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Simplified things.

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master
Per A Brodtkorb 8 years ago
parent 5a948ffbbc
commit 3f767d6024
1 changed files with 59 additions and 28 deletions
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wafo/padua.py
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@ -85,7 +85,8 @@ Contents.m : Contents file for Matlab
from __future__ import absolute_import, division
import numpy as np
from numpy.fft import fft
from .dctpack import dct
from wafo.dctpack import dct
from wafo.polynomial import map_from_interval, map_to_interval
# from scipy.fftpack.realtransforms import dct
@ -97,8 +98,8 @@ class _ExampleFunctions(object):
----------
x,y : array-like
evaluate the function in the points (x,y)
id : scalar int (default 0)
id defining which test function to use. Options are
i : scalar int (default 0)
defining which test function to use. Options are
0: franke
1: half_sphere
2: poly_degree20
@ -132,10 +133,11 @@ class _ExampleFunctions(object):
Maple: 0.40696958949155611906
'''
exp = np.exp
return (3. / 4 * exp(-((9. * x - 2)**2 + (9. * y - 2)**2) / 4) +
3. / 4 * exp(-(9. * x + 1)**2 / 49 - (9. * y + 1) / 10) +
1. / 2 * exp(-((9. * x - 7)**2 + (9. * y - 3)**2) / 4) -
1. / 5 * exp(-(9. * x - 4)**2 - (9. * y - 7)**2))
x9, y9 = 9. * x, 9. * y
return (3. / 4 * exp(-((x9 - 2)**2 + (y9 - 2)**2) / 4) +
3. / 4 * exp(-(x9 + 1)**2 / 49 - (y9 + 1) / 10) +
1. / 2 * exp(-((x9 - 7)**2 + (y9 - 3)**2) / 4) -
1. / 5 * exp(-(x9 - 4)**2 - (y9 - 7)**2))
@staticmethod
def half_sphere(x, y):
@ -192,7 +194,7 @@ class _ExampleFunctions(object):
The value of the definite integral on the square [-1,1] x [-1,1]
is 4.
'''
return np.ones(np.shape(x))
return np.ones(np.shape(x+y))
@staticmethod
def exp_xy(x, y):
@ -274,12 +276,12 @@ class _ExampleFunctions(object):
arg_z = 1. / arg_z
return np.exp(-arg_z)
def __call__(self, x, y, id=0): # @ReservedAssignment
def __call__(self, x, y, i=0):
s = self
test_function = [s.franke, s.half_sphere, s.poly_degree20, s.exp_fun1,
s.exp_fun100, s.cos30, s.constant, s.exp_xy, s.runge,
s.abs_cubed, s.gauss, s.exp_inv]
return test_function[id](x, y)
return test_function[i](x, y)
example_functions = _ExampleFunctions()
@ -315,8 +317,8 @@ def padua_points(n, domain=(-1, 1, -1, 1)):
a, b, c, d = domain
t0 = [np.pi] if n == 0 else np.linspace(0, np.pi, n + 1)
t1 = np.linspace(0, np.pi, n + 2)
zn = (a + b + (b - a) * np.cos(t0)) / 2
zn1 = (c + d + (d - c) * np.cos(t1)) / 2
zn = map_to_interval(np.cos(t0), a, b)
zn1 = map_to_interval(np.cos(t1), c, d)
Pad1, Pad2 = np.meshgrid(zn, zn1)
ix = _find_m(n)
@ -357,6 +359,30 @@ def padua_fit(Pad, fun, *args):
coefficents: coefficient matrix
abs_err : interpolation error estimate
Example
------
>>> import numpy as np
>>> import wafo.padua as wp
>>> domain = [0, 1, 0, 1]
>>> a, b, c, d = domain
>>> points = wp.padua_points(21, domain)
