def assert_array_compare(comparison, x, y, err_msg='', verbose=True,
header=''):
from numpy.core import asarray
x = asarray(x)
y = asarray(y)
try:
cond = (x.shape==() or y.shape==()) or x.shape == y.shape
if not cond:
msg = build_err_msg([x, y],
err_msg
+ '\n(shapes %s, %s mismatch)' % (x.shape,
y.shape),
verbose=verbose, header=header,
names=('x', 'y'))
assert cond, msg
val = comparison(x,y)
if isinstance(val, bool):
cond = val
reduced = [0]
else:
reduced = val.ravel()
cond = reduced.all()
reduced = reduced.tolist()
if not cond:
match = 100-100.0*reduced.count(1)/len(reduced)
msg = build_err_msg([x, y],
err_msg
+ '\n(mismatch %s%%)' % (match,),
verbose=verbose, header=header,
names=('x', 'y'))
assert cond, msg
except ValueError:
msg = build_err_msg([x, y], err_msg, verbose=verbose, header=header,
names=('x', 'y'))
raise ValueError(msg)
def check_conservation(initial_condition, flux):
"""!
@brief Check whether the Riemman intergral of the solution is equal to the integral of the initial data, i.e. check if the numerical
method preserved the conservation property of the linear advection, up to the floating point error.
@param initial_condition One of the valid initial conditions. @see initial_conditions
@param flux One of the valid fluxes. @see fluxes
"""
# Path to the hdf5 file containing results of computations
computations_database = h5py.File(database_path, "r")
# Path to the data group containing the computational data
group_path = initial_condition + "/" + flux
for k in range (6, 17):
dataset_initial_path = initial_condition + "/k = " + str(k) + " initial_data"
dataset_path = group_path + "/k = " + str(k)
h = 2**(-k)
initial_data = asarray(computations_database[dataset_initial_path])
computed_solution = asarray(computations_database[dataset_path])
# Conservation error is the absolute value of the difference between the integrals of the initial data and the computed solution
conservation_error = abs(numpy.sum(initial_data) - numpy.sum(computed_solution))*h
# Criterion of preserving conservation must take into account the floating point error accumulation during the computation
# Experience shows that 0.1 is the upper bound for the floating point error for a conservative numerical method.
if (conservation_error > 1e-1):
print("ERROR: \n\t" + dataset_path + ": conservation is not preserved")
print("\tConservation floating-point error: {0:.1e}".format(conservation_error))
def assert_array_compare(comparison, x, y, err_msg='', verbose=True,
header=''):
from numpy.core import asarray, isnan, any
from numpy import isreal, iscomplex
x = asarray(x)
y = asarray(y)
def isnumber(x):
return x.dtype.char in '?bhilqpBHILQPfdgFDG'
try:
cond = (x.shape==() or y.shape==()) or x.shape == y.shape
if not cond:
msg = build_err_msg([x, y],
err_msg
+ '\n(shapes %s, %s mismatch)' % (x.shape,
y.shape),
verbose=verbose, header=header,
names=('x', 'y'))
if not cond :
raise AssertionError(msg)
if (isnumber(x) and isnumber(y)) and (any(isnan(x)) or any(isnan(y))):
# Handling nan: we first check that x and y have the nan at the
# same locations, and then we mask the nan and do the comparison as
# usual.
xnanid = isnan(x)
ynanid = isnan(y)
try:
assert_array_equal(xnanid, ynanid)
except AssertionError:
msg = build_err_msg([x, y],
err_msg
+ '\n(x and y nan location mismatch %s, ' \
'%s mismatch)' % (xnanid, ynanid),
verbose=verbose, header=header,
names=('x', 'y'))
val = comparison(x[~xnanid], y[~ynanid])
else:
val = comparison(x,y)
if isinstance(val, bool):
cond = val
reduced = [0]
else:
reduced = val.ravel()
cond = reduced.all()
reduced = reduced.tolist()
if not cond:
match = 100-100.0*reduced.count(1)/len(reduced)
msg = build_err_msg([x, y],
err_msg
+ '\n(mismatch %s%%)' % (match,),
verbose=verbose, header=header,
names=('x', 'y'))
if not cond :
raise AssertionError(msg)
except ValueError:
msg = build_err_msg([x, y], err_msg, verbose=verbose, header=header,
names=('x', 'y'))
raise ValueError(msg)
def tensorsolve(a, b, axes=None):
"""Solves the tensor equation a x = b for x
where it is assumed that all the indices of x are summed over in
the product.
a can be N-dimensional. x will have the dimensions of A subtracted from
the dimensions of b.
"""
a = asarray(a)
b = asarray(b)
an = a.ndim
if axes is not None:
allaxes = range(0, an)
for k in axes:
allaxes.remove(k)
allaxes.insert(an, k)
a = a.transpose(allaxes)
oldshape = a.shape[-(an-b.ndim):]
prod = 1
for k in oldshape:
prod *= k
a = a.reshape(-1, prod)
b = b.ravel()
res = solve(a, b)
res.shape = oldshape
return res
def tensorsolve(a, b, axes=None):
"""Solve the tensor equation a x = b for x
It is assumed that all indices of x are summed over in the product,
together with the rightmost indices of a, similarly as in
tensordot(a, x, axes=len(b.shape)).
Parameters
----------
a : array-like, shape b.shape+Q
Coefficient tensor. Shape Q of the rightmost indices of a must
be such that a is 'square', ie., prod(Q) == prod(b.shape).
b : array-like, any shape
Right-hand tensor.
axes : tuple of integers
Axes in a to reorder to the right, before inversion.
If None (default), no reordering is done.
Returns
-------
x : array, shape Q
Examples
--------
>>> from numpy import *
>>> a = eye(2*3*4)
>>> a.shape = (2*3,4, 2,3,4)
>>> b = random.randn(2*3,4)
>>> x = linalg.tensorsolve(a, b)
>>> x.shape
(2, 3, 4)
>>> allclose(tensordot(a, x, axes=3), b)
True
"""
a = asarray(a)
b = asarray(b)
an = a.ndim
if axes is not None:
allaxes = range(0, an)
for k in axes:
allaxes.remove(k)
allaxes.insert(an, k)
a = a.transpose(allaxes)
oldshape = a.shape[-(an-b.ndim):]
prod = 1
for k in oldshape:
prod *= k
a = a.reshape(-1, prod)
b = b.ravel()
res = wrap(solve(a, b))
res.shape = oldshape
return res
def array_equal(cls, mx, other_mx):
try:
mx, other_mx = asarray(mx), asarray(other_mx)
except:
return False
if mx.shape != other_mx.shape:
return False
both_equal = np.equal(mx, other_mx)
both_nan = np.logical_and(np.isnan(mx), np.isnan(other_mx))
return bool(np.logical_or(both_equal, both_nan).all())
def fftshift(x,axes=None):
"""
Shift zero-frequency component to center of spectrum.
This function swaps half-spaces for all axes listed (defaults to all).
If len(x) is even then the Nyquist component is y[0].
Parameters
----------
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to shift. Default is None which shifts all axes.
See Also
--------
ifftshift
"""
tmp = asarray(x)
ndim = len(tmp.shape)
if axes is None:
axes = range(ndim)
y = tmp
for k in axes:
n = tmp.shape[k]
p2 = (n+1)/2
mylist = concatenate((arange(p2,n),arange(p2)))
y = take(y,mylist,k)
return y
def det(a):
"""Compute the determinant of a matrix
Parameters
----------
a : array-like, shape (M, M)
Returns
-------
det : float or complex
Determinant of a
Notes
-----
The determinant is computed via LU factorization, LAPACK routine z/dgetrf.
"""
a = asarray(a)
_assertRank2(a)
_assertSquareness(a)
t, result_t = _commonType(a)
a = _fastCopyAndTranspose(t, a)
n = a.shape[0]
if isComplexType(t):
lapack_routine = lapack_lite.zgetrf
else:
lapack_routine = lapack_lite.dgetrf
pivots = zeros((n,), fortran_int)
results = lapack_routine(n, n, a, n, pivots, 0)
info = results['info']
if (info < 0):
raise TypeError, "Illegal input to Fortran routine"
elif (info > 0):
return 0.0
sign = add.reduce(pivots != arange(1, n+1)) % 2
return (1.-2.*sign)*multiply.reduce(diagonal(a), axis=-1)
def rfft(a, n=None, axis=-1):
"""
Compute the one-dimensional fft for real input.
