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pep8 + some small refactorings

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Per.Andreas.Brodtkorb 10 years ago
parent f1b07d74b4
commit 0c185600cb
4 changed files with 460 additions and 469 deletions
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pywafo/src/wafo/integrate.py
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@ -1,16 +1,15 @@
from __future__ import division
import warnings
import copy
import numpy as np
from numpy import pi, sqrt, ones, zeros # @UnresolvedImport
from scipy import integrate as intg
import scipy.special.orthogonal as ort
from scipy import special as sp
from wafo.plotbackend import plotbackend as plt
from scipy.integrate import simps, trapz # @UnusedImport
from wafo.misc import is_numlike
from scipy.integrate import simps, trapz
from wafo.demos import humps
_EPS = np.finfo(float).eps
_POINTS_AND_WEIGHTS = {}
__all__ = ['dea3', 'clencurt', 'romberg',
@ -69,53 +68,29 @@ def dea3(v0, v1, v2):
"Lecture Notes in Math.", vol. 584,
Springer-Verlag, New York, 1977.
'''
E0, E1, E2 = np.atleast_1d(v0, v1, v2)
abs = np.abs # @ReservedAssignment
max = np.maximum # @ReservedAssignment
ten = 10.0
one = ones(1)
small = np.finfo(float).eps # 1.0e-16 #spacing(one)
delta2 = E2 - E1
delta1 = E1 - E0
err2 = abs(delta2)
err1 = abs(delta1)
tol2 = max(abs(E2), abs(E1)) * small
tol1 = max(abs(E1), abs(E0)) * small
result = zeros(E0.shape)
abserr = result.copy()
converged = (err1 <= tol1) & (err2 <= tol2).ravel()
k0, = converged.nonzero()
if k0.size > 0:
#%C IF E0, E1 AND E2 ARE EQUAL TO WITHIN MACHINE
#%C ACCURACY, CONVERGENCE IS ASSUMED.
result[k0] = E2[k0]
abserr[k0] = err1[k0] + err2[k0] + E2[k0] * small * ten
k1, = (1 - converged).nonzero()
if k1.size > 0:
delta2, delta1 = E2 - E1, E1 - E0
err2, err1 = abs(delta2), abs(delta1)
tol2, tol1 = max(abs(E2), abs(E1)) * _EPS, max(abs(E1), abs(E0)) * _EPS
with warnings.catch_warnings():
# ignore division by zero and overflow
warnings.simplefilter("ignore")
ss = one / delta2[k1] - one / delta1[k1]
smallE2 = (abs(ss * E1[k1]) <= 1.0e-3).ravel()
k2 = k1[smallE2.nonzero()]
if k2.size > 0:
result[k2] = E2[k2]
abserr[k2] = err1[k2] + err2[k2] + E2[k2] * small * ten
k4, = (1 - smallE2).nonzero()
warnings.simplefilter("ignore") # ignore division by zero and overflow
ss = 1.0 / delta2 - 1.0 / delta1
smalle2 = (abs(ss * E1) <= 1.0e-3).ravel()
result = 1.0 * E2
abserr = err1 + err2 + E2 * _EPS * 10.0
converged = (err1 <= tol1) & (err2 <= tol2).ravel() | smalle2
k4, = (1 - converged).nonzero()
if k4.size > 0:
k3 = k1[k4]
result[k3] = E1[k3] + one / ss[k4]
abserr[k3] = err1[k3] + err2[k3] + abs(result[k3] - E2[k3])
result[k4] = E1[k4] + 1.0 / ss[k4]
abserr[k4] = err1[k4] + err2[k4] + abs(result[k4] - E2[k4])
return result, abserr
def clencurt(fun, a, b, n0=5, trace=False, *args):
def clencurt(fun, a, b, n0=5, trace=False, args=()):
'''
Numerical evaluation of an integral, Clenshaw-Curtis method.
@ -163,7 +138,7 @@ def clencurt(fun, a, b, n0=5, trace=False, *args):
Numerische Matematik, Vol. 2, pp. 197--205
'''
#% make sure n is even
# make sure n is even
n = 2 * n0
a, b = np.atleast_1d(a, b)
a_shape = a.shape
@ -290,17 +265,15 @@ def romberg(fun, a, b, releps=1e-3, abseps=1e-3):
fp[i] = 4 * fp[i - 1]
# Richardson extrapolation
for k in xrange(i):
# rom(2,k+1)=(fp(k)*rom(2,k)-rom(1,k))/(fp(k)-1)
rom[two, k + 1] = rom[two, k] + \
(rom[two, k] - rom[one, k]) / (fp[k] - 1)
Ih1 = Ih2
Ih2 = Ih4
Ih4 = rom[two, i]
if (2 <= i):
[res, abserr] = dea3(Ih1, Ih2, Ih4)
res, abserr = dea3(Ih1, Ih2, Ih4)
# Ih4 = res
if (abserr <= max(abseps, releps * abs(res))):
break
@ -386,8 +359,8 @@ def h_roots(n, method='newton'):
L[kp1, :] = PIM4
for j in xrange(1, n + 1):
#%Loop up the recurrence relation to get the Hermite
#%polynomials evaluated at z.
# Loop up the recurrence relation to get the Hermite
# polynomials evaluated at z.
km1 = k0
k0 = kp1
kp1 = np.mod(kp1 + 1, 3)
@ -468,7 +441,6 @@ def j_roots(n, alpha, beta, method='newton'):
if not method.startswith('n'):
[x, w] = ort.j_roots(n, alpha, beta)
else:
max_iter = 10
releps = 3e-14
@ -505,8 +477,9 @@ def j_roots(n, alpha, beta, method='newton'):
# We next compute pp, the derivatives with a standard
# relation involving the polynomials of one lower order.
pp = (n * (alpha - beta - tmp * z) * L[kp1, :] +
2 * (n + alpha) * (n + beta) * L[k0, :]) / (tmp * (1 - z ** 2))
pp = ((n * (alpha - beta - tmp * z) * L[kp1, :] +
2 * (n + alpha) * (n + beta) * L[k0, :]) /
(tmp * (1 - z ** 2)))
dz = L[kp1, :] / pp
z = z - dz # Newton's formula.
