def __init__(self, iava, dims, dtype="float64"):
ncp = get_array_module(iava)
# check non-unique pairs (works only with numpy arrays)
_checkunique(to_numpy(iava))
# define dimension of data
ndims = len(dims)
self.dims = dims
self.dimsd = [len(iava[1])] + list(dims[2:])
# find indices and weights
self.iava_t = ncp.floor(iava[0]).astype(int)
self.iava_b = self.iava_t + 1
self.weights_tb = iava[0] - self.iava_t
self.iava_l = ncp.floor(iava[1]).astype(int)
self.iava_r = self.iava_l + 1
self.weights_lr = iava[1] - self.iava_l
# expand dims to weights for nd-arrays
if ndims > 2:
for _ in range(ndims - 2):
self.weights_tb = ncp.expand_dims(self.weights_tb, axis=-1)
self.weights_lr = ncp.expand_dims(self.weights_lr, axis=-1)
self.shape = (np.prod(np.array(self.dimsd)), np.prod(np.array(self.dims)))
self.dtype = np.dtype(dtype)
self.explicit = False
def FISTA(Op, data, niter, eps=0.1, alpha=None, eigsiter=None, eigstol=0,
tol=1e-10, returninfo=False, show=False, threshkind='soft',
perc=None, callback=None):
r"""Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
Solve an optimization problem with :math:`L1` regularization function given
the operator ``Op`` and data ``y``. The operator can be real or complex,
and should ideally be either square :math:`N=M` or underdetermined
:math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
niter : :obj:`int`
Number of iterations
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size (:math:`\alpha \le 1/\lambda_{max}(\mathbf{Op}^H\mathbf{Op})`
guarantees convergence. If ``None``, the maximum eigenvalue is
estimated and the optimal step size is chosen. If provided, the
condition will not be checked internally).
eigsiter : :obj:`int`, optional
Number of iterations for eigenvalue estimation if ``alpha=None``
eigstol : :obj:`float`, optional
Tolerance for eigenvalue estimation if ``alpha=None``
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
returninfo : :obj:`bool`, optional
Return info of FISTA solver
show : :obj:`bool`, optional
Display iterations log
threshkind : :obj:`str`, optional
Kind of thresholding ('hard', 'soft', 'half', 'soft-percentile', or
'half-percentile' - 'soft' used as default)
perc : :obj:`float`, optional
Percentile, as percentage of values to be kept by thresholding (to be
provided when thresholding is soft-percentile or half-percentile)
callback : :obj:`callable`, optional
Function with signature (``callback(x)``) to call after each iteration
where ``x`` is the current model vector
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
Raises
------
NotImplementedError
If ``threshkind`` is different from hard, soft, half, soft-percentile,
or half-percentile
ValueError
If ``perc=None`` when ``threshkind`` is soft-percentile or
half-percentile
See Also
--------
OMP: Orthogonal Matching Pursuit (OMP).
ISTA: Iterative Shrinkage-Thresholding Algorithm (ISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_2^2 +
\epsilon ||\mathbf{x}||_p
using the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) [1]_,
where :math:`p=0, 1, 1/2`. This is a modified version of ISTA solver with
improved convergence properties and limited additional computational cost.
Similarly to the ISTA solver, the choice of the thresholding algorithm to
apply at every iteration is based on the choice of :math:`p`.
.. [1] Beck, A., and Teboulle, M., “A Fast Iterative Shrinkage-Thresholding
Algorithm for Linear Inverse Problems”, SIAM Journal on
Imaging Sciences, vol. 2, pp. 183-202. 2009.