Return the n point discrete Fourier transform of the real valued
array a. n defaults to the length of a. n is the length of the
input, not the output.
Parameters
----------
a : array
input array with real data type
n : int
length of the fft
axis : int
axis over which to compute the fft
Notes
-----
The returned array will be the nonnegative frequency terms of the
Hermite-symmetric, complex transform of the real array. So for an 8-point
transform, the frequencies in the result are [ 0, 1, 2, 3, 4]. The first
term will be real, as will the last if n is even. The negative frequency
terms are not needed because they are the complex conjugates of the
positive frequency terms. (This is what I mean when I say
Hermite-symmetric.)
This is most efficient for n a power of two.
"""
a = asarray(a).astype(float)
return _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftf, _real_fft_cache)
def rfftn(a, s=None, axes=None):
"""
Compute the n-dimensional fft of a real array.
Parameters
----------
a : array (real)
input array
s : sequence (int)
shape of the fft
axis : int
axis over which to compute the fft
Notes
-----
A real transform as rfft is performed along the axis specified by the last
element of axes, then complex transforms as fft are performed along the
other axes.
"""
a = asarray(a).astype(float)
s, axes = _cook_nd_args(a, s, axes)
a = rfft(a, s[-1], axes[-1])
for ii in range(len(axes) - 1):
a = fft(a, s[ii], axes[ii])
return a
def hfft(a, n=None, axis=-1):
"""
Compute the fft of a signal which spectrum has Hermitian symmetry.
Parameters
----------
a : array
input array
n : int
length of the hfft
axis : int
axis over which to compute the hfft
See also
--------
rfft
ihfft
Notes
-----
These are a pair analogous to rfft/irfft, but for the
opposite case: here the signal is real in the frequency domain and has
Hermite symmetry in the time domain. So here it's hermite_fft for which
you must supply the length of the result if it is to be odd.
ihfft(hfft(a), len(a)) == a
within numerical accuracy.
"""
a = asarray(a).astype(complex)
if n is None:
n = (shape(a)[axis] - 1) * 2
return irfft(conjugate(a), n, axis) * n
def ifftshift(x,axes=None):
"""
Inverse of fftshift.
Parameters
----------
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to calculate. Defaults to None which is over all axes.
See Also
--------
fftshift
"""
tmp = asarray(x)
ndim = len(tmp.shape)
if axes is None:
axes = range(ndim)
y = tmp
for k in axes:
n = tmp.shape[k]
p2 = n-(n+1)/2
mylist = concatenate((arange(p2,n),arange(p2)))
y = take(y,mylist,k)
return y
def _raw_fft(a, n=None, axis=-1, init_function=fftpack.cffti,
work_function=fftpack.cfftf, fft_cache = _fft_cache ):
a = asarray(a)
if n == None: n = a.shape[axis]
if n < 1: raise ValueError("Invalid number of FFT data points (%d) specified." % n)
try:
wsave = fft_cache[n]
except(KeyError):
wsave = init_function(n)
fft_cache[n] = wsave
if a.shape[axis] != n:
s = list(a.shape)
if s[axis] > n:
index = [slice(None)]*len(s)
index[axis] = slice(0,n)
a = a[index]
else:
index = [slice(None)]*len(s)
index[axis] = slice(0,s[axis])
s[axis] = n
z = zeros(s, a.dtype.char)
z[index] = a
a = z
if axis != -1:
a = swapaxes(a, axis, -1)
r = work_function(a, wsave)
if axis != -1:
r = swapaxes(r, axis, -1)
return r
def ihfft(a, n=None, axis=-1):
"""
Compute the inverse fft of a signal whose spectrum has Hermitian symmetry.
Parameters
----------
a : array_like
Input array.
n : int, optional
Length of the ihfft.
axis : int, optional
Axis over which to compute the ihfft.
See also
--------
rfft, hfft
Notes
-----
These are a pair analogous to rfft/irfft, but for the
opposite case: here the signal is real in the frequency domain and has
Hermite symmetry in the time domain. So here it's hermite_fft for which
you must supply the length of the result if it is to be odd.
ihfft(hfft(a), len(a)) == a
within numerical accuracy.
"""
a = asarray(a).astype(float)
if n is None:
n = shape(a)[axis]
return conjugate(rfft(a, n, axis))/n
def irfftn(a, s=None, axes=None):
"""
Compute the n-dimensional inverse fft of a real array.
Parameters
----------
a : array (real)
input array
s : sequence (int)
shape of the inverse fft
axis : int
axis over which to compute the inverse fft
Notes
-----
The transform implemented in ifftn is applied along
all axes but the last, then the transform implemented in irfft is performed
along the last axis. As with irfft, the length of the result along that
axis must be specified if it is to be odd.
"""
a = asarray(a).astype(complex)
s, axes = _cook_nd_args(a, s, axes, invreal=1)
for ii in range(len(axes) - 1):
a = ifft(a, s[ii], axes[ii])
a = irfft(a, s[-1], axes[-1])
return a
def tensorinv(a, ind=2):
"""
Find the 'inverse' of a N-d array
The result is an inverse corresponding to the operation
tensordot(a, b, ind), ie.,
x == tensordot(tensordot(tensorinv(a), a, ind), x, ind)
== tensordot(tensordot(a, tensorinv(a), ind), x, ind)
for all x (up to floating-point accuracy).
Parameters
----------
a : array_like
Tensor to 'invert'. Its shape must 'square', ie.,
prod(a.shape[:ind]) == prod(a.shape[ind:])
ind : integer > 0
How many of the first indices are involved in the inverse sum.
Returns
-------
b : array, shape a.shape[:ind]+a.shape[ind:]
Raises LinAlgError if a is singular or not square
Examples
--------
>>> a = np.eye(4*6)
>>> a.shape = (4,6,8,3)
>>> ainv = np.linalg.tensorinv(a, ind=2)
>>> ainv.shape
(8, 3, 4, 6)
>>> b = np.random.randn(4,6)
>>> np.allclose(np.tensordot(ainv, b), np.linalg.tensorsolve(a, b))
True
>>> a = np.eye(4*6)
>>> a.shape = (24,8,3)
>>> ainv = np.linalg.tensorinv(a, ind=1)
>>> ainv.shape
(8, 3, 24)
>>> b = np.random.randn(24)
>>> np.allclose(np.tensordot(ainv, b, 1), np.linalg.tensorsolve(a, b))
True
"""
a = asarray(a)
oldshape = a.shape
prod = 1
if ind > 0:
invshape = oldshape[ind:] + oldshape[:ind]
for k in oldshape[ind:]:
prod *= k
else:
raise ValueError, "Invalid ind argument."
a = a.reshape(prod, -1)
ia = inv(a)
return ia.reshape(*invshape)
def _raw_fftnd(a, s=None, axes=None, function=fft):
a = asarray(a)
s, axes = _cook_nd_args(a, s, axes)
itl = range(len(axes))
itl.reverse()
for ii in itl:
a = function(a, n=s[ii], axis=axes[ii])
return a
def fftshift(x, axes=None):
"""
Shift the zero-frequency component to the center of the spectrum.
This function swaps half-spaces for all axes listed (defaults to all).
Note that ``y[0]`` is the Nyquist component only if ``len(x)`` is even.
Parameters
----------
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to shift. Default is None, which shifts all axes.
Returns
-------
y : ndarray
The shifted array.
See Also
--------
ifftshift : The inverse of `fftshift`.
Examples
--------
>>> freqs = np.fft.fftfreq(10, 0.1)
>>> freqs
array([ 0., 1., 2., 3., 4., -5., -4., -3., -2., -1.])
>>> np.fft.fftshift(freqs)
array([-5., -4., -3., -2., -1., 0., 1., 2., 3., 4.])
Shift the zero-frequency component only along the second axis:
>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
>>> np.fft.fftshift(freqs, axes=(1,))
array([[ 2., 0., 1.],
[-4., 3., 4.],
[-1., -3., -2.]])
"""
tmp = asarray(x)
ndim = len(tmp.shape)
if axes is None:
axes = list(range(ndim))
elif isinstance(axes, (int, nt.integer)):
axes = (axes,)
y = tmp
for k in axes:
n = tmp.shape[k]
p2 = (n+1)//2
mylist = concatenate((arange(p2,n),arange(p2)))
y = take(y,mylist,k)
return y
def fftshift(x, axes=None):
"""
Shift the zero-frequency component to the center of the spectrum.
