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-rw-r--r--numpy/polynomial/chebyshev.py4
-rw-r--r--numpy/polynomial/hermite.py4
-rw-r--r--numpy/polynomial/hermite_e.py4
-rw-r--r--numpy/polynomial/laguerre.py4
-rw-r--r--numpy/polynomial/legendre.py4
-rw-r--r--numpy/polynomial/polynomial.py4
-rw-r--r--numpy/polynomial/polyutils.py103
7 files changed, 86 insertions, 41 deletions
diff --git a/numpy/polynomial/chebyshev.py b/numpy/polynomial/chebyshev.py
index 093eb0048..0cd9c4d23 100644
--- a/numpy/polynomial/chebyshev.py
+++ b/numpy/polynomial/chebyshev.py
@@ -1468,7 +1468,7 @@ def chebvander2d(x, y, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander2d(chebvander, x, y, deg)
+ return pu._vander_nd_flat((chebvander, chebvander), (x, y), deg)
def chebvander3d(x, y, z, deg):
@@ -1522,7 +1522,7 @@ def chebvander3d(x, y, z, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander3d(chebvander, x, y, z, deg)
+ return pu._vander_nd_flat((chebvander, chebvander, chebvander), (x, y, z), deg)
def chebfit(x, y, deg, rcond=None, full=False, w=None):
diff --git a/numpy/polynomial/hermite.py b/numpy/polynomial/hermite.py
index 0011fa3b7..9b1aea239 100644
--- a/numpy/polynomial/hermite.py
+++ b/numpy/polynomial/hermite.py
@@ -1193,7 +1193,7 @@ def hermvander2d(x, y, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander2d(hermvander, x, y, deg)
+ return pu._vander_nd_flat((hermvander, hermvander), (x, y), deg)
def hermvander3d(x, y, z, deg):
@@ -1247,7 +1247,7 @@ def hermvander3d(x, y, z, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander3d(hermvander, x, y, z, deg)
+ return pu._vander_nd_flat((hermvander, hermvander, hermvander), (x, y, z), deg)
def hermfit(x, y, deg, rcond=None, full=False, w=None):
diff --git a/numpy/polynomial/hermite_e.py b/numpy/polynomial/hermite_e.py
index b1cc2d3ab..c5a0a05a2 100644
--- a/numpy/polynomial/hermite_e.py
+++ b/numpy/polynomial/hermite_e.py
@@ -1186,7 +1186,7 @@ def hermevander2d(x, y, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander2d(hermevander, x, y, deg)
+ return pu._vander_nd_flat((hermevander, hermevander), (x, y), deg)
def hermevander3d(x, y, z, deg):
@@ -1240,7 +1240,7 @@ def hermevander3d(x, y, z, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander3d(hermevander, x, y, z, deg)
+ return pu._vander_nd_flat((hermevander, hermevander, hermevander), (x, y, z), deg)
def hermefit(x, y, deg, rcond=None, full=False, w=None):
diff --git a/numpy/polynomial/laguerre.py b/numpy/polynomial/laguerre.py
index 7e7e45ca1..538a1d449 100644
--- a/numpy/polynomial/laguerre.py
+++ b/numpy/polynomial/laguerre.py
@@ -1193,7 +1193,7 @@ def lagvander2d(x, y, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander2d(lagvander, x, y, deg)
+ return pu._vander_nd_flat((lagvander, lagvander), (x, y), deg)
def lagvander3d(x, y, z, deg):
@@ -1247,7 +1247,7 @@ def lagvander3d(x, y, z, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander3d(lagvander, x, y, z, deg)
+ return pu._vander_nd_flat((lagvander, lagvander, lagvander), (x, y, z), deg)
def lagfit(x, y, deg, rcond=None, full=False, w=None):
diff --git a/numpy/polynomial/legendre.py b/numpy/polynomial/legendre.py
index 281982d0b..c11824761 100644
--- a/numpy/polynomial/legendre.py
+++ b/numpy/polynomial/legendre.py
@@ -1229,7 +1229,7 @@ def legvander2d(x, y, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander2d(legvander, x, y, deg)
+ return pu._vander_nd_flat((legvander, legvander), (x, y), deg)
def legvander3d(x, y, z, deg):
@@ -1283,7 +1283,7 @@ def legvander3d(x, y, z, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander3d(legvander, x, y, z, deg)
+ return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg)
def legfit(x, y, deg, rcond=None, full=False, w=None):
diff --git a/numpy/polynomial/polynomial.py b/numpy/polynomial/polynomial.py
index 3f0a902cf..315ea1495 100644
--- a/numpy/polynomial/polynomial.py
+++ b/numpy/polynomial/polynomial.py
@@ -1133,7 +1133,7 @@ def polyvander2d(x, y, deg):
polyvander, polyvander3d, polyval2d, polyval3d
"""
- return pu._vander2d(polyvander, x, y, deg)
+ return pu._vander_nd_flat((polyvander, polyvander), (x, y), deg)
def polyvander3d(x, y, z, deg):
@@ -1187,7 +1187,7 @@ def polyvander3d(x, y, z, deg):
.. versionadded:: 1.7.0
"""
- return pu._vander3d(polyvander, x, y, z, deg)
+ return pu._vander_nd_flat((polyvander, polyvander, polyvander), (x, y, z), deg)
def polyfit(x, y, deg, rcond=None, full=False, w=None):
diff --git a/numpy/polynomial/polyutils.py b/numpy/polynomial/polyutils.py
index 35b24d1ab..5dcfa7a7a 100644
--- a/numpy/polynomial/polyutils.py
+++ b/numpy/polynomial/polyutils.py
@@ -46,6 +46,7 @@ Functions
from __future__ import division, absolute_import, print_function
import operator
+import functools
import warnings
import numpy as np
@@ -415,45 +416,89 @@ def mapdomain(x, old, new):
return off + scl*x
-def _vander2d(vander_f, x, y, deg):
- """
- Helper function used to implement the ``<type>vander2d`` functions.
