def uniquely_named_symbol(xname,
exprs=(),
compare=str,
modify=None,
**assumptions):
"""Return a symbol which, when printed, will have a name unique
from any other already in the expressions given. The name is made
unique by appending numbers (default) but this can be
customized with the keyword 'modify'.
Parameters
==========
xname : a string or a Symbol
compare : a single arg function that takes a symbol and returns
a string to be compared with xname (the default is the str
function which indicates how the name will look when it
is printed, e.g. this includes underscores that appear on
Dummy symbols). When ``xname`` is a Symbol, ``compare`` will
be used to supply the value for ``xname``.
modify : a single arg function that changes its string argument
in some way (the default is to append numbers)
Examples
========
>>> from sympy.core.symbol import uniquely_named_symbol
>>> from sympy.abc import x
>>> uniquely_named_symbol('x', x)
x0
"""
def numbered_string_incr(s, start=0):
if not s:
return str(start)
i = len(s) - 1
while i != -1:
if not s[i].isdigit():
break
i -= 1
n = str(int(s[i + 1:] or start - 1) + 1)
return s[:i + 1] + n
default = None
if is_sequence(xname):
xname, default = xname
x = compare(xname)
if not exprs:
return _symbol(x, default, **assumptions)
if not is_sequence(exprs):
exprs = [exprs]
names = set().union(
[i.name for e in exprs for i in e.atoms(Symbol)] +
[i.func.name for e in exprs for i in e.atoms(AppliedUndef)])
if modify is None:
modify = numbered_string_incr
while any(x == compare(s) for s in names):
x = modify(x)
return _symbol(x, default, **assumptions)
def test_iterable_is_sequence():
ordered = [list(), tuple(), Tuple(), Matrix([[]])]
unordered = [set()]
not_sympy_iterable = [{}, '', '']
assert all(is_sequence(i) for i in ordered)
assert all(not is_sequence(i) for i in unordered)
assert all(iterable(i) for i in ordered + unordered)
assert all(not iterable(i) for i in not_sympy_iterable)
assert all(iterable(i, exclude=None) for i in not_sympy_iterable)
def _eval_subs(self, old, new):
from sympy.geometry.point import Point, Point3D
if is_sequence(old) or is_sequence(new):
if isinstance(self, Point3D):
old = Point3D(old)
new = Point3D(new)
else:
old = Point(old)
new = Point(new)
return self._subs(old, new)
def __new__(cls, function, limits):
fun = sympify(function)
if not is_sequence(fun) or len(fun) != 2:
raise ValueError("Function argument should be (x(t), y(t)) "
"but got %s" % str(function))
if not is_sequence(limits) or len(limits) != 3:
raise ValueError("Limit argument should be (t, tmin, tmax) "
"but got %s" % str(limits))
return GeometryEntity.__new__(cls, Tuple(*fun), Tuple(*limits))
def sequence(seq, limits=None):
"""
Returns appropriate sequence object.
Explanation
===========
If ``seq`` is a SymPy sequence, returns :class:`SeqPer` object
otherwise returns :class:`SeqFormula` object.
Examples
========
>>> from sympy import sequence
>>> from sympy.abc import n
>>> sequence(n**2, (n, 0, 5))
SeqFormula(n**2, (n, 0, 5))
>>> sequence((1, 2, 3), (n, 0, 5))
SeqPer((1, 2, 3), (n, 0, 5))
See Also
========
sympy.series.sequences.SeqPer
sympy.series.sequences.SeqFormula
"""
seq = sympify(seq)
if is_sequence(seq, Tuple):
return SeqPer(seq, limits)
else:
return SeqFormula(seq, limits)
def _interpret_args(args):
interval_wrong_order = "PlotInterval %s was given before any function(s)."
interpret_error = "Could not interpret %s as a function or interval."
functions, intervals = [], []
if isinstance(args[0], GeometryEntity):
for coords in list(args[0].arbitrary_point()):
functions.append(coords)
intervals.append(PlotInterval.try_parse(args[0].plot_interval()))
else:
for a in args:
i = PlotInterval.try_parse(a)
if i is not None:
if len(functions) == 0:
raise ValueError(interval_wrong_order % (str(i)))
else:
intervals.append(i)
else:
if is_sequence(a, include=str):
raise ValueError(interpret_error % (str(a)))
try:
f = sympify(a)
functions.append(f)
except TypeError:
raise ValueError(interpret_error % str(a))
return functions, intervals
def __new__(cls, label, shape=None, *, offset=S.Zero, strides=None, **kw_args):
from sympy.matrices.matrices import MatrixBase
from sympy.tensor.array.ndim_array import NDimArray
assumptions, kw_args = _filter_assumptions(kw_args)
if isinstance(label, str):
label = Symbol(label, **assumptions)
elif isinstance(label, Symbol):
assumptions = label._merge(assumptions)
elif isinstance(label, (MatrixBase, NDimArray)):
return label
elif isinstance(label, Iterable):
return _sympify(label)
else:
label = _sympify(label)
if is_sequence(shape):
shape = Tuple(*shape)
elif shape is not None:
shape = Tuple(shape)
if shape is not None:
obj = Expr.__new__(cls, label, shape)
else:
obj = Expr.__new__(cls, label)
obj._shape = shape
obj._offset = offset
obj._strides = strides
obj._name = str(label)
IndexedBase._set_assumptions(obj, assumptions)
return obj
def __setitem__(self, i, args):
"""
Parses and adds a PlotMode to the function
list.
