def restore(variables, mystring):
if list_or_tuple_or_ndarray(variables):
vars = get_variables(mystring,'_')
indices = [int(v.strip('_')) for v in vars]
for i in range(len(vars)):
mystring = mystring.replace(vars[i],variables[indices[i]])
return mystring
def restore(variables, mystring):
if list_or_tuple_or_ndarray(variables):
vars = get_variables(mystring,'_')
indices = [int(v.strip('_')) for v in vars]
for i in range(len(vars)):
mystring = mystring.replace(vars[i],variables[indices[i]])
return mystring
def _solve_nonlinear(constraints, variables='x', target=None, **kwds):
"""Build a constraints function given a string of nonlinear constraints.
Returns a constraints function.
Inputs:
constraints -- a string of symbolic constraints, with one constraint
equation per line. Constraints must be equality constraints only.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> constraints = '''x1 = x3*3. + x0*x2'''
>>> print _solve_nonlinear(constraints)
x0 = (x1 - 3.0*x3)/x2
>>> constraints = '''
... spread([x0,x1]) - 1.0 = mean([x0,x1])
... mean([x0,x1,x2]) = x2'''
>>> print _solve_nonlinear(constraints)
x0 = -0.5 + 0.5*x2
x1 = 0.5 + 1.5*x2
Additional Inputs:
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
target -- list providing the order for which the variables will be solved.
If there are "N" constraint equations, the first "N" variables given
will be selected as the dependent variables. By default, increasing
order is used.
For example:
>>> constraints = '''
... spread([x0,x1]) - 1.0 = mean([x0,x1])
... mean([x0,x1,x2]) = x2'''
>>> print _solve_nonlinear(constraints, target=['x1'])
x1 = -0.833333333333333 + 0.166666666666667*x2
x0 = -0.5 + 0.5*x2
Further Inputs:
locals -- a dictionary of additional variables used in the symbolic
constraints equations, and their desired values.
"""
nvars = None
permute = False # if True, return all permutations
warn = True # if True, don't supress warnings
verbose = False # if True, print details from _classify_variables
#-----------------------undocumented-------------------------------
permute = kwds['permute'] if 'permute' in kwds else permute
warn = kwds['warn'] if 'warn' in kwds else warn
verbose = kwds['verbose'] if 'verbose' in kwds else verbose
#------------------------------------------------------------------
if target in [None, False]:
target = []
elif isinstance(target, str):
target = target.split(',')
else:
target = list(target) # not the best for ndarray, but should work
from mystic.symbolic import replace_variables, get_variables
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
constraints = replace_variables(constraints, variables, '_')
varname = '_'
ndim = len(variables)
else:
varname = variables # varname used below instead of variables
myvar = get_variables(constraints, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
# create function to replace "_" with original variables
def restore(variables, mystring):
if list_or_tuple_or_ndarray(variables):
vars = get_variables(mystring,'_')
indices = [int(v.strip('_')) for v in vars]
for i in range(len(vars)):
mystring = mystring.replace(vars[i],variables[indices[i]])
return mystring
locals = kwds['locals'] if 'locals' in kwds else None
if locals is None: locals = {}
eqns = constraints.splitlines()
# Remove empty strings:
actual_eqns = []
for j in range(len(eqns)):
if eqns[j].strip():
actual_eqns.append(eqns[j].strip())
orig_eqns = actual_eqns[:]
neqns = len(actual_eqns)
xperms = [varname+str(i) for i in range(ndim)]
if target:
[target.remove(i) for i in target if i not in xperms]
[target.append(i) for i in xperms if i not in target]
_target = []
[_target.append(i) for i in target if i not in _target]
target = _target
target = tuple(target)
xperms = list(permutations(xperms)) #XXX: takes a while if nvars is ~10
if target: # Try the suggested order first.