This function swaps half-spaces for all axes listed (defaults to all).
Note that ``y[0]`` is the Nyquist component only if ``len(x)`` is even.
Parameters
----------
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to shift. Default is None, which shifts all axes.
Returns
-------
y : ndarray
The shifted array.
See Also
--------
ifftshift : The inverse of `fftshift`.
Examples
--------
>>> freqs = np.fft.fftfreq(10, 0.1)
>>> freqs
array([ 0., 1., 2., ..., -3., -2., -1.])
>>> np.fft.fftshift(freqs)
array([-5., -4., -3., -2., -1., 0., 1., 2., 3., 4.])
Shift the zero-frequency component only along the second axis:
>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
>>> np.fft.fftshift(freqs, axes=(1,))
array([[ 2., 0., 1.],
[-4., 3., 4.],
[-1., -3., -2.]])
"""
x = asarray(x)
if axes is None:
axes = tuple(range(x.ndim))
shift = [dim // 2 for dim in x.shape]
elif isinstance(axes, integer_types):
shift = x.shape[axes] // 2
else:
shift = [x.shape[ax] // 2 for ax in axes]
return roll(x, shift, axes)
def ihfft(a, n=None, axis=-1):
"""
Compute the inverse FFT of a signal which has Hermitian symmetry.
Parameters
----------
a : array_like
Input array.
n : int, optional
Length of the inverse FFT.
Number of points along transformation axis in the input to use.
If `n` is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If `n` is not given,
the length of the input along the axis specified by `axis` is used.
axis : int, optional
Axis over which to compute the inverse FFT. If not given, the last
axis is used.
Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
If `n` is even, the length of the transformed axis is ``(n/2)+1``.
If `n` is odd, the length is ``(n+1)/2``.
See also
--------
hfft, irfft
Notes
-----
`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
opposite case: here the signal has Hermitian symmetry in the time domain
and is real in the frequency domain. So here it's `hfft` for which
you must supply the length of the result if it is to be odd:
``ihfft(hfft(a), len(a)) == a``, within numerical accuracy.
Examples
--------
>>> spectrum = np.array([ 15, -4, 0, -1, 0, -4])
>>> np.fft.ifft(spectrum)
array([ 1.+0.j, 2.-0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.-0.j])
>>> np.fft.ihfft(spectrum)
array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j])
"""
a = asarray(a).astype(float)
if n is None:
n = shape(a)[axis]
return conjugate(rfft(a, n, axis))/n
def byte_bounds(a):
"""
Returns pointers to the end-points of an array.
Parameters
----------
a : ndarray
Input array. It must conform to the Python-side of the array
interface.
Returns
-------
(low, high) : tuple of 2 integers
The first integer is the first byte of the array, the second
integer is just past the last byte of the array. If `a` is not
contiguous it will not use every byte between the (`low`, `high`)
values.
Examples
--------
>>> I = np.eye(2, dtype='f'); I.dtype
dtype('float32')
>>> low, high = np.byte_bounds(I)
>>> high - low == I.size*I.itemsize
True
>>> I = np.eye(2, dtype='G'); I.dtype
dtype('complex192')
>>> low, high = np.byte_bounds(I)
>>> high - low == I.size*I.itemsize
True
"""
ai = a.__array_interface__
a_data = ai['data'][0]
astrides = ai['strides']
ashape = ai['shape']
bytes_a = asarray(a).dtype.itemsize
a_low = a_high = a_data
if astrides is None:
# contiguous case
a_high += a.size * bytes_a
else:
for shape, stride in zip(ashape, astrides):
if stride < 0:
a_low += (shape-1)*stride
else:
a_high += (shape-1)*stride
a_high += bytes_a
return a_low, a_high
def rfftn(a, s=None, axes=None):
"""rfftn(a, s=None, axes=None)
The n-dimensional discrete Fourier transform of a real array a. A real
transform as rfft is performed along the axis specified by the last
element of axes, then complex transforms as fft are performed along the
other axes."""
a = asarray(a).astype(float)
s, axes = _cook_nd_args(a, s, axes)
a = rfft(a, s[-1], axes[-1])
for ii in range(len(axes)-1):
a = fft(a, s[ii], axes[ii])
return a
def hfft(a, n=None, axis=-1):
"""
Compute the FFT of a signal whose spectrum has Hermitian symmetry.
Parameters
----------
a : array_like
The input array.
n : int, optional
The length of the FFT.
axis : int, optional
The axis over which to compute the FFT, assuming Hermitian symmetry
of the spectrum. Default is the last axis.
Returns
-------
out : ndarray
The transformed input.
See also
--------
rfft : Compute the one-dimensional FFT for real input.
ihfft : The inverse of `hfft`.
Notes
-----
`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
opposite case: here the signal is real in the frequency domain and has
Hermite symmetry in the time domain. So here it's `hfft` for which
you must supply the length of the result if it is to be odd:
``ihfft(hfft(a), len(a)) == a``, within numerical accuracy.
Examples
--------
>>> signal = np.array([[1, 1.j], [-1.j, 2]])
>>> np.conj(signal.T) - signal # check Hermitian symmetry
array([[ 0.-0.j, 0.+0.j],
[ 0.+0.j, 0.-0.j]])
>>> freq_spectrum = np.fft.hfft(signal)
>>> freq_spectrum
array([[ 1., 1.],
[ 2., -2.]])
"""
a = asarray(a).astype(complex)
if n is None:
n = (shape(a)[axis] - 1) * 2
return irfft(conjugate(a), n, axis) * n
def original_fftshift(x, axes=None):
""" How fftshift was implemented in v1.14"""
tmp = asarray(x)
ndim = tmp.ndim
if axes is None:
axes = list(range(ndim))
elif isinstance(axes, integer_types):
axes = (axes,)
y = tmp
for k in axes:
n = tmp.shape[k]
p2 = (n + 1) // 2
mylist = concatenate((arange(p2, n), arange(p2)))
y = take(y, mylist, k)
return y
def irfftn(a, s=None, axes=None):
"""irfftn(a, s=None, axes=None)
The inverse of rfftn. The transform implemented in ifft is
applied along all axes but the last, then the transform implemented in
irfft is performed along the last axis. As with
irfft, the length of the result along that axis must be
specified if it is to be odd."""
a = asarray(a).astype(complex)
s, axes = _cook_nd_args(a, s, axes, invreal=1)
for ii in range(len(axes)-1):
a = ifft(a, s[ii], axes[ii])
a = irfft(a, s[-1], axes[-1])
return a
def ifftshift(x,axes=None):
""" ifftshift(x,axes=None) - > y
Inverse of fftshift.
"""
tmp = asarray(x)
ndim = len(tmp.shape)
if axes is None:
axes = range(ndim)
y = tmp
for k in axes:
n = tmp.shape[k]
p2 = n-(n+1)/2
mylist = concatenate((arange(p2,n),arange(p2)))
y = take(y,mylist,k)
return y
def ihfft(a, n=None, axis=-1):
"""hfft(a, n=None, axis=-1)
ihfft(a, n=None, axis=-1)
These are a pair analogous to rfft/irfft, but for the
opposite case: here the signal is real in the frequency domain and has
Hermite symmetry in the time domain. So here it's hfft for which
you must supply the length of the result if it is to be odd.
ihfft(hfft(a), len(a)) == a
within numerical accuracy."""
a = asarray(a).astype(float)
if n == None:
n = shape(a)[axis]
return conjugate(rfft(a, n, axis))/n
def _raw_fft(a, n=None, axis=-1, init_function=fftpack.cffti,
work_function=fftpack.cfftf, fft_cache=_fft_cache):
a = asarray(a)
axis = normalize_axis_index(axis, a.ndim)
if n is None:
n = a.shape[axis]
if n < 1:
raise ValueError("Invalid number of FFT data points (%d) specified."
