def check_and_set_idx(ids, idx, prefix):
""" Reconciles passed-in IDs and indices and returns indices, as well as unique IDs
in the order specified by the indices. If only IDs supplied, returns the sort-arg
as the index. If only indices supplied, returns None for IDs. If both supplied,
checks that the correspondence is unique and returns unique IDs in the sort order of
the associated index.
:param np.ndarray ids: array of IDs
:param np.ndarray[int] idx: array of indices
:param str prefix: variable name (for error logging)
:return: unique IDs and indices (passed in or derived from the IDs)
:rtype: np.ndarray, np.ndarray
"""
if ids is None and idx is None:
raise ValueError('Both {}_ids and {}_idx cannot be None'.format(prefix, prefix))
if ids is None:
return None, np.asarray_chkfinite(idx)
if idx is None:
return np.unique(ids, return_inverse=True)
else:
ids = np.asarray(ids)
idx = np.asarray_chkfinite(idx)
if len(idx) != len(ids):
raise ValueError('{}_ids ({}) and {}_idx ({}) must have the same length'.format(
prefix, len(ids), prefix, len(idx)))
uniq_idx, idx_sort_index = np.unique(idx, return_index=True)
# make sure each unique index corresponds to a unique id
if not all(len(set(ids[idx == i])) == 1 for i in uniq_idx):
raise ValueError("Each index must correspond to a unique {}_id".format(prefix))
return ids[idx_sort_index], idx
def orthogonal_procrustes(A, B, check_finite=True):
"""
Compute the matrix solution of the orthogonal Procrustes problem.
Given matrices A and B of equal shape, find an orthogonal matrix R
that most closely maps A to B [1]_.
Note that unlike higher level Procrustes analyses of spatial data,
this function only uses orthogonal transformations like rotations
and reflections, and it does not use scaling or translation.
Parameters
----------
A : (M, N) array_like
Matrix to be mapped.
B : (M, N) array_like
Target matrix.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
R : (N, N) ndarray
The matrix solution of the orthogonal Procrustes problem.
Minimizes the Frobenius norm of dot(A, R) - B, subject to
dot(R.T, R) == I.
scale : float
Sum of the singular values of ``dot(A.T, B)``.
Raises
------
ValueError
If the input arrays are incompatibly shaped.
This may also be raised if matrix A or B contains an inf or nan
and check_finite is True, or if the matrix product AB contains
an inf or nan.
References
----------
.. [1] Peter H. Schonemann, "A generalized solution of the orthogonal
Procrustes problem", Psychometrica -- Vol. 31, No. 1, March, 1996.
"""
if check_finite:
A = np.asarray_chkfinite(A)
B = np.asarray_chkfinite(B)
else:
A = np.asanyarray(A)
B = np.asanyarray(B)
if A.ndim != 2:
raise ValueError('expected ndim to be 2, but observed %s' % A.ndim)
if A.shape != B.shape:
raise ValueError('the shapes of A and B differ (%s vs %s)' % (
A.shape, B.shape))
# Be clever with transposes, with the intention to save memory.
u, w, vt = svd(B.T.dot(A).T)
R = u.dot(vt)
scale = w.sum()
return R, scale
def __init__(self, correct, student_ids=None, item_ids=None, student_idx=None,
item_idx=None, is_held_out=None, num_students=None, num_items=None,
**bn_learner_kwargs):
"""
:param np.ndarray[bool] correct: a 1D array of correctness values
:param np.ndarray|None student_ids: student identifiers for each interaction; if no student
indices provided, sort order of these ids determines theta indices.
:param np.ndarray|None item_ids: item identifiers for each interaction; if no item indices
are provided, sort order of these ids determines item indices.
:param np.ndarray[int]|None student_idx: a 1D array mapping `correct` to student index
:param np.ndarray[int]|None item_idx: a 1D array mapping `correct` to item index
:param np.ndarray[bool] is_held_out: a 1D array indicating whether the interaction should be
held out from training (if not all zeros, a held_out test node will be added to learner)
:param int|None num_students: optional number of students. Default is one plus
the maximum index.
:param int|None num_items: optional number of items. Default is one plus
the maximum index.
:param bn_learner_kwargs: arguments to be passed on to the BayesNetLearner init
"""
# convert pandas Series to np.ndarray and check argument dimensions
correct = np.asarray_chkfinite(correct, dtype=bool)
student_ids, student_idx = check_and_set_idx(student_ids, student_idx, 'student')
item_ids, item_idx = check_and_set_idx(item_ids, item_idx, 'item')
if len(correct) != len(student_idx) or len(correct) != len(item_idx):
raise ValueError("number of elements in correct ({}), student_idx ({}), and item_idx"
"({}) must be the same".format(len(correct), len(student_idx),
len(item_idx)))
if is_held_out is not None and (
len(is_held_out) != len(correct) or is_held_out.dtype != bool):
raise ValueError("held_out ({}) must be None or an array of bools the same length as "
"correct ({})".format(len(is_held_out), len(correct)))
self.num_students = set_or_check_min(num_students, np.max(student_idx) + 1, 'num_students')
self.num_items = set_or_check_min(num_items, np.max(item_idx) + 1, 'num_items')
theta_node = DefaultGaussianNode(THETAS_KEY, self.num_students, ids=student_ids)
offset_node = DefaultGaussianNode(OFFSET_COEFFS_KEY, self.num_items, ids=item_ids)
nodes = [theta_node, offset_node]
# add response nodes (train/test if there is held-out data; else just the train set)
if is_held_out is not None and np.sum(is_held_out):
if np.sum(is_held_out) == len(is_held_out):
raise ValueError("some interactions must be not held out")
is_held_out = np.asarray_chkfinite(is_held_out, dtype=bool)
node_names = (TRAIN_RESPONSES_KEY, TEST_RESPONSES_KEY)
response_idxs = (np.logical_not(is_held_out), is_held_out)
else:
node_names = (TRAIN_RESPONSES_KEY,)
response_idxs = (np.ones_like(correct, dtype=bool),)
for node_name, response_idx in zip(node_names, response_idxs):
cpd = OnePOCPD(item_idx=item_idx[response_idx], theta_idx=student_idx[response_idx],
num_thetas=self.num_students, num_items=self.num_items)
param_nodes = {THETAS_KEY: theta_node, OFFSET_COEFFS_KEY: offset_node}
nodes.append(Node(name=node_name, data=correct[response_idx], cpd=cpd,
solver_pars=SolverPars(learn=False), param_nodes=param_nodes,
held_out=(node_name == TEST_RESPONSES_KEY)))
# store leaf nodes for learning
super(OnePOLearner, self).__init__(nodes=nodes, **bn_learner_kwargs)
def __init__( self, griddata, lo, hi, maps=None, copy=True, verbose=1,
order=1, prefilter=False ):
griddata = np.asanyarray( griddata )
dim = griddata.ndim # - (griddata.shape[-1] == 1) # ??
assert dim >= 2, griddata.shape
self.dim = dim
if np.isscalar(lo):
lo *= np.ones(dim)
if np.isscalar(hi):
hi *= np.ones(dim)
self.loclip = lo = np.asarray_chkfinite( lo ).copy()
self.hiclip = hi = np.asarray_chkfinite( hi ).copy()
assert lo.shape == (dim,), lo.shape
assert hi.shape == (dim,), hi.shape
self.copy = copy
self.verbose = verbose
self.order = order
if order > 1 and 0 < prefilter < 1: # 1/3: Mitchell-Netravali = 1/3 B + 2/3 fit
exactfit = spline_filter( griddata ) # see Unser
griddata += prefilter * (exactfit - griddata)
prefilter = False
self.griddata = griddata
self.prefilter = (prefilter == True)
if maps is None:
maps = [None,] * len(lo)
self.maps = maps
self.nmap = 0
if len(maps) > 0:
assert len(maps) == dim, "maps must have len %d, not %d" % (
dim, len(maps))
# linear maps (map None): Xcol -= lo *= scale -> [0, n-1]
# nonlinear: np.interp e.g. [50 52 62 63] -> [0 1 2 3]
self._lo = np.zeros(dim)
self._scale = np.ones(dim)
for j, (map, n, l, h) in enumerate( zip( maps, griddata.shape, lo, hi )):
## print "test: j map n l h:", j, map, n, l, h
if map is None or callable(map):
self._lo[j] = l
if h > l:
self._scale[j] = (n - 1) / (h - l) # _map lo -> 0, hi -> n - 1
else:
self._scale[j] = 0 # h <= l: X[:,j] -> 0
continue
self.maps[j] = map = np.asanyarray(map)
self.nmap += 1
assert len(map) == n, "maps[%d] must have len %d, not %d" % (
j, n, len(map) )
mlo, mhi = map.min(), map.max()
if not (l <= mlo <= mhi <= h):
print "Warning: Intergrid maps[%d] min %.3g max %.3g " \
"are outside lo %.3g hi %.3g" % (
j, mlo, mhi, l, h )
def cho_solve(c_and_lower, b, overwrite_b=False, check_finite=True):
"""Solve the linear equations A x = b, given the Cholesky factorization of A.
Parameters
----------
(c, lower) : tuple, (array, bool)
Cholesky factorization of a, as given by cho_factor
b : array
Right-hand side
overwrite_b : bool, optional
Whether to overwrite data in b (may improve performance)
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
x : array
The solution to the system A x = b
See also
--------
cho_factor : Cholesky factorization of a matrix
Examples
--------
>>> from scipy.linalg import cho_factor, cho_solve
>>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]])
>>> c, low = cho_factor(A)
>>> x = cho_solve((c, low), [1, 1, 1, 1])
>>> np.allclose(A @ x - [1, 1, 1, 1], np.zeros(4))
True
"""
(c, lower) = c_and_lower
if check_finite:
b1 = asarray_chkfinite(b)
c = asarray_chkfinite(c)
else:
b1 = asarray(b)
c = asarray(c)
if c.ndim != 2 or c.shape[0] != c.shape[1]:
raise ValueError("The factored matrix c is not square.")
if c.shape[1] != b1.shape[0]:
raise ValueError("incompatible dimensions.")
overwrite_b = overwrite_b or _datacopied(b1, b)
potrs, = get_lapack_funcs(('potrs',), (c, b1))
x, info = potrs(c, b1, lower=lower, overwrite_b=overwrite_b)
if info != 0:
raise ValueError('illegal value in %d-th argument of internal potrs'
% -info)
return x
def cho_solve_banded(cb_and_lower, b, overwrite_b=False, check_finite=True):
"""Solve the linear equations A x = b, given the Cholesky factorization of A.
Parameters
----------
(cb, lower) : tuple, (array, bool)
`cb` is the Cholesky factorization of A, as given by cholesky_banded.
`lower` must be the same value that was given to cholesky_banded.
b : array
Right-hand side
overwrite_b : bool
If True, the function will overwrite the values in `b`.
check_finite : boolean, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
x : array
The solution to the system A x = b
See also
--------
cholesky_banded : Cholesky factorization of a banded matrix
Notes
-----
.. versionadded:: 0.8.0
"""
(cb, lower) = cb_and_lower
if check_finite:
cb = asarray_chkfinite(cb)
b = asarray_chkfinite(b)
else:
cb = asarray(cb)
b = asarray(b)
# Validate shapes.
if cb.shape[-1] != b.shape[0]:
raise ValueError("shapes of cb and b are not compatible.")
pbtrs, = get_lapack_funcs(('pbtrs',), (cb, b))
x, info = pbtrs(cb, b, lower=lower, overwrite_b=overwrite_b)
if info > 0:
raise LinAlgError("%d-th leading minor not positive definite" % info)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal pbtrs'
% -info)
return x
def rq(a, overwrite_a=False, lwork=None):
"""Compute RQ decomposition of a square real matrix.
Calculate the decomposition :lm:`A = R Q` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, M)
Square real matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
econ : boolean
Returns
-------
R : double array, shape (M, N) or (K, N) for econ==True
Size K = min(M, N)
Q : double or complex array, shape (M, M) or (M, K) for econ==True
Raises LinAlgError if decomposition fails
"""
# TODO: implement support for non-square and complex arrays
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M,N = a1.shape
if M != N:
raise ValueError('expected square matrix')
if issubclass(a1.dtype.type, complexfloating):
raise ValueError('expected real (non-complex) matrix')
overwrite_a = overwrite_a or (_datanotshared(a1, a))
gerqf, = get_lapack_funcs(('gerqf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
rq, tau, work, info = gerqf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
rq, tau, work, info = gerqf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf'
% -info)
gemm, = get_blas_funcs(('gemm',), (rq,))
t = rq.dtype.char
R = special_matrices.triu(rq)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
k = min(M, N)
for i in range(k):
v = zeros((M,), t)
v[N-k+i] = 1
v[0:N-k+i] = rq[M-k+i, 0:N-k+i]
H = gemm(-tau[i], v, v, 1+0j, ident, trans_b=2)
Q = gemm(1, Q, H)
return R, Q
def det(a, overwrite_a=0):
"""Compute the determinant of a matrix
Parameters
----------
a : array, shape (M, M)
Returns
-------
det : float or complex
Determinant of a
Notes
-----
The determinant is computed via LU factorization, LAPACK routine z/dgetrf.
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError, "expected square matrix"
overwrite_a = overwrite_a or (a1 is not a and not hasattr(a, "__array__"))
fdet, = get_flinalg_funcs(("det",), (a1,))
a_det, info = fdet(a1, overwrite_a=overwrite_a)
if info < 0:
raise ValueError, "illegal value in %-th argument of internal det.getrf" % (-info)
return a_det
def find_disjoint_biclusters(self, biclusters_number=50):
data = np.asarray_chkfinite(self.matrix)
data[data == 0] = 0.000001
coclustering = SpectralCoclustering(n_clusters=biclusters_number, random_state=0)
coclustering.fit(data)
biclusters = set()
for i in range(biclusters_number):
rows, columns = coclustering.get_indices(i)
row_set = set(rows)
columns_set = set(columns)
if len(row_set) > 0 and len(columns_set) > 0:
density = self._calculate_box_cluster_density(row_set, columns_set)
odd_columns = set()
for column in columns_set:
col_density = self._calculate_column_density(column, row_set)
if col_density < density / 4:
odd_columns.add(column)
columns_set.difference_update(odd_columns)
if len(columns_set) == 0:
continue
odd_rows = set()
for row in row_set:
row_density = self._calculate_row_density(row, columns_set)
if row_density < density / 4:
odd_rows.add(row)
row_set.difference_update(odd_rows)
if len(row_set) > 0 and len(columns_set) > 0:
density = self._calculate_box_cluster_density(row_set, columns_set)
biclusters.add(Bicluster(row_set, columns_set, density))
return biclusters
def det(a, overwrite_a=False):
"""Compute the determinant of a matrix
Parameters
----------
a : array, shape (M, M)
Returns
-------
det : float or complex
Determinant of a
Notes
-----
The determinant is computed via LU factorization, LAPACK routine z/dgetrf.
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or _datacopied(a1, a)
fdet, = get_flinalg_funcs(('det',), (a1,))
a_det, info = fdet(a1, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'det.getrf' % -info)
return a_det
def lu(a,permute_l=0,overwrite_a=0):
"""Return LU decompostion of a matrix.
Inputs:
a -- An M x N matrix.
permute_l -- Perform matrix multiplication p * l [disabled].