>>> C0f, abs_error = wp.padua_fit(points, wp.example_functions.franke)
>>> x1 = np.linspace(a, b, 100)
>>> x2 = np.linspace(c, d, 101)
>>> val = wp.padua_val(x1, x2, C0f, domain, use_meshgrid=True)
>>> X1, X2 = np.meshgrid(x1, x2)
>>> true_val = wp.example_functions.franke(X1, X2)
>>> np.allclose(val, true_val, atol=10*abs_error)
True
>>> np.allclose(np.abs(val-true_val).max(), 0.0073174614275738296)
True
>>> np.allclose(abs_error, 0.0022486904061664046)
True
import matplotlib.pyplot as plt
plt.contour(x1, x2, val)
'''
N = np.shape(Pad)[1]
@ -387,7 +413,6 @@ def padua_fit(Pad, fun, *args):
def paduavals2coefs(f):
useFFTwhenNisMoreThan = 100
m = len(f)
n = int(round(-1.5 + np.sqrt(.25 + 2 * m)))
x = padua_points(n)
@ -401,16 +426,15 @@ def paduavals2coefs(f):
G = np.zeros(idx.max() + 1)
G[idx] = 4 * w * f
if (n < useFFTwhenNisMoreThan):
use_dct = 100 < n
if use_dct:
C = np.rot90(dct(dct(G.T).T)) # , axis=1)
else:
t1 = np.r_[0:n + 1].reshape(-1, 1)
Tn1 = np.cos(t1 * t1.T * np.pi / n)
t2 = np.r_[0:n + 2].reshape(-1, 1)
Tn2 = np.cos(t2 * t2.T * np.pi / (n + 1))
C = np.dot(Tn2, np.dot(G, Tn1))
else:
# dct = @(c) chebtech2.coeffs2vals(c);
C = np.rot90(dct(dct(G.T).T)) # , axis=1)
C[0] = .5 * C[0]
C[:, 1] = .5 * C[:, 1]
@ -427,7 +451,7 @@ def paduavals2coefs(f):
def padua_fit2(Pad, fun, *args):
N = np.shape(Pad)[1]
# recover the degree n from N = (n+1)(n+2)/2
_n = int(round(-3 + np.sqrt(1 + 8 * N)) / 2)
# _n = int(round(-3 + np.sqrt(1 + 8 * N)) / 2)
C0f = fun(Pad[0], Pad[1], *args)
return paduavals2coefs(C0f)
@ -477,21 +501,28 @@ def padua_val(X, Y, coefficients, domain=(-1, 1, -1, 1), use_meshgrid=False):
fxy : array-like
evaluation of the interpolation polynomial at the target points
'''
def transform(tn, x, a, b):
xn = map_from_interval(x, a, b).clip(min=-1, max=1).reshape(1, -1)
tx = np.cos(tn * np.arccos(xn)) * np.sqrt(2)
tx[0] = 1
return tx
X, Y = np.atleast_1d(X, Y)
original_shape = X.shape
min, max = np.minimum, np.maximum # @ReservedAssignment
a, b, c, d = domain
n = np.shape(coefficients)[1]
X1 = min(max(2 * (X.ravel() - a) / (b - a) - 1, -1), 1).reshape(1, -1)
X2 = min(max(2 * (Y.ravel() - c) / (d - c) - 1, -1), 1).reshape(1, -1)
tn = np.r_[0:n][:, None]
TX1 = np.cos(tn * np.arccos(X1))
TX2 = np.cos(tn * np.arccos(X2))
TX1[1:n + 1] = TX1[1:n + 1] * np.sqrt(2)
TX2[1:n + 1] = TX2[1:n + 1] * np.sqrt(2)
tx1 = transform(tn, X.ravel(), a, b)
tx2 = transform(tn, Y.ravel(), c, d)
if use_meshgrid: # eval on meshgrid points
return np.dot(TX1.T, np.dot(coefficients, TX2)).T
return np.dot(tx1.T, np.dot(coefficients, tx2)).T
# scattered points
val = np.sum(np.dot(TX1.T, coefficients) * TX2.T, axis=1)
val = np.sum(np.dot(tx1.T, coefficients) * tx2.T, axis=1)
return val.reshape(original_shape)
if __name__ == '__main__':
from wafo.testing import test_docstrings
test_docstrings(__file__)

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