# Coverage for pygeodesy/fmath.py : 97%

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''' if (r > 0 and p > 0) or \ (r < 0 and p < 0): # signs match t, p = _2sum(s, p * 2) if not p: s = t return s def _2sum(a, b): # by .testFmath '''(INTERNAL) Precision C{2sum} of M{a + b}. ''' s = a + b if not isfinite(s): raise _OverflowError(unstr(_2sum.__name__, a, b), txt=str(s)) if abs(a) < abs(b): a, b = b, a return s, b - (s - a) class Fsum(object): '''Precision summation similar to standard Python function C{math.fsum}. Unlike C{math.fsum}, this class accumulates the values and provides intermediate, precision running sums. Accumulation may continue after intermediate summations. @note: Handling of exceptions, C{inf}, C{INF}, C{nan} and C{NAN} values is different from C{math.fsum}. @see: U{Hettinger<https://GitHub.com/ActiveState/code/blob/master/recipes/Python/ 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, U{Kahan<https://WikiPedia.org/wiki/Kahan_summation_algorithm>}, U{Klein<https://Link.Springer.com/article/10.1007/s00607-005-0139-x>}, Python 2.6+ file I{Modules/mathmodule.c} and the issue log U{Full precision summation<https://Bugs.Python.org/issue2819>}. ''' _fsum2_ = None _n = 0 _ps = [] def __init__(self, *starts): '''Initialize a new accumulator with one or more start values. @arg starts: No, one or more start values (C{scalar}s). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{starts}} value. @raise ValueError: Invalid or non-finite B{C{starts}} value. ''' self._n = 0 self._ps = [] if starts: self.fadd(starts) def __add__(self, other): '''Sum of this and an other instance or a scalar. @arg other: L{Fsum} instance or C{scalar}. @return: The sum, a new instance (L{Fsum}). @see: Method L{Fsum.__iadd__}. ''' f = self.fcopy() f += other return f # self.fcopy().__iadd__(other) def __iadd__(self, other): '''Add a scalar or an other instance to this instance. @arg other: L{Fsum} instance or C{scalar}. @return: This instance, updated (L{Fsum}). @raise TypeError: Invalid B{C{other}} type. @see: Method L{Fsum.fadd}. ''' if isscalar(other): self.fadd_(other) elif other is self: self.fmul(2) elif isinstance(other, Fsum): self.fadd(other._ps) else: raise _TypeError(_SPACE_(self, '+=', repr(other))) return self def __imul__(self, other): '''Multiply this instance by a scalar or an other instance. @arg other: L{Fsum} instance or C{scalar}. @return: This instance, updated (L{Fsum}). @raise TypeError: Invalid B{C{other}} type. @see: Method L{Fsum.fmul}. ''' if isscalar(other): self.fmul(other) elif isinstance(other, Fsum): ps = list(other._ps) # copy if ps: s = self.fcopy() self.fmul(ps.pop()) while ps: # self += s * ps.pop() p = s.fcopy() p.fmul(ps.pop()) self.fadd(p._ps) else: self._ps = [] # zero self._fsum2_ = None else: raise _TypeError(_SPACE_(self, '*=', repr(other))) return self def __isub__(self, other): '''Subtract a scalar or an other instance from this instance. @arg other: L{Fsum} instance or C{scalar}. @return: This instance, updated (L{Fsum}). @raise TypeError: Invalid B{C{other}} type. @see: Method L{Fsum.fadd}. ''' if isscalar(other): self.fadd_(-other) elif other is self: self._ps = [] # zero self._fsum2_ = None elif isinstance(other, Fsum): self.fadd(-p for p in other._ps) else: raise _TypeError(_SPACE_(self, '-=', repr(other))) return self def __len__(self): '''Return the number of accumulated values. ''' return self._n def __mul__(self, other): '''Product of this and an other instance or a scalar. @arg other: L{Fsum} instance or C{scalar}. @return: The product, a new instance (L{Fsum}). @see: Method L{Fsum.__imul__}. ''' f = self.fcopy() f *= other return f def __str__(self): from pygeodesy.named import classname return Fmt.PAREN(classname(self, prefixed=True), NN) def __sub__(self, other): '''Difference of this and an other instance or a scalar. @arg other: L{Fsum} instance or C{scalar}. @return: The difference, a new instance (L{Fsum}). @see: Method L{Fsum.