% n)
# We have to ensure that only a single thread can access a wsave array
# at any given time. Thus we remove it from the cache and insert it
# again after it has been used. Multiple threads might create multiple
# copies of the wsave array. This is intentional and a limitation of
# the current C code.
wsave = fft_cache.pop_twiddle_factors(n)
if wsave is None:
wsave = init_function(n)
if a.shape[axis] != n:
s = list(a.shape)
if s[axis] > n:
index = [slice(None)]*len(s)
index[axis] = slice(0, n)
a = a[tuple(index)]
else:
index = [slice(None)]*len(s)
index[axis] = slice(0, s[axis])
s[axis] = n
z = zeros(s, a.dtype.char)
z[tuple(index)] = a
a = z
if axis != a.ndim - 1:
a = swapaxes(a, axis, -1)
r = work_function(a, wsave)
if axis != a.ndim - 1:
r = swapaxes(r, axis, -1)
# As soon as we put wsave back into the cache, another thread could pick it
# up and start using it, so we must not do this until after we're
# completely done using it ourselves.
fft_cache.put_twiddle_factors(n, wsave)
return r
def det(a):
"""
Compute the determinant of an array.
Parameters
----------
a : array_like, shape (M, M)
Input array.
Returns
-------
det : ndarray
Determinant of `a`.
Notes
-----
The determinant is computed via LU factorization using the LAPACK
routine z/dgetrf.
Examples
--------
The determinant of a 2-D array [[a, b], [c, d]] is ad - bc:
>>> a = np.array([[1, 2], [3, 4]])
>>> np.linalg.det(a)
-2.0
"""
a = asarray(a)
_assertRank2(a)
_assertSquareness(a)
t, result_t = _commonType(a)
a = _fastCopyAndTranspose(t, a)
n = a.shape[0]
if isComplexType(t):
lapack_routine = lapack_lite.zgetrf
else:
lapack_routine = lapack_lite.dgetrf
pivots = zeros((n,), fortran_int)
results = lapack_routine(n, n, a, n, pivots, 0)
info = results['info']
if (info < 0):
raise TypeError, "Illegal input to Fortran routine"
elif (info > 0):
return 0.0
sign = add.reduce(pivots != arange(1, n+1)) % 2
return (1.-2.*sign)*multiply.reduce(diagonal(a), axis=-1)
def ifftshift(x, axes=None):
"""
The inverse of `fftshift`. Although identical for even-length `x`, the
functions differ by one sample for odd-length `x`.
Parameters
----------
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to calculate. Defaults to None, which shifts all axes.
Returns
-------
y : ndarray
The shifted array.
See Also
--------
fftshift : Shift zero-frequency component to the center of the spectrum.
Examples
--------
>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
>>> np.fft.ifftshift(np.fft.fftshift(freqs))
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
"""
tmp = asarray(x)
ndim = len(tmp.shape)
if axes is None:
axes = list(range(ndim))
elif isinstance(axes, integer_types):
axes = (axes,)
y = tmp
for k in axes:
n = tmp.shape[k]
p2 = n-(n+1)//2
mylist = concatenate((arange(p2, n), arange(p2)))
y = take(y, mylist, k)
return y
def __div__(self, other):
return self._rc(divide(self.array, asarray(other)))
def fft(a, n=None, axis=-1, norm=None):
"""
Compute the one-dimensional discrete Fourier Transform.
This function computes the one-dimensional *n*-point discrete Fourier
Transform (DFT) with the efficient Fast Fourier Transform (FFT)
algorithm [CT].
Parameters
----------
a : array_like
Input array, can be complex.
n : int, optional
Length of the transformed axis of the output.
If `n` is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If `n` is not given,
the length of the input along the axis specified by `axis` is used.
axis : int, optional
Axis over which to compute the FFT. If not given, the last axis is
used.
norm : {None, "ortho"}, optional
.. versionadded:: 1.10.0
Normalization mode (see `numpy.fft`). Default is None.
Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
Raises
------
IndexError
if `axes` is larger than the last axis of `a`.
See Also
--------
numpy.fft : for definition of the DFT and conventions used.
ifft : The inverse of `fft`.
fft2 : The two-dimensional FFT.
fftn : The *n*-dimensional FFT.
rfftn : The *n*-dimensional FFT of real input.
fftfreq : Frequency bins for given FFT parameters.
Notes
-----
FFT (Fast Fourier Transform) refers to a way the discrete Fourier
Transform (DFT) can be calculated efficiently, by using symmetries in the
calculated terms. The symmetry is highest when `n` is a power of 2, and
the transform is therefore most efficient for these sizes.
The DFT is defined, with the conventions used in this implementation, in
the documentation for the `numpy.fft` module.
References
----------
.. [CT] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
machine calculation of complex Fourier series," *Math. Comput.*
19: 297-301.
Examples
--------
>>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
array([-2.33486982e-16+1.14423775e-17j, 8.00000000e+00-1.25557246e-15j,
2.33486982e-16+2.33486982e-16j, 0.00000000e+00+1.22464680e-16j,
-1.14423775e-17+2.33486982e-16j, 0.00000000e+00+5.20784380e-16j,
1.14423775e-17+1.14423775e-17j, 0.00000000e+00+1.22464680e-16j])
In this example, real input has an FFT which is Hermitian, i.e., symmetric
in the real part and anti-symmetric in the imaginary part, as described in
the `numpy.fft` documentation:
>>> import matplotlib.pyplot as plt
>>> t = np.arange(256)
>>> sp = np.fft.fft(np.sin(t))
>>> freq = np.fft.fftfreq(t.shape[-1])
>>> plt.plot(freq, sp.real, freq, sp.imag)
[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
>>> plt.show()
"""
a = asarray(a)
if n is None:
n = a.shape[axis]
inv_norm = 1
if norm is not None and _unitary(norm):
inv_norm = sqrt(n)
output = _raw_fft(a, n, axis, False, True, inv_norm)
return output
def norm(x, ord=None):
""" norm(x, ord=None) -> n
Matrix or vector norm.
Inputs:
x -- a rank-1 (vector) or rank-2 (matrix) array
ord -- the order of the norm.
Comments:
For arrays of any rank, if ord is None:
calculate the square norm (Euclidean norm for vectors,
Frobenius norm for matrices)
For vectors ord can be any real number including Inf or -Inf.
ord = Inf, computes the maximum of the magnitudes
ord = -Inf, computes minimum of the magnitudes
ord is finite, computes sum(abs(x)**ord,axis=0)**(1.0/ord)
For matrices ord can only be one of the following values:
ord = 2 computes the largest singular value
ord = -2 computes the smallest singular value
ord = 1 computes the largest column sum of absolute values
ord = -1 computes the smallest column sum of absolute values
ord = Inf computes the largest row sum of absolute values
ord = -Inf computes the smallest row sum of absolute values
ord = 'fro' computes the frobenius norm sqrt(sum(diag(X.H * X),axis=0))
For values ord < 0, the result is, strictly speaking, not a
mathematical 'norm', but it may still be useful for numerical purposes.
"""
x = asarray(x)
nd = len(x.shape)
if ord is None: # check the default case first and handle it immediately
return sqrt(add.reduce((x.conj() * x).ravel().real))
if nd == 1:
if ord == Inf:
return abs(x).max()
elif ord == -Inf:
return abs(x).min()
elif ord == 1:
return abs(x).sum() # special case for speedup
elif ord == 2:
return sqrt(
((x.conj() * x).real).sum()) # special case for speedup
else:
return ((abs(x)**ord).sum())**(1.0 / ord)
elif nd == 2:
if ord == 2:
return svd(x, compute_uv=0).max()
elif ord == -2:
return svd(x, compute_uv=0).min()
elif ord == 1:
return abs(x).sum(axis=0).max()
elif ord == Inf:
return abs(x).sum(axis=1).max()
elif ord == -1:
return abs(x).sum(axis=0).min()
elif ord == -Inf:
return abs(x).sum(axis=1).min()
elif ord in ['fro', 'f']:
return sqrt(add.reduce((x.conj() * x).real.ravel()))
else:
raise ValueError, "Invalid norm order for matrices."
else:
raise ValueError, "Improper number of dimensions to norm."
def hfft(a, n=None, axis=-1):
"""
Compute the FFT of a signal which has Hermitian symmetry (real spectrum).
Parameters
----------
a : array_like
The input array.
n : int, optional
Length of the transformed axis of the output.
For `n` output points, ``n//2+1`` input points are necessary. If the
input is longer than this, it is cropped. If it is shorter than this,
it is padded with zeros. If `n` is not given, it is determined from
the length of the input along the axis specified by `axis`.
axis : int, optional
Axis over which to compute the FFT. If not given, the last
axis is used.
Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
The length of the transformed axis is `n`, or, if `n` is not given,
``2*(m-1)`` where ``m`` is the length of the transformed axis of the
input. To get an odd number of output points, `n` must be specified.
Raises
------
IndexError
If `axis` is larger than the last axis of `a`.
See also
--------
rfft : Compute the one-dimensional FFT for real input.
ihfft : The inverse of `hfft`.
Notes
-----
`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
opposite case: here the signal has Hermitian symmetry in the time domain
and is real in the frequency domain. So here it's `hfft` for which
you must supply the length of the result if it is to be odd:
``ihfft(hfft(a), len(a)) == a``, within numerical accuracy.
Examples
--------
>>> signal = np.array([1, 2, 3, 4, 3, 2])
>>> np.fft.fft(signal)
array([ 15.+0.j, -4.+0.j, 0.+0.j, -1.-0.j, 0.+0.j, -4.+0.j])
>>> np.fft.hfft(signal[:4]) # Input first half of signal
array([ 15., -4., 0., -1., 0., -4.])
>>> np.fft.hfft(signal, 6) # Input entire signal and truncate
array([ 15., -4., 0., -1., 0., -4.])
>>> signal = np.array([[1, 1.j], [-1.j, 2]])
>>> np.conj(signal.T) - signal # check Hermitian symmetry
array([[ 0.-0.j, 0.+0.j],
[ 0.+0.j, 0.-0.j]])
>>> freq_spectrum = np.fft.hfft(signal)
>>> freq_spectrum
array([[ 1., 1.],
[ 2., -2.]])
"""
a = asarray(a).astype(complex)
if n is None:
n = (shape(a)[axis] - 1) * 2
return irfft(conjugate(a), n, axis) * n
def irfftn(a, s=None, axes=None):
"""
Compute the inverse of the N-dimensional FFT of real input.
This function computes the inverse of the N-dimensional discrete
Fourier Transform for real input over any number of axes in an
M-dimensional array by means of the Fast Fourier Transform (FFT). In
other words, ``irfftn(rfftn(a), a.shape) == a`` to within numerical
accuracy. (The ``a.shape`` is necessary like ``len(a)`` is for `irfft`,
and for the same reason.)
The input should be ordered in the same way as is returned by `rfftn`,
i.e. as for `irfft` for the final transformation axis, and as for `ifftn`
along all the other axes.
Parameters
----------
a : array_like
Input array.
s : sequence of ints, optional
Shape (length of each transformed axis) of the output
(``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.). `s` is also the
number of input points used along this axis, except for the last axis,
where ``s[-1]//2+1`` points of the input are used.
Along any axis, if the shape indicated by `s` is smaller than that of
the input, the input is cropped. If it is larger, the input is padded
with zeros. If `s` is not given, the shape of the input (along the
axes specified by `axes`) is used.
axes : sequence of ints, optional
Axes over which to compute the inverse FFT. If not given, the last
`len(s)` axes are used, or all axes if `s` is also not specified.
Repeated indices in `axes` means that the inverse transform over that
axis is performed multiple times.
Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axes
indicated by `axes`, or by a combination of `s` or `a`,
as explained in the parameters section above.
The length of each transformed axis is as given by the corresponding
element of `s`, or the length of the input in every axis except for the
last one if `s` is not given. In the final transformed axis the length
of the output when `s` is not given is ``2*(m-1)`` where `m` is the
length of the final transformed axis of the input. To get an odd
number of output points in the final axis, `s` must be specified.
Raises
------
ValueError
If `s` and `axes` have different length.
IndexError
If an element of `axes` is larger than than the number of axes of `a`.
See Also
--------
rfftn : The forward n-dimensional FFT of real input,
of which `ifftn` is the inverse.
fft : The one-dimensional FFT, with definitions and conventions used.
irfft : The inverse of the one-dimensional FFT of real input.
irfft2 : The inverse of the two-dimensional FFT of real input.
Notes
-----
See `fft` for definitions and conventions used.
See `rfft` for definitions and conventions used for real input.
Examples
--------
>>> a = np.zeros((3, 2, 2))
>>> a[0, 0, 0] = 3 * 2 * 2
>>> np.fft.irfftn(a)
array([[[ 1., 1.],
[ 1., 1.]],
[[ 1., 1.],
[ 1., 1.]],
[[ 1., 1.],
[ 1., 1.]]])
"""
a = asarray(a).astype(complex)
s, axes = _cook_nd_args(a, s, axes, invreal=1)
for ii in range(len(axes)-1):
a = ifft(a, s[ii], axes[ii])
a = irfft(a, s[-1], axes[-1])
return a
def rfft(a, n=None, axis=-1):
"""
Compute the one-dimensional discrete Fourier Transform for real input.
This function computes the one-dimensional *n*-point discrete Fourier
Transform (DFT) of a real-valued array by means of an efficient algorithm
called the Fast Fourier Transform (FFT).
Parameters
----------
a : array_like
Input array
n : int, optional
Number of points along transformation axis in the input to use.
If `n` is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If `n` is not given,
the length of the input (along the axis specified by `axis`) is used.
axis : int, optional
Axis over which to compute the FFT. If not given, the last axis is
used.
Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
If `n` is even, the length of the transformed axis is ``(n/2)+1``.
If `n` is odd, the length is ``(n+1)/2``.
Raises
------
IndexError
If `axis` is larger than the last axis of `a`.
See Also
--------
numpy.fft : For definition of the DFT and conventions used.
irfft : The inverse of `rfft`.
fft : The one-dimensional FFT of general (complex) input.
fftn : The *n*-dimensional FFT.
rfftn : The *n*-dimensional FFT of real input.
Notes
-----
When the DFT is computed for purely real input, the output is
Hermite-symmetric, i.e. the negative frequency terms are just the complex
conjugates of the corresponding positive-frequency terms, and the
negative-frequency terms are therefore redundant. This function does not
compute the negative frequency terms, and the length of the transformed
axis of the output is therefore ``n//2+1``.
When ``A = rfft(a)`` and fs is the sampling frequency, ``A[0]`` contains
the zero-frequency term 0*fs, which is real due to Hermitian symmetry.
If `n` is even, ``A[-1]`` contains the term representing both positive
and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely
real. If `n` is odd, there is no term at fs/2; ``A[-1]`` contains
the largest positive frequency (fs/2*(n-1)/n), and is complex in the
general case.
If the input `a` contains an imaginary part, it is silently discarded.
Examples
--------
>>> np.fft.fft([0, 1, 0, 0])
array([ 1.+0.j, 0.-1.j, -1.+0.j, 0.+1.j])
>>> np.fft.rfft([0, 1, 0, 0])
array([ 1.+0.j, 0.-1.j, -1.+0.j])
Notice how the final element of the `fft` output is the complex conjugate
of the second element, for real input. For `rfft`, this symmetry is
exploited to compute only the non-negative frequency terms.
"""
a = asarray(a).astype(float)
return _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftf, _real_fft_cache)
def rfftn(a, s=None, axes=None):
"""
Compute the N-dimensional discrete Fourier Transform for real input.
This function computes the N-dimensional discrete Fourier Transform over
any number of axes in an M-dimensional real array by means of the Fast
Fourier Transform (FFT). By default, all axes are transformed, with the
real transform performed over the last axis, while the remaining
transforms are complex.
Parameters
----------
a : array_like
Input array, taken to be real.
s : sequence of ints, optional
Shape (length along each transformed axis) to use from the input.
(``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.).
The final element of `s` corresponds to `n` for ``rfft(x, n)``, while
for the remaining axes, it corresponds to `n` for ``fft(x, n)``.
Along any axis, if the given shape is smaller than that of the input,
the input is cropped. If it is larger, the input is padded with zeros.
if `s` is not given, the shape of the input (along the axes specified
by `axes`) is used.
axes : sequence of ints, optional
Axes over which to compute the FFT. If not given, the last ``len(s)``
axes are used, or all axes if `s` is also not specified.
Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axes
indicated by `axes`, or by a combination of `s` and `a`,
as explained in the parameters section above.
The length of the last axis transformed will be ``s[-1]//2+1``,
while the remaining transformed axes will have lengths according to
`s`, or unchanged from the input.
Raises
------
ValueError
If `s` and `axes` have different length.
IndexError
If an element of `axes` is larger than than the number of axes of `a`.