@ -592,7 +565,7 @@ def la_roots(n, alpha=0, method='newton'):
kp1 = 1
k = slice(len(z))
for _its in xrange(max_iter):
#%Newton's method carried out simultaneously on the roots.
# Newton's method carried out simultaneously on the roots.
L[k0, k] = 0.
L[kp1, k] = 1.
@ -606,16 +579,16 @@ def la_roots(n, alpha=0, method='newton'):
L[kp1, k] = ((2 * jj - 1 + alpha - z[k]) * L[
k0, k] - (jj - 1 + alpha) * L[km1, k]) / jj
# end
#%L now contains the desired Laguerre polynomials.
#%We next compute pp, the derivatives with a standard
#% relation involving the polynomials of one lower order.
# L now contains the desired Laguerre polynomials.
# We next compute pp, the derivatives with a standard
# relation involving the polynomials of one lower order.
Lp[k] = L[k0, k]
pp[k] = (n * L[kp1, k] - (n + alpha) * Lp[k]) / z[k]
dz[k] = L[kp1, k] / pp[k]
z[k] = z[k] - dz[k] # % Newton?s formula.
#%k = find((abs(dz) > releps.*z))
# k = find((abs(dz) > releps.*z))
if not np.any(abs(dz) > releps):
break
@ -886,8 +859,7 @@ def qrule(n, wfun=1, alpha=0, beta=0):
return bp, wf
def gaussq(fun, a, b, reltol=1e-3, abstol=1e-3, alpha=0, beta=0, wfun=1,
trace=False, args=None):
class _Gaussq(object):
'''
Numerically evaluate integral, Gauss quadrature.
@ -896,9 +868,9 @@ def gaussq(fun, a, b, reltol=1e-3, abstol=1e-3, alpha=0, beta=0, wfun=1,
fun : callable
a,b : array-like
lower and upper integration limits, respectively.
reltol, abstol : real scalars, optional
releps, abseps : real scalars, optional
relative and absolute tolerance, respectively.
(default reltol=abstool=1e-3).
(default releps=abseps=1e-3).
wfun : scalar integer, optional
defining the weight function, p(x). (default wfun = 1)
1 : p(x) = 1 a =-1, b = 1 Gauss-Legendre
@ -967,178 +939,137 @@ def gaussq(fun, a, b, reltol=1e-3, abstol=1e-3, alpha=0, beta=0, wfun=1,
qrule
gaussq2d
'''
global _POINTS_AND_WEIGHTS
max_iter = 11
gn = 2
if not hasattr(fun, '__call__'):
raise ValueError('Function must be callable')
A, B = np.atleast_1d(a, b)
a_shape = np.atleast_1d(A.shape)
b_shape = np.atleast_1d(B.shape)
# make sure the integration limits have correct size
if np.prod(a_shape) == 1:
A = A * ones(b_shape)
a_shape = b_shape
elif np.prod(b_shape) == 1:
B = B * ones(a_shape)
elif any(a_shape != b_shape):
raise ValueError('The integration limits must have equal size!')
if args is None:
num_parameters = 0
else:
num_parameters = len(args)
P0 = copy.deepcopy(args)
isvector1 = zeros(num_parameters)
nk = np.prod(a_shape) # % # of integrals we have to compute
for ix in xrange(num_parameters):
if is_numlike(P0[ix]):
p0_shape = np.shape(P0[ix])
Np0 = np.prod(p0_shape)
isvector1[ix] = (Np0 > 1)
if isvector1[ix]:
if nk == 1:
a_shape = p0_shape
nk = Np0
A = A * ones(a_shape)
B = B * ones(a_shape)
elif nk != Np0:
raise ValueError('The input must have equal size!')
P0[ix].shape = (-1, 1) # make sure it is a column
k = np.arange(nk)
val = zeros(nk)
val_old = zeros(nk)
abserr = zeros(nk)
# setup mapping parameters
A.shape = (-1, 1)
B.shape = (-1, 1)
jacob = (B - A) / 2
shift = 1
if wfun == 1: # Gauss-legendre
dx = jacob
elif wfun == 2 or wfun == 3:
shift = 0
jacob = ones((nk, 1))
A = zeros((nk, 1))
def _get_dx(self, wfun, jacob, alpha, beta):
if wfun in [1, 2, 3, 7]:
dx = jacob
elif wfun == 4:
dx = jacob ** (alpha + beta + 1)
elif wfun == 5:
dx = ones((nk, 1))
dx = ones((np.size(jacob), 1))
elif wfun == 6:
dx = jacob ** 2
elif wfun == 7:
shift = 0
jacob = jacob * 2
dx = jacob
elif wfun == 8:
shift = 0
jacob = jacob * 2
dx = sqrt(jacob)
elif wfun == 9:
shift = 0
jacob = jacob * 2
dx = sqrt(jacob) ** 3
else:
raise ValueError('unknown option')
return dx.ravel()
dx = dx.ravel()
if trace:
x_trace = [0, ] * max_iter
y_trace = [0, ] * max_iter
if num_parameters > 0:
ix_vec, = np.where(isvector1)
if len(ix_vec):
P1 = copy.copy(P0)
def _points_and_weights(self, gn, wfun, alpha, beta):
global _POINTS_AND_WEIGHTS
name = 'wfun%d_%d_%g_%g' % (wfun, gn, alpha, beta)
x_and_w = _POINTS_AND_WEIGHTS.setdefault(name, [])
if len(x_and_w) == 0:
x_and_w.extend(qrule(gn, wfun, alpha, beta))
xn, w = x_and_w
return xn, w
def _initialize_trace(self, max_iter):
if self.trace:
self.x_trace = [0] * max_iter
self.y_trace = [0] * max_iter
def _plot_trace(self, x, y):
if self.trace:
self.x_trace.append(x.ravel())
self.y_trace.append(y.ravel())
hfig = plt.plot(x, y, 'r.')