Outputs:
p,l,u -- LU decomposition matrices of a [permute_l=0]
pl,u -- LU decomposition matrices of a [permute_l=1]
Definitions:
a = p * l * u [permute_l=0]
a = pl * u [permute_l=1]
p - An M x M permutation matrix
l - An M x K lower triangular or trapezoidal matrix
with unit-diagonal
u - An K x N upper triangular or trapezoidal matrix
K = min(M,N)
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError, 'expected matrix'
overwrite_a = overwrite_a or (_datanotshared(a1,a))
flu, = get_flinalg_funcs(('lu',),(a1,))
p,l,u,info = flu(a1,permute_l=permute_l,overwrite_a = overwrite_a)
if info<0: raise ValueError,\
'illegal value in %-th argument of internal lu.getrf'%(-info)
if permute_l:
return l,u
return p,l,u
def _cholesky(a, lower=False, overwrite_a=False, clean=True,
check_finite=True):
"""Common code for cholesky() and cho_factor()."""
a1 = asarray_chkfinite(a) if check_finite else asarray(a)
a1 = atleast_2d(a1)
# Dimension check
if a1.ndim != 2:
raise ValueError('Input array needs to be 2 dimensional but received '
'a {}d-array.'.format(a1.ndim))
# Squareness check
if a1.shape[0] != a1.shape[1]:
raise ValueError('Input array is expected to be square but has '
'the shape: {}.'.format(a1.shape))
# Quick return for square empty array
if a1.size == 0:
return a1.copy(), lower
overwrite_a = overwrite_a or _datacopied(a1, a)
potrf, = get_lapack_funcs(('potrf',), (a1,))
c, info = potrf(a1, lower=lower, overwrite_a=overwrite_a, clean=clean)
if info > 0:
raise LinAlgError("%d-th leading minor of the array is not positive "
"definite" % info)
if info < 0:
raise ValueError('LAPACK reported an illegal value in {}-th argument'
'on entry to "POTRF".'.format(-info))
return c, lower
def __init__(self, data, segment_iterator, uri=None):
# make sure data does not contain NaN nor inf
data = np.asarray_chkfinite(data)
# make sure segment_iterator is actually one of those
if not isinstance(segment_iterator, (SlidingWindow, Timeline)):
raise TypeError("segment_iterator must 'Timeline' or "
"'SlidingWindow'.")
# make sure it iterates the correct number of segments
try:
N = len(segment_iterator)
except Exception:
# an exception is raised by `len(sliding_window)`
# in case sliding window has infinite end.
# this is acceptable, no worry...
pass
else:
n = data.shape[0]
if n != N:
raise ValueError("mismatch between number of segments (%d) "
"and number of feature vectors (%d)." % (N, n))
super(BasePrecomputedSegmentFeature, self).__init__(uri=uri)
self.__data = data
self._segment_iterator = segment_iterator
def schur(a,output='real',lwork=None,overwrite_a=0):
"""Compute Schur decomposition of matrix a.
Description:
Return T, Z such that a = Z * T * (Z**H) where Z is a
unitary matrix and T is either upper-triangular or quasi-upper
triangular for output='real'
"""
if not output in ['real','complex','r','c']:
raise ValueError, "argument must be 'real', or 'complex'"
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError, 'expected square matrix'
typ = a1.dtype.char
if output in ['complex','c'] and typ not in ['F','D']:
if typ in _double_precision:
a1 = a1.astype('D')
typ = 'D'
else:
a1 = a1.astype('F')
typ = 'F'
overwrite_a = overwrite_a or (_datanotshared(a1,a))
gees, = get_lapack_funcs(('gees',),(a1,))
if lwork is None or lwork == -1:
# get optimal work array
result = gees(lambda x: None,a,lwork=-1)
lwork = result[-2][0]
result = gees(lambda x: None,a,lwork=result[-2][0],overwrite_a=overwrite_a)
info = result[-1]
if info<0: raise ValueError,\
'illegal value in %-th argument of internal gees'%(-info)
elif info>0: raise LinAlgError, "Schur form not found. Possibly ill-conditioned."
return result[0], result[-3]
def orthogonal_procrustes(A, ref_matrix, reflection=False):
# Adaptation of scipy.linalg.orthogonal_procrustes -> https://github.com/scipy/scipy/blob/v0.16.0/scipy/linalg/_procrustes.py#L14
# Info here: http://compgroups.net/comp.soft-sys.matlab/procrustes-analysis-without-reflection/896635
# goal is to find unitary matrix R with det(R) > 0 such that ||A*R - ref_matrix||^2 is minimized
from scipy.linalg.decomp_svd import svd # Singular Value Decomposition, factors matrices
from scipy.linalg import det
import numpy as np
A = np.asarray_chkfinite(A)
ref_matrix = np.asarray_chkfinite(ref_matrix)
if A.ndim != 2:
raise ValueError('expected ndim to be 2, but observed %s' % A.ndim)
if A.shape != ref_matrix.shape:
raise ValueError('the shapes of A and ref_matrix differ (%s vs %s)' % (A.shape, ref_matrix.shape))
u, w, vt = svd(ref_matrix.T.dot(A).T)
# Goal: minimize ||A*R - ref||^2, switch to trace
# trace((A*R-ref).T*(A*R-ref)), now we distribute
# trace(R'*A'*A*R) + trace(ref.T*ref) - trace((A*R).T*ref) - trace(ref.T*(A*R)), trace doesn't care about order, so re-order
# trace(R*R.T*A.T*A) + trace(ref.T*ref) - trace(R.T*A.T*ref) - trace(ref.T*A*R), simplify
# trace(A.T*A) + trace(ref.T*ref) - 2*trace(ref.T*A*R)
# Thus, to minimize we want to maximize trace(ref.T * A * R)
# u*w*v.T = (ref.T*A).T
# ref.T * A = w * u.T * v
# trace(ref.T * A * R) = trace (w * u.T * v * R)
# differences minimized when trace(ref.T * A * R) is maximized, thus when trace(u.T * v * R) is maximized
# This occurs when u.T * v * R = I (as u, v and R are all unitary matrices so max is 1)
# R is a rotation matrix so R.T = R^-1
# u.T * v * I = R^-1 = R.T
# R = u * v.T
# Thus, R = u.dot(vt)
R = u.dot(vt) # Get the rotation matrix, including reflections
if not reflection and det(R) < 0: # If we don't want reflection
# To remove reflection, we change the sign of the rightmost column of u (or v) and the scalar associated
# with that column
u[:,-1] *= -1
w[-1] *= -1
R = u.dot(vt)
scale = w.sum() # Get the scaled difference
return R,scale
def expm_cond(A, check_finite=True):
"""
Relative condition number of the matrix exponential in the Frobenius norm.
Parameters
----------
A : 2d array_like
Square input matrix with shape (N, N).
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
kappa : float
The relative condition number of the matrix exponential
in the Frobenius norm
Notes
-----
A faster estimate for the condition number in the 1-norm
has been published but is not yet implemented in scipy.
.. versionadded:: 0.14.0
See also
--------
expm : Compute the exponential of a matrix.
expm_frechet : Compute the Frechet derivative of the matrix exponential.
Examples
--------
>>> from scipy.linalg import expm_cond
>>> A = np.array([[-0.3, 0.2, 0.6], [0.6, 0.3, -0.1], [-0.7, 1.2, 0.9]])
>>> k = expm_cond(A)
>>> k
1.7787805864469866
"""
if check_finite:
A = np.asarray_chkfinite(A)
else:
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected a square matrix')
X = scipy.linalg.expm(A)
K = expm_frechet_kronform(A, check_finite=False)
# The following norm choices are deliberate.
# The norms of A and X are Frobenius norms,
# and the norm of K is the induced 2-norm.
A_norm = scipy.linalg.norm(A, 'fro')
X_norm = scipy.linalg.norm(X, 'fro')
K_norm = scipy.linalg.norm(K, 2)
kappa = (K_norm * A_norm) / X_norm
return kappa
def qr_old(a, overwrite_a=False, lwork=None, check_finite=True):
"""Compute QR decomposition of a matrix.
Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
check_finite : boolean, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
Q : float or complex array, shape (M, M)
R : float or complex array, shape (M, N)
Size K = min(M, N)
Raises LinAlgError if decomposition fails
"""
if check_finite:
a1 = numpy.asarray_chkfinite(a)
else:
a1 = numpy.asarray(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M,N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
geqrf, = get_lapack_funcs(('geqrf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
qr, tau, work, info = geqrf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
qr, tau, work, info = geqrf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf'
% -info)
gemm, = get_blas_funcs(('gemm',), (qr,))
t = qr.dtype.char
R = numpy.triu(qr)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
for i in range(min(M, N)):
v = zeros((M,), t)
v[i] = 1
v[i+1:M] = qr[i+1:M, i]
H = gemm(-tau[i], v, v, 1+0j, ident, trans_b=2)
Q = gemm(1, Q, H)
return Q, R
def _cho_inv_batch(a, check_finite=True):
"""
Invert the matrices a_i, using a Cholesky factorization of A, where
a_i resides in the last two dimensions of a and the other indices describe
the index i.
Overwrites the data in a.
Parameters
----------
a : array
Array of matrices to invert, where the matrices themselves are stored
in the last two dimensions.
check_finite : boolean, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
x : array
Array of inverses of the matrices a_i
See also
--------
scipy.linalg.cholesky : Cholesky factorization of a matrix
"""
if check_finite:
a1 = asarray_chkfinite(a)
else:
a1 = asarray(a)
if len(a1.shape) < 2 or a1.shape[-2] != a1.shape[-1]:
raise ValueError('expected square matrix in last two dimensions')
potrf, potri = get_lapack_funcs(('potrf','potri'), (a1,))
tril_idx = np.tril_indices(a.shape[-2], k=-1)
triu_idx = np.triu_indices(a.shape[-2], k=1)
for index in np.ndindex(a1.shape[:-2]):
# Cholesky decomposition
a1[index], info = potrf(a1[index], lower=True, overwrite_a=False,
clean=False)
if info > 0:
raise LinAlgError("%d-th leading minor not positive definite"
% info)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal'
' potrf' % -info)
# Inversion
a1[index], info = potri(a1[index], lower=True, overwrite_c=False)
if info > 0:
raise LinAlgError("the inverse could not be computed")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal'
' potrf' % -info)
# Make symmetric (dpotri only fills in the lower triangle)
a1[index][triu_idx] = a1[index][tril_idx]
return a1
def pinv(a, cond=None, rcond=None, return_rank=False, check_finite=True):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using a least-squares
solver.
Parameters
----------
a : array, shape (M, N)
Matrix to be pseudo-inverted.
cond, rcond : float, optional
Cutoff for 'small' singular values in the least-squares solver.
Singular values smaller than ``rcond * largest_singular_value``
are considered zero.
return_rank : bool, optional
if True, return the effective rank of the matrix
check_finite : boolean, optional
Whether to check the input matrixes contain only finite numbers.
Disabling may give a performance gain, but may result to problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
B : array, shape (N, M)
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if return_rank == True
Raises
------
LinAlgError
If computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv(a)
>>> np.allclose(a, dot(a, dot(B, a)))
True
>>> np.allclose(B, dot(B, dot(a, B)))
True
"""
if check_finite:
a = np.asarray_chkfinite(a)
else:
a = np.asarray(a)
b = np.identity(a.shape[0], dtype=a.dtype)
if rcond is not None:
cond = rcond
x, resids, rank, s = lstsq(a, b, cond=cond)
if return_rank:
return x, rank
else:
return x
def det(a, overwrite_a=False, check_finite=True):
"""
Compute the determinant of a matrix
The determinant of a square matrix is a value derived arithmetically
from the coefficients of the matrix.
The determinant for a 3x3 matrix, for example, is computed as follows::
a b c
d e f = A
g h i
det(A) = a*e*i +b*f*g + c*d*h - c*e*g - b*d*i - a*f*h
Parameters
----------
a : array_like, shape (M, M)
A square matrix.
overwrite_a : bool
Allow overwriting data in a (may enhance performance).
check_finite : boolean, optional
Whether to check the input matrixes contain only finite numbers.
Disabling may give a performance gain, but may result to problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
det : float or complex
Determinant of `a`.
Notes
-----
The determinant is computed via LU factorization, LAPACK routine z/dgetrf.
Examples
--------
>>> a = np.array([[1,2,3],[4,5,6],[7,8,9]])
>>> linalg.det(a)
0.0
>>> a = np.array([[0,2,3],[4,5,6],[7,8,9]])
>>> linalg.det(a)
3.0
"""
if check_finite:
a1 = np.asarray_chkfinite(a)
else:
a1 = np.asarray(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or _datacopied(a1, a)
fdet, = get_flinalg_funcs(('det',), (a1,))
a_det, info = fdet(a1, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'det.getrf' % -info)
return a_det
def __init__( self, griddata, lo, hi, maps=[], copy=True, verbose=1,
order=1, prefilter=False ):
griddata = np.asanyarray( griddata )
dim = griddata.ndim # - (griddata.shape[-1] == 1) # ??
assert dim >= 2, griddata.shape
self.dim = dim
if np.isscalar(lo):
lo *= np.ones(dim)
if np.isscalar(hi):
hi *= np.ones(dim)
assert lo.shape == (dim,), lo.shape
assert hi.shape == (dim,), hi.shape
self.loclip = lo = np.asarray_chkfinite( lo ).copy()
self.hiclip = hi = np.asarray_chkfinite( hi ).copy()
self.copy = copy
self.verbose = verbose
self.order = order
if order > 1 and 0 < prefilter < 1: # 1/3: Mitchell-Netravali = 1/3 B + 2/3 fit
exactfit = spline_filter( griddata ) # see Unser
griddata += prefilter * (exactfit - griddata)
prefilter = False
self.griddata = griddata
self.prefilter = (prefilter == True)
self.nmap = 0
if maps != []: # sanity check maps --
assert len(maps) == dim, "maps must have len %d, not %d" % (
dim, len(maps))
for j, (map, n, l, h) in enumerate( zip( maps, griddata.shape, lo, hi )):
if map is None or callable(map):
lo[j], hi[j] = 0, 1
continue
maps[j] = map = np.asanyarray(map)
self.nmap += 1
assert len(map) == n, "maps[%d] must have len %d, not %d" % (
j, n, len(map) )
mlo, mhi = map.min(), map.max()
if not (l <= mlo <= mhi <= h):
print "Warning: Intergrid maps[%d] min %.3g max %.3g " \
"are outside lo %.3g hi %.3g" % (
j, mlo, mhi, l, h )
self.maps = maps
self._lo = lo # caller's lo for linear, 0 nonlinear maps
shape_1 = np.array( self.griddata.shape, float ) - 1
self._linearmap = shape_1 / np.where( hi > lo, hi - lo, np.inf ) # 25jun
def _pinvh(a, cond=None, rcond=None, lower=True):
"""Compute the (Moore-Penrose) pseudo-inverse of a hermetian matrix.
Calculate a generalized inverse of a symmetric matrix using its
eigenvalue decomposition and including all 'large' eigenvalues.
Parameters
----------
a : array, shape (N, N)
Real symmetric or complex hermetian matrix to be pseudo-inverted
cond, rcond : float or None
Cutoff for 'small' eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are considered
zero.
If None or -1, suitable machine precision is used.
lower : boolean
Whether the pertinent array data is taken from the lower or upper
triangle of a. (Default: lower)
Returns
-------
B : array, shape (N, N)
Raises
------
LinAlgError
If eigenvalue does not converge
Examples
--------
>>> import numpy as np
>>> a = np.random.randn(9, 6)
>>> a = np.dot(a, a.T)
>>> B = _pinvh(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
"""
a = np.asarray_chkfinite(a)
s, u = linalg.eigh(a, lower=lower)
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
# unlike svd case, eigh can lead to negative eigenvalues
above_cutoff = (abs(s) > cond * np.max(abs(s)))
psigma_diag = np.zeros_like(s)
psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]
return np.dot(u * psigma_diag, np.conjugate(u).T)
def cho_solve(clow, b):
"""Solve a previously factored symmetric system of equations.
First input is a tuple (LorU, lower) which is the output to cho_factor.
Second input is the right-hand side.
"""
c, lower = clow
c = asarray_chkfinite(c)
_assert_squareness(c)
b = asarray_chkfinite(b)
potrs, = get_lapack_funcs(('potrs',),(c,))
b, info = potrs(c,b,lower)
if info < 0:
msg = "Argument %d to lapack's ?potrs() has an illegal value." % info
raise TypeError, msg
if info > 0:
msg = "Unknown error occured int ?potrs(): error code = %d" % info
raise TypeError, msg
return b
def cho_solve(c_and_lower, b, overwrite_b=False, check_finite=True):
"""Solve the linear equations A x = b, given the Cholesky factorization of A.