"""
if not (isinstance(i, (SYMPY_INTS, Integer)) and i >= 0):
raise ValueError("Function index must " "be an integer >= 0.")
if isinstance(args, PlotObject):
f = args
else:
if (not is_sequence(args)) or isinstance(args, GeometryEntity):
args = [args]
if len(args) == 0:
return # no arguments given
kwargs = dict(bounds_callback=self.adjust_all_bounds)
f = PlotMode(*args, **kwargs)
if f:
self._render_lock.acquire()
self._functions[i] = f
self._render_lock.release()
else:
raise ValueError("Failed to parse '%s'." %
', '.join(str(a) for a in args))
def _(func):
for t in types:
if not is_sequence(t):
t = (
t,
) # for convenience, allow passing `type` to mean `(type,)`
cls.register(*t, **kwargs)(func)
def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False):
if subs and is_sequence(subs):
raise TypeError('subs must be given as a dictionary')
x, y = self.args
return f_t(x, y)
def wrapper(*args):
out = func(*args)
# out can be array(1.0) or [array(1.0), array(2.0)]
if is_sequence(out):
return [o[()] if is_0d[i] else o for i, o in enumerate(out)]
else:
return out[()]
def _eval_args(cls, args):
# _eval_args has the right logic for the controls argument.
controls = args[0]
gate = args[1]
if not is_sequence(controls):
controls = (controls,)
controls = UnitaryOperator._eval_args(controls)
_validate_targets_controls(chain(controls, gate.targets))
return (Tuple(*controls), gate)
def __getitem__(self, indices, **kw_args):
if is_sequence(indices):
# Special case needed because M[*my_tuple] is a syntax error.
if self.shape and len(self.shape) != len(indices):
raise IndexException("Rank mismatch.")
return Indexed(self, *indices, **kw_args)
else:
if self.shape and len(self.shape) != 1:
raise IndexException("Rank mismatch.")
return Indexed(self, indices, **kw_args)
def __new__(cls, *args, **kwargs):
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.matrices import zeros
from sympy.matrices.matrices import MatrixBase
from sympy.utilities.iterables import is_sequence
isMat = lambda i: getattr(i, 'is_Matrix', False)
if len(args) != 1 or \
not is_sequence(args[0]) or \
len(set([isMat(r) for r in args[0]])) != 1:
raise ValueError(
filldedent('''
expecting a sequence of 1 or more rows
containing Matrices.'''))
rows = args[0] if args else []
if not isMat(rows):
if rows and isMat(rows[0]):
rows = [rows] # rows is not list of lists or []
# regularity check
# same number of matrices in each row
blocky = ok = len(set([len(r) for r in rows])) == 1
if ok:
# same number of rows for each matrix in a row
for r in rows:
ok = len(set([i.rows for i in r])) == 1
if not ok:
break
blocky = ok
# same number of cols for each matrix in each col
for c in range(len(rows[0])):
ok = len(set([rows[i][c].cols
for i in range(len(rows))])) == 1
if not ok:
break
if not ok:
# same total cols in each row
ok = len(set([sum([i.cols for i in r]) for r in rows])) == 1
if blocky and ok:
raise ValueError(
filldedent('''
Although this matrix is comprised of blocks,
the blocks do not fill the matrix in a
size-symmetric fashion. To create a full matrix
from these arguments, pass them directly to
Matrix.'''))
raise ValueError(
filldedent('''
When there are not the same number of rows in each
row's matrices or there are not the same number of
total columns in each row, the matrix is not a
block matrix. If this matrix is known to consist of
blocks fully filling a 2-D space then see
Matrix.irregular.'''))
mat = ImmutableDenseMatrix(rows, evaluate=False)
obj = Basic.__new__(cls, mat)
return obj
def _process_limits(func, limits):
"""
Limits should be of the form (x, start, stop).
x should be a symbol. Both start and stop should be bounded.
