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diff --git a/venv/lib/python3.9/site-packages/numpy/polynomial/polyutils.py b/venv/lib/python3.9/site-packages/numpy/polynomial/polyutils.py new file mode 100644 index 00000000..48291389 --- /dev/null +++ b/venv/lib/python3.9/site-packages/numpy/polynomial/polyutils.py @@ -0,0 +1,789 @@ +""" +Utility classes and functions for the polynomial modules. + +This module provides: error and warning objects; a polynomial base class; +and some routines used in both the `polynomial` and `chebyshev` modules. + +Warning objects +--------------- + +.. autosummary:: + :toctree: generated/ + + RankWarning raised in least-squares fit for rank-deficient matrix. + +Functions +--------- + +.. autosummary:: + :toctree: generated/ + + as_series convert list of array_likes into 1-D arrays of common type. + trimseq remove trailing zeros. + trimcoef remove small trailing coefficients. + getdomain return the domain appropriate for a given set of abscissae. + mapdomain maps points between domains. + mapparms parameters of the linear map between domains. + +""" +import operator +import functools +import warnings + +import numpy as np + +from numpy.core.multiarray import dragon4_positional, dragon4_scientific +from numpy.core.umath import absolute + +__all__ = [ + 'RankWarning', 'as_series', 'trimseq', + 'trimcoef', 'getdomain', 'mapdomain', 'mapparms', + 'format_float'] + +# +# Warnings and Exceptions +# + +class RankWarning(UserWarning): + """Issued by chebfit when the design matrix is rank deficient.""" + pass + +# +# Helper functions to convert inputs to 1-D arrays +# +def trimseq(seq): + """Remove small Poly series coefficients. + + Parameters + ---------- + seq : sequence + Sequence of Poly series coefficients. This routine fails for + empty sequences. + + Returns + ------- + series : sequence + Subsequence with trailing zeros removed. If the resulting sequence + would be empty, return the first element. The returned sequence may + or may not be a view. + + Notes + ----- + Do not lose the type info if the sequence contains unknown objects. + + """ + if len(seq) == 0: + return seq + else: + for i in range(len(seq) - 1, -1, -1): + if seq[i] != 0: + break + return seq[:i+1] + + +def as_series(alist, trim=True): + """ + Return argument as a list of 1-d arrays. + + The returned list contains array(s) of dtype double, complex double, or + object. A 1-d argument of shape ``(N,)`` is parsed into ``N`` arrays of + size one; a 2-d argument of shape ``(M,N)`` is parsed into ``M`` arrays + of size ``N`` (i.e., is "parsed by row"); and a higher dimensional array + raises a Value Error if it is not first reshaped into either a 1-d or 2-d + array. + + Parameters + ---------- + alist : array_like + A 1- or 2-d array_like + trim : boolean, optional + When True, trailing zeros are removed from the inputs. + When False, the inputs are passed through intact. + + Returns + ------- + [a1, a2,...] : list of 1-D arrays + A copy of the input data as a list of 1-d arrays. + + Raises + ------ + ValueError + Raised when `as_series` cannot convert its input to 1-d arrays, or at + least one of the resulting arrays is empty. + + Examples + -------- + >>> from numpy.polynomial import polyutils as pu + >>> a = np.arange(4) + >>> pu.as_series(a) + [array([0.]), array([1.]), array([2.]), array([3.])] + >>> b = np.arange(6).reshape((2,3)) + >>> pu.as_series(b) + [array([0., 1., 2.]), array([3., 4., 5.])] + + >>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16))) + [array([1.]), array([0., 1., 2.]), array([0., 1.])] + + >>> pu.as_series([2, [1.1, 0.]]) + [array([2.]), array([1.1])] + + >>> pu.as_series([2, [1.1, 0.]], trim=False) + [array([2.]), array([1.1, 0. ])] + + """ + arrays = [np.array(a, ndmin=1, copy=False) for a in alist] + if min([a.size for a in arrays]) == 0: + raise ValueError("Coefficient array is empty") + if any(a.ndim != 1 for a in arrays): + raise ValueError("Coefficient array is not 1-d") + if trim: + arrays = [trimseq(a) for a in arrays] + + if any(a.dtype == np.dtype(object) for a in arrays): + ret = [] + for a in arrays: + if a.dtype != np.dtype(object): + tmp = np.empty(len(a), dtype=np.dtype(object)) + tmp[:] = a[:] + ret.append(tmp) + else: + ret.append(a.copy()) + else: + try: + dtype = np.common_type(*arrays) + except Exception as e: + raise ValueError("Coefficient arrays have no common type") from e + ret = [np.array(a, copy=True, dtype=dtype) for a in arrays] + return ret + + +def trimcoef(c, tol=0): + """ + Remove "small" "trailing" coefficients from a polynomial. + + "Small" means "small in absolute value" and is controlled by the + parameter `tol`; "trailing" means highest order coefficient(s), e.g., in + ``[0, 1, 1, 0, 0]`` (which represents ``0 + x + x**2 + 0*x**3 + 0*x**4``) + both the 3-rd and 4-th order coefficients would be "trimmed." + + Parameters + ---------- + c : array_like + 1-d array of coefficients, ordered from lowest order to highest. + tol : number, optional + Trailing (i.e., highest order) elements with absolute value less + than or equal to `tol` (default value is zero) are removed. + + Returns + ------- + trimmed : ndarray + 1-d array with trailing zeros removed. If the resulting series + would be empty, a series containing a single zero is returned. + + Raises + ------ + ValueError + If `tol` < 0 + + See Also + -------- + trimseq + + Examples + -------- + >>> from numpy.polynomial import polyutils as pu + >>> pu.trimcoef((0,0,3,0,5,0,0)) + array([0., 0., 3., 0., 5.]) + >>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed + array([0.]) + >>> i = complex(0,1) # works for complex + >>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3) + array([0.0003+0.j , 0.001 -0.001j]) + + """ + if tol < 0: + raise ValueError("tol must be non-negative") + + [c] = as_series([c]) + [ind] = np.nonzero(np.abs(c) > tol) + if len(ind) == 0: + return c[:1]*0 + else: + return c[:ind[-1] + 1].copy() + +def getdomain(x): + """ + Return a domain suitable for given abscissae. + + Find a domain suitable for a polynomial or Chebyshev series + defined at the values supplied. + + Parameters + ---------- + x : array_like + 1-d array of abscissae whose domain will be determined. + + Returns + ------- + domain : ndarray + 1-d array containing two values. If the inputs are complex, then + the two returned points are the lower left and upper right corners + of the smallest rectangle (aligned with the axes) in the complex + plane containing the points `x`. If the inputs are real, then the + two points are the ends of the smallest interval containing the + points `x`. + + See Also + -------- + mapparms, mapdomain + + Examples + -------- + >>> from numpy.polynomial import polyutils as pu + >>> points = np.arange(4)**2 - 5; points + array([-5, -4, -1, 4]) + >>> pu.getdomain(points) + array([-5., 4.]) + >>> c = np.exp(complex(0,1)*np.pi*np.arange(12)/6) # unit circle + >>> pu.getdomain(c) + array([-1.-1.j, 1.+1.j]) + + """ + [x] = as_series([x], trim=False) + if x.dtype.char in np.typecodes['Complex']: + rmin, rmax = x.real.min(), x.real.max() + imin, imax = x.imag.min(), x.imag.max() + return np.array((complex(rmin, imin), complex(rmax, imax))) + else: + return np.array((x.min(), x.max())) + +def mapparms(old, new): + """ + Linear map parameters between domains. + + Return the parameters of the linear map ``offset + scale*x`` that maps + `old` to `new` such that ``old[i] -> new[i]``, ``i = 0, 1``. + + Parameters + ---------- + old, new : array_like + Domains. Each domain must (successfully) convert to a 1-d array + containing precisely two values. + + Returns + ------- + offset, scale : scalars + The map ``L(x) = offset + scale*x`` maps the first domain to the + second. + + See Also + -------- + getdomain, mapdomain + + Notes + ----- + Also works for complex numbers, and thus can be used to calculate the + parameters required to map any line in the complex plane to any other + line therein. + + Examples + -------- + >>> from numpy.polynomial import polyutils as pu + >>> pu.mapparms((-1,1),(-1,1)) + (0.0, 1.0) + >>> pu.mapparms((1,-1),(-1,1)) + (-0.0, -1.0) + >>> i = complex(0,1) + >>> pu.mapparms((-i,-1),(1,i)) + ((1+1j), (1-0j)) + + """ + oldlen = old[1] - old[0] + newlen = new[1] - new[0] + off = (old[1]*new[0] - old[0]*new[1])/oldlen + scl = newlen/oldlen + return off, scl + +def mapdomain(x, old, new): + """ + Apply linear map to input points. + + The linear map ``offset + scale*x`` that maps the domain `old` to + the domain `new` is applied to the points `x`. + + Parameters + ---------- + x : array_like + Points to be mapped. If `x` is a subtype of ndarray the subtype + will be preserved. + old, new : array_like + The two domains that determine the map. Each must (successfully) + convert to 1-d arrays containing precisely two values. + + Returns + ------- + x_out : ndarray + Array of points of the same shape as `x`, after application of the + linear map between the two domains. + + See Also + -------- + getdomain, mapparms + + Notes + ----- + Effectively, this implements: + + .. math:: + x\\_out = new[0] + m(x - old[0]) + + where + + .. math:: + m = \\frac{new[1]-new[0]}{old[1]-old[0]} + + Examples + -------- + >>> from numpy.polynomial import polyutils as pu + >>> old_domain = (-1,1) + >>> new_domain = (0,2*np.pi) + >>> x = np.linspace(-1,1,6); x + array([-1. , -0.6, -0.2, 0.2, 0.6, 1. ]) + >>> x_out = pu.mapdomain(x, old_domain, new_domain); x_out + array([ 0. , 1.25663706, 2.51327412, 3.76991118, 5.02654825, # may vary + 6.28318531]) + >>> x - pu.mapdomain(x_out, new_domain, old_domain) + array([0., 0., 0., 0., 0., 0.]) + + Also works for complex numbers (and thus can be used to map any line in + the complex plane to any other line therein). + + >>> i = complex(0,1) + >>> old = (-1 - i, 1 + i) + >>> new = (-1 + i, 1 - i) + >>> z = np.linspace(old[0], old[1], 6); z + array([-1. -1.j , -0.6-0.6j, -0.2-0.2j, 0.2+0.2j, 0.6+0.6j, 1. +1.j ]) + >>> new_z = pu.mapdomain(z, old, new); new_z + array([-1.0+1.j , -0.6+0.6j, -0.2+0.2j, 0.2-0.2j, 0.6-0.6j, 1.0-1.j ]) # may vary + + """ + x = np.asanyarray(x) + off, scl = mapparms(old, new) + return off + scl*x + + +def _nth_slice(i, ndim): + sl = [np.newaxis] * ndim + sl[i] = slice(None) + return tuple(sl) + + +def _vander_nd(vander_fs, points, degrees): + r""" + A generalization of the Vandermonde matrix for N dimensions + + The result is built by combining the results of 1d Vandermonde matrices, + + .. math:: + W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{V_k(x_k)[i_0, \ldots, i_M, j_k]} + + where + + .. math:: + N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\ + M &= \texttt{points[k].ndim} \\ + V_k &= \texttt{vander\_fs[k]} \\ + x_k &= \texttt{points[k]} \\ + 0 \le j_k &\le \texttt{degrees[k]} + + Expanding the one-dimensional :math:`V_k` functions gives: + + .. math:: + W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{B_{k, j_k}(x_k[i_0, \ldots, i_M])} + + where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along + dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`. + + Parameters + ---------- + vander_fs : Sequence[function(array_like, int) -> ndarray] + The 1d vander function to use for each axis, such as ``polyvander`` + points : Sequence[array_like] + Arrays of point coordinates, all of the same shape. The dtypes + will be converted to either float64 or complex128 depending on + whether any of the elements are complex. Scalars are converted to + 1-D arrays. + This must be the same length as `vander_fs`. + degrees : Sequence[int] + The maximum degree (inclusive) to use for each axis. + This must be the same length as `vander_fs`. + + Returns + ------- + vander_nd : ndarray + An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``. + """ + n_dims = len(vander_fs) + if n_dims != len(points): + raise ValueError( + f"Expected {n_dims} dimensions of sample points, got {len(points)}") + if n_dims != len(degrees): + raise ValueError( + f"Expected {n_dims} dimensions of degrees, got {len(degrees)}") + if n_dims == 0: + raise ValueError("Unable to guess a dtype or shape when no points are given") + + # convert to the same shape and type + points = tuple(np.array(tuple(points), copy=False) + 0.0) + + # produce the vandermonde matrix for each dimension, placing the last + # axis of each in an independent trailing axis of the output + vander_arrays = ( + vander_fs[i](points[i], degrees[i])[(...,) + _nth_slice(i, n_dims)] + for i in range(n_dims) + ) + + # we checked this wasn't empty already, so no `initial` needed + return functools.reduce(operator.mul, vander_arrays) + + +def _vander_nd_flat(vander_fs, points, degrees): + """ + Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis + + Used to implement the public ``<type>vander<n>d`` functions. + """ + v = _vander_nd(vander_fs, points, degrees) + return v.reshape(v.shape[:-len(degrees)] + (-1,)) + + +def _fromroots(line_f, mul_f, roots): + """ + Helper function used to implement the ``<type>fromroots`` functions. + + Parameters + ---------- + line_f : function(float, float) -> ndarray + The ``<type>line`` function, such as ``polyline`` + mul_f : function(array_like, array_like) -> ndarray + The ``<type>mul`` function, such as ``polymul`` + roots + See the ``<type>fromroots`` functions for more detail + """ + if len(roots) == 0: + return np.ones(1) + else: + [roots] = as_series([roots], trim=False) + roots.sort() + p = [line_f(-r, 1) for r in roots] + n = len(p) + while n > 1: + m, r = divmod(n, 2) + tmp = [mul_f(p[i], p[i+m]) for i in range(m)] + if r: + tmp[0] = mul_f(tmp[0], p[-1]) + p = tmp + n = m + return p[0] + + +def _valnd(val_f, c, *args): + """ + Helper function used to implement the ``<type>val<n>d`` functions. + + Parameters + ---------- + val_f : function(array_like, array_like, tensor: bool) -> array_like + The ``<type>val`` function, such as ``polyval`` + c, args + See the ``<type>val<n>d`` functions for more detail + """ + args = [np.asanyarray(a) for a in args] + shape0 = args[0].shape + if not all((a.shape == shape0 for a in args[1:])): + if len(args) == 3: + raise ValueError('x, y, z are incompatible') + elif len(args) == 2: + raise ValueError('x, y are incompatible') + else: + raise ValueError('ordinates are incompatible') + it = iter(args) + x0 = next(it) + + # use tensor on only the first + c = val_f(x0, c) + for xi in it: + c = val_f(xi, c, tensor=False) + return c + + +def _gridnd(val_f, c, *args): + """ + Helper function used to implement the ``<type>grid<n>d`` functions. + + Parameters + ---------- + val_f : function(array_like, array_like, tensor: bool) -> array_like + The ``<type>val`` function, such as ``polyval`` + c, args + See the ``<type>grid<n>d`` functions for more detail + """ + for xi in args: + c = val_f(xi, c) + return c + + +def _div(mul_f, c1, c2): + """ + Helper function used to implement the ``<type>div`` functions. + + Implementation uses repeated subtraction of c2 multiplied by the nth basis. + For some polynomial types, a more efficient approach may be possible. + + Parameters + ---------- + mul_f : function(array_like, array_like) -> array_like + The ``<type>mul`` function, such as ``polymul`` + c1, c2 + See the ``<type>div`` functions for more detail + """ + # c1, c2 are trimmed copies + [c1, c2] = as_series([c1, c2]) + if c2[-1] == 0: + raise ZeroDivisionError() + + lc1 = len(c1) + lc2 = len(c2) + if lc1 < lc2: + return c1[:1]*0, c1 + elif lc2 == 1: + return c1/c2[-1], c1[:1]*0 + else: + quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype) + rem = c1 + for i in range(lc1 - lc2, - 1, -1): + p = mul_f([0]*i + [1], c2) + q = rem[-1]/p[-1] + rem = rem[:-1] - q*p[:-1] + quo[i] = q + return quo, trimseq(rem) + + +def _add(c1, c2): + """ Helper function used to implement the ``<type>add`` functions. """ + # c1, c2 are trimmed copies + [c1, c2] = as_series([c1, c2]) + if len(c1) > len(c2): + c1[:c2.size] += c2 + ret = c1 + else: + c2[:c1.size] += c1 + ret = c2 + return trimseq(ret) + + +def _sub(c1, c2): + """ Helper function used to implement the ``<type>sub`` functions. """ + # c1, c2 are trimmed copies + [c1, c2] = as_series([c1, c2]) + if len(c1) > len(c2): + c1[:c2.size] -= c2 + ret = c1 + else: + c2 = -c2 + c2[:c1.size] += c1 + ret = c2 + return trimseq(ret) + + +def _fit(vander_f, x, y, deg, rcond=None, full=False, w=None): + """ + Helper function used to implement the ``<type>fit`` functions. + + Parameters + ---------- + vander_f : function(array_like, int) -> ndarray + The 1d vander function, such as ``polyvander`` + c1, c2 + See the ``<type>fit`` functions for more detail + """ + x = np.asarray(x) + 0.0 + y = np.asarray(y) + 0.0 + deg = np.asarray(deg) + + # check arguments. + if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0: + raise TypeError("deg must be an int or non-empty 1-D array of int") + if deg.min() < 0: + raise ValueError("expected deg >= 0") + if