plt.setp(hfig, 'color', 'b')
# Break out of the iteration loop for three reasons:
# 1) the last update is very small (compared to int and to reltol)
# 2) There are more than 11 iterations. This should NEVER happen.
def _plot_final_trace(self):
if self.trace > 0:
plt.clf()
plt.plot(np.hstack(self.x_trace), np.hstack(self.y_trace), '+')
for ix in xrange(max_iter):
x_and_w = 'wfun%d_%d_%g_%g' % (wfun, gn, alpha, beta)
if x_and_w in _POINTS_AND_WEIGHTS:
xn, w = _POINTS_AND_WEIGHTS[x_and_w]
def _get_jacob(self, wfun, A, B):
if wfun in [2, 3]:
nk = np.size(A)
jacob = ones((nk, 1))
else:
xn, w = qrule(gn, wfun, alpha, beta)
_POINTS_AND_WEIGHTS[x_and_w] = (xn, w)
# calculate the x values
x = (xn + shift) * jacob[k, :] + A[k, :]
# calculate function values y=fun(x,p1,p2,....,pn)
if num_parameters > 0:
if len(ix_vec):
#% Expand vector to the correct size
for iy in ix_vec:
P1[iy] = P0[iy][k, :]
jacob = (B - A) * 0.5
if wfun in [7, 8, 9]:
jacob = jacob * 2
return jacob
y = fun(x, **P1)
def _warn(self, k, a_shape):
nk = len(k)
if nk > 1:
if (nk == np.prod(a_shape)):
tmptxt = 'All integrals did not converge'
else:
y = fun(x, **P0)
tmptxt = '%d integrals did not converge' % (nk, )
tmptxt = tmptxt + '--singularities likely!'
else:
y = fun(x)
tmptxt = 'Integral did not converge--singularity likely!'
warnings.warn(tmptxt)
def _initialize(self, wfun, a, b, args):
args = np.broadcast_arrays(*np.atleast_1d(a, b, *args))
a_shape = args[0].shape
args = map(lambda x: np.reshape(x, (-1, 1)), args)
A, B = args[:2]
args = args[2:]
if wfun in [2, 3]:
A = zeros((A.size, 1))
return A, B, args, a_shape
def __call__(self, fun, a, b, releps=1e-3, abseps=1e-3, alpha=0, beta=0,
wfun=1, trace=False, args=(), max_iter=11):
self.trace = trace
gn = 2
val[k] = np.sum(w * y, axis=1) * dx[k] # do the integration sum(y.*w)
A, B, args, a_shape = self._initialize(wfun, a, b, args)
if trace:
x_trace.append(x.ravel())
y_trace.append(y.ravel())
jacob = self._get_jacob(wfun, A, B)
shift = int(wfun in [1, 4, 5, 6])
dx = self._get_dx(wfun, jacob, alpha, beta)
hfig = plt.plot(x, y, 'r.')
# hold on
# drawnow,shg
# if trace>1:
# pause
self._initialize_trace(max_iter)
plt.setp(hfig, 'color', 'b')
# Break out of the iteration loop for three reasons:
# 1) the last update is very small (compared to int and to releps)
# 2) There are more than 11 iterations. This should NEVER happen.
dtype = np.result_type(fun((A+B)*0.5, *args))
nk = np.prod(a_shape) # # of integrals we have to compute
k = np.arange(nk)
opts = (nk, dtype)
val, val_old, abserr = zeros(*opts), ones(*opts), zeros(*opts)
for i in xrange(max_iter):
xn, w = self._points_and_weights(gn, wfun, alpha, beta)
x = (xn + shift) * jacob[k, :] + A[k, :]
pi = [xi[k, :] for xi in args]
y = fun(x, *pi)
self._plot_trace(x, y)
val[k] = np.sum(w * y, axis=1) * dx[k] # do the integration
if any(np.isnan(val)):
val[np.isnan(val)] = val_old[np.isnan(val)]
if 1 < i:
abserr[k] = abs(val_old[k] - val[k]) # absolute tolerance
if ix > 1:
k, = np.where(abserr > np.maximum(abs(reltol * val), abstol))
# abserr > abs(abstol))%indices to integrals which
# did not converge
k, = np.where(abserr > np.maximum(abs(releps * val), abseps))
nk = len(k) # of integrals we have to compute again
if nk:
val_old[k] = val[k]
else:
if nk == 0:
break
val_old[k] = val[k]
gn *= 2 # double the # of basepoints and weights
else:
if nk > 1:
if (nk == np.prod(a_shape)):
tmptxt = 'All integrals did not converge'
else:
tmptxt = '%d integrals did not converge' % (nk,)
else:
tmptxt = 'Integral did not converge--singularity likely!'
warnings.warn(tmptxt + '--singularities likely!')
self._warn(k, a_shape)
# make sure int is the same size as the integration limits
val.shape = a_shape
abserr.shape = a_shape
if trace > 0:
plt.clf()
plt.plot(np.hstack(x_trace), np.hstack(y_trace), '+')
self._plot_final_trace()
return val, abserr
gaussq = _Gaussq()
def richardson(Q, k):
# license BSD
# Richardson extrapolation with parameter estimation
c = np.real((Q[k - 1] - Q[k - 2]) / (Q[k] - Q[k - 1])) - 1.
#% The lower bound 0.07 admits the singularity x.^-0.9
# The lower bound 0.07 admits the singularity x.^-0.9
c = max(c, 0.07)
R = Q[k] + (Q[k] - Q[k - 1]) / c
return R
@ -1197,7 +1128,7 @@ def quadgr(fun, a, b, abseps=1e-5, max_iter=17):
else:
reverse = False
#% Infinite limits
# Infinite limits
if np.isinf(a) | np.isinf(b):
# Check real limits
if ~ np.isreal(a) | ~np.isreal(b) | np.isnan(a) | np.isnan(b):
@ -1235,14 +1166,9 @@ def quadgr(fun, a, b, abseps=1e-5, max_iter=17):
xq = np.hstack((xq, -xq))
wq = np.hstack((wq, wq))
nq = len(xq)
# iscomplex = (np.iscomplex(a) | np.iscomplex(b)).any()
# if iscomplex:
# dtype = np.complex128
# else:
dtype = np.float64
dtype = np.result_type(fun(a), fun(b))
# Initiate vectors
# max_iter = 17 # Max number of iterations
Q0 = zeros(max_iter, dtype=dtype) # Quadrature
Q1 = zeros(max_iter, dtype=dtype) # First Richardson extrapolation
Q2 = zeros(max_iter, dtype=dtype) # Second Richardson extrapolation
@ -1270,7 +1196,7 @@ def quadgr(fun, a, b, abseps=1e-5, max_iter=17):
elif k >= 3:
Q1[k] = richardson(Q0, k)
#% Estimate absolute error
# Estimate absolute error
if k >= 6:
Qv = np.hstack((Q0[k], Q1[k], Q2[k]))
Qw = np.hstack((Q0[k - 1], Q1[k - 1], Q2[k - 1]))
@ -1306,7 +1232,16 @@ def quadgr(fun, a, b, abseps=1e-5, max_iter=17):
return Q, err
def qdemo(f, a, b):
def boole(y, x):
a, b = x[0], x[-1]
n = len(x)
h = (b - a) / (n - 1)
return (2 * h / 45) * (7 * (y[0] + y[-1]) + 12 * np.sum(y[2:n - 1:4]) +
32 * np.sum(y[1:n - 1:2]) +
14 * np.sum(y[4:n - 3:4]))
def qdemo(f, a, b, kmax=9, plot_error=False):
'''
Compares different quadrature rules.