Explanation
===========
* If x is not given, x is determined from func.
* If limits is None. Limit of the form (x, -pi, pi) is returned.
Examples
========
>>> from sympy.series.fourier import _process_limits as pari
>>> from sympy.abc import x
>>> pari(x**2, (x, -2, 2))
(x, -2, 2)
>>> pari(x**2, (-2, 2))
(x, -2, 2)
>>> pari(x**2, None)
(x, -pi, pi)
"""
def _find_x(func):
free = func.free_symbols
if len(free) == 1:
return free.pop()
elif not free:
return Dummy('k')
else:
raise ValueError(
" specify dummy variables for %s. If the function contains"
" more than one free symbol, a dummy variable should be"
" supplied explicitly e.g. FourierSeries(m*n**2, (n, -pi, pi))"
% func)
x, start, stop = None, None, None
if limits is None:
x, start, stop = _find_x(func), -pi, pi
if is_sequence(limits, Tuple):
if len(limits) == 3:
x, start, stop = limits
elif len(limits) == 2:
x = _find_x(func)
start, stop = limits
if not isinstance(x, Symbol) or start is None or stop is None:
raise ValueError('Invalid limits given: %s' % str(limits))
unbounded = [S.NegativeInfinity, S.Infinity]
if start in unbounded or stop in unbounded:
raise ValueError("Both the start and end value should be bounded")
return sympify((x, start, stop))
def __new__(cls, *args, **kwargs):
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.matrices import zeros
from sympy.matrices.matrices import MatrixBase
from sympy.utilities.iterables import is_sequence
isMat = lambda i: getattr(i, 'is_Matrix', False)
if len(args) != 1 or \
not is_sequence(args[0]) or \
len(set([isMat(r) for r in args[0]])) != 1:
raise ValueError(filldedent('''
expecting a sequence of 1 or more rows
containing Matrices.'''))
rows = args[0] if args else []
if not isMat(rows):
if rows and isMat(rows[0]):
rows = [rows] # rows is not list of lists or []
# regularity check
# same number of matrices in each row
blocky = ok = len(set([len(r) for r in rows])) == 1
if ok:
# same number of rows for each matrix in a row
for r in rows:
ok = len(set([i.rows for i in r])) == 1
if not ok:
break
blocky = ok
# same number of cols for each matrix in each col
for c in range(len(rows[0])):
ok = len(set([rows[i][c].cols
for i in range(len(rows))])) == 1
if not ok:
break
if not ok:
# same total cols in each row
ok = len(set([
sum([i.cols for i in r]) for r in rows])) == 1
if blocky and ok:
raise ValueError(filldedent('''
Although this matrix is comprised of blocks,
the blocks do not fill the matrix in a
size-symmetric fashion. To create a full matrix
from these arguments, pass them directly to
Matrix.'''))
raise ValueError(filldedent('''
When there are not the same number of rows in each
row's matrices or there are not the same number of
total columns in each row, the matrix is not a
block matrix. If this matrix is known to consist of
blocks fully filling a 2-D space then see
Matrix.irregular.'''))
mat = ImmutableDenseMatrix(rows, evaluate=False)
obj = Basic.__new__(cls, mat)
return obj
def __new__(cls, periodical, limits=None):
periodical = sympify(periodical)
def _find_x(periodical):
free = periodical.free_symbols
if len(periodical.free_symbols) == 1:
return free.pop()
else:
return Dummy('k')
x, start, stop = None, None, None
if limits is None:
x, start, stop = _find_x(periodical), 0, S.Infinity
if is_sequence(limits, Tuple):
if len(limits) == 3:
x, start, stop = limits
elif len(limits) == 2:
x = _find_x(periodical)
start, stop = limits
if not isinstance(x, (Symbol, Idx)) or start is None or stop is None:
raise ValueError('Invalid limits given: %s' % str(limits))
if start is S.NegativeInfinity and stop is S.Infinity:
raise ValueError("Both the start and end value"
"cannot be unbounded")
limits = sympify((x, start, stop))
if is_sequence(periodical, Tuple):
periodical = sympify(tuple(flatten(periodical)))
else:
raise ValueError("invalid period %s should be something "
"like e.g (1, 2) " % periodical)
if Interval(limits[1], limits[2]) is S.EmptySet:
return S.EmptySequence
return Basic.__new__(cls, periodical, limits)
def _set_color(self, v):
try:
if v is not None:
if is_sequence(v):
v = ColorScheme(*v)
else:
v = ColorScheme(v)
if repr(v) == repr(self._color):
return
self._on_change_color(v)
self._color = v
except Exception as e:
raise RuntimeError("Color change failed. " "Reason: %s" % (str(e)))
def sfield(exprs, *symbols, **options):
"""Construct a field deriving generators and domain
from options and input expressions.