See Also
--------
irfftn : The inverse of `rfftn`, i.e. the inverse of the n-dimensional FFT
of real input.
fft : The one-dimensional FFT, with definitions and conventions used.
rfft : The one-dimensional FFT of real input.
fftn : The n-dimensional FFT.
rfft2 : The two-dimensional FFT of real input.
Notes
-----
The transform for real input is performed over the last transformation
axis, as by `rfft`, then the transform over the remaining axes is
performed as by `fftn`. The order of the output is as for `rfft` for the
final transformation axis, and as for `fftn` for the remaining
transformation axes.
See `fft` for details, definitions and conventions used.
Examples
--------
>>> a = np.ones((2, 2, 2))
>>> np.fft.rfftn(a)
array([[[ 8.+0.j, 0.+0.j],
[ 0.+0.j, 0.+0.j]],
[[ 0.+0.j, 0.+0.j],
[ 0.+0.j, 0.+0.j]]])
>>> np.fft.rfftn(a, axes=(2, 0))
array([[[ 4.+0.j, 0.+0.j],
[ 4.+0.j, 0.+0.j]],
[[ 0.+0.j, 0.+0.j],
[ 0.+0.j, 0.+0.j]]])
"""
a = asarray(a).astype(float)
s, axes = _cook_nd_args(a, s, axes)
a = rfft(a, s[-1], axes[-1])
for ii in range(len(axes)-1):
a = fft(a, s[ii], axes[ii])
return a
def __add__(self, other):
return self._rc(self.array + asarray(other))
def irfft(a, n=None, axis=-1, norm=None):
"""
Compute the inverse of the n-point DFT for real input.
This function computes the inverse of the one-dimensional *n*-point
discrete Fourier Transform of real input computed by `rfft`.
In other words, ``irfft(rfft(a), len(a)) == a`` to within numerical
accuracy. (See Notes below for why ``len(a)`` is necessary here.)
The input is expected to be in the form returned by `rfft`, i.e. the
real zero-frequency term followed by the complex positive frequency terms
in order of increasing frequency. Since the discrete Fourier Transform of
real input is Hermitian-symmetric, the negative frequency terms are taken
to be the complex conjugates of the corresponding positive frequency terms.
Parameters
----------
a : array_like
The input array.
n : int, optional
Length of the transformed axis of the output.
For `n` output points, ``n//2+1`` input points are necessary. If the
input is longer than this, it is cropped. If it is shorter than this,
it is padded with zeros. If `n` is not given, it is taken to be
``2*(m-1)`` where ``m`` is the length of the input along the axis
specified by `axis`.
axis : int, optional
Axis over which to compute the inverse FFT. If not given, the last
axis is used.
norm : {None, "ortho"}, optional
.. versionadded:: 1.10.0
Normalization mode (see `numpy.fft`). Default is None.
Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
The length of the transformed axis is `n`, or, if `n` is not given,
``2*(m-1)`` where ``m`` is the length of the transformed axis of the
input. To get an odd number of output points, `n` must be specified.
Raises
------
IndexError
If `axis` is larger than the last axis of `a`.
See Also
--------
numpy.fft : For definition of the DFT and conventions used.
rfft : The one-dimensional FFT of real input, of which `irfft` is inverse.
fft : The one-dimensional FFT.
irfft2 : The inverse of the two-dimensional FFT of real input.
irfftn : The inverse of the *n*-dimensional FFT of real input.
Notes
-----
Returns the real valued `n`-point inverse discrete Fourier transform
of `a`, where `a` contains the non-negative frequency terms of a
Hermitian-symmetric sequence. `n` is the length of the result, not the
input.
If you specify an `n` such that `a` must be zero-padded or truncated, the
extra/removed values will be added/removed at high frequencies. One can
thus resample a series to `m` points via Fourier interpolation by:
``a_resamp = irfft(rfft(a), m)``.
The correct interpretation of the hermitian input depends on the length of
the original data, as given by `n`. This is because each input shape could
correspond to either an odd or even length signal. By default, `irfft`
assumes an even output length which puts the last entry at the Nyquist
frequency; aliasing with its symmetric counterpart. By Hermitian symmetry,
the value is thus treated as purely real. To avoid losing information, the
correct length of the real input **must** be given.
Examples
--------
>>> np.fft.ifft([1, -1j, -1, 1j])
array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]) # may vary
>>> np.fft.irfft([1, -1j, -1])
array([0., 1., 0., 0.])
Notice how the last term in the input to the ordinary `ifft` is the
complex conjugate of the second term, and the output has zero imaginary
part everywhere. When calling `irfft`, the negative frequencies are not
specified, and the output array is purely real.
"""
a = asarray(a)
if n is None:
n = (a.shape[axis] - 1) * 2
inv_norm = n
if norm is not None and _unitary(norm):
inv_norm = sqrt(n)
output = _raw_fft(a, n, axis, True, False, inv_norm)
return output
def __sub__(self, other):
return self._rc(self.array - asarray(other))
def __rsub__(self, other):
return self._rc(asarray(other) - self.array)
def lstsq(a, b, rcond=-1):
"""returns x,resids,rank,s
where x minimizes 2-norm(|b - Ax|)
resids is the sum square residuals
rank is the rank of A
s is the rank of the singular values of A in descending order
If b is a matrix then x is also a matrix with corresponding columns.
If the rank of A is less than the number of columns of A or greater than
the number of rows, then residuals will be returned as an empty array
otherwise resids = sum((b-dot(A,x)**2).
Singular values less than s[0]*rcond are treated as zero.
"""
import math
a = asarray(a)
b, wrap = _makearray(b)
one_eq = len(b.shape) == 1
if one_eq:
b = b[:, newaxis]
_assertRank2(a, b)
m = a.shape[0]
n = a.shape[1]
n_rhs = b.shape[1]
ldb = max(n, m)
if m != b.shape[0]:
raise LinAlgError, 'Incompatible dimensions'
t, result_t = _commonType(a, b)
real_t = _linalgRealType(t)
bstar = zeros((ldb, n_rhs), t)
bstar[:b.shape[0], :n_rhs] = b.copy()
a, bstar = _fastCopyAndTranspose(t, a, bstar)
s = zeros((min(m, n), ), real_t)
nlvl = max(0, int(math.log(float(min(m, n)) / 2.)) + 1)
iwork = zeros((3 * min(m, n) * nlvl + 11 * min(m, n), ), fortran_int)
if isComplexType(t):
lapack_routine = lapack_lite.zgelsd
lwork = 1
rwork = zeros((lwork, ), real_t)
work = zeros((lwork, ), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond, 0,
work, -1, rwork, iwork, 0)
lwork = int(abs(work[0]))
rwork = zeros((lwork, ), real_t)
a_real = zeros((m, n), real_t)
bstar_real = zeros((
ldb,
n_rhs,
), real_t)
results = lapack_lite.dgelsd(m, n, n_rhs, a_real, m, bstar_real, ldb,
s, rcond, 0, rwork, -1, iwork, 0)
lrwork = int(rwork[0])
work = zeros((lwork, ), t)
rwork = zeros((lrwork, ), real_t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond, 0,
work, lwork, rwork, iwork, 0)
else:
lapack_routine = lapack_lite.dgelsd
lwork = 1
work = zeros((lwork, ), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond, 0,
work, -1, iwork, 0)
lwork = int(work[0])
work = zeros((lwork, ), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond, 0,
work, lwork, iwork, 0)
if results['info'] > 0:
raise LinAlgError, 'SVD did not converge in Linear Least Squares'
resids = array([], t)
if one_eq:
x = array(ravel(bstar)[:n], dtype=result_t, copy=True)
if results['rank'] == n and m > n:
resids = array([sum((ravel(bstar)[n:])**2)], dtype=result_t)
else:
x = array(transpose(bstar)[:n, :], dtype=result_t, copy=True)
if results['rank'] == n and m > n:
resids = sum((transpose(bstar)[n:, :])**2, axis=0).astype(result_t)
st = s[:min(n, m)].copy().astype(_realType(result_t))
return wrap(x), resids, results['rank'], st
def hfft(a, n=None, axis=-1, norm=None):
"""
Compute the FFT of a signal that has Hermitian symmetry, i.e., a real
spectrum.