"""
if not threshkind in ['hard', 'soft', 'half', 'hard-percentile',
'soft-percentile', 'half-percentile']:
raise NotImplementedError('threshkind should be hard, soft, half,'
'hard-percentile, soft-percentile, '
'or half-percentile')
if threshkind in ['hard-percentile',
'soft-percentile',
'half-percentile'] and perc is None:
raise ValueError('Provide a percentile when choosing hard-percentile,'
'soft-percentile, or half-percentile thresholding')
# choose thresholding function
if threshkind == 'soft':
threshf = _softthreshold
elif threshkind == 'hard':
threshf = _hardthreshold
elif threshkind == 'half':
threshf = _halfthreshold
elif threshkind == 'hard-percentile':
threshf = _hardthreshold_percentile
elif threshkind == 'soft-percentile':
threshf = _softthreshold_percentile
else:
threshf = _halfthreshold_percentile
# identify backend to use
ncp = get_array_module(data)
if show:
tstart = time.time()
print('FISTA optimization (%s thresholding)\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' % (threshkind,
Op.shape[0],
Op.shape[1],
eps, tol, niter))
# step size
if alpha is None:
if not isinstance(Op, LinearOperator):
Op = LinearOperator(Op, explicit=False)
# compute largest eigenvalues of Op^H * Op
Op1 = LinearOperator(Op.H * Op, explicit=False)
if get_module_name(ncp) == 'numpy':
maxeig = np.abs(Op1.eigs(neigs=1, symmetric=True, niter=eigsiter,
**dict(tol=eigstol, which='LM')))[0]
else:
maxeig = np.abs(power_iteration(Op1, niter=eigsiter,
tol=eigstol, dtype=Op1.dtype,
backend='cupy')[0])
alpha = 1. / maxeig
# define threshold
thresh = eps * alpha * 0.5
if show:
if perc is None:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
else:
print('alpha = %10e\tperc = %.1f' % (alpha, perc))
print('-----------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm xupdate'
print(head1)
# initialize model and cost function
xinv = ncp.zeros(int(Op.shape[1]), dtype=Op.dtype)
zinv = xinv.copy()
t = 1
if returninfo:
cost = np.zeros(niter + 1)
# iterate
for iiter in range(niter):
xinvold = xinv.copy()
# compute residual
resz = data - Op.matvec(zinv)
# compute gradient
grad = alpha * Op.rmatvec(resz)
# update inverted model
xinv_unthesh = zinv + grad
if perc is None:
xinv = threshf(xinv_unthesh, thresh)
else:
xinv = threshf(xinv_unthesh, 100 - perc)
# update auxiliary coefficients
told = t
t = (1. + np.sqrt(1. + 4. * t ** 2)) / 2.
zinv = xinv + ((told - 1.) / t) * (xinv - xinvold)
# model update
xupdate = np.linalg.norm(xinv - xinvold)
if returninfo or show:
costdata = 0.5 * np.linalg.norm(data - Op.matvec(xinv)) ** 2
costreg = eps * np.linalg.norm(xinv, ord=1)
if returninfo:
cost[iiter] = costdata + costreg
# run callback
if callback is not None:
callback(xinv)
if show:
if iiter < 10 or niter - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e %10.3e %9.3e %10.3e' % \
(iiter + 1, to_numpy(xinv[:2])[0], costdata,
costdata + costreg, xupdate)
print(msg)
# check tolerance
if xupdate < tol:
niter = iiter
break
# get values pre-threshold at locations where xinv is different from zero
# xinv = np.where(xinv != 0, xinv_unthesh, xinv)
if show:
print('\nIterations = %d Total time (s) = %.2f'
% (niter, time.time() - tstart))
print('---------------------------------------------------------\n')
if returninfo:
return xinv, niter, cost[:niter]
else:
return xinv, niter
def dottest(Op,
nr,
nc,
tol=1e-6,
complexflag=0,
raiseerror=True,
verb=False,
backend='numpy'):
r"""Dot test.