Parameters
----------
(c, lower) : tuple, (array, bool)
Cholesky factorization of a, as given by cho_factor
b : array
Right-hand side
check_finite : boolean, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
x : array
The solution to the system A x = b
See also
--------
cho_factor : Cholesky factorization of a matrix
"""
(c, lower) = c_and_lower
if check_finite:
b1 = asarray_chkfinite(b)
c = asarray_chkfinite(c)
else:
b1 = asarray(b)
c = asarray(c)
if c.ndim != 2 or c.shape[0] != c.shape[1]:
raise ValueError("The factored matrix c is not square.")
if c.shape[1] != b1.shape[0]:
raise ValueError("incompatible dimensions.")
overwrite_b = overwrite_b or _datacopied(b1, b)
potrs, = get_lapack_funcs(('potrs',), (c, b1))
x, info = potrs(c, b1, lower=lower, overwrite_b=overwrite_b)
if info != 0:
raise ValueError('illegal value in %d-th argument of internal potrs'
% -info)
return x
def pinv2(a, cond=None, rcond=None):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : array, shape (M, N)
Matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' singular values.
Singular values smaller than ``rcond*largest_singular_value``
are considered zero.
If None or -1, suitable machine precision is used.
Returns
-------
B : array, shape (N, M)
The pseudo-inverse of matrix `a`.
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> np.allclose(a, dot(a, dot(B, a)))
True
>>> np.allclose(B, dot(B, dot(a, B)))
True
"""
a = np.asarray_chkfinite(a)
u, s, vh = decomp_svd.svd(a)
t = u.dtype.char
if rcond is not None:
cond = rcond
if cond in [None,-1]:
eps = np.finfo(np.float).eps
feps = np.finfo(np.single).eps
_array_precision = {'f': 0, 'd': 1, 'F': 0, 'D': 1}
cond = {0: feps*1e3, 1: eps*1e6}[_array_precision[t]]
m, n = a.shape
cutoff = cond*np.maximum.reduce(s)
psigma = np.zeros((m, n), t)
for i in range(len(s)):
if s[i] > cutoff:
psigma[i,i] = 1.0/np.conjugate(s[i])
#XXX: use lapack/blas routines for dot
return np.transpose(np.conjugate(np.dot(np.dot(u,psigma),vh)))
def lu_factor(a, overwrite_a=False, check_finite=True):
"""
Compute pivoted LU decomposition of a matrix.
The decomposition is::
A = P L U
where P is a permutation matrix, L lower triangular with unit
diagonal elements, and U upper triangular.
Parameters
----------
a : (M, M) array_like
Matrix to decompose
overwrite_a : bool, optional
Whether to overwrite data in A (may increase performance)
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
lu : (N, N) ndarray
Matrix containing U in its upper triangle, and L in its lower triangle.
The unit diagonal elements of L are not stored.
piv : (N,) ndarray
Pivot indices representing the permutation matrix P:
row i of matrix was interchanged with row piv[i].
See also
--------
lu_solve : solve an equation system using the LU factorization of a matrix
Notes
-----
This is a wrapper to the ``*GETRF`` routines from LAPACK.
"""
if check_finite:
a1 = asarray_chkfinite(a)
else:
a1 = asarray(a)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
getrf, = get_lapack_funcs(('getrf',), (a1,))
lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of '
'internal getrf (lu_factor)' % -info)
if info > 0:
warn("Diagonal number %d is exactly zero. Singular matrix." % info,
RuntimeWarning)
return lu, piv
def norm(a, ord=None):
# Differs from numpy only in non-finite handling and the use of
# blas
a = np.asarray_chkfinite(a)
if ord in (None, 2) and (a.ndim == 1) and (a.dtype.char in 'fdFD'):
# use blas for fast and stable euclidean norm
func_name = _nrm2_prefix.get(a.dtype.char, 'd') + 'nrm2'
nrm2 = getattr(fblas, func_name)
return nrm2(a)
return np.linalg.norm(a, ord=ord)
def pinv(a, cond=None, rcond=None):
""" pinv(a, rcond=None) -> a_pinv
Compute generalized inverse of A using least-squares solver.
"""
a = asarray_chkfinite(a)
b = numpy.identity(a.shape[0], dtype=a.dtype)
if rcond is not None:
cond = rcond
return lstsq(a, b, cond=cond)[0]
def lu_solve(a_lu_pivots,b):
"""Solve a previously factored system. First input is a tuple (lu, pivots)
which is the output to lu_factor. Second input is the right hand side.
"""
a_lu, pivots = a_lu_pivots
a_lu = asarray_chkfinite(a_lu)
pivots = asarray_chkfinite(pivots)
b = asarray_chkfinite(b)
_assert_squareness(a_lu)
getrs, = get_lapack_funcs(('getrs',),(a_lu,))
b, info = getrs(a_lu,pivots,b)
if info < 0:
msg = "Argument %d to lapack's ?getrs() has an illegal value." % info
raise TypeError, msg
if info > 0:
msg = "Unknown error occured int ?getrs(): error code = %d" % info
raise TypeError, msg
return b
def rq(a,overwrite_a=0,lwork=None):
"""RQ decomposition of an M x N matrix a.
Description:
Find an upper-triangular matrix r and a unitary (orthogonal)
matrix q such that r * q = a
Inputs:
a -- the matrix
overwrite_a=0 -- if non-zero then discard the contents of a,
i.e. a is used as a work array if possible.
lwork=None -- >= shape(a)[1]. If None (or -1) compute optimal
work array size.
Outputs:
r, q -- matrices such that r * q = a
"""
# TODO: implement support for non-square and complex arrays
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError, 'expected matrix'
M,N = a1.shape
if M != N:
raise ValueError, 'expected square matrix'
if issubclass(a1.dtype.type,complexfloating):
raise ValueError, 'expected real (non-complex) matrix'
overwrite_a = overwrite_a or (_datanotshared(a1,a))
gerqf, = get_lapack_funcs(('gerqf',),(a1,))
if lwork is None or lwork == -1:
# get optimal work array
rq,tau,work,info = gerqf(a1,lwork=-1,overwrite_a=1)
lwork = work[0]
rq,tau,work,info = gerqf(a1,lwork=lwork,overwrite_a=overwrite_a)
if info<0: raise ValueError, \
'illegal value in %-th argument of internal geqrf'%(-info)
gemm, = get_blas_funcs(('gemm',),(rq,))
t = rq.dtype.char
R = basic.triu(rq)
Q = numpy.identity(M,dtype=t)
ident = numpy.identity(M,dtype=t)
zeros = numpy.zeros
k = min(M,N)
for i in range(k):
v = zeros((M,),t)
v[N-k+i] = 1
v[0:N-k+i] = rq[M-k+i,0:N-k+i]
H = gemm(-tau[i],v,v,1+0j,ident,trans_b=2)
Q = gemm(1,Q,H)
return R, Q
def pinvh(a, cond=None, rcond=None, lower=True):
"""Compute the (Moore-Penrose) pseudo-inverse of a hermetian matrix.
Calculate a generalized inverse of a symmetric matrix using its
eigenvalue decomposition and including all 'large' eigenvalues.
Parameters
----------
a : array, shape (N, N)
Real symmetric or complex hermetian matrix to be pseudo-inverted
cond : float or None, default None
Cutoff for 'small' eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are considered
zero.
If None or -1, suitable machine precision is used.
rcond : float or None, default None (deprecated)
Cutoff for 'small' eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are considered
zero.
If None or -1, suitable machine precision is used.
lower : boolean
Whether the pertinent array data is taken from the lower or upper
triangle of a. (Default: lower)
Returns
-------
B : array, shape (N, N)
Raises
------
LinAlgError
If eigenvalue does not converge
Examples
--------
>>> import numpy as np
>>> a = np.random.randn(9, 6)
>>> a = np.dot(a, a.T)
>>> B = pinvh(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
"""
a = np.asarray_chkfinite(a)
s, u = linalg.eigh(a, lower=lower)
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
# unlike svd case, eigh can lead to negative eigenvalues
above_cutoff = (abs(s) > cond * np.max(abs(s)))
psigma_diag = np.zeros_like(s)
psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]
return np.dot(u * psigma_diag, np.conjugate(u).T)
def inv(a, overwrite_a=0):
"""Compute the inverse of a matrix.
Parameters
----------
a : array-like, shape (M, M)
Matrix to be inverted
Returns
-------
ainv : array-like, shape (M, M)
Inverse of the matrix a
Raises LinAlgError if a is singular
Examples
--------
>>> a = array([[1., 2.], [3., 4.]])
>>> inv(a)
array([[-2. , 1. ],
[ 1.5, -0.5]])
>>> dot(a, inv(a))
array([[ 1., 0.],
[ 0., 1.]])
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError, 'expected square matrix'
overwrite_a = overwrite_a or (a1 is not a and not hasattr(a, '__array__'))
#XXX: I found no advantage or disadvantage of using finv.
## finv, = get_flinalg_funcs(('inv',),(a1,))
## if finv is not None:
## a_inv,info = finv(a1,overwrite_a=overwrite_a)
## if info==0:
## return a_inv
## if info>0: raise LinAlgError, "singular matrix"
## if info<0: raise ValueError,\
## 'illegal value in %-th argument of internal inv.getrf|getri'%(-info)
getrf, getri = get_lapack_funcs(('getrf', 'getri'), (a1, ))
#XXX: C ATLAS versions of getrf/i have rowmajor=1, this could be
# exploited for further optimization. But it will be probably
# a mess. So, a good testing site is required before trying
# to do that.
if getrf.module_name[:7] == 'clapack' != getri.module_name[:7]:
# ATLAS 3.2.1 has getrf but not getri.
lu, piv, info = getrf(transpose(a1),
rowmajor=0,
overwrite_a=overwrite_a)
lu = transpose(lu)
else:
lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
if info == 0:
if getri.module_name[:7] == 'flapack':
lwork = calc_lwork.getri(getri.prefix, a1.shape[0])
lwork = lwork[1]
# XXX: the following line fixes curious SEGFAULT when
# benchmarking 500x500 matrix inverse. This seems to
# be a bug in LAPACK ?getri routine because if lwork is
# minimal (when using lwork[0] instead of lwork[1]) then
# all tests pass. Further investigation is required if
# more such SEGFAULTs occur.
lwork = int(1.01 * lwork)
inv_a, info = getri(lu, piv, lwork=lwork, overwrite_lu=1)
else: # clapack
inv_a, info = getri(lu, piv, overwrite_lu=1)
if info > 0: raise LinAlgError, "singular matrix"
if info < 0: raise ValueError,\
'illegal value in %-th argument of internal getrf|getri'%(-info)
return inv_a
def nnls_predotted(A_dot_A, A_dot_b, tol=1e-8):
"""
Solve ``argmin_x || Ax - b ||_2`` for ``x>=0``. This version may be superior to the FORTRAN implementation when ``A`` has more rows than
columns, and especially when ``A`` is sparse.
Note that the arguments and return values differ from the FORTRAN implementation; in particular, this implementation does not expect the actual
design matrix ``A`` nor the RHS vector ``b``, but rather ``A.T.dot(A)`` and ``A.T.dot(b)``. These are smaller than the original ``A`` and ``b``
iff ``A`` has more rows than columns.
This function also does not return the residual. The squared residual ``|| Ax-b ||^2`` may be calculated efficiently as:
``b.dot(b) + x.dot(A_dot_A.dot(x) - 2*A_dot_b)``
where ``x`` is the output of this function
Parameters
----------
A_dot_A : ndarray
Square matrix corresponding to ``A.T.dot(A)``, where ``A`` is as shown above.
A_dot_b : ndarray
Vector corresponding to ``A.T.dot(b)``, where ``A`` and ``b`` are as shown above.
Returns
-------
x : ndarray
Solution vector.
"""
A_dot_A = np.asarray_chkfinite(A_dot_A)
A_dot_b = np.asarray_chkfinite(A_dot_b)
if len(A_dot_A.shape) != 2:
raise ValueError("expected matrix")
if len(A_dot_b.shape) != 1:
raise ValueError("expected vector")
nvar = A_dot_A.shape[0]
if nvar != A_dot_A.shape[1]:
raise ValueError("expected square matrix")
if nvar != A_dot_b.shape[0]:
raise ValueError("incompatible dimensions")
P_bool = np.zeros(nvar, np.bool)
x = np.zeros(nvar, dtype=A_dot_A.dtype)
s = np.empty_like(x)
w = A_dot_b
while not P_bool.all() and w.max() > tol:
j_idx = w[~P_bool].argmax()
newly_allowed = np.flatnonzero(~P_bool)[j_idx]
P_bool[newly_allowed] = True
s[:] = 0
currPs = np.flatnonzero(P_bool)
if len(currPs) > 1:
s[currPs] = np.linalg.solve(
A_dot_A[currPs[:, None], currPs[None, :]], A_dot_b[currPs])
else:
currP = currPs[0]
s[currP] = A_dot_b[currP] / A_dot_A[currP, currP]
s_P_l_0 = (s[currPs] < 0)
while s_P_l_0.any():
currPs_s_P_l_0 = currPs[s_P_l_0]
alpha = (x[currPs_s_P_l_0] /
(x[currPs_s_P_l_0] - s[currPs_s_P_l_0])).min()
x += alpha * (s - x)
P_bool[currPs] = (x[currPs] > tol)
s[:] = 0
currPs = np.flatnonzero(P_bool)
if len(currPs) > 1:
s[currPs] = np.linalg.solve(
A_dot_A[currPs[:, None], currPs[None, :]], A_dot_b[currPs])
elif len(currPs) == 1:
currP = currPs[0]
s[currP] = A_dot_b[currP] / A_dot_A[currP, currP]
s_P_l_0 = (s[currPs] < 0)
x[:] = s[:]
if x[newly_allowed] == 0:
break # avoid infinite loop
w = A_dot_b - A_dot_A.dot(x)
return x
def eggholder(x):
x = np.asarray_chkfinite(x)
return (-(x[1] + 47) * np.sin(np.sqrt(abs(x[0] / 2 + (x[1] + 47)))) -
x[0] * np.sin(np.sqrt(abs(x[0] - (x[1] + 47)))))
def norm(a, ord=None, axis=None, keepdims=False, check_finite=True):
"""
Matrix or vector norm.
This function is able to return one of seven different matrix norms,
or one of an infinite number of vector norms (described below), depending
on the value of the ``ord`` parameter.
Parameters
----------
a : (M,) or (M, N) array_like
Input array. If `axis` is None, `a` must be 1D or 2D.
ord : {non-zero int, inf, -inf, 'fro'}, optional
Order of the norm (see table under ``Notes``). inf means NumPy's
`inf` object
axis : {int, 2-tuple of ints, None}, optional
If `axis` is an integer, it specifies the axis of `a` along which to
compute the vector norms. If `axis` is a 2-tuple, it specifies the
axes that hold 2-D matrices, and the matrix norms of these matrices
are computed. If `axis` is None then either a vector norm (when `a`
is 1-D) or a matrix norm (when `a` is 2-D) is returned.
keepdims : bool, optional
If this is set to True, the axes which are normed over are left in the
result as dimensions with size one. With this option the result will
broadcast correctly against the original `a`.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
n : float or ndarray
Norm of the matrix or vector(s).
Notes
-----
For values of ``ord <= 0``, the result is, strictly speaking, not a
mathematical 'norm', but it may still be useful for various 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))
0 -- sum(x != 0)
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)
===== ============================ ==========================
The Frobenius norm is given by [1]_:
:math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
The ``axis`` and ``keepdims`` arguments are passed directly to
``numpy.linalg.norm`` and are only usable if they are supported
by the version of numpy in use.
References
----------
.. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples
--------
>>> from scipy.linalg import norm
>>> a = np.arange(9) - 4.0
>>> a
array([-4., -3., -2., -1., 0., 1., 2., 3., 4.])
>>> b = a.reshape((3, 3))
>>> b
array([[-4., -3., -2.],
[-1., 0., 1.],
[ 2., 3., 4.]])