Parameters
----------
a : array_like
The input array.
n : int, optional
Length of the transformed axis of the output. For `n` output
points, ``n//2 + 1`` input points are necessary. If the input is
longer than this, it is cropped. If it is shorter than this, it is
padded with zeros. If `n` is not given, it is taken to be ``2*(m-1)``
where ``m`` is the length of the input along the axis specified by
`axis`.
axis : int, optional
Axis over which to compute the FFT. If not given, the last
axis is used.
norm : {None, "ortho"}, optional
Normalization mode (see `numpy.fft`). Default is None.
.. versionadded:: 1.10.0
Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
The length of the transformed axis is `n`, or, if `n` is not given,
``2*m - 2`` where ``m`` is the length of the transformed axis of
the input. To get an odd number of output points, `n` must be
specified, for instance as ``2*m - 1`` in the typical case,
Raises
------
IndexError
If `axis` is larger than the last axis of `a`.
See also
--------
rfft : Compute the one-dimensional FFT for real input.
ihfft : The inverse of `hfft`.
Notes
-----
`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
opposite case: here the signal has Hermitian symmetry in the time
domain and is real in the frequency domain. So here it's `hfft` for
which you must supply the length of the result if it is to be odd.
* even: ``ihfft(hfft(a, 2*len(a) - 2) == a``, within roundoff error,
* odd: ``ihfft(hfft(a, 2*len(a) - 1) == a``, within roundoff error.
The correct interpretation of the hermitian input depends on the length of
the original data, as given by `n`. This is because each input shape could
correspond to either an odd or even length signal. By default, `hfft`
assumes an even output length which puts the last entry at the Nyquist
frequency; aliasing with its symmetric counterpart. By Hermitian symmetry,
the value is thus treated as purely real. To avoid losing information, the
shape of the full signal **must** be given.
Examples
--------
>>> signal = np.array([1, 2, 3, 4, 3, 2])
>>> np.fft.fft(signal)
array([15.+0.j, -4.+0.j, 0.+0.j, -1.-0.j, 0.+0.j, -4.+0.j]) # may vary
>>> np.fft.hfft(signal[:4]) # Input first half of signal
array([15., -4., 0., -1., 0., -4.])
>>> np.fft.hfft(signal, 6) # Input entire signal and truncate
array([15., -4., 0., -1., 0., -4.])
>>> signal = np.array([[1, 1.j], [-1.j, 2]])
>>> np.conj(signal.T) - signal # check Hermitian symmetry
array([[ 0.-0.j, -0.+0.j], # may vary
[ 0.+0.j, 0.-0.j]])
>>> freq_spectrum = np.fft.hfft(signal)
>>> freq_spectrum
array([[ 1., 1.],
[ 2., -2.]])
"""
a = asarray(a)
if n is None:
n = (a.shape[axis] - 1) * 2
unitary = _unitary(norm)
return irfft(conjugate(a), n, axis) * (sqrt(n) if unitary else n)
def __mul__(self, other):
return self._rc(multiply(self.array, asarray(other)))
def __setslice__(self, i, j, value):
self.array[i:j] = asarray(value, self.dtype)
def __setitem__(self, index, value):
self.array[index] = asarray(value, self.dtype)
def irfft(a, n=None, axis=-1):
"""
Compute the inverse of the n-point DFT for real input.
This function computes the inverse of the one-dimensional *n*-point
discrete Fourier Transform of real input computed by `rfft`.
In other words, ``irfft(rfft(a), len(a)) == a`` to within numerical
accuracy. (See Notes below for why ``len(a)`` is necessary here.)
The input is expected to be in the form returned by `rfft`, i.e. the
real zero-frequency term followed by the complex positive frequency terms
in order of increasing frequency. Since the discrete Fourier Transform of
real input is Hermite-symmetric, the negative frequency terms are taken
to be the complex conjugates of the corresponding positive frequency terms.
Parameters
----------
a : array_like
The input array.
n : int, optional
Length of the transformed axis of the output.
For `n` output points, ``n//2+1`` input points are necessary. If the
input is longer than this, it is cropped. If it is shorter than this,
it is padded with zeros. If `n` is not given, it is determined from
the length of the input (along the axis specified by `axis`).
axis : int, optional
Axis over which to compute the inverse FFT.
Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
The length of the transformed axis is `n`, or, if `n` is not given,
``2*(m-1)`` where `m` is the length of the transformed axis of the
input. To get an odd number of output points, `n` must be specified.
Raises
------
IndexError
If `axis` is larger than the last axis of `a`.
See Also
--------
numpy.fft : For definition of the DFT and conventions used.
rfft : The one-dimensional FFT of real input, of which `irfft` is inverse.
fft : The one-dimensional FFT.
irfft2 : The inverse of the two-dimensional FFT of real input.
irfftn : The inverse of the *n*-dimensional FFT of real input.
Notes
-----
Returns the real valued `n`-point inverse discrete Fourier transform
of `a`, where `a` contains the non-negative frequency terms of a
Hermite-symmetric sequence. `n` is the length of the result, not the
input.
If you specify an `n` such that `a` must be zero-padded or truncated, the
extra/removed values will be added/removed at high frequencies. One can
thus resample a series to `m` points via Fourier interpolation by:
``a_resamp = irfft(rfft(a), m)``.
Examples
--------
>>> np.fft.ifft([1, -1j, -1, 1j])
array([ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j])
>>> np.fft.irfft([1, -1j, -1])
array([ 0., 1., 0., 0.])
Notice how the last term in the input to the ordinary `ifft` is the
complex conjugate of the second term, and the output has zero imaginary
part everywhere. When calling `irfft`, the negative frequencies are not
specified, and the output array is purely real.
"""
a = asarray(a).astype(complex)
if n is None:
n = (shape(a)[axis] - 1) * 2
return _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftb,
_real_fft_cache) / n
def __rpow__(self, other):
return self._rc(power(asarray(other), self.array))
def ifft(a, n=None, axis=-1):
"""
Compute the one-dimensional inverse discrete Fourier Transform.
This function computes the inverse of the one-dimensional *n*-point
discrete Fourier transform computed by `fft`. In other words,
``ifft(fft(a)) == a`` to within numerical accuracy.
For a general description of the algorithm and definitions,
see `numpy.fft`.
The input should be ordered in the same way as is returned by `fft`,
i.e., ``a[0]`` should contain the zero frequency term,
``a[1:n/2+1]`` should contain the positive-frequency terms, and
``a[n/2+1:]`` should contain the negative-frequency terms, in order of
decreasingly negative frequency. See `numpy.fft` for details.
Parameters
----------
a : array_like
Input array, can be complex.
n : int, optional
Length of the transformed axis of the output.
If `n` is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If `n` is not given,
the length of the input (along the axis specified by `axis`) is used.
See notes about padding issues.
axis : int, optional
Axis over which to compute the inverse DFT. If not given, the last
axis is used.
Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
Raises
------
IndexError
If `axes` is larger than the last axis of `a`.
See Also
--------
numpy.fft : An introduction, with definitions and general explanations.
fft : The one-dimensional (forward) FFT, of which `ifft` is the inverse
ifft2 : The two-dimensional inverse FFT.
ifftn : The n-dimensional inverse FFT.
Notes
-----
If the input parameter `n` is larger than the size of the input, the input
is padded by appending zeros at the end. Even though this is the common
approach, it might lead to surprising results. If a different padding is
desired, it must be performed before calling `ifft`.
Examples
--------
>>> np.fft.ifft([0, 4, 0, 0])
array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j])
Create and plot a band-limited signal with random phases:
>>> import matplotlib.pyplot as plt
>>> t = np.arange(400)
>>> n = np.zeros((400,), dtype=complex)
>>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,)))
>>> s = np.fft.ifft(n)
>>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
>>> plt.legend(('real', 'imaginary'))
<matplotlib.legend.Legend object at 0x...>
>>> plt.show()
"""
a = asarray(a).astype(complex)
if n is None:
n = shape(a)[axis]
return _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftb, _fft_cache) / n
def __pow__(self, other):
return self._rc(power(self.array, asarray(other)))
def __setitem__(self, index, value):
self.weights[index] = asarray(value, self.weights.dtype)
def ifft(a, n=None, axis=-1, norm=None):
"""
Compute the one-dimensional inverse discrete Fourier Transform.
This function computes the inverse of the one-dimensional *n*-point
discrete Fourier transform computed by `fft`. In other words,
``ifft(fft(a)) == a`` to within numerical accuracy.