xperms.remove(target)
xperms.insert(0, target)
complete_list = []
constraints_function_list = []
# Some of the permutations will give the same answer;
# look into reducing the number of repeats?
for perm in xperms:
# Sort the list actual_eqns so any equation containing x0 is first, etc.
sorted_eqns = []
actual_eqns_copy = orig_eqns[:]
usedvars = []
for variable in perm: # range(ndim):
for eqn in actual_eqns_copy:
if eqn.find(variable) != -1:
sorted_eqns.append(eqn)
actual_eqns_copy.remove(eqn)
usedvars.append(variable)
break
if actual_eqns_copy: # Append the remaining equations
for item in actual_eqns_copy:
sorted_eqns.append(item)
actual_eqns = sorted_eqns
# Append the remaining variables to usedvars
tempusedvar = usedvars[:]
tempusedvar.sort()
nmissing = ndim - len(tempusedvar)
for m in range(nmissing):
usedvars.append(varname + str(len(tempusedvar) + m))
for i in range(neqns):
# Trying to use xi as a pivot. Loop through the equations
# looking for one containing xi.
_target = usedvars[i]
for eqn in actual_eqns[i:]:
invertedstring = _solve_single(eqn, variables=varname, target=_target, warn=warn)
if invertedstring:
warn = False
break
# substitute into the remaining equations. the equations' order
# in the list newsystem is like in a linear coefficient matrix.
newsystem = ['']*neqns
j = actual_eqns.index(eqn)
newsystem[j] = eqn
othereqns = actual_eqns[:j] + actual_eqns[j+1:]
for othereqn in othereqns:
expression = invertedstring.split("=")[1]
fixed = othereqn.replace(_target, '(' + expression + ')')
k = actual_eqns.index(othereqn)
newsystem[k] = fixed
actual_eqns = newsystem
# Invert so that it can be fed properly to generate_constraint
simplified = []
for eqn in actual_eqns:
_target = usedvars[actual_eqns.index(eqn)]
mysoln = _solve_single(eqn, variables=varname, target=_target, warn=warn)
if mysoln: simplified.append(mysoln)
simplified = restore(variables, '\n'.join(simplified).rstrip())
if permute:
complete_list.append(simplified)
continue
if verbose:
print _classify_variables(simplified, variables, ndim)
return simplified
warning='Warning: an error occurred in building the constraints.'
if warn: print warning
if verbose:
print _classify_variables(simplified, variables, ndim)
if permute: #FIXME: target='x3,x1' may order correct, while 'x1,x3' doesn't
filter = []; results = []
for i in complete_list:
_eqs = '\n'.join(sorted(i.split('\n')))
if _eqs and (_eqs not in filter):
filter.append(_eqs)
results.append(i)
return tuple(results) #FIXME: somehow 'rhs = xi' can be in results
return simplified
def _solve_linear(constraints, variables='x', target=None, **kwds):
"""Solve a system of symbolic linear constraints equations.
Inputs:
constraints -- a string of symbolic constraints, with one constraint
equation per line. Constraints must be equality constraints only.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> constraints = '''
... x0 - x2 = 2.
... x2 = x3*2.'''
>>> print _solve_linear(constraints)
x2 = 2.0*x3
x0 = 2.0 + 2.0*x3
Additional Inputs:
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
target -- list providing the order for which the variables will be solved.
If there are "N" constraint equations, the first "N" variables given
will be selected as the dependent variables. By default, increasing
order is used.
For example:
>>> constraints = '''
... x0 - x2 = 2.
... x2 = x3*2.'''
>>> print _solve_linear(constraints, target=['x3','x2'])
x3 = -1.0 + 0.5*x0
x2 = -2.0 + x0
Further Inputs:
locals -- a dictionary of additional variables used in the symbolic
constraints equations, and their desired values.