x.ndim != 1: + raise TypeError("expected 1D vector for x") + if x.size == 0: + raise TypeError("expected non-empty vector for x") + if y.ndim < 1 or y.ndim > 2: + raise TypeError("expected 1D or 2D array for y") + if len(x) != len(y): + raise TypeError("expected x and y to have same length") + + if deg.ndim == 0: + lmax = deg + order = lmax + 1 + van = vander_f(x, lmax) + else: + deg = np.sort(deg) + lmax = deg[-1] + order = len(deg) + van = vander_f(x, lmax)[:, deg] + + # set up the least squares matrices in transposed form + lhs = van.T + rhs = y.T + if w is not None: + w = np.asarray(w) + 0.0 + if w.ndim != 1: + raise TypeError("expected 1D vector for w") + if len(x) != len(w): + raise TypeError("expected x and w to have same length") + # apply weights. Don't use inplace operations as they + # can cause problems with NA. + lhs = lhs * w + rhs = rhs * w + + # set rcond + if rcond is None: + rcond = len(x)*np.finfo(x.dtype).eps + + # Determine the norms of the design matrix columns. + if issubclass(lhs.dtype.type, np.complexfloating): + scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1)) + else: + scl = np.sqrt(np.square(lhs).sum(1)) + scl[scl == 0] = 1 + + # Solve the least squares problem. + c, resids, rank, s = np.linalg.lstsq(lhs.T/scl, rhs.T, rcond) + c = (c.T/scl).T + + # Expand c to include non-fitted coefficients which are set to zero + if deg.ndim > 0: + if c.ndim == 2: + cc = np.zeros((lmax+1, c.shape[1]), dtype=c.dtype) + else: + cc = np.zeros(lmax+1, dtype=c.dtype) + cc[deg] = c + c = cc + + # warn on rank reduction + if rank != order and not full: + msg = "The fit may be poorly conditioned" + warnings.warn(msg, RankWarning, stacklevel=2) + + if full: + return c, [resids, rank, s, rcond] + else: + return c + + +def _pow(mul_f, c, pow, maxpower): + """ + Helper function used to implement the ``<type>pow`` functions. + + Parameters + ---------- + mul_f : function(array_like, array_like) -> ndarray + The ``<type>mul`` function, such as ``polymul`` + c : array_like + 1-D array of array of series coefficients + pow, maxpower + See the ``<type>pow`` functions for more detail + """ + # c is a trimmed copy + [c] = as_series([c]) + power = int(pow) + if power != pow or power < 0: + raise ValueError("Power must be a non-negative integer.") + elif maxpower is not None and power > maxpower: + raise ValueError("Power is too large") + elif power == 0: + return np.array([1], dtype=c.dtype) + elif power == 1: + return c + else: + # This can be made more efficient by using powers of two + # in the usual way. + prd = c + for i in range(2, power + 1): + prd = mul_f(prd, c) + return prd + + +def _deprecate_as_int(x, desc): + """ + Like `operator.index`, but emits a deprecation warning when passed a float + + Parameters + ---------- + x : int-like, or float with integral value + Value to interpret as an integer + desc : str + description to include in any error message + + Raises + ------ + TypeError : if x is a non-integral float or non-numeric + DeprecationWarning : if x is an integral float + """ + try: + return operator.index(x) + except TypeError as e: + # Numpy 1.17.0, 2019-03-11 + try: + ix = int(x) + except TypeError: + pass + else: + if ix == x: + warnings.warn( + f"In future, this will raise TypeError, as {desc} will " + "need to be an integer not just an integral float.", + DeprecationWarning, + stacklevel=3 + ) + return ix + + raise TypeError(f"{desc} must be an integer") from e + + +def format_float(x, parens=False): + if not np.issubdtype(type(x), np.floating): + return str(x) + + opts = np.get_printoptions() + + if np.isnan(x): + return opts['nanstr'] + elif np.isinf(x): + return opts['infstr'] + + exp_format = False + if x != 0: + a = absolute(x) + if a >= 1.e8 or a < 10**min(0, -(opts['precision']-1)//2): + exp_format = True + + trim, unique = '0', True + if opts['floatmode'] == 'fixed': + trim, unique = 'k', False + + if exp_format: + s = dragon4_scientific(x, precision=opts['precision'], + unique=unique, trim=trim, + sign=opts['sign'] == '+') + if parens: + s = '(' + s + ')' + else: + s = dragon4_positional(x, precision=opts['precision'], + fractional=True, + unique=unique, trim=trim, + sign=opts['sign'] == '+') + return s |