,1])
>>> _boolrelextrema(testdata, np.greater, axis=0)
array([False, False, True, False, False], dtype=bool)
"""
if((int(order) != order) or (order < 1)):
raise ValueError('Order must be an int >= 1')
datalen = data.shape[axis]
locs = np.arange(0, datalen)
results = np.ones(data.shape, dtype=bool)
main = data.take(locs, axis=axis, mode=mode)
for shift in xrange(1, order + 1):
plus = data.take(locs + shift, axis=axis, mode=mode)
minus = data.take(locs - shift, axis=axis, mode=mode)
results &= comparator(main, plus)
results &= comparator(main, minus)
if(~results.any()):
return results
return results