"""
nvars = None
permute = False # if True, return all permutations
warn = True # if True, don't supress warnings
verbose = False # if True, print debug info
#-----------------------undocumented-------------------------------
permute = kwds['permute'] if 'permute' in kwds else permute
warn = kwds['warn'] if 'warn' in kwds else warn
verbose = kwds['verbose'] if 'verbose' in kwds else verbose
#------------------------------------------------------------------
if target in [None, False]:
target = []
elif isinstance(target, str):
target = target.split(',')
else:
target = list(target) # not the best for ndarray, but should work
from mystic.symbolic import replace_variables, get_variables
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
_constraints = replace_variables(constraints, variables, '_')
varname = '_'
ndim = len(variables)
for i in range(len(target)):
if variables.count(target[i]):
target[i] = replace_variables(target[i],variables,markers='_')
else:
_constraints = constraints
varname = variables # varname used below instead of variables
myvar = get_variables(constraints, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
# create function to replace "_" with original variables
def restore(variables, mystring):
if list_or_tuple_or_ndarray(variables):
vars = get_variables(mystring,'_')
indices = [int(v.strip('_')) for v in vars]
for i in range(len(vars)):
mystring = mystring.replace(vars[i],variables[indices[i]])
return mystring
# default is _locals with sympy imported
_locals = {}
locals = kwds['locals'] if 'locals' in kwds else None
if locals is None: locals = {}
# if sympy not installed, return original constraints
try:
code = """from sympy import Eq, Symbol;"""
code += """from sympy import solve as symsol;"""
code = compile(code, '<string>', 'exec')
exec code in _locals
except ImportError: # Equation will not be simplified."
if warn: print "Warning: sympy not installed."
return constraints
# default is _locals with numpy and math imported
# numpy throws an 'AttributeError', but math passes error to sympy
code = """from numpy import *; from math import *;""" # prefer math
code += """from numpy import mean as average;""" # use np.mean not average
code += """from numpy import var as variance;""" # look like mystic.math
code += """from numpy import ptp as spread;""" # look like mystic.math
code = compile(code, '<string>', 'exec')
exec code in _locals
_locals.update(locals) #XXX: allow this?
code,left,right,xlist,neqns = _prepare_sympy(_constraints, varname, ndim)
eqlist = ""
for i in range(1, neqns+1):
eqn = 'eq' + str(i)
eqlist += eqn + ","
code += eqn + '= Eq(' + left[i-1] + ',' + right[i-1] + ')\n'
eqlist = eqlist.rstrip(',')
# get full list of variables in 'targeted' order
xperms = xlist.split(',')[:-1]
targeted = target[:]
[targeted.remove(i) for i in targeted if i not in xperms]
[targeted.append(i) for i in xperms if i not in targeted]
_target = []
[_target.append(i) for i in targeted if i not in _target]
targeted = _target
targeted = tuple(targeted)
if permute:
# Form constraints equations for each permutation.
# This will change the order of the x variables passed to symsol()
# to get different variables solved for.
xperms = list(permutations(xperms)) #XXX: takes a while if nvars is ~10
if target: # put the tuple with the 'targeted' order first
xperms.remove(targeted)
xperms.insert(0, targeted)
else:
xperms = [tuple(targeted)]
solns = []
for perm in xperms:
_code = code
xlist = ','.join(perm).rstrip(',') #XXX: if not all, use target ?
# solve dependent xi: symsol([linear_system], [x0,x1,...,xi,...,xn])
# returns: {x0: f(xn,...), x1: f(xn,...), ...}
_code += 'soln = symsol([' + eqlist + '], [' + xlist + '])'
#XXX: need to convert/check soln similarly as in _solve_single ?
if verbose: print _code
_code = compile(_code, '<string>', 'exec')
try:
exec _code in globals(), _locals
soln = _locals['soln']
if not soln:
if warn: print "Warning: could not simplify equation."
soln = {}
except NotImplementedError: # catch 'multivariate' error
if warn: print "Warning: could not simplify equation."
soln = {}
except NameError, error: # catch when variable is not defined
if warn: print "Warning:", error
soln = {}
if verbose: print soln
solved = ""
for key, value in soln.iteritems():
solved += str(key) + ' = ' + str(value) + '\n'
if solved: solns.append( restore(variables, solved.rstrip()) )
def _solve_single(constraint, variables='x', target=None, **kwds):
"""Solve a symbolic constraints equation for a single variable.
Inputs:
constraint -- a string of symbolic constraints. Only a single constraint
equation should be provided, and must be an equality constraint.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> equation = "x1 - 3. = x0*x2"
>>> print _solve_single(equation)
x0 = -(3.0 - x1)/x2
Additional Inputs:
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
target -- list providing the order for which the variables will be solved.