>>> norm(a)
7.745966692414834
>>> norm(b)
7.745966692414834
>>> norm(b, 'fro')
7.745966692414834
>>> norm(a, np.inf)
4
>>> norm(b, np.inf)
9
>>> norm(a, -np.inf)
0
>>> norm(b, -np.inf)
2
>>> norm(a, 1)
20
>>> norm(b, 1)
7
>>> norm(a, -1)
-4.6566128774142013e-010
>>> norm(b, -1)
6
>>> norm(a, 2)
7.745966692414834
>>> norm(b, 2)
7.3484692283495345
>>> norm(a, -2)
0
>>> norm(b, -2)
1.8570331885190563e-016
>>> norm(a, 3)
5.8480354764257312
>>> norm(a, -3)
0
"""
# Differs from numpy only in non-finite handling and the use of blas.
if check_finite:
a = np.asarray_chkfinite(a)
else:
a = np.asarray(a)
# Only use optimized norms if axis and keepdims are not specified.
if a.dtype.char in 'fdFD' and axis is None and not keepdims:
if ord in (None, 2) and (a.ndim == 1):
# use blas for fast and stable euclidean norm
nrm2 = get_blas_funcs('nrm2', dtype=a.dtype)
return nrm2(a)
if a.ndim == 2 and axis is None and not keepdims:
# Use lapack for a couple fast matrix norms.
# For some reason the *lange frobenius norm is slow.
lange_args = None
# Make sure this works if the user uses the axis keywords
# to apply the norm to the transpose.
if ord == 1:
if np.isfortran(a):
lange_args = '1', a
elif np.isfortran(a.T):
lange_args = 'i', a.T
elif ord == np.inf:
if np.isfortran(a):
lange_args = 'i', a
elif np.isfortran(a.T):
lange_args = '1', a.T
if lange_args:
lange = get_lapack_funcs('lange', dtype=a.dtype)
return lange(*lange_args)
# Filter out the axis and keepdims arguments if they aren't used so they
# are never inadvertently passed to a version of numpy that doesn't
# support them.
if axis is not None:
if keepdims:
return np.linalg.norm(a, ord=ord, axis=axis, keepdims=keepdims)
return np.linalg.norm(a, ord=ord, axis=axis)
return np.linalg.norm(a, ord=ord)
def moments(timeseries: np.ndarray,
bw: float = None,
bins: np.ndarray = None,
power: int = 6,
lag: list = [1],
correction: bool = True,
norm: bool = False,
kernel: callable = None,
tol: float = 1e-10,
conv_method: str = 'auto',
verbose: bool = False) -> np.ndarray:
r"""
Estimates the moments of the Kramers─Moyal expansion from a timeseries using
a Nadaraya─Watson kernel estimator method. These later can be turned into
the drift and diffusion coefficients after normalisation.
Parameters
----------
timeseries: np.ndarray
A 1-dimensional timeseries.
bw: float
Desired bandwidth of the kernel. A value of 1 occupies the full space of
the bin space. Recommended are values ``0.005 < bw < 0.4``.
bins: np.ndarray (default ``None``)
The number of bins for each dimension, defaults to ``np.array([5000])``.
This is the underlying space for the Kramers─Moyal conditional moments.
power: int (default ``6``)
Upper limit of the the Kramers─Moyal conditional moments to calculate.
It will generate all Kramers─Moyal conditional moments up to power.
lag: list (default ``1``)
Calculates the Kramers─Moyal conditional moments at each indicated lag,
i.e., for ``timeseries[::lag[]]``. Defaults to ``1``, the shortest
timestep in the data.
corrections: bool (default ``True``)
Implements the second-order corrections of the Kramers─Moyal conditional
moments directly
norm: bool (default ``False``)
Sets the normalisation. ``False`` returns the Kramers─Moyal conditional
moments, and ``True`` returns the Kramers─Moyal coefficients.
kernel: callable (default ``None``)
Kernel used to convolute with the Kramers─Moyal conditional moments. To
select example an Epanechnikov kernel use
kernel = kernels.epanechnikov
If None the Epanechnikov kernel will be used.
tol: float (default ``1e-10``)
Round to zero absolute values smaller than ``tol``, after convolutions.
conv_method: str (default ``auto``)
A string indicating which method to use to calculate the convolution.
docs.scipy.org/doc/scipy/reference/generated/scipy.signal.convolve.
verbose: bool (default ``False``)
If ``True`` will report on the bandwidth used.
Returns
-------
edges: np.ndarray
The bin edges with shape (D,bins.shape) of the calculated moments.
moments: np.ndarray
The calculated moments from the Kramers─Moyal expansion of the
timeseries at each lag. To extract the selected orders of the moments,
use ``moments[i,:,j]``, with ``i`` the order according to powers, ``j``
the lag (if any given).
"""
timeseries = np.asarray_chkfinite(timeseries, dtype=float)
if len(timeseries.shape) == 1:
timeseries = timeseries.reshape(-1, 1)
assert len(timeseries.shape) == 2, "Timeseries must be 1-dimensional"
assert timeseries.shape[0] > 0, "No data in timeseries"
if bins is None:
bins = np.array([5000])
if lag is None:
lag = [1]
powers = np.linspace(0, power, power + 1).astype(int)
if len(powers.shape) == 1:
powers = powers.reshape(-1, 1)
if bw is None:
bw = silvermans_rule(timeseries)
elif callable(bw):
bw = bw(timeseries)
assert bw > 0.0, "Bandwidth must be > 0"
if kernel is None:
kernel = epanechnikov
assert kernel in _kernels, "Kernel not found"
if verbose == True:
print(r'bandwidth = {:f}'.format(bw) +
r', bins = {:d}'.format(bins[0]))
edges, moments = _moments(timeseries, bins, powers, lag, kernel, bw, tol,
conv_method)
if correction == True:
moments = corrections(m=moments, power=power)
if norm == True:
for i in range(power):
moments = moments / float(factorial(i))
return (edges, moments)
def norm(a, ord=None):
"""
Matrix or vector norm.
This function is able to return one of seven different matrix norms,
or one of an infinite number of vector norms (described below), depending
on the value of the ``ord`` parameter.
Parameters
----------
x : (M,) or (M, N) array_like
Input array.
ord : {non-zero int, inf, -inf, 'fro'}, optional
Order of the norm (see table under ``Notes``). inf means numpy's
`inf` object.
Returns
-------
norm : float
Norm of the matrix or vector.
Notes
-----
For values of ``ord <= 0``, the result is, strictly speaking, not a
mathematical 'norm', but it may still be useful for various 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))
0 -- sum(x != 0)
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)
===== ============================ ==========================
The Frobenius norm is given by [1]_:
:math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
References
----------
.. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples
--------
>>> from scipy.linalg import norm
>>> a = np.arange(9) - 4
>>> a
array([-4, -3, -2, -1, 0, 1, 2, 3, 4])
>>> b = a.reshape((3, 3))
>>> b
array([[-4, -3, -2],
[-1, 0, 1],
[ 2, 3, 4]])
>>> norm(a)
7.745966692414834
>>> norm(b)
7.745966692414834
>>> norm(b, 'fro')
7.745966692414834
>>> norm(a, np.inf)
4
>>> norm(b, np.inf)
9
>>> norm(a, -np.inf)
0
>>> norm(b, -np.inf)
2
>>> norm(a, 1)
20
>>> norm(b, 1)
7
>>> norm(a, -1)
-4.6566128774142013e-010
>>> norm(b, -1)
6
>>> norm(a, 2)
7.745966692414834
>>> norm(b, 2)
7.3484692283495345
>>> norm(a, -2)
nan
>>> norm(b, -2)
1.8570331885190563e-016
>>> norm(a, 3)
5.8480354764257312
>>> norm(a, -3)
nan
"""
# Differs from numpy only in non-finite handling and the use of
# blas
a = np.asarray_chkfinite(a)
if ord in (None, 2) and (a.ndim == 1) and (a.dtype.char in 'fdFD'):
# use blas for fast and stable euclidean norm
func_name = _nrm2_prefix.get(a.dtype.char, 'd') + 'nrm2'
nrm2 = getattr(blas, func_name)
return nrm2(a)
return np.linalg.norm(a, ord=ord)
def rastrigin(x): # rast.m
x = np.asarray_chkfinite(x)
n = len(x)
return 10 * n + sum(x**2 - 10 * cos(2 * pi * x))
# In[41]: a=np.asfarray([2, 3]) a # ## 29. numpy.asarray_chkfinite # Convert a list into an array. # If all elements are finite asarray_chkfinite is identical to asarray. # In[42]: a = [90,80,70] np.asarray_chkfinite(a,dtype=float) # ## 30. numpy.asscalar # Convert an array of size 1 to its scalar equivalent. # In[43]: np.asscalar(np.array([10000000])) # ## 31. numpy.concatenate # Join a sequence of arrays along an existing axis. # In[44]:
def rq(a, overwrite_a=False, lwork=None, mode='full', check_finite=True):
"""
Compute RQ decomposition of a matrix.
Calculate the decomposition ``A = R Q`` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : (M, N) array_like
Matrix to be decomposed
overwrite_a : bool, optional
Whether data in a is overwritten (may improve performance)
lwork : int, optional
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
mode : {'full', 'r', 'economic'}, optional
Determines what information is to be returned: either both Q and R
('full', default), only R ('r') or both Q and R but computed in
economy-size ('economic', see Notes).
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
R : float or complex ndarray
Of shape (M, N) or (M, K) for ``mode='economic'``. ``K = min(M, N)``.
Q : float or complex ndarray
Of shape (N, N) or (K, N) for ``mode='economic'``. Not returned
if ``mode='r'``.
Raises
------
LinAlgError
If decomposition fails.
Notes
-----
This is an interface to the LAPACK routines sgerqf, dgerqf, cgerqf, zgerqf,
sorgrq, dorgrq, cungrq and zungrq.
If ``mode=economic``, the shapes of Q and R are (K, N) and (M, K) instead
of (N,N) and (M,N), with ``K=min(M,N)``.
Examples
--------
>>> from scipy import linalg
>>> a = np.random.randn(6, 9)
>>> r, q = linalg.rq(a)
>>> np.allclose(a, r @ q)
True
>>> r.shape, q.shape
((6, 9), (9, 9))
>>> r2 = linalg.rq(a, mode='r')
>>> np.allclose(r, r2)
True
>>> r3, q3 = linalg.rq(a, mode='economic')
>>> r3.shape, q3.shape
((6, 6), (6, 9))
"""
if mode not in ['full', 'r', 'economic']:
raise ValueError(
"Mode argument should be one of ['full', 'r', 'economic']")
if check_finite:
a1 = numpy.asarray_chkfinite(a)
else:
a1 = numpy.asarray(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M, N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
gerqf, = get_lapack_funcs(('gerqf', ), (a1, ))
rq, tau = safecall(gerqf,
'gerqf',
a1,
lwork=lwork,
overwrite_a=overwrite_a)
if not mode == 'economic' or N < M:
R = numpy.triu(rq, N - M)
else:
R = numpy.triu(rq[-M:, -M:])
if mode == 'r':
return R
gor_un_grq, = get_lapack_funcs(('orgrq', ), (rq, ))
if N < M:
Q, = safecall(gor_un_grq,
"gorgrq/gungrq",
rq[-N:],
tau,
lwork=lwork,
overwrite_a=1)
elif mode == 'economic':
Q, = safecall(gor_un_grq,
"gorgrq/gungrq",
rq,
tau,
lwork=lwork,
overwrite_a=1)
else:
rq1 = numpy.empty((N, N), dtype=rq.dtype)
rq1[-M:] = rq
Q, = safecall(gor_un_grq,
"gorgrq/gungrq",
rq1,
tau,
lwork=lwork,
overwrite_a=1)
return R, Q
def curve_fit(f,
xdata,
ydata,
p0=None,
sigma=None,
absolute_sigma=False,
check_finite=True,
bounds=(-np.inf, np.inf),
method=None,
jac=None,
**kwargs):
"""
Use non-linear least squares to fit a function, f, to data.
Assumes ``ydata = f(xdata, *params) + eps``
Parameters
----------
f : callable
The model function, f(x, ...). It must take the independent
variable as the first argument and the parameters to fit as
separate remaining arguments.
xdata : array_like or object
The independent variable where the data is measured.
Should usually be an M-length sequence or an (k,M)-shaped array for
functions with k predictors, but can actually be any object.
ydata : array_like
The dependent data, a length M array - nominally ``f(xdata, ...)``.
p0 : array_like, optional
Initial guess for the parameters (length N). If None, then the
initial values will all be 1 (if the number of parameters for the
function can be determined using introspection, otherwise a
ValueError is raised).
sigma : None or M-length sequence or MxM array, optional
Determines the uncertainty in `ydata`. If we define residuals as
``r = ydata - f(xdata, *popt)``, then the interpretation of `sigma`
depends on its number of dimensions:
- A 1-d `sigma` should contain values of standard deviations of
errors in `ydata`. In this case, the optimized function is
``chisq = sum((r / sigma) ** 2)``.
- A 2-d `sigma` should contain the covariance matrix of
errors in `ydata`. In this case, the optimized function is
``chisq = r.T @ inv(sigma) @ r``.
.. versionadded:: 0.19
None (default) is equivalent of 1-d `sigma` filled with ones.
absolute_sigma : bool, optional
If True, `sigma` is used in an absolute sense and the estimated parameter
covariance `pcov` reflects these absolute values.
If False, only the relative magnitudes of the `sigma` values matter.
The returned parameter covariance matrix `pcov` is based on scaling
`sigma` by a constant factor. This constant is set by demanding that the
reduced `chisq` for the optimal parameters `popt` when using the
*scaled* `sigma` equals unity. In other words, `sigma` is scaled to
match the sample variance of the residuals after the fit.
Mathematically,
``pcov(absolute_sigma=False) = pcov(absolute_sigma=True) * chisq(popt)/(M-N)``
check_finite : bool, optional
If True, check that the input arrays do not contain nans of infs,
and raise a ValueError if they do. Setting this parameter to
False may silently produce nonsensical results if the input arrays
do contain nans. Default is True.
bounds : 2-tuple of array_like, optional
Lower and upper bounds on parameters. Defaults to no bounds.
Each element of the tuple must be either an array with the length equal
to the number of parameters, or a scalar (in which case the bound is
taken to be the same for all parameters.) Use ``np.inf`` with an
appropriate sign to disable bounds on all or some parameters.
.. versionadded:: 0.17
method : {'lm', 'trf', 'dogbox'}, optional
Method to use for optimization. See `least_squares` for more details.
Default is 'lm' for unconstrained problems and 'trf' if `bounds` are
provided. The method 'lm' won't work when the number of observations
is less than the number of variables, use 'trf' or 'dogbox' in this
case.
.. versionadded:: 0.17
jac : callable, string or None, optional
Function with signature ``jac(x, ...)`` which computes the Jacobian
matrix of the model function with respect to parameters as a dense
array_like structure. It will be scaled according to provided `sigma`.
If None (default), the Jacobian will be estimated numerically.
String keywords for 'trf' and 'dogbox' methods can be used to select
a finite difference scheme, see `least_squares`.
.. versionadded:: 0.18
kwargs
Keyword arguments passed to `leastsq` for ``method='lm'`` or
`least_squares` otherwise.
Returns
-------
popt : array
Optimal values for the parameters so that the sum of the squared
residuals of ``f(xdata, *popt) - ydata`` is minimized
pcov : 2d array
The estimated covariance of popt. The diagonals provide the variance
of the parameter estimate. To compute one standard deviation errors
on the parameters use ``perr = np.sqrt(np.diag(pcov))``.
How the `sigma` parameter affects the estimated covariance
depends on `absolute_sigma` argument, as described above.
If the Jacobian matrix at the solution doesn't have a full rank, then
'lm' method returns a matrix filled with ``np.inf``, on the other hand
'trf' and 'dogbox' methods use Moore-Penrose pseudoinverse to compute
the covariance matrix.
Raises
------
ValueError
if either `ydata` or `xdata` contain NaNs, or if incompatible options
are used.
RuntimeError
if the least-squares minimization fails.
OptimizeWarning
if covariance of the parameters can not be estimated.
See Also
--------
least_squares : Minimize the sum of squares of nonlinear functions.
scipy.stats.linregress : Calculate a linear least squares regression for
two sets of measurements.
Notes
-----
With ``method='lm'``, the algorithm uses the Levenberg-Marquardt algorithm
through `leastsq`. Note that this algorithm can only deal with
unconstrained problems.