For a general description of the algorithm and definitions,
see `numpy.fft`.
The input should be ordered in the same way as is returned by `fft`,
i.e.,
* ``a[0]`` should contain the zero frequency term,
* ``a[1:n//2]`` should contain the positive-frequency terms,
* ``a[n//2 + 1:]`` should contain the negative-frequency terms, in
increasing order starting from the most negative frequency.
For an even number of input points, ``A[n//2]`` represents the sum of
the values at the positive and negative Nyquist frequencies, as the two
are aliased together. See `numpy.fft` for details.
Parameters
----------
a : array_like
Input array, can be complex.
n : int, optional
Length of the transformed axis of the output.
If `n` is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If `n` is not given,
the length of the input along the axis specified by `axis` is used.
See notes about padding issues.
axis : int, optional
Axis over which to compute the inverse DFT. If not given, the last
axis is used.
norm : {None, "ortho"}, optional
.. versionadded:: 1.10.0
Normalization mode (see `numpy.fft`). Default is None.
Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
Raises
------
IndexError
If `axes` is larger than the last axis of `a`.
See Also
--------
numpy.fft : An introduction, with definitions and general explanations.
fft : The one-dimensional (forward) FFT, of which `ifft` is the inverse
ifft2 : The two-dimensional inverse FFT.
ifftn : The n-dimensional inverse FFT.
Notes
-----
If the input parameter `n` is larger than the size of the input, the input
is padded by appending zeros at the end. Even though this is the common
approach, it might lead to surprising results. If a different padding is
desired, it must be performed before calling `ifft`.
Examples
--------
>>> np.fft.ifft([0, 4, 0, 0])
array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j]) # may vary
Create and plot a band-limited signal with random phases:
>>> import matplotlib.pyplot as plt
>>> t = np.arange(400)
>>> n = np.zeros((400,), dtype=complex)
>>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,)))
>>> s = np.fft.ifft(n)
>>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
[<matplotlib.lines.Line2D object at ...>, <matplotlib.lines.Line2D object at ...>]
>>> plt.legend(('real', 'imaginary'))
<matplotlib.legend.Legend object at ...>
>>> plt.show()
"""
a = asarray(a)
if n is None:
n = a.shape[axis]
if norm is not None and _unitary(norm):
inv_norm = sqrt(max(n, 1))
else:
inv_norm = n
output = _raw_fft(a, n, axis, False, False, inv_norm)
return output
def __rdiv__(self, other):
return self._rc(divide(asarray(other), self.array))
def irfftn(a, s=None, axes=None, norm=None):
"""
Compute the inverse of the N-dimensional FFT of real input.
This function computes the inverse of the N-dimensional discrete
Fourier Transform for real input over any number of axes in an
M-dimensional array by means of the Fast Fourier Transform (FFT). In
other words, ``irfftn(rfftn(a), a.shape) == a`` to within numerical
accuracy. (The ``a.shape`` is necessary like ``len(a)`` is for `irfft`,
and for the same reason.)
The input should be ordered in the same way as is returned by `rfftn`,
i.e. as for `irfft` for the final transformation axis, and as for `ifftn`
along all the other axes.
Parameters
----------
a : array_like
Input array.
s : sequence of ints, optional
Shape (length of each transformed axis) of the output
(``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.). `s` is also the
number of input points used along this axis, except for the last axis,
where ``s[-1]//2+1`` points of the input are used.
Along any axis, if the shape indicated by `s` is smaller than that of
the input, the input is cropped. If it is larger, the input is padded
with zeros. If `s` is not given, the shape of the input along the axes
specified by axes is used. Except for the last axis which is taken to be
``2*(m-1)`` where ``m`` is the length of the input along that axis.
axes : sequence of ints, optional
Axes over which to compute the inverse FFT. If not given, the last
`len(s)` axes are used, or all axes if `s` is also not specified.
Repeated indices in `axes` means that the inverse transform over that
axis is performed multiple times.
norm : {None, "ortho"}, optional
.. versionadded:: 1.10.0
Normalization mode (see `numpy.fft`). Default is None.
Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axes
indicated by `axes`, or by a combination of `s` or `a`,
as explained in the parameters section above.
The length of each transformed axis is as given by the corresponding
element of `s`, or the length of the input in every axis except for the
last one if `s` is not given. In the final transformed axis the length
of the output when `s` is not given is ``2*(m-1)`` where ``m`` is the
length of the final transformed axis of the input. To get an odd
number of output points in the final axis, `s` must be specified.
Raises
------
ValueError
If `s` and `axes` have different length.
IndexError
If an element of `axes` is larger than than the number of axes of `a`.
See Also
--------
rfftn : The forward n-dimensional FFT of real input,
of which `ifftn` is the inverse.
fft : The one-dimensional FFT, with definitions and conventions used.
irfft : The inverse of the one-dimensional FFT of real input.
irfft2 : The inverse of the two-dimensional FFT of real input.
Notes
-----
See `fft` for definitions and conventions used.
See `rfft` for definitions and conventions used for real input.
The correct interpretation of the hermitian input depends on the shape of
the original data, as given by `s`. This is because each input shape could
correspond to either an odd or even length signal. By default, `irfftn`
assumes an even output length which puts the last entry at the Nyquist
frequency; aliasing with its symmetric counterpart. When performing the
final complex to real transform, the last value is thus treated as purely
real. To avoid losing information, the correct shape of the real input
**must** be given.
Examples
--------
>>> a = np.zeros((3, 2, 2))
>>> a[0, 0, 0] = 3 * 2 * 2
>>> np.fft.irfftn(a)
array([[[1., 1.],
[1., 1.]],
[[1., 1.],
[1., 1.]],
[[1., 1.],
[1., 1.]]])
"""
a = asarray(a)
s, axes = _cook_nd_args(a, s, axes, invreal=1)
for ii in range(len(axes) - 1):
a = ifft(a, s[ii], axes[ii], norm)
a = irfft(a, s[-1], axes[-1], norm)
return a
def _makearray(a):
new = asarray(a)
wrap = getattr(a, "__array_wrap__", new.__array_wrap__)
return new, wrap
def norm(x, ord=None):
"""
Matrix or vector norm.
Parameters
----------
x : array_like, shape (M,) or (M, N)
Input array.
ord : {int, 1, -1, 2, -2, inf, -inf, 'fro'}
Order of the norm (see table under ``Notes``).
Returns
-------
n : float
Norm of the matrix or vector
Notes
-----
For values ord < 0, the result is, strictly speaking, not a
mathematical 'norm', but it may still be useful for numerical
purposes.
The following norms can be calculated:
===== ============================ ==========================
ord norm for matrices norm for vectors
===== ============================ ==========================
None Frobenius norm 2-norm
'fro' Frobenius norm --
inf max(sum(abs(x), axis=1)) max(abs(x))
-inf min(sum(abs(x), axis=1)) min(abs(x))
1 max(sum(abs(x), axis=0)) as below
-1 min(sum(abs(x), axis=0)) as below
2 2-norm (largest sing. value) as below
-2 smallest singular value as below
other -- sum(abs(x)**ord)**(1./ord)
===== ============================ ==========================
"""
x = asarray(x)
nd = len(x.shape)
if ord is None: # check the default case first and handle it immediately
return sqrt(add.reduce((x.conj() * x).ravel().real))
if nd == 1:
if ord == Inf:
return abs(x).max()
elif ord == -Inf:
return abs(x).min()
elif ord == 1:
return abs(x).sum() # special case for speedup
elif ord == 2:
return sqrt(
((x.conj() * x).real).sum()) # special case for speedup
else:
return ((abs(x)**ord).sum())**(1.0 / ord)
elif nd == 2:
if ord == 2:
return svd(x, compute_uv=0).max()
elif ord == -2:
return svd(x, compute_uv=0).min()
elif ord == 1:
return abs(x).sum(axis=0).max()
elif ord == Inf:
return abs(x).sum(axis=1).max()
elif ord == -1:
return abs(x).sum(axis=0).min()
elif ord == -Inf:
return abs(x).sum(axis=1).min()
elif ord in ['fro', 'f']:
return sqrt(add.reduce((x.conj() * x).real.ravel()))
else:
raise ValueError, "Invalid norm order for matrices."
else:
raise ValueError, "Improper number of dimensions to norm."