@ -1333,122 +1268,116 @@ def qdemo(f, a, b):
>>> import numpy as np
>>> qdemo(np.exp,0,3)
true value = 19.08553692
ftn Trapezoid Simpsons Booles
evals approx error approx error approx error
3, 22.5366862979, 3.4511493747, 19.5061466023, 0.4206096791, 19.4008539142, 0.3153169910
5, 19.9718950387, 0.8863581155, 19.1169646189, 0.0314276957, 19.0910191534, 0.0054822302
9, 19.3086731081, 0.2231361849, 19.0875991312, 0.0020622080, 19.0856414320, 0.0001045088
17, 19.1414188470, 0.0558819239, 19.0856674267, 0.0001305035, 19.0855386464, 0.0000017232
33, 19.0995135407, 0.0139766175, 19.0855451052, 0.0000081821, 19.0855369505, 0.0000000273
65, 19.0890314614, 0.0034945382, 19.0855374350, 0.0000005118, 19.0855369236, 0.0000000004
129, 19.0864105817, 0.0008736585, 19.0855369552, 0.0000000320, 19.0855369232, 0.0000000000
257, 19.0857553393, 0.0002184161, 19.0855369252, 0.0000000020, 19.0855369232, 0.0000000000
513, 19.0855915273, 0.0000546041, 19.0855369233, 0.0000000001, 19.0855369232, 0.0000000000
ftn Clenshaw Chebychev Gauss-L
evals approx error approx error approx error
3, 19.5061466023, 0.4206096791, 0.0000000000, 1.0000000000, 19.0803304585, 0.0052064647
5, 19.0834145766, 0.0021223465, 0.0000000000, 1.0000000000, 19.0855365951, 0.0000003281
9, 19.0855369150, 0.0000000082, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
17, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
33, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
65, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
129, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
257, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
513, 19.0855369232, 0.0000000000, 0.0000000000, 1.0000000000, 19.0855369232, 0.0000000000
ftn, Boole, Chebychev
evals approx error approx error
3, 19.4008539142, 0.3153169910, 19.5061466023, 0.4206096791
5, 19.0910191534, 0.0054822302, 19.0910191534, 0.0054822302
9, 19.0856414320, 0.0001045088, 19.0855374134, 0.0000004902
17, 19.0855386464, 0.0000017232, 19.0855369232, 0.0000000000
33, 19.0855369505, 0.0000000273, 19.0855369232, 0.0000000000
65, 19.0855369236, 0.0000000004, 19.0855369232, 0.0000000000
129, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
257, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
513, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
ftn, Clenshaw-Curtis, Gauss-Legendre
evals approx error approx error
3, 19.5061466023, 0.4206096791, 19.0803304585, 0.0052064647
5, 19.0834145766, 0.0021223465, 19.0855365951, 0.0000003281
9, 19.0855369150, 0.0000000082, 19.0855369232, 0.0000000000
17, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
33, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
65, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
129, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
257, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
513, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000
ftn, Simps, Trapz
evals approx error approx error
3, 19.5061466023, 0.4206096791, 22.5366862979, 3.4511493747
5, 19.1169646189, 0.0314276957, 19.9718950387, 0.8863581155
9, 19.0875991312, 0.0020622080, 19.3086731081, 0.2231361849
17, 19.0856674267, 0.0001305035, 19.1414188470, 0.0558819239
33, 19.0855451052, 0.0000081821, 19.0995135407, 0.0139766175
65, 19.0855374350, 0.0000005118, 19.0890314614, 0.0034945382
129, 19.0855369552, 0.0000000320, 19.0864105817, 0.0008736585
257, 19.0855369252, 0.0000000020, 19.0857553393, 0.0002184161
513, 19.0855369233, 0.0000000001, 19.0855915273, 0.0000546041
'''
# use quad8 with small tolerance to get "true" value
#true1 = quad8(f,a,b,1e-10)
#[true tol]= gaussq(f,a,b,1e-12)
#[true tol] = agakron(f,a,b,1e-13)
true_val, _tol = intg.quad(f, a, b)
print('true value = %12.8f' % (true_val,))
kmax = 9
neval = zeros(kmax, dtype=int)
qt = zeros(kmax)
qs = zeros(kmax)
qb = zeros(kmax)
qc = zeros(kmax)
qc2 = zeros(kmax)
qg = zeros(kmax)
et = ones(kmax)
es = ones(kmax)
eb = ones(kmax)
ec = ones(kmax)
ec2 = ones(kmax)
ec3 = ones(kmax)
eg = ones(kmax)
vals_dic = {}
err_dic = {}
# try various approximations
methods = [trapz, simps, boole, ]
for k in xrange(kmax):
n = 2 ** (k + 1) + 1
neval[k] = n
h = (b - a) / (n - 1)
x = np.linspace(a, b, n)
y = f(x)
# trapezoid approximation
q = np.trapz(y, x)
# h*( (y(1)+y(n))/2 + sum(y(2:n-1)) )
qt[k] = q
