def check_4(self,f=f1,per=0,s=0,a=0,b=2*pi,N=20,xb=None,xe=None,
ia=0,ib=2*pi,dx=0.2*pi):
if xb is None:
xb = a
if xe is None:
xe = b
x = a+(b-a)*arange(N+1,dtype=float)/float(N) # nodes
x1 = a + (b-a)*arange(1,N,dtype=float)/float(N-1) # middle points of the nodes
v, _ = f(x),f(x1)
put(" u = %s N = %d" % (repr(round(dx,3)),N))
put(" k : [x(u), %s(x(u))] Error of splprep Error of splrep " % (f(0,None)))
for k in range(1,6):
tckp,u = splprep([x,v],s=s,per=per,k=k,nest=-1)
tck = splrep(x,v,s=s,per=per,k=k)
uv = splev(dx,tckp)
err1 = abs(uv[1]-f(uv[0]))
err2 = abs(splev(uv[0],tck)-f(uv[0]))
assert_(err1 < 1e-2)
assert_(err2 < 1e-2)
put(" %d : %s %.1e %.1e" %
(k,repr([round(z,3) for z in uv]),
err1,
err2))
put("Derivatives of parametric cubic spline at u (first function):")
k = 3
tckp,u = splprep([x,v],s=s,per=per,k=k,nest=-1)
for d in range(1,k+1):
uv = splev(dx,tckp,d)
put(" %s " % (repr(uv[0])))
def test_1d_shape(self):
x = [1,2,3,4,5]
y = [4,5,6,7,8]
tck = splrep(x, y)
z = splev([1], tck)
assert_equal(z.shape, (1,))
z = splev(1, tck)
assert_equal(z.shape, ())
def test_2d_shape(self):
x = [1, 2, 3, 4, 5]
y = [4, 5, 6, 7, 8]
tck = splrep(x, y)
t = np.array([[1.0, 1.5, 2.0, 2.5], [3.0, 3.5, 4.0, 4.5]])
z = splev(t, tck)
z0 = splev(t[0], tck)
z1 = splev(t[1], tck)
assert_equal(z, np.row_stack((z0, z1)))
def test_splantider_vs_splint(self):
# Check antiderivative vs. FITPACK
spl2 = splantider(self.spl)
# no extrapolation, splint assumes function is zero outside
# range
xx = np.linspace(0, 1, 20)
for x1 in xx:
for x2 in xx:
y1 = splint(x1, x2, self.spl)
y2 = splev(x2, spl2) - splev(x1, spl2)
assert_allclose(y1, y2)
def test_splder_vs_splev(self):
# Check derivative vs. FITPACK
for n in range(3+1):
# Also extrapolation!
xx = np.linspace(-1, 2, 2000)
if n == 3:
# ... except that FITPACK extrapolates strangely for
# order 0, so let's not check that.
xx = xx[(xx >= 0) & (xx <= 1)]
dy = splev(xx, self.spl, n)
spl2 = splder(self.spl, n)
dy2 = splev(xx, spl2)
if n == 1:
assert_allclose(dy, dy2, rtol=2e-6)
else:
assert_allclose(dy, dy2)
def test_smoke_splprep_splrep_splev(self, N, k):
a, b, dx = 0, 2. * np.pi, 0.2 * np.pi
x = np.linspace(a, b, N + 1) # nodes
v = np.sin(x)
tckp, u = splprep([x, v], s=0, per=0, k=k, nest=-1)
uv = splev(dx, tckp)
err1 = abs(uv[1] - np.sin(uv[0]))
assert err1 < 1e-2
tck = splrep(x, v, s=0, per=0, k=k)
err2 = abs(splev(uv[0], tck) - np.sin(uv[0]))
assert err2 < 1e-2
# Derivatives of parametric cubic spline at u (first function)
if k == 3:
tckp, u = splprep([x, v], s=0, per=0, k=k, nest=-1)
for d in range(1, k + 1):
uv = splev(dx, tckp, d)
def test_splev_der_k():
# regression test for gh-2188: splev(x, tck, der=k) gives garbage or crashes
# for x outside of knot range
# test case from gh-2188
tck = (np.array([0., 0., 2.5,
2.5]), np.array([-1.56679978, 2.43995873, 0., 0.]), 1)
t, c, k = tck
x = np.array([-3, 0, 2.5, 3])
# an explicit form of the linear spline
assert_allclose(splev(x, tck), c[0] + (c[1] - c[0]) * x / t[2])
assert_allclose(splev(x, tck, 1), (c[1] - c[0]) / t[2])
# now check a random spline vs splder
np.random.seed(1234)
x = np.sort(np.random.random(30))
y = np.random.random(30)
t, c, k = splrep(x, y)
x = [t[0] - 1., t[-1] + 1.]