Parameters
==========
exprs : py:class:`~.Expr` or sequence of :py:class:`~.Expr` (sympifiable)
symbols : sequence of :py:class:`~.Symbol`/:py:class:`~.Expr`
options : keyword arguments understood by :py:class:`~.Options`
Examples
========
>>> from sympy import exp, log, symbols, sfield
>>> x = symbols("x")
>>> K, f = sfield((x*log(x) + 4*x**2)*exp(1/x + log(x)/3)/x**2)
>>> K
Rational function field in x, exp(1/x), log(x), x**(1/3) over ZZ with lex order
>>> f
(4*x**2*(exp(1/x)) + x*(exp(1/x))*(log(x)))/((x**(1/3))**5)
"""
single = False
if not is_sequence(exprs):
exprs, single = [exprs], True
exprs = list(map(sympify, exprs))
opt = build_options(symbols, options)
numdens = []
for expr in exprs:
numdens.extend(expr.as_numer_denom())
reps, opt = _parallel_dict_from_expr(numdens, opt)
if opt.domain is None:
# NOTE: this is inefficient because construct_domain() automatically
# performs conversion to the target domain. It shouldn't do this.
coeffs = sum([list(rep.values()) for rep in reps], [])
opt.domain, _ = construct_domain(coeffs, opt=opt)
_field = FracField(opt.gens, opt.domain, opt.order)
fracs = []
for i in range(0, len(reps), 2):
fracs.append(_field(tuple(reps[i:i + 2])))
if single:
return (_field, fracs[0])
else:
return (_field, fracs)
def __contains__(self, point):
if is_sequence(point):
if len(point) == 2:
z1 = 1
else:
z1 = point[2]
x1, y1 = point[:2]
elif isinstance(point, EllipticCurvePoint):
x1, y1, z1 = point.x, point.y, point.z
else:
raise ValueError('Invalid point.')
if self.characteristic == 0 and z1 == 0:
return True
return self._eq.subs({self.x: x1, self.y: y1, self.z: z1})
def preprocess(cls, gens):
if isinstance(gens, Basic):
gens = (gens, )
elif len(gens) == 1 and is_sequence(gens[0]):
gens = gens[0]
if gens == (None, ):
gens = ()
elif has_dups(gens):
raise GeneratorsError("duplicated generators: %s" % str(gens))
elif any(gen.is_commutative is False for gen in gens):
raise GeneratorsError("non-commutative generators: %s" % str(gens))
return tuple(gens)
def _parse_symbols(symbols):
if not symbols:
return tuple()
if isinstance(symbols, str):
return _symbols(symbols, seq=True)
elif isinstance(symbols, Expr or FreeGroupElement):
return (symbols, )
elif is_sequence(symbols):
if all(isinstance(s, str) for s in symbols):
return _symbols(symbols)
elif all(isinstance(s, Expr) for s in symbols):
return symbols
raise ValueError("The type of `symbols` must be one of the following: "
"a str, Symbol/Expr or a sequence of "
"one of these types")
def _randrange(seed=None):
"""Return a randrange generator. ``seed`` can be
o None - return randomly seeded generator
o int - return a generator seeded with the int
o list - the values to be returned will be taken from the list
in the order given; the provided list is not modified.
Examples
========
>>> from sympy.core.random import _randrange
>>> rr = _randrange()
>>> rr(1000) # doctest: +SKIP
999
>>> rr = _randrange(3)
>>> rr(1000) # doctest: +SKIP
238
>>> rr = _randrange([0, 5, 1, 3, 4])
>>> rr(3), rr(3)
(0, 1)
"""
if seed is None:
return randrange
elif isinstance(seed, int):
rng.seed(seed)
return randrange
elif is_sequence(seed):
seed = list(seed) # make a copy
seed.reverse()
def give(a, b=None, seq=seed):
if b is None:
a, b = 0, a
a, b = as_int(a), as_int(b)
w = b - a
if w < 1:
raise ValueError('_randrange got empty range')
try:
x = seq.pop()
except IndexError:
raise ValueError('_randrange sequence was too short')
if a <= x < b:
return x
else:
return give(a, b, seq)
return give
else:
raise ValueError('_randrange got an unexpected seed')
def _eval_args(cls, args):
targets = args[0]
if not is_sequence(targets):
targets = (targets, )
targets = Gate._eval_args(targets)
_validate_targets_controls(targets)
mat = args[1]
if not isinstance(mat, MatrixBase):
raise TypeError('Matrix expected, got: %r' % mat)
#make sure this matrix is of a Basic type
mat = _sympify(mat)
dim = 2**len(targets)
if not all(dim == shape for shape in mat.shape):
raise IndexError(
'Number of targets must match the matrix size: %r %r' %
(targets, mat))
return (targets, mat)
def __new__(cls, label, range=None, **kw_args):
if isinstance(label, str):
label = Symbol(label, integer=True)
label, range = list(map(sympify, (label, range)))
if label.is_Number:
if not label.is_integer:
raise TypeError("Index is not an integer number.")