et[k] = abs(q - true_val)
# Simpson approximation
q = intg.simps(y, x)
#(h/3)*( y(1)+y(n) + 4*sum(y(2:2:n-1)) + 2*sum(y(3:2:n-2)) )
qs[k] = q
es[k] = abs(q - true_val)
# Boole's rule
#q = boole(x,y)
q = (2 * h / 45) * (7 * (y[0] + y[-1]) + 12 * np.sum(y[2:n - 1:4])
+ 32 * np.sum(y[1:n - 1:2]) +
14 * np.sum(y[4:n - 3:4]))
qb[k] = q
eb[k] = abs(q - true_val)
# Clenshaw-Curtis
[q, ec3[k]] = clencurt(f, a, b, (n - 1) / 2)
qc[k] = q
ec[k] = abs(q - true_val)
# Chebychev
for method in methods:
name = method.__name__.title()
q = method(y, x)
vals_dic.setdefault(name, []).append(q)
err_dic.setdefault(name, []).append(abs(q - true_val))
name = 'Clenshaw-Curtis'
q, _ec3 = clencurt(f, a, b, (n - 1) / 2)
vals_dic.setdefault(name, []).append(q[0])
err_dic.setdefault(name, []).append(abs(q[0] - true_val))
name = 'Chebychev'
ck = np.polynomial.chebyshev.chebfit(x, y, deg=min(n-1, 36))
cki = np.polynomial.chebyshev.chebint(ck)
q = np.polynomial.chebyshev.chebval(x[-1], cki)
vals_dic.setdefault(name, []).append(q)
err_dic.setdefault(name, []).append(abs(q - true_val))
# ck = chebfit(f,n,a,b)
# q = chebval(b,chebint(ck,a,b),a,b)
# qc2[k] = q; ec2[k] = abs(q - true)
# Gauss-Legendre quadrature
name = 'Gauss-Legendre' # quadrature
q = intg.fixed_quad(f, a, b, n=n)[0]
# [x, w]=qrule(n,1)
# x = (b-a)/2*x + (a+b)/2 % Transform base points X.
# w = (b-a)/2*w % Adjust weigths.
# q = sum(feval(f,x)*w)
qg[k] = q
eg[k] = abs(q - true_val)
#% display results
formats = ['%4.0f, ', ] + ['%10.10f, ', ] * 6
formats[-1] = formats[-1].split(',')[0]
data = np.vstack((neval, qt, et, qs, es, qb, eb)).T
print(' ftn Trapezoid Simpson''s Boole''s') # @IgnorePep8
print('evals approx error approx error approx error') # @IgnorePep8
for k in xrange(kmax):
tmp = data[k].tolist()
print(''.join(fi % t for fi, t in zip(formats, tmp)))
vals_dic.setdefault(name, []).append(q)
err_dic.setdefault(name, []).append(abs(q - true_val))
# display results
data = np.vstack((neval, qc, ec, qc2, ec2, qg, eg)).T
print(' ftn Clenshaw Chebychev Gauss-L') # @IgnorePep8
print('evals approx error approx error approx error') # @IgnorePep8
names = sorted(vals_dic.keys())
num_cols = 2
formats = ['%4.0f, ', ] + ['%10.10f, ', ] * num_cols * 2
formats[-1] = formats[-1].split(',')[0]
formats_h = ['%4s, ', ] + ['%20s, ', ] * num_cols
formats_h[-1] = formats_h[-1].split(',')[0]
headers = ['evals'] + ['%12s %12s' % ('approx', 'error')] * num_cols
while len(names) > 0:
print(''.join(fi % t for fi, t in zip(formats_h,
['ftn'] + names[:num_cols])))
print(' '.join(headers))
data = [neval]
for name in names[:num_cols]:
data.append(vals_dic[name])
data.append(err_dic[name])
data = np.vstack(tuple(data)).T
for k in xrange(kmax):
tmp = data[k].tolist()
print(''.join(fi % t for fi, t in zip(formats, tmp)))
if plot_error:
plt.figure(0)
for name in names[:num_cols]:
plt.loglog(neval, err_dic[name], label=name)
plt.loglog(neval, np.vstack((et, es, eb, ec, ec2, eg)).T)
names = names[num_cols:]
if plot_error:
plt.xlabel('number of function evaluations')
plt.ylabel('error')
plt.legend(
('Trapezoid', 'Simpsons', 'Booles', 'Clenshaw', 'Chebychev', 'Gauss-L')) # @IgnorePep8
# ec3'
plt.legend()
plt.show('hold')
def main():
@ -1493,6 +1422,9 @@ def test_docstrings():
import doctest
doctest.testmod()
if __name__ == '__main__':
test_docstrings()
# qdemo(np.exp, 0, 3, plot_error=True)
# plt.show('hold')
# main()

22
pywafo/src/wafo/interpolate.py
Unescape Escape View File

@ -1,19 +1,8 @@
#-------------------------------------------------------------------------
# Name: module1
# Purpose:
#
# Author: pab
#
# Created: 30.12.2008
# Copyright: (c) pab 2008
# Licence: <your licence>
#-------------------------------------------------------------------------
#!/usr/bin/env python
from __future__ import division
import numpy as np
import scipy.signal
#import scipy.special as spec
import scipy.sparse.linalg # @UnusedImport
# import scipy.sparse.linalg # @UnusedImport
import scipy.sparse as sparse
from numpy.ma.core import ones, zeros, prod, sin
from numpy import diff, pi, inf # @UnresolvedImport
@ -270,7 +259,7 @@ def sgolay2d(z, window_size, order, derivative=None):
np.fliplr(Z[-half_size:, half_size + 1:2 * half_size + 1]) - band)
# solve system and convolve
if derivative == None:
if derivative is None:
m = np.linalg.pinv(A)[0].reshape((window_size, -1))
return scipy.signal.fftconvolve(Z, m, mode='valid')
elif derivative == 'col':
@ -554,10 +543,9 @@ class SmoothSpline(PPform):
zrs = zeros(nd)
if p < 1:
# faster than yi-6*(1-p)*Q*u
ai = (y - (6 * (1 - p) * D *
diff(vstack([zrs,
diff(vstack([zrs, u, zrs]), axis=0) * dx1,
zrs]), axis=0)).T).T
Qu = D * diff(vstack([zrs, diff(vstack([zrs, u, zrs]),
axis=0) * dx1, zrs]), axis=0)
ai = (y - (6 * (1 - p) * Qu).T).T
else:
ai = y.reshape(n, -1)

151
pywafo/src/wafo/spectrum/core.py
Unescape Escape View File

@ -285,7 +285,7 @@ def plotspec(specdata, linetype='b-', flag=1):
# txt = ''.join(txt)
# if (flag == 3):
# plotbackend.subplot(2, 1, 1)
# if (flag == 1) or (flag == 3):#% Plot in normal scale
# if (flag == 1) or (flag == 3):# Plot in normal scale
# plotbackend.plot(np.vstack([Fp, Fp]),
# np.vstack([zeros(len(indm)), data.take(indm)]),
# ':', label=txt)
@ -1120,11 +1120,11 @@ class SpecData1D(PlotData):
Hw2 = Hw1
# end
#%Hw1(end)
#%maxS*1e-3
#%if Hw1[-1]*S.data>maxS*1e-3,
#% warning('The Nyquist frequency of the spectrum may be too low')
#%end
# Hw1(end)
# maxS*1e-3
# if Hw1[-1]*S.data>maxS*1e-3,
# warning('The Nyquist frequency of the spectrum may be too low')
# end
SL.date = now() # datestr(now)
# if nargout>1
@ -1354,11 +1354,11 @@ class SpecData1D(PlotData):
B_up = hstack([un + XtInf, XdInf, 0])
B_lo = hstack([un, 0, -XdInf])
#%INFIN = [1 1 0]
# INFIN = [1 1 0]
# BIG = zeros((Ntime+2,Ntime+2))
ex = zeros(Ntime + 2, dtype=float)
#%CC = 2*pi*sqrt(-R(1,1)/R(1,3))*exp(un^2/(2*R(1,1)))
#% XcScale = log(CC)
# CC = 2*pi*sqrt(-R(1,1)/R(1,3))*exp(un^2/(2*R(1,1)))
# XcScale = log(CC)
opts['xcscale'] = log(
2 * pi * sqrt(-R[0, 0] / R[0, 2])) + (un ** 2 / (2 * R[0, 0]))
@ -1376,7 +1376,7 @@ class SpecData1D(PlotData):
indI[2] = Nt
indI[3] = Ntd - 1
#% positive wave period
# positive wave period
BIG = self._covinput_t_pdf(pt, R)
tmp = rind(BIG, ex[:Ntdc], B_lo, B_up, indI, xc, Nt)
@ -1450,14 +1450,14 @@ class SpecData1D(PlotData):
Scd = array([[0, R[pt, 1]], [-R[pt, 1], 0]])
if pt > 1:
#%cov(Xt)
# cov(Xt)
# Cov(X(tn),X(ts)) = r(ts-tn) = r(|ts-tn|)
Stt = toeplitz(R[:pt - 1, 0])
#%cov(Xc,Xt)
# cov(Xc,Xt)
# Cov(X(tn),X(ts)) = r(ts-tn) = r(|ts-tn|)
Sct = R[1:pt, 0]
Sct = vstack((Sct, Sct[::-1]))
#%Cov(Xd,Xt)
# Cov(Xd,Xt)
# Cov(X'(t1),X(ts)) = -r'(ts-t1) = r(|s-t|)
Sdt = -R[1:pt, 1]
Sdt = vstack((Sdt, -Sdt[::-1]))
@ -1623,7 +1623,6 @@ class SpecData1D(PlotData):
'Discretization levels must be larger than zero')
# Transform amplitudes to Gaussian levels:
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
h = linspace(*paramu)
if defnr > 1: # level v separated Max2min densities
@ -2045,9 +2044,9 @@ class SpecData1D(PlotData):
# end %do
# close(h11)
err = sqrt(err)
#% goto 800
# goto 800
else: # def_nr>3
#%200 continue
# 200 continue
# waitTxt = sprintf('Please wait ...(start at: %s)',datestr(now))
# h11 = fwaitbar(0,[],waitTxt)
tnold = -1
@ -2095,14 +2094,14 @@ class SpecData1D(PlotData):
# end
elif def_nr == 5:
if (Nx == 1):
#% Joint density (Tdm,TMm) given the Max and the min.
#% Note the density is not scaled to unity
# Joint density (Tdm,TMm) given the Max and the min.
# Note the density is not scaled to unity
pdf[0, tn - ts, tn] = fxind[0] # %*CC
err[0, tn - ts, tn] = err0[0] ** 2
terr[0, tn - ts, tn] = terr0[0]
else:
#% 5, gives level u separated Max2min and wave period from
#% the crossing of level u to the min (M,m,Tdm).
# 5, gives level u separated Max2min and wave period from
# the crossing of level u to the min (M,m,Tdm).
IJ = 0
for i in range(1, Nx1): # = 2:Nx1
@ -2120,11 +2119,11 @@ class SpecData1D(PlotData):
# end % SELECT
# end% enddo
else: # % exploit symmetry
#%300 Symmetry
# 300 Symmetry
for ts in range(1, Ntd // 2): # = 2:floor(Ntd//2)
#% Using the symmetry since U = 0 and the transformation is
#% linear.
#% positive wave period
# Using the symmetry since U = 0 and the transformation is
# linear.
# positive wave period
BIG[:Ntdc, :Ntdc] = covinput(BIG[:Ntdc, :Ntdc],
R0, R1, R2, R3, R4,
tn, ts, tnold)
@ -2132,11 +2131,11 @@ class SpecData1D(PlotData):
a_lo, a_up, indI, xc, Nt, *opt0)
#[fxind,err0] = rind(BIG(1:Ntdc,1:Ntdc),ex,a_lo,a_up,indI, xc,Nt,opt0{:})
#%tnold = tn
# tnold = tn
if (Nx == 1): # % THEN
#% Joint density of (TMd,TMm),(Tdm,TMm) given the max and
#% the min.