+def _nth_slice(i, ndim):
+ sl = [np.newaxis] * ndim
+ sl[i] = slice(None)
+ return tuple(sl)
+
+
+def _vander_nd(vander_fs, points, degrees):
+ r"""
+ A generalization of the Vandermonde matrix for N dimensions
+
+ The result is built by combining the results of 1d Vandermonde matrices,
+
+ .. math::
+ W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{V_k(x_k)[i_0, \ldots, i_M, j_k]}
+
+ where
+
+ .. math::
+ N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\
+ M &= \texttt{points[k].ndim} \\
+ V_k &= \texttt{vander\_fs[k]} \\
+ x_k &= \texttt{points[k]} \\
+ 0 \le j_k &\le \texttt{degrees[k]}
+
+ Expanding the one-dimensional :math:`V_k` functions gives:
+
+ .. math::
+ W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{B_{k, j_k}(x_k[i_0, \ldots, i_M])}
+
+ where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along
+ dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`.
Parameters
----------
- vander_f : function(array_like, int) -> ndarray
- The 1d vander function, such as ``polyvander``
- x, y, deg :
- See the ``<type>vander2d`` functions for more detail
+ vander_fs : Sequence[function(array_like, int) -> ndarray]
+ The 1d vander function to use for each axis, such as ``polyvander``
+ points : Sequence[array_like]
+ Arrays of point coordinates, all of the same shape. The dtypes
+ will be converted to either float64 or complex128 depending on
+ whether any of the elements are complex. Scalars are converted to
+ 1-D arrays.
+ This must be the same length as `vander_fs`.
+ degrees : Sequence[int]
+ The maximum degree (inclusive) to use for each axis.
+ This must be the same length as `vander_fs`.
+
+ Returns
+ -------
+ vander_nd : ndarray
+ An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``.
"""
- degx, degy = deg
- x, y = np.array((x, y), copy=False) + 0.0
+ n_dims = len(vander_fs)
+ if n_dims != len(points):
+ raise ValueError(
+ "Expected {} dimensions of sample points, got {}".format(n_dims, len(points)))
+ if n_dims != len(degrees):
+ raise ValueError(
+ "Expected {} dimensions of degrees, got {}".format(n_dims, len(degrees)))
+ if n_dims == 0:
+ raise ValueError("Unable to guess a dtype or shape when no points are given")
+
+ # convert to the same shape and type
+ points = tuple(np.array(tuple(points), copy=False) + 0.0)
- vx = vander_f(x, degx)
- vy = vander_f(y, degy)
- v = vx[..., None]*vy[..., None,:]
- return v.reshape(v.shape[:-2] + (-1,))
+ # produce the vandermonde matrix for each dimension, placing the last
+ # axis of each in an independent trailing axis of the output
+ vander_arrays = (
+ vander_fs[i](points[i], degrees[i])[(...,) + _nth_slice(i, n_dims)]
+ for i in range(n_dims)
+ )
+ # we checked this wasn't empty already, so no `initial` needed
+ return functools.reduce(operator.mul, vander_arrays)
-def _vander3d(vander_f, x, y, z, deg):
+
+def _vander_nd_flat(vander_fs, points, degrees):
"""
- Helper function used to implement the ``<type>vander3d`` functions.
+ Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis
- Parameters
- ----------
- vander_f : function(array_like, int) -> ndarray
- The 1d vander function, such as ``polyvander``
- x, y, z, deg :
- See the ``<type>vander3d`` functions for more detail
+ Used to implement the public ``<type>vander<n>d`` functions.
"""
- degx, degy, degz = deg
- x, y, z = np.array((x, y, z), copy=False) + 0.0
-
- vx = vander_f(x, degx)
- vy = vander_f(y, degy)
- vz = vander_f(z, degz)
- v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]
- return v.reshape(v.shape[:-3] + (-1,))
+ v = _vander_nd(vander_fs, points, degrees)
+ return v.reshape(v.shape[:-len(degrees)] + (-1,))
def _fromroots(line_f, mul_f, roots):
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