return label
if not label.is_integer:
raise TypeError("Idx object requires an integer label.")
elif is_sequence(range):
if len(range) != 2:
raise ValueError(
filldedent("""
Idx range tuple must have length 2, but got %s""" %
len(range)))
for bound in range:
if (bound.is_integer is False and bound is not S.Infinity
and bound is not S.NegativeInfinity):
raise TypeError("Idx object requires integer bounds.")
args = label, Tuple(*range)
elif isinstance(range, Expr):
if range is not S.Infinity and fuzzy_not(range.is_integer):
raise TypeError("Idx object requires an integer dimension.")
args = label, Tuple(0, range - 1)
elif range:
raise TypeError(
filldedent("""
The range must be an ordered iterable or
integer SymPy expression."""))
else:
args = label,
obj = Expr.__new__(cls, *args, **kw_args)
obj._assumptions["finite"] = True
obj._assumptions["real"] = True
return obj
def __new__(cls, p1, a=None, b=None, **kwargs):
p1 = Point3D(p1, dim=3)
if a and b:
p2 = Point(a, dim=3)
p3 = Point(b, dim=3)
if Point3D.are_collinear(p1, p2, p3):
raise ValueError('Enter three non-collinear points')
a = p1.direction_ratio(p2)
b = p1.direction_ratio(p3)
normal_vector = tuple(Matrix(a).cross(Matrix(b)))
else:
a = kwargs.pop('normal_vector', a)
evaluate = kwargs.get('evaluate', True)
if is_sequence(a) and len(a) == 3:
normal_vector = Point3D(a).args if evaluate else a
else:
raise ValueError(filldedent('''
Either provide 3 3D points or a point with a
normal vector expressed as a sequence of length 3'''))
if all(coord.is_zero for coord in normal_vector):
raise ValueError('Normal vector cannot be zero vector')
return GeometryEntity.__new__(cls, p1, normal_vector, **kwargs)
def __new__(cls, formula, limits=None):
formula = sympify(formula)
def _find_x(formula):
free = formula.free_symbols
if len(free) == 1:
return free.pop()
elif not free:
return Dummy('k')
else:
raise ValueError(
" specify dummy variables for %s. If the formula contains"
" more than one free symbol, a dummy variable should be"
" supplied explicitly e.g., SeqFormula(m*n**2, (n, 0, 5))"
% formula)
x, start, stop = None, None, None
if limits is None:
x, start, stop = _find_x(formula), 0, S.Infinity
if is_sequence(limits, Tuple):
if len(limits) == 3:
x, start, stop = limits
elif len(limits) == 2:
x = _find_x(formula)
start, stop = limits
if not isinstance(x, (Symbol, Idx)) or start is None or stop is None:
raise ValueError('Invalid limits given: %s' % str(limits))
if start is S.NegativeInfinity and stop is S.Infinity:
raise ValueError("Both the start and end value "
"cannot be unbounded")
limits = sympify((x, start, stop))
if Interval(limits[1], limits[2]) is S.EmptySet:
return S.EmptySequence
return Basic.__new__(cls, formula, limits)
def copyin_list(self, key, value):
"""Copy in elements from a list.
Parameters
==========
key : slice
The section of this matrix to replace.
value : iterable
The iterable to copy values from.
Examples
========
>>> from sympy import eye
>>> I = eye(3)
>>> I[:2, 0] = [1, 2] # col
>>> I
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
>>> I[1, :2] = [[3, 4]]
>>> I
Matrix([
[1, 0, 0],
[3, 4, 0],
[0, 0, 1]])
See Also
========
copyin_matrix
"""
if not is_sequence(value):
raise TypeError("`value` must be an ordered iterable, not %s." %
type(value))
return self.copyin_matrix(key, type(self)(value))
def __qsympify_sequence_helper(seq):
"""
Helper function for _qsympify_sequence
This function does the actual work.