Generate random vectors :math:`\mathbf{u}` and :math:`\mathbf{v}`
and perform dot-test to verify the validity of forward and adjoint
operators. This test can help to detect errors in the operator
implementation.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Linear operator to test.
nr : :obj:`int`
Number of rows of operator (i.e., elements in data)
nc : :obj:`int`
Number of columns of operator (i.e., elements in model)
tol : :obj:`float`, optional
Dottest tolerance
complexflag : :obj:`bool`, optional
generate random vectors with real (0) or complex numbers
(1: only model, 2: only data, 3:both)
raiseerror : :obj:`bool`, optional
Raise error or simply return ``False`` when dottest fails
verb : :obj:`bool`, optional
Verbosity
backend : :obj:`str`, optional
Backend used for dot test computations (``numpy`` or ``cupy``). This
parameter will be used to choose how to create the random vectors.
Raises
------
ValueError
If dot-test is not verified within chosen tolerance.
Notes
-----
A dot-test is mathematical tool used in the development of numerical
linear operators.
More specifically, a correct implementation of forward and adjoint for
a linear operator should verify the following *equality*
within a numerical tolerance:
.. math::
(\mathbf{Op}*\mathbf{u})^H*\mathbf{v} =
\mathbf{u}^H*(\mathbf{Op}^H*\mathbf{v})
"""
ncp = get_module(backend)
if complexflag in (0, 2):
u = ncp.random.randn(nc)
else:
u = ncp.random.randn(nc) + 1j * ncp.random.randn(nc)
if complexflag in (0, 1):
v = ncp.random.randn(nr)
else:
v = ncp.random.randn(nr) + 1j * ncp.random.randn(nr)
y = Op.matvec(u) # Op * u
x = Op.rmatvec(v) # Op'* v
if complexflag == 0:
yy = ncp.dot(y, v) # (Op * u)' * v
xx = ncp.dot(u, x) # u' * (Op' * v)
else:
yy = ncp.vdot(y, v) # (Op * u)' * v
xx = ncp.vdot(u, x) # u' * (Op' * v)
# convert back to numpy (in case cupy arrays where used), make into a numpy
# array and extract the first element. This is ugly but allows to handle
# complex numbers in subsequent prints also when using cupy arrays.
xx, yy = np.array([to_numpy(xx)])[0], np.array([to_numpy(yy)])[0]
# evaluate if dot test is passed
if complexflag == 0:
if np.abs((yy - xx) / ((yy + xx + 1e-15) / 2)) < tol:
if verb:
print('Dot test passed, v^T(Opu)=%f - u^T(Op^Tv)=%f' %
(yy, xx))
return True
else:
if raiseerror:
raise ValueError(
'Dot test failed, v^T(Opu)=%f - u^T(Op^Tv)=%f' % (yy, xx))
if verb:
print('Dot test failed, v^T(Opu)=%f - u^T(Op^Tv)=%f' %
(yy, xx))
return False
else:
checkreal = np.abs((np.real(yy) - np.real(xx)) /
((np.real(yy) + np.real(xx) + 1e-15) / 2)) < tol
checkimag = np.abs((np.real(yy) - np.real(xx)) /
((np.real(yy) + np.real(xx) + 1e-15) / 2)) < tol
if checkreal and checkimag:
if verb:
print('Dot test passed, v^T(Opu)=%f%+fi - u^T(Op^Tv)=%f%+fi' %
(yy.real, yy.imag, xx.real, xx.imag))
return True
else:
if raiseerror:
raise ValueError('Dot test failed, v^H(Opu)=%f%+fi '
'- u^H(Op^Hv)=%f%+fi' %
(yy.real, yy.imag, xx.real, xx.imag))
if verb:
print('Dot test failed, v^H(Opu)=%f%+fi - u^H(Op^Hv)=%f%+fi' %
(yy.real, yy.imag, xx.real, xx.imag))
return False
def ISTA(Op, data, niter, eps=0.1, alpha=None, eigsiter=None, eigstol=0,
tol=1e-10, monitorres=False, returninfo=False, show=False,
threshkind='soft', perc=None, callback=None):
r"""Iterative Shrinkage-Thresholding Algorithm (ISTA).