Box constraints can be handled by methods 'trf' and 'dogbox'. Refer to
the docstring of `least_squares` for more information.
Examples
--------
>>> import matplotlib.pyplot as plt
>>> from scipy.optimize import curve_fit
>>> def func(x, a, b, c):
... return a * np.exp(-b * x) + c
Define the data to be fit with some noise:
>>> xdata = np.linspace(0, 4, 50)
>>> y = func(xdata, 2.5, 1.3, 0.5)
>>> np.random.seed(1729)
>>> y_noise = 0.2 * np.random.normal(size=xdata.size)
>>> ydata = y + y_noise
>>> plt.plot(xdata, ydata, 'b-', label='data')
Fit for the parameters a, b, c of the function `func`:
>>> popt, pcov = curve_fit(func, xdata, ydata)
>>> popt
array([ 2.55423706, 1.35190947, 0.47450618])
>>> plt.plot(xdata, func(xdata, *popt), 'r-',
... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))
Constrain the optimization to the region of ``0 <= a <= 3``,
``0 <= b <= 1`` and ``0 <= c <= 0.5``:
>>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 1., 0.5]))
>>> popt
array([ 2.43708906, 1. , 0.35015434])
>>> plt.plot(xdata, func(xdata, *popt), 'g--',
... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))
>>> plt.xlabel('x')
>>> plt.ylabel('y')
>>> plt.legend()
>>> plt.show()
"""
if p0 is None:
# determine number of parameters by inspecting the function
from scipy._lib._util import getargspec_no_self as _getargspec
args, varargs, varkw, defaults = _getargspec(f)
if len(args) < 2:
raise ValueError("Unable to determine number of fit parameters.")
n = len(args) - 1
else:
p0 = np.atleast_1d(p0)
n = p0.size
lb, ub = prepare_bounds(bounds, n)
if p0 is None:
p0 = _initialize_feasible(lb, ub)
bounded_problem = np.any((lb > -np.inf) | (ub < np.inf))
if method is None:
if bounded_problem:
method = 'trf'
else:
method = 'lm'
if method == 'lm' and bounded_problem:
raise ValueError("Method 'lm' only works for unconstrained problems. "
"Use 'trf' or 'dogbox' instead.")
# optimization may produce garbage for float32 inputs, cast them to float64
# NaNs can not be handled
if check_finite:
ydata = np.asarray_chkfinite(ydata, float)
else:
ydata = np.asarray(ydata, float)
if isinstance(xdata, (list, tuple, np.ndarray)):
# `xdata` is passed straight to the user-defined `f`, so allow
# non-array_like `xdata`.
if check_finite:
xdata = np.asarray_chkfinite(xdata, float)
else:
xdata = np.asarray(xdata, float)
if ydata.size == 0:
raise ValueError("`ydata` must not be empty!")
# Determine type of sigma
if sigma is not None:
sigma = np.asarray(sigma)
# if 1-d, sigma are errors, define transform = 1/sigma
if sigma.shape == (ydata.size, ):
transform = 1.0 / sigma
# if 2-d, sigma is the covariance matrix,
# define transform = L such that L L^T = C
elif sigma.shape == (ydata.size, ydata.size):
try:
# scipy.linalg.cholesky requires lower=True to return L L^T = A
transform = cholesky(sigma, lower=True)
except LinAlgError:
raise ValueError("`sigma` must be positive definite.")
else:
raise ValueError("`sigma` has incorrect shape.")
else:
transform = None
func = _wrap_func(f, xdata, ydata, transform)
if callable(jac):
jac = _wrap_jac(jac, xdata, transform)
elif jac is None and method != 'lm':
jac = '2-point'
if method == 'lm':
# Remove full_output from kwargs, otherwise we're passing it in twice.
return_full = kwargs.pop('full_output', False)
res = leastsq(func, p0, Dfun=jac, full_output=1, **kwargs)
popt, pcov, infodict, errmsg, ier = res
ysize = len(infodict['fvec'])
cost = np.sum(infodict['fvec']**2)
if ier not in [1, 2, 3, 4]:
raise RuntimeError("Optimal parameters not found: " + errmsg)
else:
# Rename maxfev (leastsq) to max_nfev (least_squares), if specified.
if 'max_nfev' not in kwargs:
kwargs['max_nfev'] = kwargs.pop('maxfev', None)
res = least_squares(func,
p0,
jac=jac,
bounds=bounds,
method=method,
**kwargs)
if not res.success:
raise RuntimeError("Optimal parameters not found: " + res.message)
ysize = len(res.fun)
cost = 2 * res.cost # res.cost is half sum of squares!
popt = res.x
# Do Moore-Penrose inverse discarding zero singular values.
_, s, VT = svd(res.jac, full_matrices=False)
threshold = np.finfo(float).eps * max(res.jac.shape) * s[0]
s = s[s > threshold]
VT = VT[:s.size]
pcov = np.dot(VT.T / s**2, VT)
return_full = False
warn_cov = False
if pcov is None:
# indeterminate covariance
pcov = zeros((len(popt), len(popt)), dtype=float)
pcov.fill(inf)
warn_cov = True
elif not absolute_sigma:
if ysize > p0.size:
s_sq = cost / (ysize - p0.size)
pcov = pcov * s_sq
else:
pcov.fill(inf)
warn_cov = True
if warn_cov:
warnings.warn('Covariance of the parameters could not be estimated',
category=OptimizeWarning)
if return_full:
return popt, pcov, infodict, errmsg, ier
else:
return popt, pcov
def qr_multiply(a,
c,
mode='right',
pivoting=False,
conjugate=False,
overwrite_a=False,
overwrite_c=False):
"""
Calculate the QR decomposition and multiply Q with a matrix.
Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
and R upper triangular. Multiply Q with a vector or a matrix c.
Parameters
----------
a : (M, N), array_like
Input array
c : array_like
Input array to be multiplied by ``q``.
mode : {'left', 'right'}, optional
``Q @ c`` is returned if mode is 'left', ``c @ Q`` is returned if
mode is 'right'.
The shape of c must be appropriate for the matrix multiplications,
if mode is 'left', ``min(a.shape) == c.shape[0]``,
if mode is 'right', ``a.shape[0] == c.shape[1]``.
pivoting : bool, optional
Whether or not factorization should include pivoting for rank-revealing
qr decomposition, see the documentation of qr.
conjugate : bool, optional
Whether Q should be complex-conjugated. This might be faster
than explicit conjugation.
overwrite_a : bool, optional
Whether data in a is overwritten (may improve performance)
overwrite_c : bool, optional
Whether data in c is overwritten (may improve performance).
If this is used, c must be big enough to keep the result,
i.e. ``c.shape[0]`` = ``a.shape[0]`` if mode is 'left'.
Returns
-------
CQ : ndarray
The product of ``Q`` and ``c``.
R : (K, N), ndarray
R array of the resulting QR factorization where ``K = min(M, N)``.
P : (N,) ndarray
Integer pivot array. Only returned when ``pivoting=True``.
Raises
------
LinAlgError
Raised if QR decomposition fails.
Notes
-----
This is an interface to the LAPACK routines ``?GEQRF``, ``?ORMQR``,
``?UNMQR``, and ``?GEQP3``.
.. versionadded:: 0.11.0
Examples
--------
>>> from scipy.linalg import qr_multiply, qr
>>> A = np.array([[1, 3, 3], [2, 3, 2], [2, 3, 3], [1, 3, 2]])
>>> qc, r1, piv1 = qr_multiply(A, 2*np.eye(4), pivoting=1)
>>> qc
array([[-1., 1., -1.],
[-1., -1., 1.],
[-1., -1., -1.],
[-1., 1., 1.]])
>>> r1
array([[-6., -3., -5. ],
[ 0., -1., -1.11022302e-16],
[ 0., 0., -1. ]])
>>> piv1
array([1, 0, 2], dtype=int32)
>>> q2, r2, piv2 = qr(A, mode='economic', pivoting=1)
>>> np.allclose(2*q2 - qc, np.zeros((4, 3)))
True
"""
if mode not in ['left', 'right']:
raise ValueError("Mode argument can only be 'left' or 'right' but "
"not '{}'".format(mode))
c = numpy.asarray_chkfinite(c)
if c.ndim < 2:
onedim = True
c = numpy.atleast_2d(c)
if mode == "left":
c = c.T
else:
onedim = False
a = numpy.atleast_2d(numpy.asarray(a)) # chkfinite done in qr
M, N = a.shape
if mode == 'left':
if c.shape[0] != min(M, N + overwrite_c * (M - N)):
raise ValueError('Array shapes are not compatible for Q @ c'
' operation: {} vs {}'.format(a.shape, c.shape))
else:
if M != c.shape[1]:
raise ValueError('Array shapes are not compatible for c @ Q'
' operation: {} vs {}'.format(c.shape, a.shape))
raw = qr(a, overwrite_a, None, "raw", pivoting)
Q, tau = raw[0]
gor_un_mqr, = get_lapack_funcs(('ormqr', ), (Q, ))
if gor_un_mqr.typecode in ('s', 'd'):
trans = "T"
else:
trans = "C"
Q = Q[:, :min(M, N)]
if M > N and mode == "left" and not overwrite_c:
if conjugate:
cc = numpy.zeros((c.shape[1], M), dtype=c.dtype, order="F")
cc[:, :N] = c.T
else:
cc = numpy.zeros((M, c.shape[1]), dtype=c.dtype, order="F")
cc[:N, :] = c
trans = "N"
if conjugate:
lr = "R"
else:
lr = "L"
overwrite_c = True
elif c.flags["C_CONTIGUOUS"] and trans == "T" or conjugate:
cc = c.T
if mode == "left":
lr = "R"
else:
lr = "L"
else:
trans = "N"
cc = c
if mode == "left":
lr = "L"
else:
lr = "R"
cQ, = safecall(gor_un_mqr,
"gormqr/gunmqr",
lr,
trans,
Q,
tau,
cc,
overwrite_c=overwrite_c)
if trans != "N":
cQ = cQ.T
if mode == "right":
cQ = cQ[:, :min(M, N)]
if onedim:
cQ = cQ.ravel()
return (cQ, ) + raw[1:]
def schur(a,
output='real',
lwork=None,
overwrite_a=False,
sort=None,
check_finite=True):
"""
Compute Schur decomposition of a matrix.
The Schur decomposition is::
A = Z T Z^H
where Z is unitary and T is either upper-triangular, or for real
Schur decomposition (output='real'), quasi-upper triangular. In
the quasi-triangular form, 2x2 blocks describing complex-valued
eigenvalue pairs may extrude from the diagonal.
Parameters
----------
a : (M, M) array_like
Matrix to decompose
output : {'real', 'complex'}, optional
Construct the real or complex Schur decomposition (for real matrices).
lwork : int, optional
Work array size. If None or -1, it is automatically computed.
overwrite_a : bool, optional
Whether to overwrite data in a (may improve performance).
sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
Specifies whether the upper eigenvalues should be sorted. A callable
may be passed that, given a eigenvalue, returns a boolean denoting
whether the eigenvalue should be sorted to the top-left (True).
Alternatively, string parameters may be used::
'lhp' Left-hand plane (x.real < 0.0)
'rhp' Right-hand plane (x.real > 0.0)
'iuc' Inside the unit circle (x*x.conjugate() <= 1.0)
'ouc' Outside the unit circle (x*x.conjugate() > 1.0)
Defaults to None (no sorting).
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
T : (M, M) ndarray
Schur form of A. It is real-valued for the real Schur decomposition.
Z : (M, M) ndarray
An unitary Schur transformation matrix for A.
It is real-valued for the real Schur decomposition.
sdim : int
If and only if sorting was requested, a third return value will
contain the number of eigenvalues satisfying the sort condition.
Raises
------
LinAlgError
Error raised under three conditions:
1. The algorithm failed due to a failure of the QR algorithm to
compute all eigenvalues.
2. If eigenvalue sorting was requested, the eigenvalues could not be
reordered due to a failure to separate eigenvalues, usually because
of poor conditioning.
3. If eigenvalue sorting was requested, roundoff errors caused the
leading eigenvalues to no longer satisfy the sorting condition.
See also
--------
rsf2csf : Convert real Schur form to complex Schur form
Examples
--------
>>> from scipy.linalg import schur, eigvals
>>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]])
>>> T, Z = schur(A)
>>> T
array([[ 2.65896708, 1.42440458, -1.92933439],
[ 0. , -0.32948354, -0.49063704],
[ 0. , 1.31178921, -0.32948354]])
>>> Z
array([[0.72711591, -0.60156188, 0.33079564],
[0.52839428, 0.79801892, 0.28976765],
[0.43829436, 0.03590414, -0.89811411]])
>>> T2, Z2 = schur(A, output='complex')
>>> T2
array([[ 2.65896708, -1.22839825+1.32378589j, 0.42590089+1.51937378j],
[ 0. , -0.32948354+0.80225456j, -0.59877807+0.56192146j],
[ 0. , 0. , -0.32948354-0.80225456j]])
>>> eigvals(T2)
array([2.65896708, -0.32948354+0.80225456j, -0.32948354-0.80225456j])
An arbitrary custom eig-sorting condition, having positive imaginary part,
which is satisfied by only one eigenvalue
>>> T3, Z3, sdim = schur(A, output='complex', sort=lambda x: x.imag > 0)
>>> sdim
1
"""
if output not in ['real', 'complex', 'r', 'c']:
raise ValueError("argument must be 'real', or 'complex'")
if check_finite:
a1 = asarray_chkfinite(a)
else:
a1 = asarray(a)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError('expected square matrix')
typ = a1.dtype.char
if output in ['complex', 'c'] and typ not in ['F', 'D']:
if typ in _double_precision:
a1 = a1.astype('D')
typ = 'D'
else:
a1 = a1.astype('F')
typ = 'F'
overwrite_a = overwrite_a or (_datacopied(a1, a))
gees, = get_lapack_funcs(('gees', ), (a1, ))
if lwork is None or lwork == -1:
# get optimal work array
result = gees(lambda x: None, a1, lwork=-1)
lwork = result[-2][0].real.astype(numpy.int)
if sort is None:
sort_t = 0
sfunction = lambda x: None
else:
sort_t = 1
if callable(sort):
sfunction = sort
elif sort == 'lhp':
sfunction = lambda x: (x.real < 0.0)
elif sort == 'rhp':
sfunction = lambda x: (x.real >= 0.0)
elif sort == 'iuc':
sfunction = lambda x: (abs(x) <= 1.0)
elif sort == 'ouc':
sfunction = lambda x: (abs(x) > 1.0)
else:
raise ValueError("'sort' parameter must either be 'None', or a "
"callable, or one of ('lhp','rhp','iuc','ouc')")
result = gees(sfunction,
a1,
lwork=lwork,
overwrite_a=overwrite_a,
sort_t=sort_t)
info = result[-1]
if info < 0:
raise ValueError('illegal value in {}-th argument of internal gees'
''.format(-info))
elif info == a1.shape[0] + 1:
raise LinAlgError('Eigenvalues could not be separated for reordering.')
elif info == a1.shape[0] + 2:
raise LinAlgError('Leading eigenvalues do not satisfy sort condition.')
elif info > 0:
raise LinAlgError("Schur form not found. Possibly ill-conditioned.")
if sort_t == 0:
return result[0], result[-3]
else:
return result[0], result[-3], result[1]
def norm(x, ord=None):
"""Matrix or vector norm.
Parameters
----------
x : array, shape (M,) or (M, N)
ord : number, or {None, 1, -1, 2, -2, inf, -inf, 'fro'}
Order of the norm:
===== ============================ ==========================
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)
===== ============================ ==========================
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.