By default, increasing order is used.
For example:
>>> equation = "x1 - 3. = x0*x2"
>>> print _solve_single(equation, target='x1')
x1 = 3.0 + x0*x2
Further Inputs:
locals -- a dictionary of additional variables used in the symbolic
constraints equations, and their desired values.
""" #XXX: an very similar version of this code is found in _solve_linear XXX#
# for now, we abort on multi-line equations or inequalities
if len(constraint.replace('==','=').split('=')) != 2:
raise NotImplementedError, "requires a single valid equation"
if ">" in constraint or "<" in constraint:
raise NotImplementedError, "cannot simplify inequalities"
nvars = None
permute = False # if True, return all permutations
warn = True # if True, don't supress warnings
verbose = False # if True, print debug info
#-----------------------undocumented-------------------------------
permute = kwds['permute'] if 'permute' in kwds else permute
warn = kwds['warn'] if 'warn' in kwds else warn
verbose = kwds['verbose'] if 'verbose' in kwds else verbose
#------------------------------------------------------------------
if target in [None, False]:
target = []
elif isinstance(target, str):
target = target.split(',')
else:
target = list(target) # not the best for ndarray, but should work
from mystic.symbolic import replace_variables, get_variables
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
constraints = replace_variables(constraint, variables, markers='_')
varname = '_'
ndim = len(variables)
for i in range(len(target)):
if variables.count(target[i]):
target[i] = replace_variables(target[i],variables,markers='_')
else:
constraints = constraint # constraints used below
varname = variables # varname used below instead of variables
myvar = get_variables(constraint, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
# create function to replace "_" with original variables
def restore(variables, mystring):
if list_or_tuple_or_ndarray(variables):
vars = get_variables(mystring,'_')
indices = [int(v.strip('_')) for v in vars]
for i in range(len(vars)):
mystring = mystring.replace(vars[i],variables[indices[i]])
return mystring
# default is _locals with sympy imported
_locals = {}
locals = kwds['locals'] if 'locals' in kwds else None
if locals is None: locals = {}
try:
code = """from sympy import Eq, Symbol;"""
code += """from sympy import solve as symsol;"""
code = compile(code, '<string>', 'exec')
exec code in _locals
except ImportError: # Equation will not be simplified."
if warn: print "Warning: sympy not installed."
return constraint
# default is _locals with numpy and math imported
# numpy throws an 'AttributeError', but math passes error to sympy
code = """from numpy import *; from math import *;""" # prefer math
code += """from numpy import mean as average;""" # use np.mean not average
code += """from numpy import var as variance;""" # look like mystic.math
code += """from numpy import ptp as spread;""" # look like mystic.math
code = compile(code, '<string>', 'exec')
exec code in _locals
_locals.update(locals) #XXX: allow this?
code,left,right,xlist,neqns = _prepare_sympy(constraints, varname, ndim)
eqlist = ""
for i in range(1, neqns+1):
eqn = 'eq' + str(i)
eqlist += eqn + ","
code += eqn + '= Eq(' + left[i-1] + ',' + right[i-1] + ')\n'
eqlist = eqlist.rstrip(',')
# get full list of variables in 'targeted' order
xperms = xlist.split(',')[:-1]
targeted = target[:]
[targeted.remove(i) for i in targeted if i not in xperms]
[targeted.append(i) for i in xperms if i not in targeted]
_target = []
[_target.append(i) for i in targeted if i not in _target]
targeted = _target
targeted = tuple(targeted)
########################################################################
# solve each xi: symsol(single_equation, [x0,x1,...,xi,...,xn])
# returns: {x0: f(xn,...), x1: f(xn,...), ..., xn: f(...,x0)}
if permute or not target: #XXX: the goal is solving *only one* equation
code += '_xlist = %s\n' % ','.join(targeted)
code += '_elist = [symsol(['+eqlist+'], [i]) for i in _xlist]\n'
code += '_elist = [i if isinstance(i, dict) else {j:i[-1][-1]} for j,i in zip(_xlist,_elist)]\n'
code += 'soln = {}\n'
code += '[soln.update(i) for i in _elist if i]\n'
else:
code += 'soln = symsol([' + eqlist + '], [' + target[0] + '])\n'
#code += 'soln = symsol([' + eqlist + '], [' + targeted[0] + '])\n'
code += 'soln = soln if isinstance(soln, dict) else {' + target[0] + ': soln[-1][-1]}\n'
########################################################################
if verbose: print code
code = compile(code, '<string>', 'exec')
try:
exec code in globals(), _locals
soln = _locals['soln']
if not soln:
if warn: print "Warning: target variable is not valid"
soln = {}
except NotImplementedError: # catch 'multivariate' error for older sympy
if warn: print "Warning: could not simplify equation."
