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""" 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
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