"""
#base case. If not a list, do Sympification
if not is_sequence(seq):
if isinstance(seq, Matrix):
return seq
elif isinstance(seq, str):
return Symbol(seq)
else:
return sympify(seq)
# base condition, when seq is QExpr and also
# is iterable.
if isinstance(seq, QExpr):
return seq
#if list, recurse on each item in the list
result = [__qsympify_sequence_helper(item) for item in seq]
return Tuple(*result)
def mplot2d(f, var, *, show=True):
"""
Plot a 2d function using matplotlib/Tk.
"""
import warnings
warnings.filterwarnings("ignore", r"Could not match \S")
p = import_module('pylab')
if not p:
sys.exit("Matplotlib is required to use mplot2d.")
if not is_sequence(f):
f = [
f,
]
for f_i in f:
x, y = sample(f_i, var)
p.plot(x, y)
p.draw()
if show:
p.show()
def extend(self, arg):
"""Adds all series from another plot.
Examples
========
Consider two ``Plot`` objects, ``p1`` and ``p2``. To add the
second plot to the first, use the ``extend`` method, like so:
.. plot::
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
>>> p1 = plot(x**2, show=False)
>>> p2 = plot(x, -x, show=False)
>>> p1.extend(p2)
>>> p1
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
[1]: cartesian line: x for x over (-10.0, 10.0)
[2]: cartesian line: -x for x over (-10.0, 10.0)
>>> p1.show()
"""
if isinstance(arg, Plot):
self._series.extend(arg._series)
elif is_sequence(arg) and all([isinstance(a, BaseSeries) for a in arg]):
self._series.extend(arg)
else:
raise TypeError("Expecting Plot or sequence of BaseSeries")
# auto legend
if len(self._series) > 1:
self.legend = True
def extract_leading_order(self, symbols, point=None):
"""
Returns the leading term and its order.
Examples
========
>>> from sympy.abc import x
>>> (x + 1 + 1/x**5).extract_leading_order(x)
((x**(-5), O(x**(-5))),)
>>> (1 + x).extract_leading_order(x)
((1, O(1)),)
>>> (x + x**2).extract_leading_order(x)
((x, O(x)),)
"""
from sympy.series.order import Order
lst = []
symbols = list(symbols if is_sequence(symbols) else [symbols])
if not point:
point = [0]*len(symbols)
seq = [(f, Order(f, *zip(symbols, point))) for f in self.args]
for ef, of in seq:
for e, o in lst:
if o.contains(of) and o != of:
of = None
break
if of is None:
continue
new_lst = [(ef, of)]
for e, o in lst:
if of.contains(o) and o != of:
continue
new_lst.append((e, o))
lst = new_lst
return tuple(lst)
def _contains(self, other):
from sympy.matrices import Matrix
from sympy.solvers.solveset import solveset, linsolve
from sympy.utilities.iterables import is_sequence, iterable, cartes
L = self.lamda
if is_sequence(other):
if not is_sequence(L.expr):
return S.false
if len(L.expr) != len(other):
raise ValueError(filldedent('''
Dimensions of other and output of Lambda are different.'''))
elif iterable(other):
raise ValueError(filldedent('''
`other` should be an ordered object like a Tuple.'''))
solns = None
if self._is_multivariate():
if not is_sequence(L.expr):
# exprs -> (numer, denom) and check again
# XXX this is a bad idea -- make the user
# remap self to desired form
return other.as_numer_denom() in self.func(
Lambda(L.variables, L.expr.as_numer_denom()), self.base_set)
eqs = [expr - val for val, expr in zip(other, L.expr)]
variables = L.variables
free = set(variables)
if all(i.is_number for i in list(Matrix(eqs).jacobian(variables))):
solns = list(linsolve([e - val for e, val in
zip(L.expr, other)], variables))
else:
syms = [e.free_symbols & free for e in eqs]
solns = {}
for i, (e, s, v) in enumerate(zip(eqs, syms, other)):
if not s:
if e != v:
return S.false
solns[vars[i]] = [v]
continue
elif len(s) == 1:
sy = s.pop()
sol = solveset(e, sy)
if sol is S.EmptySet:
return S.false
elif isinstance(sol, FiniteSet):
solns[sy] = list(sol)
else:
raise NotImplementedError
else:
raise NotImplementedError
solns = cartes(*[solns[s] for s in variables])
else:
x = L.variables[0]
if isinstance(L.expr, Expr):
# scalar -> scalar mapping
solnsSet = solveset(L.expr - other, x)
if solnsSet.is_FiniteSet:
solns = list(solnsSet)
else:
msgset = solnsSet
else:
# scalar -> vector
for e, o in zip(L.expr, other):
solns = solveset(e - o, x)
if solns is S.EmptySet:
return S.false
for soln in solns:
try:
if soln in self.base_set:
break # check next pair
except TypeError:
if self.base_set.contains(soln.evalf()):
break
else:
return S.false # never broke so there was no True
return S.true
if solns is None:
raise NotImplementedError(filldedent('''
Determining whether %s contains %s has not
been implemented.''' % (msgset, other)))
for soln in solns:
try:
if soln in self.base_set:
return S.true
except TypeError:
return self.base_set.contains(soln.evalf())
return S.false
def banded(*args, **kwargs):
"""Returns a SparseMatrix from the given dictionary describing
the diagonals of the matrix. The keys are positive for upper
diagonals and negative for those below the main diagonal. The
values may be:
* expressions or single-argument functions,
* lists or tuples of values,
* matrices
Unless dimensions are given, the size of the returned matrix will
be large enough to contain the largest non-zero value provided.