#% Note that the density is not scaled to unity
# Joint density of (TMd,TMm),(Tdm,TMm) given the max and
# the min.
# Note that the density is not scaled to unity
pdf[0, ts, tn] = fxind[0] # %*CC
err[0, ts, tn] = err0[0] ** 2
err[0, ts, tn] = terr0(1)
@ -2145,12 +2144,12 @@ class SpecData1D(PlotData):
err[0, tn - ts, tn] = err0[0] ** 2
terr[0, tn - ts, tn] = terr0[0]
# end
#%GOTO 350
# GOTO 350
else:
IJ = 0
if def_nr == 4:
#% 4, gives level u separated Max2min and wave period from
#% Max to the crossing of level u (M,m,TMd).
# 4, gives level u separated Max2min and wave period from
# Max to the crossing of level u (M,m,TMd).
for i in range(1, Nx1):
J = IJ + Nx1
pdf[1:Nx1, i, ts] = pdf[
@ -2160,7 +2159,7 @@ class SpecData1D(PlotData):
terr[1:Nx1, i, ts] = terr[
1:Nx1, i, ts] + (terr0[IJ:J] * dt)
if (ts < tn - ts):
#% exploiting the symmetry
# exploiting the symmetry
# %*CC
pdf[i, 1:Nx1, tn - ts] = pdf[i,
1:Nx1, tn - ts] + fxind[IJ:J] * dt
@ -2172,8 +2171,8 @@ class SpecData1D(PlotData):
IJ = J
# end %do
elif def_nr == 5:
#% 5, gives level u separated Max2min and wave period
#% from the crossing of level u to min (M,m,Tdm).
# 5, gives level u separated Max2min and wave period
# from the crossing of level u to min (M,m,Tdm).
for i in range(1, Nx1): # = 2:Nx1,
J = IJ + Nx1
pdf[1:Nx1, i, tn - ts] = pdf[1:Nx1,
@ -2195,7 +2194,7 @@ class SpecData1D(PlotData):
# end %do
# end %END SELECT
# end
#%350
# 350
# end %do
# end
# waitTxt = sprintf('%s Ready: %d of %d',datestr(now),tn,Ntime)
@ -2206,7 +2205,7 @@ class SpecData1D(PlotData):
err = sqrt(err)
# end % if
#%Nx1,size(pdf) def Ntime
# Nx1,size(pdf) def Ntime
if (Nx > 1): # % THEN
IJ = 1
if (def_nr > 2 or def_nr == 1):
@ -2267,14 +2266,14 @@ class SpecData1D(PlotData):
# Cov(Xt,Xc)
# for
i = np.arange(tn - 2) # 1:tn-2
#%j = abs(i+1-ts)
#%BIG(i,N) = -sign(R1(j+1),R1(j+1)*dble(ts-i-1)) %cov(X'(ti+1),X(ts))
# j = abs(i+1-ts)
# BIG(i,N) = -sign(R1(j+1),R1(j+1)*dble(ts-i-1)) %cov(X'(ti+1),X(ts))
j = i + 1 - ts
tau = abs(j)
#%BIG(i,N) = abs(R1(tau)).*sign(R1(tau).*j.')
# BIG(i,N) = abs(R1(tau)).*sign(R1(tau).*j.')
BIG[i, N] = R1[tau] * sign(j)
#%end do
#%Cov(Xc)
# end do
# Cov(Xc)
BIG[N, N] = R0[0] # cov(X(ts),X(ts))
BIG[tn + shft + 1, N] = -R1[ts] # cov(X'(t1),X(ts))
BIG[tn + shft + 2, N] = R1[tn - ts] # cov(X'(tn),X(ts))
@ -2284,20 +2283,20 @@ class SpecData1D(PlotData):
BIG[tn - 1, N] = R2[ts] # %cov(X''(t1),X(ts))
BIG[tn, N] = R2[tn - ts] # %cov(X''(tn),X(ts))
#%ADD a level u crossing at ts
# ADD a level u crossing at ts
#%Cov(Xt,Xd)
#%for
# Cov(Xt,Xd)
# for
i = np.arange(tn - 2) # 1:tn-2
j = abs(i + 1 - ts)
BIG[i, tn + shft] = -R2[j] # %cov(X'(ti+1),X'(ts))
#%end do
#%Cov(Xd)
# end do
# Cov(Xd)
BIG[tn + shft, tn + shft] = -R2[0] # %cov(X'(ts),X'(ts))
BIG[tn - 1, tn + shft] = R3[ts] # %cov(X''(t1),X'(ts))
BIG[tn, tn + shft] = -R3[tn - ts] # %cov(X''(tn),X'(ts))
#%Cov(Xd,Xc)
# Cov(Xd,Xc)
BIG[tn + shft, N] = 0.0 # %cov(X'(ts),X(ts))
# % cov(X'(ts),X'(t1))
BIG[tn + shft, tn + shft + 1] = -R2[ts]
@ -2308,19 +2307,19 @@ class SpecData1D(PlotData):
BIG[tn + shft, tn + shft + 4] = -R1[tn - ts]
if (tnold == tn):
#% A previous call to covinput with tn==tnold has been made
#% need only to update row and column N and tn+1 of big:
# A previous call to covinput with tn==tnold has been made
# need only to update row and column N and tn+1 of big:
return BIG
#% % make lower triangular part equal to upper and then return
#% for j=1:tn+shft
#% BIG(N,j) = BIG(j,N)
#% BIG(tn+shft,j) = BIG(j,tn+shft)
#% end
#% for j=tn+shft+1:N-1
#% BIG(N,j) = BIG(j,N)
#% BIG(j,tn+shft) = BIG(tn+shft,j)
#% end
#% return
# % make lower triangular part equal to upper and then return
# for j=1:tn+shft
# BIG(N,j) = BIG(j,N)
# BIG(tn+shft,j) = BIG(j,tn+shft)
# end
# for j=tn+shft+1:N-1
# BIG(N,j) = BIG(j,N)
# BIG(j,tn+shft) = BIG(tn+shft,j)
# end
# return
# end %if
# %tnold = tn
else:
@ -2329,11 +2328,11 @@ class SpecData1D(PlotData):
# end %if
if (tn > 2):
#%for i=1:tn-2
#%cov(Xt)
#% for j=i:tn-2
#% BIG(i,j) = -R2(j-i+1) % cov(X'(ti+1),X'(tj+1))
#% end %do
# for i=1:tn-2
# cov(Xt)
# for j=i:tn-2
# BIG(i,j) = -R2(j-i+1) % cov(X'(ti+1),X'(tj+1))
# end %do
# % cov(Xt) = % cov(X'(ti+1),X'(tj+1))
BIG[:tn - 2, :tn - 2] = toeplitz(-R2[:tn - 2])
@ -2346,17 +2345,17 @@ class SpecData1D(PlotData):
# cov(X'(ti+1),X(tn))
BIG[:tn - 2, tn + shft + 3] = -R1[tn - 2:0:-1]
#%Cov(Xt,Xd)
# Cov(Xt,Xd)
BIG[:tn - 2, tn - 2] = R3[1:tn - 1] # cov(X'(ti+1),X''(t1))
BIG[:tn - 2, tn - 1] = -R3[tn - 2:0:-1] # cov(X'(ti+1),X''(tn))
#%end %do
# end %do
# end
#%cov(Xd)
# cov(Xd)
BIG[tn - 2, tn - 2] = R4[0]
BIG[tn - 2, tn - 1] = R4[tn - 1] # cov(X''(t1),X''(tn))
BIG[tn - 1, tn - 1] = R4[0]
#%cov(Xc)
# cov(Xc)
BIG[tn + shft + 2, tn + shft + 2] = R0[0] # cov(X(t1),X(t1))
# cov(X(t1),X(tn))
BIG[tn + shft + 2, tn + shft + 3] = R0[tn - 1]
@ -3016,7 +3015,7 @@ class SpecData1D(PlotData):
Vol. 3, pp.1-15
"""
#% default options
# default options
if h is None:
h = self.h
@ -3055,7 +3054,7 @@ class SpecData1D(PlotData):
else:
raise ValueError('Unknown option!')