"""
x = asarray_chkfinite(x)
if ord is None: # check the default case first and handle it immediately
return sqrt(add.reduce(real((conjugate(x) * x).ravel())))
nd = len(x.shape)
Inf = numpy.Inf
if nd == 1:
if ord == Inf:
return numpy.amax(abs(x))
elif ord == -Inf:
return numpy.amin(abs(x))
elif ord == 1:
return numpy.sum(abs(x), axis=0) # special case for speedup
elif ord == 2:
return sqrt(numpy.sum(real((conjugate(x) * x)),
axis=0)) # special case for speedup
else:
return numpy.sum(abs(x)**ord, axis=0)**(1.0 / ord)
elif nd == 2:
if ord == 2:
return numpy.amax(decomp.svd(x, compute_uv=0))
elif ord == -2:
return numpy.amin(decomp.svd(x, compute_uv=0))
elif ord == 1:
return numpy.amax(numpy.sum(abs(x), axis=0))
elif ord == Inf:
return numpy.amax(numpy.sum(abs(x), axis=1))
elif ord == -1:
return numpy.amin(numpy.sum(abs(x), axis=0))
elif ord == -Inf:
return numpy.amin(numpy.sum(abs(x), axis=1))
elif ord in ['fro', 'f']:
return sqrt(add.reduce(real((conjugate(x) * x).ravel())))
else:
raise ValueError, "Invalid norm order for matrices."
else:
raise ValueError, "Improper number of dimensions to norm."
def timeshifted_split(inputs,
lagtime: int,
chunksize: int = 1000,
stride: int = 1,
n_splits: Optional[int] = None,
shuffle: bool = False,
random_state: Optional[np.random.RandomState] = None):
r""" Utility function which splits input trajectories into pairs of timeshifted data :math:`(X_t, X_{t+\tau})`.
In case multiple trajectories are provided, the timeshifted pairs are always within the same trajectory.
Parameters
----------
inputs : (T, n) ndarray or list of (T_i, n) ndarrays
Input trajectory or trajectories. In case multiple trajectories are provided, they must have the same dimension
in the second axis but may be of variable length.
lagtime : int
The lag time :math:`\tau` used to produce timeshifted blocks.
chunksize : int, default=1000
The chunk size, i.e., the maximal length of the blocks.
stride: int, default=1
Optional stride which is applied *after* creating a tau-shifted version of the dataset.
n_splits : int, optional, default=None
Alternative to chunksize - this determines the number of timeshifted blocks that is drawn from each provided
trajectory. Supersedes whatever was provided as chunksize.
shuffle : bool, default=False
Whether to shuffle the data prior to splitting it.
random_state : np.random.RandomState, default=None
When shuffling this can be used to set a specific random state.
Returns
-------
iterable : Generator
A Python generator which can be iterated.
Examples
--------
Using chunksize:
>>> data = np.array([0, 1, 2, 3, 4, 5, 6])
>>> for X, Y in timeshifted_split(data, lagtime=1, chunksize=4):
... print(X, Y)
[0 1 2 3] [1 2 3 4]
[4 5] [5 6]
Using n_splits:
>>> data = np.array([0, 1, 2, 3, 4, 5, 6])
>>> for X, Y in timeshifted_split(data, lagtime=1, n_splits=2):
... print(X, Y)
[0 1 2] [1 2 3]
[3 4 5] [4 5 6]
"""
if lagtime < 0:
raise ValueError('lagtime has to be positive')
if int(chunksize) < 0:
raise ValueError('chunksize has to be positive')
if shuffle and random_state is None:
random_state = np.random.RandomState()
if not isinstance(inputs, list):
if isinstance(inputs, tuple):
inputs = list(inputs)
inputs = [inputs]
if not all(len(data) > lagtime for data in inputs):
too_short_inputs = [
i for i, x in enumerate(inputs) if len(x) < lagtime
]
raise ValueError(
f'Input contained to short (smaller than lagtime({lagtime}) at following '
f'indices: {too_short_inputs}')
for data in inputs:
data = np.asarray_chkfinite(data)
data_lagged = data[lagtime:][::stride]
if lagtime > 0:
data = data[:-lagtime][::stride]
else:
data = data[
0::stride] # otherwise data is empty as slice over `data[:-0]`
ix = np.arange(len(data)) # iota range over data
if shuffle:
random_state.shuffle(ix)
if n_splits is not None:
assert n_splits >= 1
for ix_split in np.array_split(ix, n_splits):
if len(ix_split) > 0:
x = data[ix_split]
if lagtime > 0:
x_lagged = data_lagged[ix_split]
yield x, x_lagged
else:
yield x
else:
break
else:
t = 0
while t < len(data):
if t == len(data_lagged):
break
x = data[ix[t:min(t + chunksize, len(data))]]
if lagtime > 0:
x_lagged = data_lagged[
ix[t:min(t + chunksize, len(data_lagged))]]
yield x, x_lagged
else:
yield x
t += chunksize
def curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False,
check_finite=True, bounds=(-np.inf, np.inf), method=None,
jac=None, **kwargs):
"""
Use non-linear least squares to fit a function, f, to data.
Assumes ``ydata = f(xdata, *params) + eps``
Parameters
----------
f : callable
The model function, f(x, ...). It must take the independent
variable as the first argument and the parameters to fit as
separate remaining arguments.
xdata : An M-length sequence or an (k,M)-shaped array
for functions with k predictors.
The independent variable where the data is measured.
ydata : M-length sequence
The dependent data --- nominally f(xdata, ...)
p0 : None, scalar, or N-length sequence, optional
Initial guess for the parameters. If None, then the initial
values will all be 1 (if the number of parameters for the function
can be determined using introspection, otherwise a ValueError
is raised).
sigma : None or M-length sequence, optional
If not None, the uncertainties in the ydata array. These are used as
weights in the least-squares problem
i.e. minimising ``np.sum( ((f(xdata, *popt) - ydata) / sigma)**2 )``
If None, the uncertainties are assumed to be 1.
absolute_sigma : bool, optional
If False, `sigma` denotes relative weights of the data points.
The returned covariance matrix `pcov` is based on *estimated*
errors in the data, and is not affected by the overall
magnitude of the values in `sigma`. Only the relative
magnitudes of the `sigma` values matter.
If True, `sigma` describes one standard deviation errors of
the input data points. The estimated covariance in `pcov` is
based on these values.
check_finite : bool, optional
If True, check that the input arrays do not contain nans of infs,
and raise a ValueError if they do. Setting this parameter to
False may silently produce nonsensical results if the input arrays
do contain nans. Default is True.
bounds : 2-tuple of array_like, optional
Lower and upper bounds on independent variables. Defaults to no bounds.
Each element of the tuple must be either an array with the length equal
to the number of parameters, or a scalar (in which case the bound is
taken to be the same for all parameters.) Use ``np.inf`` with an
appropriate sign to disable bounds on all or some parameters.
.. versionadded:: 0.17
method : {'lm', 'trf', 'dogbox'}, optional
Method to use for optimization. See `least_squares` for more details.
Default is 'lm' for unconstrained problems and 'trf' if `bounds` are
provided. The method 'lm' won't work when the number of observations
is less than the number of variables, use 'trf' or 'dogbox' in this
case.
.. versionadded:: 0.17
jac : callable, string or None, optional
Function with signature ``jac(x, ...)`` which computes the Jacobian
matrix of the model function with respect to parameters as a dense
array_like structure. It will be scaled according to provided `sigma`.
If None (default), the Jacobian will be estimated numerically.
String keywords for 'trf' and 'dogbox' methods can be used to select
a finite difference scheme, see `least_squares`.
.. versionadded:: 0.18
kwargs
Keyword arguments passed to `leastsq` for ``method='lm'`` or
`least_squares` otherwise.
Returns
-------
popt : array
Optimal values for the parameters so that the sum of the squared error
of ``f(xdata, *popt) - ydata`` is minimized
pcov : 2d array
The estimated covariance of popt. The diagonals provide the variance
of the parameter estimate. To compute one standard deviation errors
on the parameters use ``perr = np.sqrt(np.diag(pcov))``.
How the `sigma` parameter affects the estimated covariance
depends on `absolute_sigma` argument, as described above.
If the Jacobian matrix at the solution doesn't have a full rank, then
'lm' method returns a matrix filled with ``np.inf``, on the other hand
'trf' and 'dogbox' methods use Moore-Penrose pseudoinverse to compute
the covariance matrix.
Raises
------
ValueError
if either `ydata` or `xdata` contain NaNs, or if incompatible options
are used.
RuntimeError
if the least-squares minimization fails.
OptimizeWarning
if covariance of the parameters can not be estimated.
See Also
--------
least_squares : Minimize the sum of squares of nonlinear functions.
stats.linregress : Calculate a linear least squares regression for two sets
of measurements.
Notes
-----
With ``method='lm'``, the algorithm uses the Levenberg-Marquardt algorithm
through `leastsq`. Note that this algorithm can only deal with
unconstrained problems.
Box constraints can be handled by methods 'trf' and 'dogbox'. Refer to
the docstring of `least_squares` for more information.
Examples
--------
>>> import numpy as np
>>> from scipy.optimize import curve_fit
>>> def func(x, a, b, c):
... return a * np.exp(-b * x) + c
>>> xdata = np.linspace(0, 4, 50)
>>> y = func(xdata, 2.5, 1.3, 0.5)
>>> ydata = y + 0.2 * np.random.normal(size=len(xdata))
>>> popt, pcov = curve_fit(func, xdata, ydata)
Constrain the optimization to the region of ``0 < a < 3``, ``0 < b < 2``
and ``0 < c < 1``:
>>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 2., 1.]))
"""
if p0 is None:
# determine number of parameters by inspecting the function
from scipy._lib._util import getargspec_no_self as _getargspec
args, varargs, varkw, defaults = _getargspec(f)
if len(args) < 2:
raise ValueError("Unable to determine number of fit parameters.")
n = len(args) - 1
else:
p0 = np.atleast_1d(p0)
n = p0.size
lb, ub = prepare_bounds(bounds, n)
if p0 is None:
p0 = _initialize_feasible(lb, ub)
bounded_problem = np.any((lb > -np.inf) | (ub < np.inf))
if method is None:
if bounded_problem:
method = 'trf'
else:
method = 'lm'
if method == 'lm' and bounded_problem:
raise ValueError("Method 'lm' only works for unconstrained problems. "
"Use 'trf' or 'dogbox' instead.")
# NaNs can not be handled
if check_finite:
ydata = np.asarray_chkfinite(ydata)
else:
ydata = np.asarray(ydata)
if isinstance(xdata, (list, tuple, np.ndarray)):
# `xdata` is passed straight to the user-defined `f`, so allow
# non-array_like `xdata`.
if check_finite:
xdata = np.asarray_chkfinite(xdata)
else:
xdata = np.asarray(xdata)
weights = 1.0 / asarray(sigma) if sigma is not None else None
func = _wrap_func(f, xdata, ydata, weights)
if callable(jac):
jac = _wrap_jac(jac, xdata, weights)
elif jac is None and method != 'lm':
jac = '2-point'
if method == 'lm':
# Remove full_output from kwargs, otherwise we're passing it in twice.
return_full = kwargs.pop('full_output', False)
res = leastsq(func, p0, Dfun=jac, full_output=1, **kwargs)
popt, pcov, infodict, errmsg, ier = res
cost = np.sum(infodict['fvec'] ** 2)
if ier not in [1, 2, 3, 4]:
raise RuntimeError("Optimal parameters not found: " + errmsg)
else:
res = least_squares(func, p0, jac=jac, bounds=bounds, method=method,
**kwargs)
if not res.success:
raise RuntimeError("Optimal parameters not found: " + res.message)
cost = 2 * res.cost # res.cost is half sum of squares!
popt = res.x
# Do Moore-Penrose inverse discarding zero singular values.
_, s, VT = svd(res.jac, full_matrices=False)
threshold = np.finfo(float).eps * max(res.jac.shape) * s[0]
s = s[s > threshold]
VT = VT[:s.size]
pcov = np.dot(VT.T / s**2, VT)
return_full = False
warn_cov = False
if pcov is None:
# indeterminate covariance
pcov = zeros((len(popt), len(popt)), dtype=float)
pcov.fill(inf)
warn_cov = True
elif not absolute_sigma:
if ydata.size > p0.size:
s_sq = cost / (ydata.size - p0.size)
pcov = pcov * s_sq
else:
pcov.fill(inf)
warn_cov = True
if warn_cov:
warnings.warn('Covariance of the parameters could not be estimated',
category=OptimizeWarning)
if return_full:
return popt, pcov, infodict, errmsg, ier
else:
return popt, pcov
def trid(x):
x = np.asarray_chkfinite(x)
return sum((x - 1)**2) - sum(x[:-1] * x[1:])
def rq(a, overwrite_a=False, lwork=None, mode='full'):
"""Compute RQ decomposition of a square real matrix.
Calculate the decomposition :lm:`A = R Q` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, M)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
mode : {'full', 'r', 'economic'}
Determines what information is to be returned: either both Q and R
('full', default), only R ('r') or both Q and R but computed in
economy-size ('economic', see Notes).
Returns
-------
R : float array, shape (M, N)
Q : float or complex array, shape (M, M)
Raises LinAlgError if decomposition fails
Examples
--------
>>> from scipy import linalg
>>> from numpy import random, dot, allclose
>>> a = random.randn(6, 9)
>>> r, q = linalg.rq(a)
>>> allclose(a, dot(r, q))
True
>>> r.shape, q.shape
((6, 9), (9, 9))
>>> r2 = linalg.rq(a, mode='r')
>>> allclose(r, r2)
True
>>> r3, q3 = linalg.rq(a, mode='economic')
>>> r3.shape, q3.shape
((6, 6), (6, 9))
"""
if not mode in ['full', 'r', 'economic']:
raise ValueError(\
"Mode argument should be one of ['full', 'r', 'economic']")
a1 = numpy.asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M, N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
gerqf, = get_lapack_funcs(('gerqf', ), (a1, ))
if lwork is None or lwork == -1:
# get optimal work array
rq, tau, work, info = gerqf(a1, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
rq, tau, work, info = gerqf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gerqf' %
-info)
if not mode == 'economic' or N < M:
R = numpy.triu(rq, N - M)
else:
R = numpy.triu(rq[-M:, -M:])
if mode == 'r':
return R
if find_best_lapack_type((a1, ))[0] in ('s', 'd'):
gor_un_grq, = get_lapack_funcs(('orgrq', ), (rq, ))
else:
gor_un_grq, = get_lapack_funcs(('ungrq', ), (rq, ))
if N < M:
# get optimal work array
Q, work, info = gor_un_grq(rq[-N:], tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_grq(rq[-N:], tau, lwork=lwork, overwrite_a=1)
elif mode == 'economic':
# get optimal work array
Q, work, info = gor_un_grq(rq, tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_grq(rq, tau, lwork=lwork, overwrite_a=1)
else:
rq1 = numpy.empty((N, N), dtype=rq.dtype)
rq1[-M:] = rq
# get optimal work array
Q, work, info = gor_un_grq(rq1, tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_grq(rq1, tau, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError("illegal value in %d-th argument of internal orgrq" %
-info)
return R, Q
def lu_factor(a, overwrite_a=False, check_finite=True):
"""
Compute pivoted LU decomposition of a matrix.
The decomposition is::
A = P L U
where P is a permutation matrix, L lower triangular with unit
diagonal elements, and U upper triangular.
Parameters
----------
a : (M, M) array_like
Matrix to decompose
overwrite_a : bool, optional
Whether to overwrite data in A (may increase performance)
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
lu : (N, N) ndarray
Matrix containing U in its upper triangle, and L in its lower triangle.
The unit diagonal elements of L are not stored.
piv : (N,) ndarray
Pivot indices representing the permutation matrix P:
row i of matrix was interchanged with row piv[i].
See also
--------
lu_solve : solve an equation system using the LU factorization of a matrix
Notes
-----
This is a wrapper to the ``*GETRF`` routines from LAPACK.
Examples
--------
>>> from scipy.linalg import lu_factor
>>> from numpy import tril, triu, allclose, zeros, eye
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
>>> lu, piv = lu_factor(A)
>>> piv
array([2, 2, 3, 3], dtype=int32)
Convert LAPACK's ``piv`` array to NumPy index and test the permutation
>>> piv_py = [2, 0, 3, 1]
>>> L, U = np.tril(lu, k=-1) + np.eye(4), np.triu(lu)
>>> np.allclose(A[piv_py] - L @ U, np.zeros((4, 4)))
True
"""
if check_finite:
a1 = asarray_chkfinite(a)
else:
a1 = asarray(a)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
getrf, = get_lapack_funcs(('getrf', ), (a1, ))
lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of '
'internal getrf (lu_factor)' % -info)
if info > 0:
warn("Diagonal number %d is exactly zero. Singular matrix." % info,
LinAlgWarning,
stacklevel=2)
return lu, piv
def eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False,
overwrite_b=False, turbo=True, eigvals=None, type=1):
"""Solve an ordinary or generalized eigenvalue problem for a complex
Hermitian or real symmetric matrix.