kwargs
======
rows : rows of the resulting matrix; computed if
not given.
cols : columns of the resulting matrix; computed if
not given.
Examples
========
>>> from sympy import banded, ones, Matrix
>>> from sympy.abc import x
If explicit values are given in tuples,
the matrix will autosize to contain all values, otherwise
a single value is filled onto the entire diagonal:
>>> banded({1: (1, 2, 3), -1: (4, 5, 6), 0: x})
Matrix([
[x, 1, 0, 0],
[4, x, 2, 0],
[0, 5, x, 3],
[0, 0, 6, x]])
A function accepting a single argument can be used to fill the
diagonal as a function of diagonal index (which starts at 0).
The size (or shape) of the matrix must be given to obtain more
than a 1x1 matrix:
>>> s = lambda d: (1 + d)**2
>>> banded(5, {0: s, 2: s, -2: 2})
Matrix([
[1, 0, 1, 0, 0],
[0, 4, 0, 4, 0],
[2, 0, 9, 0, 9],
[0, 2, 0, 16, 0],
[0, 0, 2, 0, 25]])
The diagonal of matrices placed on a diagonal will coincide
with the indicated diagonal:
>>> vert = Matrix([1, 2, 3])
>>> banded({0: vert}, cols=3)
Matrix([
[1, 0, 0],
[2, 1, 0],
[3, 2, 1],
[0, 3, 2],
[0, 0, 3]])
>>> banded(4, {0: ones(2)})
Matrix([
[1, 1, 0, 0],
[1, 1, 0, 0],
[0, 0, 1, 1],
[0, 0, 1, 1]])
Errors are raised if the designated size will not hold
all values an integral number of times. Here, the rows
are designated as odd (but an even number is required to
hold the off-diagonal 2x2 ones):
>>> banded({0: 2, 1: ones(2)}, rows=5)
Traceback (most recent call last):
...
ValueError:
sequence does not fit an integral number of times in the matrix
And here, an even number of rows is given...but the square
matrix has an even number of columns, too. As we saw
in the previous example, an odd number is required:
>>> banded(4, {0: 2, 1: ones(2)}) # trying to make 4x4 and cols must be odd
Traceback (most recent call last):
...
ValueError:
sequence does not fit an integral number of times in the matrix
A way around having to count rows is to enclosing matrix elements
in a tuple and indicate the desired number of them to the right:
>>> banded({0: 2, 2: (ones(2),)*3})
Matrix([
[2, 0, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 1, 0, 0, 0, 0],
[0, 0, 2, 0, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 0, 2, 0, 1, 1],
[0, 0, 0, 0, 0, 2, 1, 1]])
An error will be raised if more than one value
is written to a given entry. Here, the ones overlap
with the main diagonal if they are placed on the
first diagonal:
>>> banded({0: (2,)*5, 1: (ones(2),)*3})
Traceback (most recent call last):
...