__isub__}. ''' f = self.fcopy() f -= other return f def fadd(self, xs): '''Accumulate more values from an iterable. @arg xs: Iterable, list, tuple, etc. (C{scalar}s). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{xs}} value. @raise ValueError: Invalid or non-finite B{C{xs}} value. ''' if isscalar(xs): # for backward compatibility xs = (xs,) ps = self._ps for i, x in enumerate(map(float, xs)): # _iter() if not isfinite(x): i = Fmt.SQUARE(xs=i) raise _ValueError(i, x, txt=_not_(_finite_)) i = 0 for p in ps: x, p = _2sum(x, p) if p: ps[i] = p i += 1 ps[i:] = [x] self._n += 1 # assert self._ps is ps self._fsum2_ = None def fadd_(self, *xs): '''Accumulate more values from positional arguments. @arg xs: Values to add (C{scalar}s), all positional. @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{xs}} value. @raise ValueError: Invalid or non-finite B{C{xs}} value. ''' self.fadd(xs) def fcopy(self, deep=False): '''Copy this instance, C{shallow} or B{C{deep}}. @return: The copy, a new instance (L{Fsum}). ''' f = _xcopy(self, deep=deep) # f._fsum2_ = self._fsum2_ # f._n = self._n f._ps = list(self._ps) # separate copy return f copy = fcopy def fmul(self, factor): '''Multiple the current, partial sum by a factor. @arg factor: The multiplier (C{scalar}). @raise TypeError: Non-scalar B{C{factor}}. @raise ValueError: Invalid or non-finite B{C{factor}}. @see: Method L{Fsum.fadd}. ''' if not isfinite(factor): raise _ValueError(factor=factor, txt=_not_(_finite_)) f, ps = float(factor), self._ps if ps: # multiply and adjust partial sums ps[:] = [p * f for p in ps] self.fadd_(ps.pop()) self._n -= 1 # assert self._ps is ps def fsub(self, iterable): '''Accumulate more values from an iterable. @arg iterable: Sequence, list, tuple, etc. (C{scalar}s). @see: Method L{Fsum.fadd}. ''' if iterable: self.fadd(-s for s in iterable) def fsub_(self, *xs): '''Accumulate more values from positional arguments. @arg xs: Values to subtract (C{scalar}s), all positional. @see: Method L{Fsum.fadd}. ''' self.fsub(xs) def fsum(self, iterable=()): '''Accumulate more values from an iterable and sum all. @kwarg iterable: Sequence, list, tuple, etc. (C{scalar}s), optional. @return: Accurate, running sum (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{iterable}} value. @raise ValueError: Invalid or non-finite B{C{iterable}} value. @note: Accumulation can continue after summation. ''' if iterable: self.fadd(iterable) ps = self._ps i = len(ps) - 1 if i < 0: s = _0_0 else: s = ps[i] while i > 0: i -= 1 s, p = _2sum(s, ps[i]) ps[i:] = [s] if p: # sum(ps) became inexact ps.append(p) if i > 0: # half-even round if signs match s = _2even(s, ps[i-1], p) break # assert self._ps is ps self._fsum2_ = s return s def fsum_(self, *xs): '''Accumulate more values from positional arguments and sum all. @arg xs: Values to add (C{scalar}s), all positional. @return: Accurate, running sum (C{float}). @see: Method L{Fsum.fsum}. @note: Accumulation can continue after summation. ''' return self.fsum(xs) def fsum2_(self, *xs): '''Accumulate more values from positional arguments, sum all and provide the sum and delta. @arg xs: Values to add (C{scalar}s), all positional. @return: 2-Tuple C{(sum, delta)} with the accurate, running C{sum} and the C{delta} with the previous running C{sum}, both (C{float}). @see: Method L{Fsum.fsum_}. @note: Accumulation can continue after summation. ''' p = self._fsum2_ if p is None: p = self.fsum() s = self.fsum(xs) # if xs else self._fsum2_ return s, s - p class Fdot(Fsum): '''Precision dot product. ''' def __init__(self, a, *b): '''New L{Fdot} precision dot product M{sum(a[i] * b[i] for i=0..len(a))}. @arg a: List, sequence, tuple, etc. (C{scalar}s). @arg b: All positional arguments (C{scalar}s). @raise OverflowError: Partial C{2sum} overflow. @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. @see: Function L{fdot} and method L{Fsum.fadd}. ''' if len(a) != len(b): raise LenError(Fdot, a=len(a), b=len(b)) Fsum.