tck2 = splder((t, c, k), k)
assert_allclose(splev(x, (t, c, k), k), splev(x, tck2))
def test_extrapolation_modes(self):
# test extrapolation modes
# * if ext=0, return the extrapolated value.
# * if ext=1, return 0
# * if ext=2, raise a ValueError
# * if ext=3, return the boundary value.
x = [1,2,3]
y = [0,2,4]
tck = splrep(x, y, k=1)
rstl = [[-2, 6], [0, 0], None, [0, 4]]
for ext in (0, 1, 3):
assert_array_almost_equal(splev([0, 4], tck, ext=ext), rstl[ext])
assert_raises(ValueError, splev, [0, 4], tck, ext=2)
def test_smoke_sproot(self):
# sproot is only implemented for k=3
a, b = 0.1, 15
x = np.linspace(a, b, 20)
v = np.sin(x)
for k in [1, 2, 4, 5]:
tck = splrep(x, v, s=0, per=0, k=k, xe=b)
with assert_raises(ValueError):
sproot(tck)
k = 3
tck = splrep(x, v, s=0, k=3)
roots = sproot(tck)
assert_allclose(splev(roots, tck), 0, atol=1e-10, rtol=1e-10)
assert_allclose(roots, np.pi * np.array([1, 2, 3, 4]), rtol=1e-3)
def check_1(self,f=f1,per=0,s=0,a=0,b=2*pi,N=20,at=0,xb=None,xe=None):
if xb is None:
xb = a
if xe is None:
xe = b
x = a+(b-a)*arange(N+1,dtype=float)/float(N) # nodes
x1 = a+(b-a)*arange(1,N,dtype=float)/float(N-1) # middle points of the nodes
v = f(x)
nk = []
def err_est(k, d):
# Assume f has all derivatives < 1
h = 1.0/float(N)
tol = 5 * h**(.75*(k-d))
if s > 0:
tol += 1e5*s
return tol
for k in range(1,6):
tck = splrep(x,v,s=s,per=per,k=k,xe=xe)
if at:
t = tck[0][k:-k]
else:
t = x1
nd = []
for d in range(k+1):
tol = err_est(k, d)
err = norm2(f(t,d)-splev(t,tck,d)) / norm2(f(t,d))
assert_(err < tol, (k, d, err, tol))
nd.append((err, tol))
nk.append(nd)
put("\nf = %s s=S_k(x;t,c) x in [%s, %s] > [%s, %s]" % (f(None),
repr(round(xb,3)),repr(round(xe,3)),
repr(round(a,3)),repr(round(b,3))))
if at:
str = "at knots"
else:
str = "at the middle of nodes"
put(" per=%d s=%s Evaluation %s" % (per,repr(s),str))
put(" k : |f-s|^2 |f'-s'| |f''-.. |f'''-. |f''''- |f'''''")
k = 1
for l in nk:
put(' %d : ' % k)
for r in l:
put(' %.1e %.1e' % r)
put('\n')
k = k+1
def check_3(self,f=f1,per=0,s=0,a=0,b=2*pi,N=20,xb=None,xe=None,
ia=0,ib=2*pi,dx=0.2*pi):
if xb is None:
xb = a
if xe is None:
xe = b
x = a+(b-a)*arange(N+1,dtype=float)/float(N) # nodes
v = f(x)
put(" k : Roots of s(x) approx %s x in [%s,%s]:" %
(f(None),repr(round(a,3)),repr(round(b,3))))
for k in range(1,6):
tck = splrep(x, v, s=s, per=per, k=k, xe=xe)
if k == 3:
roots = sproot(tck)
assert_allclose(splev(roots, tck), 0, atol=1e-10, rtol=1e-10)
assert_allclose(roots, pi*array([1, 2, 3, 4]), rtol=1e-3)
put(' %d : %s' % (k, repr(roots.tolist())))
else:
assert_raises(ValueError, sproot, tck)
def check_1(self,
per=0,
s=0,
a=0,
b=2 * np.pi,
at_nodes=False,
xb=None,
xe=None):
if xb is None:
xb = a
if xe is None:
xe = b
N = 20
# nodes and middle points of the nodes
x = np.linspace(a, b, N + 1)
x1 = a + (b - a) * np.arange(1, N, dtype=float) / float(N - 1)
v = f1(x)
def err_est(k, d):
# Assume f has all derivatives < 1
h = 1.0 / N
tol = 5 * h**(.75 * (k - d))
if s > 0:
tol += 1e5 * s
return tol
for k in range(1, 6):
tck = splrep(x, v, s=s, per=per, k=k, xe=xe)
tt = tck[0][k:-k] if at_nodes else x1
nd = []
for d in range(k + 1):
tol = err_est(k, d)
err = norm2(f1(tt, d) - splev(tt, tck, d)) / norm2(f1(tt, d))
assert err < tol