Find eigenvalues w and optionally eigenvectors v of matrix a, where
b is positive definite::
a v[:,i] = w[i] b v[:,i]
v[i,:].conj() a v[:,i] = w[i]
v[i,:].conj() b v[:,i] = 1
Parameters
----------
a : array, shape (M, M)
A complex Hermitian or real symmetric matrix whose eigenvalues and
eigenvectors will be computed.
b : array, shape (M, M)
A complex Hermitian or real symmetric definite positive matrix in.
If omitted, identity matrix is assumed.
lower : boolean
Whether the pertinent array data is taken from the lower or upper
triangle of a. (Default: lower)
eigvals_only : boolean
Whether to calculate only eigenvalues and no eigenvectors.
(Default: both are calculated)
turbo : boolean
Use divide and conquer algorithm (faster but expensive in memory,
only for generalized eigenvalue problem and if eigvals=None)
eigvals : tuple (lo, hi)
Indexes of the smallest and largest (in ascending order) eigenvalues
and corresponding eigenvectors to be returned: 0 <= lo < hi <= M-1.
If omitted, all eigenvalues and eigenvectors are returned.
type: integer
Specifies the problem type to be solved:
type = 1: a v[:,i] = w[i] b v[:,i]
type = 2: a b v[:,i] = w[i] v[:,i]
type = 3: b a v[:,i] = w[i] v[:,i]
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
overwrite_b : boolean
Whether to overwrite data in b (may improve performance)
Returns
-------
w : real array, shape (N,)
The N (1<=N<=M) selected eigenvalues, in ascending order, each
repeated according to its multiplicity.
(if eigvals_only == False)
v : complex array, shape (M, N)
The normalized selected eigenvector corresponding to the
eigenvalue w[i] is the column v[:,i]. Normalization:
type 1 and 3: v.conj() a v = w
type 2: inv(v).conj() a inv(v) = w
type = 1 or 2: v.conj() b v = I
type = 3 : v.conj() inv(b) v = I
Raises LinAlgError if eigenvalue computation does not converge,
an error occurred, or b matrix is not definite positive. Note that
if input matrices are not symmetric or hermitian, no error is reported
but results will be wrong.
See Also
--------
eig : eigenvalues and right eigenvectors for non-symmetric arrays
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
if iscomplexobj(a1):
cplx = True
else:
cplx = False
if b is not None:
b1 = asarray_chkfinite(b)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError('expected square matrix')
if b1.shape != a1.shape:
raise ValueError("wrong b dimensions %s, should "
"be %s" % (str(b1.shape), str(a1.shape)))
if iscomplexobj(b1):
cplx = True
else:
cplx = cplx or False
else:
b1 = None
# Set job for fortran routines
_job = (eigvals_only and 'N') or 'V'
# port eigenvalue range from python to fortran convention
if eigvals is not None:
lo, hi = eigvals
if lo < 0 or hi >= a1.shape[0]:
raise ValueError('The eigenvalue range specified is not valid.\n'
'Valid range is [%s,%s]' % (0, a1.shape[0]-1))
lo += 1
hi += 1
eigvals = (lo, hi)
# set lower
if lower:
uplo = 'L'
else:
uplo = 'U'
# fix prefix for lapack routines
if cplx:
pfx = 'he'
else:
pfx = 'sy'
# Standard Eigenvalue Problem
# Use '*evr' routines
# FIXME: implement calculation of optimal lwork
# for all lapack routines
if b1 is None:
(evr,) = get_lapack_funcs((pfx+'evr',), (a1,))
if eigvals is None:
w, v, info = evr(a1, uplo=uplo, jobz=_job, range="A", il=1,
iu=a1.shape[0], overwrite_a=overwrite_a)
else:
(lo, hi)= eigvals
w_tot, v, info = evr(a1, uplo=uplo, jobz=_job, range="I",
il=lo, iu=hi, overwrite_a=overwrite_a)
w = w_tot[0:hi-lo+1]
# Generalized Eigenvalue Problem
else:
# Use '*gvx' routines if range is specified
if eigvals is not None:
(gvx,) = get_lapack_funcs((pfx+'gvx',), (a1,b1))
(lo, hi) = eigvals
w_tot, v, ifail, info = gvx(a1, b1, uplo=uplo, iu=hi,
itype=type,jobz=_job, il=lo,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
w = w_tot[0:hi-lo+1]
# Use '*gvd' routine if turbo is on and no eigvals are specified
elif turbo:
(gvd,) = get_lapack_funcs((pfx+'gvd',), (a1,b1))
v, w, info = gvd(a1, b1, uplo=uplo, itype=type, jobz=_job,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
# Use '*gv' routine if turbo is off and no eigvals are specified
else:
(gv,) = get_lapack_funcs((pfx+'gv',), (a1,b1))
v, w, info = gv(a1, b1, uplo=uplo, itype= type, jobz=_job,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
# Check if we had a successful exit
if info == 0:
if eigvals_only:
return w
else:
return w, v
elif info < 0:
raise LinAlgError("illegal value in %i-th argument of internal"
" fortran routine." % (-info))
elif info > 0 and b1 is None:
raise LinAlgError("unrecoverable internal error.")
# The algorithm failed to converge.
elif info > 0 and info <= b1.shape[0]:
if eigvals is not None:
raise LinAlgError("the eigenvectors %s failed to"
" converge." % nonzero(ifail)-1)
else:
raise LinAlgError("internal fortran routine failed to converge: "
"%i off-diagonal elements of an "
"intermediate tridiagonal form did not converge"
" to zero." % info)
# This occurs when b is not positive definite
else:
raise LinAlgError("the leading minor of order %i"
" of 'b' is not positive definite. The"
" factorization of 'b' could not be completed"
" and no eigenvalues or eigenvectors were"
" computed." % (info-b1.shape[0]))
def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True):
"""Solve an equation system, a x = b, given the LU factorization of a
Parameters
----------
(lu, piv)
Factorization of the coefficient matrix a, as given by lu_factor
b : array
Right-hand side
trans : {0, 1, 2}, optional
Type of system to solve:
===== =========
trans system
===== =========
0 a x = b
1 a^T x = b
2 a^H x = b
===== =========
overwrite_b : bool, optional
Whether to overwrite data in b (may increase performance)
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
x : array
Solution to the system
See also
--------
lu_factor : LU factorize a matrix
Examples
--------
>>> from scipy.linalg import lu_factor, lu_solve
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
>>> b = np.array([1, 1, 1, 1])
>>> lu, piv = lu_factor(A)
>>> x = lu_solve((lu, piv), b)
>>> np.allclose(A @ x - b, np.zeros((4,)))
True
"""
(lu, piv) = lu_and_piv
if check_finite:
b1 = asarray_chkfinite(b)
else:
b1 = asarray(b)
overwrite_b = overwrite_b or _datacopied(b1, b)
if lu.shape[0] != b1.shape[0]:
raise ValueError("incompatible dimensions.")
getrs, = get_lapack_funcs(('getrs', ), (lu, b1))
x, info = getrs(lu, piv, b1, trans=trans, overwrite_b=overwrite_b)
if info == 0:
return x
raise ValueError('illegal value in %d-th argument of internal gesv|posv' %
-info)
def eig_banded(a_band, lower=False, eigvals_only=False, overwrite_a_band=False,
select='a', select_range=None, max_ev = 0):
"""Solve real symmetric or complex hermitian band matrix eigenvalue problem.
Find eigenvalues w and optionally right eigenvectors v of a::
a v[:,i] = w[i] v[:,i]
v.H v = identity
The matrix a is stored in ab either in lower diagonal or upper
diagonal ordered form:
ab[u + i - j, j] == a[i,j] (if upper form; i <= j)
ab[ i - j, j] == a[i,j] (if lower form; i >= j)
Example of ab (shape of a is (6,6), u=2)::
upper form:
* * a02 a13 a24 a35
* a01 a12 a23 a34 a45
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 * *
Cells marked with * are not used.
Parameters
----------
a_band : array, shape (M, u+1)
Banded matrix whose eigenvalues to calculate
lower : boolean
Is the matrix in the lower form. (Default is upper form)
eigvals_only : boolean
Compute only the eigenvalues and no eigenvectors.
(Default: calculate also eigenvectors)
overwrite_a_band:
Discard data in a_band (may enhance performance)
select: {'a', 'v', 'i'}
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max)
Range of selected eigenvalues
max_ev : integer
For select=='v', maximum number of eigenvalues expected.
For other values of select, has no meaning.
In doubt, leave this parameter untouched.
Returns
-------
w : array, shape (M,)
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
v : double or complex double array, shape (M, M)
The normalized eigenvector corresponding to the eigenvalue w[i] is
the column v[:,i].
Raises LinAlgError if eigenvalue computation does not converge
"""
if eigvals_only or overwrite_a_band:
a1 = asarray_chkfinite(a_band)
overwrite_a_band = overwrite_a_band or (_datacopied(a1, a_band))
else:
a1 = array(a_band)
if issubclass(a1.dtype.type, inexact) and not isfinite(a1).all():
raise ValueError("array must not contain infs or NaNs")
overwrite_a_band = 1
if len(a1.shape) != 2:
raise ValueError('expected two-dimensional array')
if select.lower() not in [0, 1, 2, 'a', 'v', 'i', 'all', 'value', 'index']:
raise ValueError('invalid argument for select')
if select.lower() in [0, 'a', 'all']:
if a1.dtype.char in 'GFD':
bevd, = get_lapack_funcs(('hbevd',), (a1,))
# FIXME: implement this somewhen, for now go with builtin values
# FIXME: calc optimal lwork by calling ?hbevd(lwork=-1)
# or by using calc_lwork.f ???
# lwork = calc_lwork.hbevd(bevd.prefix, a1.shape[0], lower)
internal_name = 'hbevd'
else: # a1.dtype.char in 'fd':
bevd, = get_lapack_funcs(('sbevd',), (a1,))
# FIXME: implement this somewhen, for now go with builtin values
# see above
# lwork = calc_lwork.sbevd(bevd.prefix, a1.shape[0], lower)
internal_name = 'sbevd'
w,v,info = bevd(a1, compute_v=not eigvals_only,
lower=lower,
overwrite_ab=overwrite_a_band)
if select.lower() in [1, 2, 'i', 'v', 'index', 'value']:
# calculate certain range only
if select.lower() in [2, 'i', 'index']:
select = 2
vl, vu, il, iu = 0.0, 0.0, min(select_range), max(select_range)
if min(il, iu) < 0 or max(il, iu) >= a1.shape[1]:
raise ValueError('select_range out of bounds')
max_ev = iu - il + 1
else: # 1, 'v', 'value'
select = 1
vl, vu, il, iu = min(select_range), max(select_range), 0, 0
if max_ev == 0:
max_ev = a_band.shape[1]
if eigvals_only:
max_ev = 1
# calculate optimal abstol for dsbevx (see manpage)
if a1.dtype.char in 'fF': # single precision
lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='f'),))
else:
lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='d'),))
abstol = 2 * lamch('s')
if a1.dtype.char in 'GFD':
bevx, = get_lapack_funcs(('hbevx',), (a1,))
internal_name = 'hbevx'
else: # a1.dtype.char in 'gfd'
bevx, = get_lapack_funcs(('sbevx',), (a1,))
internal_name = 'sbevx'
# il+1, iu+1: translate python indexing (0 ... N-1) into Fortran
# indexing (1 ... N)
w, v, m, ifail, info = bevx(a1, vl, vu, il+1, iu+1,
compute_v=not eigvals_only,
mmax=max_ev,
range=select, lower=lower,
overwrite_ab=overwrite_a_band,
abstol=abstol)
# crop off w and v
w = w[:m]
if not eigvals_only:
v = v[:, :m]
if info < 0:
raise ValueError('illegal value in %d-th argument of internal %s'
% (-info, internal_name))
if info > 0:
raise LinAlgError("eig algorithm did not converge")
if eigvals_only:
return w
return w, v
def _qz(A,
B,
output='real',
lwork=None,
sort=None,
overwrite_a=False,
overwrite_b=False,
check_finite=True):
if sort is not None:
# Disabled due to segfaults on win32, see ticket 1717.
raise ValueError("The 'sort' input of qz() has to be None and will be "
"removed in a future release. Use ordqz instead.")
if output not in ['real', 'complex', 'r', 'c']:
raise ValueError("argument must be 'real', or 'complex'")
if check_finite:
a1 = asarray_chkfinite(A)
b1 = asarray_chkfinite(B)
else:
a1 = np.asarray(A)
b1 = np.asarray(B)
a_m, a_n = a1.shape
b_m, b_n = b1.shape
if not (a_m == a_n == b_m == b_n):
raise ValueError("Array dimensions must be square and agree")
typa = a1.dtype.char
if output in ['complex', 'c'] and typa not in ['F', 'D']:
if typa in _double_precision:
a1 = a1.astype('D')
typa = 'D'
else:
a1 = a1.astype('F')
typa = 'F'
typb = b1.dtype.char
if output in ['complex', 'c'] and typb not in ['F', 'D']:
if typb in _double_precision:
b1 = b1.astype('D')
typb = 'D'
else:
b1 = b1.astype('F')
typb = 'F'
overwrite_a = overwrite_a or (_datacopied(a1, A))
overwrite_b = overwrite_b or (_datacopied(b1, B))
gges, = get_lapack_funcs(('gges', ), (a1, b1))
if lwork is None or lwork == -1:
# get optimal work array size
result = gges(lambda x: None, a1, b1, lwork=-1)
lwork = result[-2][0].real.astype(np.int)
sfunction = lambda x: None
result = gges(sfunction,
a1,
b1,
lwork=lwork,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b,
sort_t=0)
info = result[-1]
if info < 0:
raise ValueError("Illegal value in argument {} of gges".format(-info))
elif info > 0 and info <= a_n:
warnings.warn("The QZ iteration failed. (a,b) are not in Schur "
"form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be "
"correct for J={},...,N".format(info - 1),
LinAlgWarning,
stacklevel=3)
elif info == a_n + 1:
raise LinAlgError("Something other than QZ iteration failed")
elif info == a_n + 2:
raise LinAlgError("After reordering, roundoff changed values of some "
"complex eigenvalues so that leading eigenvalues "
"in the Generalized Schur form no longer satisfy "
"sort=True. This could also be due to scaling.")
elif info == a_n + 3:
raise LinAlgError("Reordering failed in <s,d,c,z>tgsen")
return result, gges.typecode
def eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False):
"""Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix::
a vr[:,i] = w[i] b vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where .H is the Hermitean conjugation.
Parameters
----------
a : array, shape (M, M)
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
b : array, shape (M, M)
Right-hand side matrix in a generalized eigenvalue problem.
If omitted, identity matrix is assumed.
left : boolean
Whether to calculate and return left eigenvectors
right : boolean
Whether to calculate and return right eigenvectors
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
overwrite_b : boolean
Whether to overwrite data in b (may improve performance)
Returns
-------
w : double or complex array, shape (M,)
The eigenvalues, each repeated according to its multiplicity.
(if left == True)
vl : double or complex array, shape (M, M)
The normalized left eigenvector corresponding to the eigenvalue w[i]
is the column v[:,i].
(if right == True)
vr : double or complex array, shape (M, M)
The normalized right eigenvector corresponding to the eigenvalue w[i]
is the column vr[:,i].