Solve an optimization problem with :math:`Lp, \quad p=0, 1/2, 1`
regularization, given the operator ``Op`` and data ``y``. The operator
can be real or complex, and should ideally be either square :math:`N=M`
or underdetermined :math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
niter : :obj:`int`
Number of iterations
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size (:math:`\alpha \le 1/\lambda_{max}(\mathbf{Op}^H\mathbf{Op})`
guarantees convergence. If ``None``, the maximum eigenvalue is
estimated and the optimal step size is chosen. If provided, the
condition will not be checked internally).
eigsiter : :obj:`float`, optional
Number of iterations for eigenvalue estimation if ``alpha=None``
eigstol : :obj:`float`, optional
Tolerance for eigenvalue estimation if ``alpha=None``
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
monitorres : :obj:`bool`, optional
Monitor that residual is decreasing
returninfo : :obj:`bool`, optional
Return info of CG solver
show : :obj:`bool`, optional
Display iterations log
threshkind : :obj:`str`, optional
Kind of thresholding ('hard', 'soft', 'half', 'hard-percentile',
'soft-percentile', or 'half-percentile' - 'soft' used as default)
perc : :obj:`float`, optional
Percentile, as percentage of values to be kept by thresholding (to be
provided when thresholding is soft-percentile or half-percentile)
callback : :obj:`callable`, optional
Function with signature (``callback(x)``) to call after each iteration
where ``x`` is the current model vector
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
Raises
------
NotImplementedError
If ``threshkind`` is different from hard, soft, half, soft-percentile,
or half-percentile
ValueError
If ``perc=None`` when ``threshkind`` is soft-percentile or
half-percentile
ValueError
If ``monitorres=True`` and residual increases
See Also
--------
OMP: Orthogonal Matching Pursuit (OMP).
FISTA: Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_2^2 +
\epsilon ||\mathbf{x}||_p
using the Iterative Shrinkage-Thresholding Algorithms (ISTA) [1]_, where
:math:`p=0, 1, 1/2`. This is a very simple iterative algorithm which
applies the following step:
.. math::
\mathbf{x}^{(i+1)} = T_{(\epsilon \alpha /2, p)} (\mathbf{x}^{(i)} +
\alpha \mathbf{Op}^H (\mathbf{d} - \mathbf{Op} \mathbf{x}^{(i)}))
where :math:`\epsilon \alpha /2` is the threshold and :math:`T_{(\tau, p)}`
is the thresholding rule. The most common variant of ISTA uses the
so-called soft-thresholding rule :math:`T(\tau, p=1)`. Alternatively an
hard-thresholding rule is used in the case of `p=0` or a half-thresholding
rule is used in the case of `p=1/2`. Finally, percentile bases thresholds
are also implemented: the damping factor is not used anymore an the
threshold changes at every iteration based on the computed percentile.
.. [1] Daubechies, I., Defrise, M., and De Mol, C., “An iterative
thresholding algorithm for linear inverse problems with a sparsity
constraint”, Communications on pure and applied mathematics, vol. 57,
pp. 1413-1457. 2004.