__init__(self) self.fadd(map(_mul, a, b)) class Fhorner(Fsum): '''Precision polynomial evaluation using the Horner form. ''' def __init__(self, x, *cs): '''New L{Fhorner} evaluation of the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}[]). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}. ''' if not isfinite(x): raise _ValueError(x=x, txt=_not_(_finite_)) if not cs: raise _ValueError(cs=cs, txt=MISSING) x, cs = float(x), list(cs) Fsum.__init__(self, cs.pop()) while cs: self.fmul(x) c = cs.pop() if c: self.fadd_(c) class Fpolynomial(Fsum): '''Precision polynomial evaluation. ''' def __init__(self, x, *cs): '''New L{Fpolynomial} evaluation of the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}[]). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fpolynomial} and method L{Fsum.fadd}. ''' if not isfinite(x): raise _ValueError(x=x, txt=_not_(_finite_)) if not cs: raise _ValueError(cs=cs, txt=MISSING) x, cs, xp = float(x), list(cs), _1_0 Fsum.__init__(self, cs.pop(0)) while cs: xp *= x c = cs.pop(0) if c: self.fadd_(xp * c) def cbrt(x): '''Compute the cubic root M{x**(1/3)}. @arg x: Value (C{scalar}). @return: Cubic root (C{float}). @see: Functions L{cbrt2} and L{sqrt3}. ''' # simpler and more accurate than Ken Turkowski's CubeRoot, see # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf> return copysign(pow(abs(x), _1_3rd), x) def cbrt2(x): '''Compute the cubic root squared M{x**(2/3)}. @arg x: Value (C{scalar}). @return: Cubic root squared (C{float}). @see: Functions L{cbrt} and L{sqrt3}. ''' return pow(abs(x), _2_3rd) # XXX pow(abs(x), _1_3rd)**2 def euclid(x, y): '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by M{max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142...}. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @return: Appoximate norm (C{float}). @see: Function L{euclid_}. ''' x, y = abs(x), abs(y) if y > x: x, y = y, x return x + y * _0_4142 # XXX _0_5 before 20.10.02 def euclid_(*xs): '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} by cascaded L{euclid}. @arg xs: X arguments, positional (C{scalar}[]). @return: Appoximate norm (C{float}). @see: Function L{euclid}. ''' e = _0_0 for x in sorted(map(abs, xs)): # e = euclid(x, e) if x > e: e, x = x, e e += x * _0_4142 return e def favg(v1, v2, f=0.5): '''Return the average of two values. @arg v1: One value (C{scalar}). @arg v2: Other value (C{scalar}). @kwarg f: Optional fraction (C{float}). @return: M{v1 + f * (v2 - v1)} (C{float}). ''' # @raise ValueError: Fraction out of range. # ''' # if not 0 <= f <= 1: # XXX restrict fraction? # raise _ValueError(fraction=f) return v1 + f * (v2 - v1) # v1 * (1 - f) + v2 * f def fdot(a, *b): '''Return the precision dot product M{sum(a[i] * b[i] for i=0..len(a))}. @arg a: List, sequence, tuple, etc. (C{scalar}s). @arg b: All positional arguments (C{scalar}s). @return: Dot product (C{float}). @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. @see: Class L{Fdot}. ''' if len(a) != len(b): raise LenError(fdot, a=len(a), b=len(b)) return fsum(map(_mul, a, b)) def fdot3(a, b, c, start=0): '''Return the precision dot product M{start + sum(a[i] * b[i] * c[i] for i=0..len(a))}. @arg a: List, sequence, tuple, etc. (C{scalar}[]). @arg b: List, sequence, tuple, etc. (C{scalar}[]). @arg c: List, sequence, tuple, etc. (C{scalar}[]). @kwarg start: Optional bias (C{scalar}). @return: Dot product (C{float}). @raise LenError: Unequal C{len(B{a})}, C{len(B{b})} and/or C{len(B{c})}. @raise OverflowError: Partial C{2sum} overflow. ''' def _mul3(a, b, c): # map function return a * b * c # PYCHOK returns if not len(a) == len(b) == len(c): raise LenError(fdot3, a=len(a), b=len(b), c=len(c)) if start: f = Fsum(start) return f.fsum(map(_mul3, a, b, c)) else: return fsum(map(_mul3, a, b, c)) def fhorner(x, *cs): '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs))} using the Horner form. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}[]). @return: Horner value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fpolynomial} and class L{Fhorner}. ''' h = Fhorner(x, *cs) return h.fsum() def fidw(xs, ds, beta=2): '''Interpolate using using U{Inverse Distance Weighting <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW). @arg xs: Known values (C{scalar}[]). @arg ds: Non-negative distances (C{scalar}[]). @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3). @return: Interpolated value C{x} (C{float}). @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}. @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} value, weighted B{C{ds}} below L{EPS}. @note: Using C{B{beta}=0} returns the mean of B{C{xs}}. ''' n, xs = len2(xs) d, ds = len2(ds) if n != d or n < 1: raise LenError(fidw, xs=n, ds=d) d, x = min(zip(ds, xs)) if d > EPS and n > 1: b = -Int_(beta=beta, low=0, high=3) if b < 0: ds = tuple(d**b for d in ds) d = fsum(ds) if d < EPS: raise _ValueError(ds=d) x = fdot(xs, *ds) / d else: x = fmean(xs) elif d < 0: i = Fmt.SQUARE(ds=ds.index(d)) raise _ValueError(i, d) return x def fmean(xs): '''Compute the accurate mean M{sum(xs[i] for i=0..len(xs)) / len(xs)}. @arg xs: Values (C{scalar}s). @return: Mean value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: No B{C{xs}} values. ''' n, xs = len2(xs) if n > 0: return fsum(xs) / n raise _ValueError(xs=xs) def fmean_(*xs): '''Compute the accurate mean M{sum(xs[i] for i=0..len(xs)) / len(xs)}. @arg xs: Values (C{scalar}s). @return: Mean value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: No B{C{xs}} values. ''' return fmean(xs) def fpolynomial(x, *cs): '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}[]). @return: Polynomial value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fhorner} and class L{Fpolynomial}. ''' p = Fpolynomial(x, *cs) return p.fsum() def fpowers(x, n, alts=0): '''Return a series of powers M{[x**i for i=1..n]}. @arg x: Value (C{scalar}). @arg n: Highest exponent (C{int}). @kwarg alts: Only alternating powers, starting with this exponent (C{int}). @return: Powers of B{C{x}} (C{float}s or C{int}s). @raise TypeError: Non-scalar B{C{x}} or B{C{n}} not C{int}. @raise ValueError: Non-finite B{C{x}} or non-positive B{C{n}}. ''' if not isfinite(x): raise _ValueError(x=x, txt=_not_(_finite_)) if not isint(n): raise _IsnotError(int.__name__, n=n) elif n < 1: raise _ValueError(n=n) xs = [x] for _ in range(1, n): xs.append(xs[-1] * x) if alts > 0: # x**2, x**4, ... # xs[alts-1::2] chokes PyChecker xs = xs[slice(alts-1, None, 2)] return xs try: from math import prod as fprod # Python 3.8 except ImportError: def fprod(iterable, start=_1_0): '''Iterable product, like C{math.prod} or C{numpy.prod}. @arg iterable: Terms to be multiplied (C{scalar}[]). @kwarg start: Initial term, also the value returned for an empty iterable (C{scalar}). @return: The product (C{float}). @see: U{NumPy.prod<https://docs.SciPy.org/doc/ numpy/reference/generated/numpy.prod.html>}. ''' return freduce(_mul, iterable, start) def frange(start, number, step=1): '''Generate a range of C{float}s. @arg start: First value (C{float}). @arg number: The number of C{float}s to generate (C{int}). @kwarg step: Increment value (C{float}). @return: A generator (C{float}s). @see: U{NumPy.prod<https://docs.SciPy.org/doc/ numpy/reference/generated/numpy.arange.html>}. ''' if not isint(number): raise _IsnotError(int.