ValueError: collision at (1, 1)
By placing a 0 at the bottom left of the 2x2 matrix of
ones, the collision is avoided:
>>> u2 = Matrix([
... [1, 1],
... [0, 1]])
>>> banded({0: [2]*5, 1: [u2]*3})
Matrix([
[2, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 0, 0, 0, 0],
[0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 0, 0],
[0, 0, 0, 0, 2, 1, 1],
[0, 0, 0, 0, 0, 0, 1]])
"""
from sympy import Dict, Dummy, SparseMatrix
try:
if len(args) not in (1, 2, 3):
raise TypeError
if not isinstance(args[-1], (dict, Dict)):
raise TypeError
if len(args) == 1:
rows = kwargs.get('rows', None)
cols = kwargs.get('cols', None)
if rows is not None:
rows = as_int(rows)
if cols is not None:
cols = as_int(cols)
elif len(args) == 2:
rows = cols = as_int(args[0])
else:
rows, cols = map(as_int, args[:2])
# fails with ValueError if any keys are not ints
_ = all(as_int(k) for k in args[-1])
except (ValueError, TypeError):
raise TypeError(filldedent(
'''unrecognized input to banded:
expecting [[row,] col,] {int: value}'''))
def rc(d):
# return row,col coord of diagonal start
r = -d if d < 0 else 0
c = 0 if r else d
return r, c
smat = {}
undone = []
tba = Dummy()
# first handle objects with size
for d, v in args[-1].items():
r, c = rc(d)
# note: only list and tuple are recognized since this
# will allow other Basic objects like Tuple
# into the matrix if so desired
if isinstance(v, (list, tuple)):
extra = 0
for i, vi in enumerate(v):
i += extra
if is_sequence(vi):
vi = SparseMatrix(vi)
smat[r + i, c + i] = vi
extra += min(vi.shape) - 1
else:
smat[r + i, c + i] = vi
elif is_sequence(v):
v = SparseMatrix(v)
rv, cv = v.shape
if rows and cols:
nr, xr = divmod(rows - r, rv)
nc, xc = divmod(cols - c, cv)
x = xr or xc
do = min(nr, nc)
elif rows:
do, x = divmod(rows - r, rv)
elif cols:
do, x = divmod(cols - c, cv)
else:
do = 1
x = 0
if x:
raise ValueError(filldedent('''
sequence does not fit an integral number of times
in the matrix'''))
j = min(v.shape)
for i in range(do):
smat[r, c] = v
r += j
c += j
elif v:
smat[r, c] = tba
undone.append((d, v))
s = SparseMatrix(None, smat) # to expand matrices
smat = s._smat
# check for dim errors here
if rows is not None and rows < s.rows:
raise ValueError('Designated rows %s < needed %s' % (rows, s.rows))
if cols is not None and cols < s.cols:
raise ValueError('Designated cols %s < needed %s' % (cols, s.cols))
if rows is cols is None:
rows = s.rows
cols = s.cols
elif rows is not None and cols is None:
cols = max(rows, s.cols)
elif cols is not None and rows is None:
rows = max(cols, s.rows)
def update(i, j, v):
# update smat and make sure there are
# no collisions
if v:
if (i, j) in smat and smat[i, j] not in (tba, v):
raise ValueError('collision at %s' % ((i, j),))
smat[i, j] = v
if undone:
for d, vi in undone:
r, c = rc(d)
v = vi if callable(vi) else lambda _: vi
i = 0
while r + i < rows and c + i < cols:
update(r + i, c + i, v(i))
i += 1
return SparseMatrix(rows, cols, smat)
def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False):
"""
Evaluate the given formula to an accuracy of n digits.
Optional keyword arguments:
subs=<dict>
Substitute numerical values for symbols, e.g.
subs={x:3, y:1+pi}. The substitutions must be given as a
dictionary.
maxn=<integer>
Allow a maximum temporary working precision of maxn digits
(default=100)
chop=<bool>
Replace tiny real or imaginary parts in subresults
by exact zeros (default=False)
strict=<bool>
Raise PrecisionExhausted if any subresult fails to evaluate
to full accuracy, given the available maxprec
(default=False)
quad=<str>
Choose algorithm for numerical quadrature. By default,
tanh-sinh quadrature is used. For oscillatory
integrals on an infinite interval, try quad='osc'.
verbose=<bool>
Print debug information (default=False)
"""
from sympy import Float, Number
n = n if n is not None else 15
if subs and is_sequence(subs):
raise TypeError('subs must be given as a dictionary')
# for sake of sage that doesn't like evalf(1)
if n == 1 and isinstance(self, Number):
from sympy.core.expr import _mag
rv = self.evalf(2, subs, maxn, chop, strict, quad, verbose)
m = _mag(rv)
rv = rv.round(1 - m)
return rv
if not evalf_table:
_create_evalf_table()
prec = dps_to_prec(n)
options = {'maxprec': max(prec, int(maxn*LG10)), 'chop': chop,
'strict': strict, 'verbose': verbose}
if subs is not None:
options['subs'] = subs
if quad is not None:
options['quad'] = quad
try:
result = evalf(self, prec + 4, options)
except NotImplementedError:
# Fall back to the ordinary evalf
v = self._eval_evalf(prec)
if v is None:
return self
try:
# If the result is numerical, normalize it
result = evalf(v, prec, options)
except NotImplementedError:
# Probably contains symbols or unknown functions
return v
re, im, re_acc, im_acc = result
if re:
p = max(min(prec, re_acc), 1)
re = Float._new(re, p)
else:
re = S.Zero
if im:
p = max(min(prec, im_acc), 1)
im = Float._new(im, p)
return re + im*S.ImaginaryUnit
else:
return re