"""
if not threshkind in ['hard', 'soft', 'half', 'hard-percentile',
'soft-percentile', 'half-percentile']:
raise NotImplementedError('threshkind should be hard, soft, half,'
'hard-percentile, soft-percentile, '
'or half-percentile')
if threshkind in ['hard-percentile',
'soft-percentile',
'half-percentile'] and perc is None:
raise ValueError('Provide a percentile when choosing hard-percentile,'
'soft-percentile, or half-percentile thresholding')
# choose thresholding function
if threshkind == 'soft':
threshf = _softthreshold
elif threshkind == 'hard':
threshf = _hardthreshold
elif threshkind == 'half':
threshf = _halfthreshold
elif threshkind == 'hard-percentile':
threshf = _hardthreshold_percentile
elif threshkind == 'soft-percentile':
threshf = _softthreshold_percentile
else:
threshf = _halfthreshold_percentile
# identify backend to use
ncp = get_array_module(data)
if show:
tstart = time.time()
print('ISTA optimization (%s thresholding)\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' % (threshkind,
Op.shape[0],
Op.shape[1],
eps, tol, niter))
# step size
if alpha is None:
if not isinstance(Op, LinearOperator):
Op = LinearOperator(Op, explicit=False)
# compute largest eigenvalues of Op^H * Op
Op1 = LinearOperator(Op.H * Op, explicit=False)
if get_module_name(ncp) == 'numpy':
maxeig = np.abs(Op1.eigs(neigs=1, symmetric=True, niter=eigsiter,
**dict(tol=eigstol, which='LM'))[0])
else:
maxeig = np.abs(power_iteration(Op1, niter=eigsiter,
tol=eigstol, dtype=Op1.dtype,
backend='cupy')[0])
alpha = 1./maxeig
# define threshold
thresh = eps * alpha * 0.5
if show:
if perc is None:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
else:
print('alpha = %10e\tperc = %.1f' % (alpha, perc))
print('-----------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm xupdate'
print(head1)
# initialize model and cost function
xinv = ncp.zeros(int(Op.shape[1]), dtype=Op.dtype)
if monitorres:
normresold = np.inf
if returninfo:
cost = np.zeros(niter+1)
# iterate
for iiter in range(niter):
xinvold = xinv.copy()
# compute residual
res = data - Op.matvec(xinv)
if monitorres:
normres = np.linalg.norm(res)
if normres > normresold:
raise ValueError('ISTA stopped at iteration %d due to '
'residual increasing, consider modifying '
'eps and/or alpha...' % iiter)
else:
normresold = normres
# compute gradient
grad = alpha * Op.rmatvec(res)
# update inverted model
xinv_unthesh = xinv + grad
if perc is None:
xinv = threshf(xinv_unthesh, thresh)
else:
xinv = threshf(xinv_unthesh, 100 - perc)
# model update
xupdate = np.linalg.norm(xinv - xinvold)
if returninfo or show:
costdata = 0.5 * np.linalg.norm(res) ** 2
costreg = eps * np.linalg.norm(xinv, ord=1)
if returninfo:
cost[iiter] = costdata + costreg
# run callback
if callback is not None:
callback(xinv)
if show:
if iiter < 10 or niter - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e %10.3e %9.3e %10.3e' % \
(iiter+1, to_numpy(xinv[:2])[0], costdata,
costdata + costreg, xupdate)
print(msg)
# check tolerance
if xupdate < tol:
logging.warning('update smaller that tolerance for '
'iteration %d' % iiter)
niter = iiter
break
# get values pre-threshold at locations where xinv is different from zero
# xinv = np.where(xinv != 0, xinv_unthesh, xinv)
if show:
print('\nIterations = %d Total time (s) = %.2f'
% (niter, time.time() - tstart))
print('---------------------------------------------------------\n')
if returninfo:
return xinv, niter, cost[:niter]
else:
return xinv, niter
def dottest(
Op,
nr=None,
nc=None,
tol=1e-6,
complexflag=0,
raiseerror=True,
verb=False,
backend="numpy",
):
r"""Dot test.
Generate random vectors :math:`\mathbf{u}` and :math:`\mathbf{v}`
and perform dot-test to verify the validity of forward and adjoint
operators. This test can help to detect errors in the operator
implementation.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Linear operator to test.
nr : :obj:`int`
Number of rows of operator (i.e., elements in data)
nc : :obj:`int`
Number of columns of operator (i.e., elements in model)
tol : :obj:`float`, optional
Dottest tolerance
complexflag : :obj:`bool`, optional
Generate random vectors with
* ``0``: Real entries for model and data
* ``1``: Complex entries for model and real entries for data
* ``2``: Real entries for model and complex entries for data
* ``3``: Complex entries for model and data
raiseerror : :obj:`bool`, optional
Raise error or simply return ``False`` when dottest fails
verb : :obj:`bool`, optional
Verbosity
backend : :obj:`str`, optional
Backend used for dot test computations (``numpy`` or ``cupy``). This
parameter will be used to choose how to create the random vectors.
