def nsolve(*args, **kwargs):
"""
Solve a nonlinear equation system numerically.
nsolve(f, [args,] x0, modules=['mpmath'], **kwargs)
f is a vector function of symbolic expressions representing the system.
args are the variables. If there is only one variable, this argument can be
omitted.
x0 is a starting vector close to a solution.
Use the modules keyword to specify which modules should be used to evaluate
the function and the Jacobian matrix. Make sure to use a module that
supports matrices. For more information on the syntax, please see the
docstring of lambdify.
Overdetermined systems are supported.
>>> from sympy import Symbol, nsolve
>>> import sympy
>>> sympy.mpmath.mp.dps = 15
>>> x1 = Symbol('x1')
>>> x2 = Symbol('x2')
>>> f1 = 3 * x1**2 - 2 * x2**2 - 1
>>> f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8
>>> print nsolve((f1, f2), (x1, x2), (-1, 1))
[-1.19287309935246]
[ 1.27844411169911]
For one-dimensional functions the syntax is simplified:
>>> from sympy import sin, nsolve
>>> from sympy.abc import x
>>> nsolve(sin(x), x, 2)
3.14159265358979
>>> nsolve(sin(x), 2)
3.14159265358979
mpmath.findroot is used, you can find there more extensive documentation,
especially concerning keyword parameters and available solvers.
"""
# interpret arguments
if len(args) == 3:
f = args[0]
fargs = args[1]
x0 = args[2]
elif len(args) == 2:
f = args[0]
fargs = None
x0 = args[1]
elif len(args) < 2:
raise TypeError("nsolve expected at least 2 arguments, got %i" % len(args))
else:
raise TypeError("nsolve expected at most 3 arguments, got %i" % len(args))
modules = kwargs.get("modules", ["mpmath"])
if isinstance(f, (list, tuple)):
f = Matrix(f).T
if not isinstance(f, Matrix):
# assume it's a sympy expression
if isinstance(f, Equality):
f = f.lhs - f.rhs
f = f.evalf()
atoms = f.atoms(Symbol)
if fargs is None:
fargs = atoms.copy().pop()
if not (len(atoms) == 1 and (fargs in atoms or fargs[0] in atoms)):
raise ValueError("expected a one-dimensional and numerical function")
# the function is much better behaved if there is no denominator
f = f.as_numer_denom()[0]
f = lambdify(fargs, f, modules)
return findroot(f, x0, **kwargs)
if len(fargs) > f.cols:
raise NotImplementedError("need at least as many equations as variables")
verbose = kwargs.get("verbose", False)
if verbose:
print "f(x):"
print f
# derive Jacobian
J = f.jacobian(fargs)
if verbose:
print "J(x):"
print J
# create functions
f = lambdify(fargs, f.T, modules)
J = lambdify(fargs, J, modules)
# solve the system numerically
x = findroot(f, x0, J=J, **kwargs)
return x
def nsolve(*args, **kwargs):
"""
Solve a nonlinear equation system numerically.
nsolve(f, [args,] x0, modules=['mpmath'], **kwargs)
f is a vector function of symbolic expressions representing the system.
args are the variables. If there is only one variable, this argument can be
omitted.
x0 is a starting vector close to a solution.
Use the modules keyword to specify which modules should be used to evaluate
the function and the Jacobian matrix. Make sure to use a module that
supports matrices. For more information on the syntax, please see the
docstring of lambdify.
Overdetermined systems are supported.
>>> from sympy import Symbol, nsolve
>>> import sympy
>>> sympy.mpmath.mp.dps = 15
>>> x1 = Symbol('x1')
>>> x2 = Symbol('x2')
>>> f1 = 3 * x1**2 - 2 * x2**2 - 1
>>> f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8
>>> print nsolve((f1, f2), (x1, x2), (-1, 1))
[-1.19287309935246]
[ 1.27844411169911]
For one-dimensional functions the syntax is simplified:
>>> from sympy import sin, nsolve
>>> from sympy.abc import x
>>> nsolve(sin(x), x, 2)
3.14159265358979
>>> nsolve(sin(x), 2)
3.14159265358979
mpmath.findroot is used, you can find there more extensive documentation,
especially concerning keyword parameters and available solvers.
"""
# interpret arguments
if len(args) == 3:
f = args[0]
fargs = args[1]
x0 = args[2]
elif len(args) == 2:
f = args[0]
fargs = None
x0 = args[1]
elif len(args) < 2:
raise TypeError('nsolve expected at least 2 arguments, got %i' %
len(args))
else:
raise TypeError('nsolve expected at most 3 arguments, got %i' %
len(args))
modules = kwargs.get('modules', ['mpmath'])
if isinstance(f, (list, tuple)):
f = Matrix(f).T
if not isinstance(f, Matrix):
# assume it's a sympy expression
if isinstance(f, Equality):
f = f.lhs - f.rhs
f = f.evalf()
atoms = f.atoms(Symbol)
if fargs is None:
fargs = atoms.copy().pop()
if not (len(atoms) == 1 and (fargs in atoms or fargs[0] in atoms)):
raise ValueError(
'expected a one-dimensional and numerical function')
# the function is much better behaved if there is no denominator
f = f.as_numer_denom()[0]
f = lambdify(fargs, f, modules)
return findroot(f, x0, **kwargs)
if len(fargs) > f.cols:
raise NotImplementedError(
'need at least as many equations as variables')
verbose = kwargs.get('verbose', False)
if verbose:
print 'f(x):'
print f
# derive Jacobian
J = f.jacobian(fargs)
if verbose:
print 'J(x):'
print J
# create functions
f = lambdify(fargs, f.T, modules)
J = lambdify(fargs, J, modules)
# solve the system numerically
x = findroot(f, x0, J=J, **kwargs)
return x