# elif method[0]== 'q': #, #% quasi method
# elif method[0]== 'q': #, # quasi method
# Fn = self.nyquist_freq()
# dw = Fn/Nw
# tmp1 =sqrt(S[:,newaxis]*S[newaxis,:])*dw
@ -3373,7 +3372,7 @@ class SpecData1D(PlotData):
w = self.args.ravel()
n = w.size
#%doInterpolate = 0
# doInterpolate = 0
# Nyquist to sampling interval factor
Cnf2dt = 0.5 if ftype == 'f' else pi # % ftype == w og ftype == k
@ -3762,8 +3761,8 @@ class SpecData1D(PlotData):
** 2 + m[2] * mij[8] / m[4] ** 3),
m_11 / m[0] ** 2 + (m[5] / m[0] ** 2) ** 2 *
mij[0] - 2 * m[5] / m[0] ** 3 * m_10,
nan,
(8 * pi / g) ** 2 * (m[2] ** 2 / (4 * m[0] ** 3) *
nan, (8 * pi / g) ** 2 *
(m[2] ** 2 / (4 * m[0] ** 3) *
mij[0] + mij[4] / m[0] - m[2] / m[0] ** 2 * mij[2]),
nan * ones(4),
m[2] ** 2 * mij[0] / (4 * m[0] ** 3 * m[4]) + mij[4] /
@ -4093,11 +4092,11 @@ class SpecData2D(PlotData):
if self.type not in two_dim_spectra:
raise ValueError('Unknown 2D spectrum type!')
if vari == None and nr <= 1:
if vari is None and nr <= 1:
vari = 'x'
elif vari == None:
elif vari is None:
vari = 'xt'
else: # % secure the mutual order ('xyt')
else: # secure the mutual order ('xyt')
vari = ''.join(sorted(vari.lower()))
Nv = len(vari)

72
pywafo/src/wafo/test/test_integrate.py
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'''
Created on 23. okt. 2014
@author: pab
'''
import unittest
import numpy as np
from numpy import exp, Inf
from numpy.testing import assert_array_almost_equal
from wafo.integrate import gaussq
class Gaussq(unittest.TestCase):
'''
1 : p(x) = 1 a =-1, b = 1 Gauss-Legendre
2 : p(x) = exp(-x^2) a =-inf, b = inf Hermite
3 : p(x) = x^alpha*exp(-x) a = 0, b = inf Laguerre
4 : p(x) = (x-a)^alpha*(b-x)^beta a =-1, b = 1 Jacobi
5 : p(x) = 1/sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 1'st kind
6 : p(x) = sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 2'nd kind
7 : p(x) = sqrt((x-a)/(b-x)), a = 0, b = 1
8 : p(x) = 1/sqrt(b-x), a = 0, b = 1
9 : p(x) = sqrt(b-x), a = 0, b = 1
'''
def test_gauss_legendre(self):
val, _err = gaussq(exp, 0, 1)
self.assertAlmostEqual(val, exp(1)-exp(0))
a, b, y = [0, 0], [1, 1], np.array([1., 2.])
val, _err = gaussq(lambda x, y: x * y, a, b, args=(y, ))
assert_array_almost_equal(val, 0.5*y)
def test_gauss_hermite(self):
f = lambda x: x
val, _err = gaussq(f, -Inf, Inf, wfun=2)
self.assertAlmostEqual(val, 0)
def test_gauss_laguerre(self):
f = lambda x: x
val, _err = gaussq(f, 0, Inf, wfun=3, alpha=1)
self.assertAlmostEqual(val, 2)
def test_gauss_jacobi(self):
f = lambda x: x
val, _err = gaussq(f, -1, 1, wfun=4, alpha=-0.5, beta=-0.5)
self.assertAlmostEqual(val, 0)
def test_gauss_wfun5_6(self):
f = lambda x: x
for i in [5, 6]:
val, _err = gaussq(f, -1, 1, wfun=i)
self.assertAlmostEqual(val, 0)
def test_gauss_wfun7(self):
f = lambda x: x
val, _err = gaussq(f, 0, 1, wfun=7)
self.assertAlmostEqual(val, 1.17809725)
def test_gauss_wfun8(self):
f = lambda x: x
val, _err = gaussq(f, 0, 1, wfun=8)
self.assertAlmostEqual(val, 1.33333333)
def test_gauss_wfun9(self):
f = lambda x: x
val, _err = gaussq(f, 0, 1, wfun=9)
self.assertAlmostEqual(val, 0.26666667)
if __name__ == "__main__":
#import sys;sys.argv = ['', 'Test.testName']
unittest.main()
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