Raises LinAlgError if eigenvalue computation does not converge
See Also
--------
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
if b is not None:
b = asarray_chkfinite(b)
if b.shape != a1.shape:
raise ValueError('a and b must have the same shape')
return _geneig(a1, b, left, right, overwrite_a, overwrite_b)
geev, = get_lapack_funcs(('geev',), (a1,))
compute_vl, compute_vr = left, right
if geev.module_name[:7] == 'flapack':
lwork = calc_lwork.geev(geev.prefix, a1.shape[0],
compute_vl, compute_vr)[1]
if geev.prefix in 'cz':
w, vl, vr, info = geev(a1, lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
else:
wr, wi, vl, vr, info = geev(a1, lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f':'F','d':'D'}[wr.dtype.char]
w = wr + _I * wi
else: # 'clapack'
if geev.prefix in 'cz':
w, vl, vr, info = geev(a1,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
else:
wr, wi, vl, vr, info = geev(a1,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f':'F','d':'D'}[wr.dtype.char]
w = wr + _I * wi
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geev'
% -info)
if info > 0:
raise LinAlgError("eig algorithm did not converge (only eigenvalues "
"with order >= %d have converged)" % info)
only_real = numpy.logical_and.reduce(numpy.equal(w.imag, 0.0))
if not (geev.prefix in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False,
check_finite=True, **kw):
"""
Use non-linear least squares to fit a function, f, to data.
Assumes ``ydata = f(xdata, *params) + eps``
Parameters
----------
f : callable
The model function, f(x, ...). It must take the independent
variable as the first argument and the parameters to fit as
separate remaining arguments.
xdata : An M-length sequence or an (k,M)-shaped array
for functions with k predictors.
The independent variable where the data is measured.
ydata : M-length sequence
The dependent data --- nominally f(xdata, ...)
p0 : None, scalar, or N-length sequence, optional
Initial guess for the parameters. If None, then the initial
values will all be 1 (if the number of parameters for the function
can be determined using introspection, otherwise a ValueError
is raised).
sigma : None or M-length sequence, optional
If not None, the uncertainties in the ydata array. These are used as
weights in the least-squares problem
i.e. minimising ``np.sum( ((f(xdata, *popt) - ydata) / sigma)**2 )``
If None, the uncertainties are assumed to be 1.
absolute_sigma : bool, optional
If False, `sigma` denotes relative weights of the data points.
The returned covariance matrix `pcov` is based on *estimated*
errors in the data, and is not affected by the overall
magnitude of the values in `sigma`. Only the relative
magnitudes of the `sigma` values matter.
If True, `sigma` describes one standard deviation errors of
the input data points. The estimated covariance in `pcov` is
based on these values.
check_finite : bool, optional
If True, check that the input arrays do not contain nans of infs,
and raise a ValueError if they do. Setting this parameter to
False may silently produce nonsensical results if the input arrays
do contain nans.
Default is True.
Returns
-------
popt : array
Optimal values for the parameters so that the sum of the squared error
of ``f(xdata, *popt) - ydata`` is minimized
pcov : 2d array
The estimated covariance of popt. The diagonals provide the variance
of the parameter estimate. To compute one standard deviation errors
on the parameters use ``perr = np.sqrt(np.diag(pcov))``.
How the `sigma` parameter affects the estimated covariance
depends on `absolute_sigma` argument, as described above.
Raises
------
OptimizeWarning
if covariance of the parameters can not be estimated.
ValueError
if ydata and xdata contain NaNs.
See Also
--------
leastsq
Notes
-----
The algorithm uses the Levenberg-Marquardt algorithm through `leastsq`.
Additional keyword arguments are passed directly to that algorithm.
Examples
--------
>>> import numpy as np
>>> from scipy.optimize import curve_fit
>>> def func(x, a, b, c):
... return a * np.exp(-b * x) + c
>>> xdata = np.linspace(0, 4, 50)
>>> y = func(xdata, 2.5, 1.3, 0.5)
>>> ydata = y + 0.2 * np.random.normal(size=len(xdata))
>>> popt, pcov = curve_fit(func, xdata, ydata)
"""
if p0 is None:
# determine number of parameters by inspecting the function
import inspect
args, varargs, varkw, defaults = inspect.getargspec(f)
if len(args) < 2:
msg = "Unable to determine number of fit parameters."
raise ValueError(msg)
if 'self' in args:
p0 = [1.0] * (len(args)-2)
else:
p0 = [1.0] * (len(args)-1)
# Check input arguments
if isscalar(p0):
p0 = array([p0])
# NaNs can not be handled
if check_finite:
ydata = np.asarray_chkfinite(ydata)
else:
ydata = np.asarray(ydata)
if isinstance(xdata, (list, tuple, np.ndarray)):
# `xdata` is passed straight to the user-defined `f`, so allow
# non-array_like `xdata`.
if check_finite:
xdata = np.asarray_chkfinite(xdata)
else:
xdata = np.asarray(xdata)
args = (xdata, ydata, f)
if sigma is None:
func = _general_function
else:
func = _weighted_general_function
args += (1.0 / asarray(sigma),)
# Remove full_output from kw, otherwise we're passing it in twice.
return_full = kw.pop('full_output', False)
res = leastsq(func, p0, args=args, full_output=1, **kw)
(popt, pcov, infodict, errmsg, ier) = res
if ier not in [1, 2, 3, 4]:
msg = "Optimal parameters not found: " + errmsg
raise RuntimeError(msg)
warn_cov = False
if pcov is None:
# indeterminate covariance
pcov = zeros((len(popt), len(popt)), dtype=float)
pcov.fill(inf)
warn_cov = True
elif not absolute_sigma:
if len(ydata) > len(p0):
s_sq = (asarray(func(popt, *args))**2).sum() / (len(ydata) - len(p0))
pcov = pcov * s_sq
else:
pcov.fill(inf)
warn_cov = True
if warn_cov:
warnings.warn('Covariance of the parameters could not be estimated',
category=OptimizeWarning)
if return_full:
return popt, pcov, infodict, errmsg, ier
else:
return popt, pcov
def lu(a, permute_l=False, overwrite_a=False, check_finite=True):
"""
Compute pivoted LU decomposition of a matrix.
The decomposition is::
A = P L U
where P is a permutation matrix, L lower triangular with unit
diagonal elements, and U upper triangular.
Parameters
----------
a : (M, N) array_like
Array to decompose
permute_l : bool
Perform the multiplication P*L (Default: do not permute)
overwrite_a : bool
Whether to overwrite data in a (may improve performance)
check_finite : boolean, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
**(If permute_l == False)**
p : (M, M) ndarray
Permutation matrix
l : (M, K) ndarray
Lower triangular or trapezoidal matrix with unit diagonal.
K = min(M, N)
u : (K, N) ndarray
Upper triangular or trapezoidal matrix
**(If permute_l == True)**
pl : (M, K) ndarray
Permuted L matrix.
K = min(M, N)
u : (K, N) ndarray
Upper triangular or trapezoidal matrix
Notes
-----
This is a LU factorization routine written for Scipy.
"""
if check_finite:
a1 = asarray_chkfinite(a)
else:
a1 = asarray(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
flu, = get_flinalg_funcs(('lu', ), (a1, ))
p, l, u, info = flu(a1, permute_l=permute_l, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of '
'internal lu.getrf' % -info)
if permute_l:
return l, u
return p, l, u
def qr_multiply(a,
c,
mode='right',
pivoting=False,
conjugate=False,
overwrite_a=False,
overwrite_c=False):
"""Calculate the QR decomposition and multiply Q with a matrix.
Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
and R upper triangular. Multiply Q with a vector or a matrix c.
.. versionadded:: 0.11
Parameters
----------
a : ndarray, shape (M, N)
Matrix to be decomposed
c : ndarray, one- or two-dimensional
calculate the product of c and q, depending on the mode:
mode : {'left', 'right'}
``dot(Q, c)`` is returned if mode is 'left',
``dot(c, Q)`` is returned if mode is 'right'.
The shape of c must be appropriate for the matrix multiplications,
if mode is 'left', ``min(a.shape) == c.shape[0]``,
if mode is 'right', ``a.shape[0] == c.shape[1]``.
pivoting : bool, optional
Whether or not factorization should include pivoting for rank-revealing
qr decomposition, see the documentation of qr.
conjugate : bool, optional
Whether Q should be complex-conjugated. This might be faster
than explicit conjugation.
overwrite_a : bool, optional
Whether data in a is overwritten (may improve performance)
overwrite_c : bool, optional
Whether data in c is overwritten (may improve performance).
If this is used, c must be big enough to keep the result,
i.e. c.shape[0] = a.shape[0] if mode is 'left'.
Returns
-------
CQ : float or complex ndarray
the product of Q and c, as defined in mode
R : float or complex ndarray
Of shape (K, N), ``K = min(M, N)``.
P : ndarray of ints
Of shape (N,) for ``pivoting=True``. Not returned if ``pivoting=False``.
Raises
------
LinAlgError
Raised if decomposition fails
Notes
-----
This is an interface to the LAPACK routines dgeqrf, zgeqrf,
dormqr, zunmqr, dgeqp3, and zgeqp3.
"""
if not mode in ['left', 'right']:
raise ValueError("Mode argument should be one of ['left', 'right']")
c = numpy.asarray_chkfinite(c)
onedim = c.ndim == 1
if onedim:
c = c.reshape(1, len(c))
if mode == "left":
c = c.T
a = numpy.asarray(a) # chkfinite done in qr
M, N = a.shape
if not (mode == "left" and (not overwrite_c and min(M, N) == c.shape[0]
or overwrite_c and M == c.shape[0])
or mode == "right" and M == c.shape[1]):
raise ValueError("objects are not aligned")
raw = qr(a, overwrite_a, None, "raw", pivoting)
Q, tau = raw[0]
if find_best_lapack_type((Q, ))[0] in ('s', 'd'):
gor_un_mqr, = get_lapack_funcs(('ormqr', ), (Q, ))
trans = "T"
else:
gor_un_mqr, = get_lapack_funcs(('unmqr', ), (Q, ))
trans = "C"
Q = Q[:, :min(M, N)]
if M > N and mode == "left" and not overwrite_c:
if conjugate:
cc = numpy.zeros((c.shape[1], M), dtype=c.dtype, order="F")
cc[:, :N] = c.T
else:
cc = numpy.zeros((M, c.shape[1]), dtype=c.dtype, order="F")
cc[:N, :] = c
trans = "N"
if conjugate:
lr = "R"
else:
lr = "L"
overwrite_c = True
elif c.flags["C_CONTIGUOUS"] and trans == "T" or conjugate:
cc = c.T
if mode == "left":
lr = "R"
else:
lr = "L"
else:
trans = "N"
cc = c
if mode == "left":
lr = "L"
else:
lr = "R"
cQ, = safecall(gor_un_mqr,
"gormqr/gunmqr",
lr,
trans,
Q,
tau,
cc,
overwrite_c=overwrite_c)
if trans != "N":
cQ = cQ.T
if mode == "right":
cQ = cQ[:, :min(M, N)]
if onedim:
cQ = cQ.ravel()
return (cQ, ) + raw[1:]
def qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False):
"""
Compute QR decomposition of a matrix.
Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : bool, optional
Whether data in a is overwritten (may improve performance)
lwork : int, optional
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
mode : {'full', 'r', 'economic', 'raw'}
Determines what information is to be returned: either both Q and R
('full', default), only R ('r') or both Q and R but computed in
economy-size ('economic', see Notes). The final option 'raw'
(added in Scipy 0.11) makes the function return two matrixes
(Q, TAU) in the internal format used by LAPACK.
pivoting : bool, optional
Whether or not factorization should include pivoting for rank-revealing
qr decomposition. If pivoting, compute the decomposition
``A P = Q R`` as above, but where P is chosen such that the diagonal
of R is non-increasing.
Returns
-------
Q : float or complex ndarray
Of shape (M, M), or (M, K) for ``mode='economic'``. Not returned
if ``mode='r'``.
R : float or complex ndarray
Of shape (M, N), or (K, N) for ``mode='economic'``. ``K = min(M, N)``.
P : integer ndarray
Of shape (N,) for ``pivoting=True``. Not returned if
``pivoting=False``.
Raises
------
LinAlgError
Raised if decomposition fails
Notes
-----
This is an interface to the LAPACK routines dgeqrf, zgeqrf,
dorgqr, zungqr, dgeqp3, and zgeqp3.
If ``mode=economic``, the shapes of Q and R are (M, K) and (K, N) instead
of (M,M) and (M,N), with ``K=min(M,N)``.
Examples
--------
>>> from scipy import random, linalg, dot, diag, all, allclose
>>> a = random.randn(9, 6)
>>> q, r = linalg.qr(a)
>>> allclose(a, np.dot(q, r))
True
>>> q.shape, r.shape
((9, 9), (9, 6))
>>> r2 = linalg.qr(a, mode='r')
>>> allclose(r, r2)
True
>>> q3, r3 = linalg.qr(a, mode='economic')
>>> q3.shape, r3.shape
((9, 6), (6, 6))
>>> q4, r4, p4 = linalg.qr(a, pivoting=True)
>>> d = abs(diag(r4))
>>> all(d[1:] <= d[:-1])
True
>>> allclose(a[:, p4], dot(q4, r4))
True
>>> q4.shape, r4.shape, p4.shape
((9, 9), (9, 6), (6,))
>>> q5, r5, p5 = linalg.qr(a, mode='economic', pivoting=True)
>>> q5.shape, r5.shape, p5.shape
((9, 6), (6, 6), (6,))
"""
# 'qr' was the old default, equivalent to 'full'. Neither 'full' nor
# 'qr' are used below.
# 'raw' is used internally by qr_multiply
if mode not in ['full', 'qr', 'r', 'economic', 'raw']:
raise ValueError(
"Mode argument should be one of ['full', 'r', 'economic', 'raw']")
a1 = numpy.asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError("expected 2D array")
M, N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
if pivoting:
geqp3, = get_lapack_funcs(('geqp3', ), (a1, ))
qr, jpvt, tau = safecall(geqp3, "geqp3", a1, overwrite_a=overwrite_a)
jpvt -= 1 # geqp3 returns a 1-based index array, so subtract 1
else:
geqrf, = get_lapack_funcs(('geqrf', ), (a1, ))
qr, tau = safecall(geqrf,
"geqrf",
a1,
lwork=lwork,
overwrite_a=overwrite_a)
if mode not in ['economic', 'raw'] or M < N:
R = numpy.triu(qr)
else:
R = numpy.triu(qr[:N, :])
if pivoting:
Rj = R, jpvt
else:
Rj = R,
if mode == 'r':
return Rj
elif mode == 'raw':
return ((qr, tau), ) + Rj
if find_best_lapack_type((a1, ))[0] in ('s', 'd'):
gor_un_gqr, = get_lapack_funcs(('orgqr', ), (qr, ))
else:
gor_un_gqr, = get_lapack_funcs(('ungqr', ), (qr, ))
if M < N:
Q, = safecall(gor_un_gqr,
"gorgqr/gungqr",
qr[:, :M],
tau,
lwork=lwork,
overwrite_a=1)
elif mode == 'economic':
Q, = safecall(gor_un_gqr,
"gorgqr/gungqr",
qr,
tau,
lwork=lwork,
overwrite_a=1)
else:
t = qr.dtype.char
qqr = numpy.empty((M, M), dtype=t)
qqr[:, :N] = qr
Q, = safecall(gor_un_gqr,
"gorgqr/gungqr",
qqr,
tau,
lwork=lwork,
overwrite_a=1)
return (Q, ) + Rj
0 a x = b
1 a^T x = b
2 a^H x = b
===== =========
Returns
-------
x : array
Solution to the system
See also
--------
lu_factor : LU factorize a matrix
"""
b1 = asarray_chkfinite(b)
overwrite_b = overwrite_b or (b1 is not b and not hasattr(b, '__array__'))
if lu.shape[0] != b1.shape[0]:
raise ValueError, "incompatible dimensions."
getrs, = get_lapack_funcs(('getrs', ), (lu, b1))
x, info = getrs(lu, piv, b1, trans=trans, overwrite_b=overwrite_b)
if info == 0:
return x
raise ValueError,\
'illegal value in %-th argument of internal gesv|posv'%(-info)
def cho_solve((c, lower), b, overwrite_b=0):
"""Solve an equation system, a x = b, given the Cholesky factorization of a
Parameters
def sphere(x):
x = np.asarray_chkfinite(x)
return sum(x**2)