Raises
------
ValueError
If dot-test is not verified within chosen tolerance.
Notes
-----
A dot-test is mathematical tool used in the development of numerical
linear operators.
More specifically, a correct implementation of forward and adjoint for
a linear operator should verify the following *equality*
within a numerical tolerance:
.. math::
(\mathbf{Op}\,\mathbf{u})^H\mathbf{v} =
\mathbf{u}^H(\mathbf{Op}^H\mathbf{v})
"""
ncp = get_module(backend)
if nr is None:
nr = Op.shape[0]
if nc is None:
nc = Op.shape[1]
assert (nr,
nc) == Op.shape, "Provided nr and nc do not match operator shape"
# make u and v vectors
if complexflag != 0:
rdtype = np.real(np.ones(1, Op.dtype)).dtype
if complexflag in (0, 2):
u = ncp.random.randn(nc).astype(Op.dtype)
else:
u = ncp.random.randn(nc).astype(
rdtype) + 1j * ncp.random.randn(nc).astype(rdtype)
if complexflag in (0, 1):
v = ncp.random.randn(nr).astype(Op.dtype)
else:
v = ncp.random.randn(nr).astype(
rdtype) + 1j * ncp.random.randn(nr).astype(rdtype)
y = Op.matvec(u) # Op * u
x = Op.rmatvec(v) # Op'* v
if getattr(Op, "clinear", True):
yy = ncp.vdot(y, v) # (Op * u)' * v
xx = ncp.vdot(u, x) # u' * (Op' * v)
else:
# Op is only R-linear, so treat complex numbers as elements of R^2
yy = ncp.dot(y.real, v.real) + ncp.dot(y.imag, v.imag)
xx = ncp.dot(u.real, x.real) + ncp.dot(u.imag, x.imag)
# convert back to numpy (in case cupy arrays were used), make into a numpy
# array and extract the first element. This is ugly but allows to handle
# complex numbers in subsequent prints also when using cupy arrays.
xx, yy = np.array([to_numpy(xx)])[0], np.array([to_numpy(yy)])[0]
# evaluate if dot test is passed
if complexflag == 0:
if np.abs((yy - xx) / ((yy + xx + 1e-15) / 2)) < tol:
if verb:
print("Dot test passed, v^T(Opu)=%f - u^T(Op^Tv)=%f" %
(yy, xx))
return True
else:
if raiseerror:
raise ValueError(
"Dot test failed, v^T(Opu)=%f - u^T(Op^Tv)=%f" % (yy, xx))
if verb:
print("Dot test failed, v^T(Opu)=%f - u^T(Op^Tv)=%f" %
(yy, xx))
return False
else:
# Check both real and imag parts
checkreal = (np.abs((np.real(yy) - np.real(xx)) /
((np.real(yy) + np.real(xx) + 1e-15) / 2)) < tol)
checkimag = (np.abs((np.imag(yy) - np.imag(xx)) /
((np.imag(yy) + np.imag(xx) + 1e-15) / 2)) < tol)
if checkreal and checkimag:
if verb:
print("Dot test passed, v^T(Opu)=%f%+fi - u^T(Op^Tv)=%f%+fi" %
(yy.real, yy.imag, xx.real, xx.imag))
return True
else:
if raiseerror:
raise ValueError("Dot test failed, v^H(Opu)=%f%+fi "
"- u^H(Op^Hv)=%f%+fi" %
(yy.real, yy.imag, xx.real, xx.imag))
if verb:
print("Dot test failed, v^H(Opu)=%f%+fi - u^H(Op^Hv)=%f%+fi" %
(yy.real, yy.imag, xx.real, xx.imag))
return False