__name__, number=number) for i in range(number): yield start + i * step try: from functools import reduce as freduce except ImportError: # PYCHOK no cover try: freduce = reduce # PYCHOK expected except NameError: # Python 3+ def freduce(f, iterable, *start): '''For missing C{functools.reduce}. ''' if start: r = v = start[0] else: r, v = 0, MISSING for v in iterable: r = f(r, v) if v is MISSING: raise _TypeError(iterable=(), start=MISSING) return r def fsum_(*xs): '''Precision summation of the positional argument vulues. @arg xs: Values to be added (C{scalar}[]). @return: Accurate L{fsum} (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{xs}} value. @raise ValueError: Invalid or non-finite B{C{xs}} value. ''' return fsum(map(float, xs)) try: from math import fsum # precision IEEE-754 sum, Python 2.6+ # make sure fsum works as expected (XXX check # float.__getformat__('float')[:4] == 'IEEE'?) if fsum_(1, 1e101, 1, -1e101) != 2: del fsum # nope, remove fsum ... raise ImportError # ... use fsum below except ImportError: # PYCHOK no cover def fsum(iterable): '''Precision summation similar to standard Python function C{math.fsum}. Exception and I{non-finite} handling differs from C{math.fsum}. @arg iterable: Values to be added (C{scalar}[]). @return: Accurate C{sum} (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{iterable}} value. @raise ValueError: Invalid or non-finite B{C{iterable}} value. @see: Class L{Fsum}. ''' f = Fsum() return f.fsum(iterable) try: _ = hypot(1, 2, 3) # new in Python 3.8+ hypot_ = hypot del _ except TypeError: # Python 3.7- def hypot_(*xs): '''Compute the norm M{sqrt(sum(x**2 for x in xs))}. @arg xs: X arguments, positional (C{scalar}[]). @return: Norm (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Invalid or no B{C{xs}} value. @see: Similar to Python 3.8+ U{math.hypot <https://docs.Python.org/3.8/library/math.html#math.hypot>}, but handling of exceptions, C{nan} and C{infinite} values is different. @note: The Python 3.8+ U{math.dist <https://docs.Python.org/3.8/library/math.html#math.dist>} Euclidian distance between 2 I{n}-dimensional points I{p1} and I{p2} can be computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))}, provided I{p1} and I{p2} have the same, non-zero length I{n}. ''' h, x2 = _h_x2(xs) return (h * sqrt(x2)) if x2 else _0_0 def _h_x2(xs): '''(INTERNAL) Helper for L{hypot_} and L{hypot2_}. ''' if xs: n, xs = len2(xs) if n > 0: h = float(max(abs(x) for x in xs)) if h > 0: if n > 1: X = Fsum(_1_0) X.fadd((x / h)**2 for x in xs) x2 = X.fsum_(-_1_0) else: x2 = _1_0 else: h = x2 = _0_0 return h, x2 raise _ValueError(xs=xs, txt=_too_(_few_)) def hypot1(x): '''Compute the norm M{sqrt(1 + x**2)}. @arg x: Argument (C{scalar}). @return: Norm (C{float}). ''' return hypot(_1_0, x) if x else _1_0 def hypot2(x, y): '''Compute the I{squared} norm M{x**2 + y**2}. @arg x: Argument (C{scalar}). @arg y: Argument (C{scalar}). @return: B{C{x}}C{**2 + }B{C{y}}C{**2} (C{float}). ''' x, y = x**2, y**2 if x < y: x, y = y, x if y: # and x x *= _1_0 + y / x return x def hypot2_(*xs): '''Compute the I{squared} norm M{sum(x**2 for x in xs)}. @arg xs: X arguments, positional (C{scalar}[]). @return: Squared norm (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Invalid or no B{C{xs}} value. @see: Function L{hypot_}. ''' h, x2 = _h_x2(xs) return (h**2 * x2) if x2 else _0_0 def sqrt3(x): '''Compute the square root, cubed M{sqrt(x)**3} or M{sqrt(x**3)}. @arg x: Value (C{scalar}). @return: Cubed square root (C{float}). @raise ValueError: Negative B{C{x}}. @see: Functions L{cbrt} and L{cbrt2}. ''' if x < 0: raise _ValueError(x=x) return pow(x, _3_2nd) if x else _0_0 # **) MIT License # # Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.