return constraint #FIXME: resolve diff with _solve_linear
except NameError, error: # catch when variable is not defined
if warn: print "Warning:", error
soln = {}
def _classify_variables(constraints, variables='x', nvars=None):
"""Takes a string of constraint equations and determines which variables
are dependent, independent, and unconstrained. Assumes there are no duplicate
equations. Returns a dictionary with keys: 'dependent', 'independent', and
'unconstrained', and with values that enumerate the variables that match
each variable type.
Inputs:
constraints -- a string of symbolic constraints, with one constraint
equation per line. Constraints must be equality constraints only.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> constraints = '''
... x0 = x4**2
... x2 = x3 + x4'''
>>> _classify_variables(constraints, nvars=5)
{'dependent':['x0','x2'], 'independent':['x3','x4'], 'unconstrained':['x1']}
>>> constraints = '''
... x0 = x4**2
... x4 - x3 = 0.
... x4 - x0 = x2'''
>>> _classify_variables(constraints, nvars=5)
{'dependent': ['x0','x2','x4'], 'independent': ['x3'], 'unconstrained': ['x1']}
Additional Inputs:
nvars -- number of variables. Includes variables not explicitly
given by the constraint equations (e.g. 'x1' in the example above).
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
"""
if ">" in constraints or "<" in constraints:
raise NotImplementedError, "cannot classify inequalities"
from mystic.symbolic import replace_variables, get_variables
#XXX: use solve? or first if not in form xi = ... ?
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
constraints = replace_variables(constraints, variables)
varname = '$'
ndim = len(variables)
else:
varname = variables # varname used below instead of variables
myvar = get_variables(constraints, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
eqns = constraints.splitlines()
indices = range(ndim)
dep = []
indep = []
for eqn in eqns: # find which variables are used
if eqn:
for var in range(ndim):
if indices.count(var) != 0:
if eqn.find(varname + str(var)) != -1:
indep.append(var)
indices.remove(var)
indep.sort()
_dep = []
for eqn in eqns: # find which variables are on the LHS
if eqn:
split = eqn.split('=')
for var in indep:
if split[0].find(varname + str(var)) != -1:
_dep.append(var)
indep.remove(var)
break
_dep.sort()
indep = _dep + indep # prefer variables found on LHS
for eqn in eqns: # find one dependent variable per equation
_dep = []
_indep = indep[:]
if eqn:
for var in _indep:
if eqn.find(varname + str(var)) != -1:
_dep.append(var)
_indep.remove(var)
if _dep:
dep.append(_dep[0])
indep.remove(_dep[0])
#FIXME: 'equivalent' equations not ignored (e.g. x2=x2; or x2=1, 2*x2=2)
"""These are good:
>>> constraints = '''
... x0 = x4**2
... x2 - x4 - x3 = 0.'''
>>> _classify_variables(constraints, nvars=5)
{'dependent': ['x0','x2'], 'independent': ['x3','x4'], 'unconstrained': ['x1']}
>>> constraints = '''
... x0 + x2 = 0.
... x0 + 2*x2 = 0.'''
>>> _classify_variables(constraints, nvars=5)
{'dependent': ['x0','x2'], 'independent': [], 'unconstrained': ['x1','x3','x4']}
This is a bug:
>>> constraints = '''
... x0 + x2 = 0.
... 2*x0 + 2*x2 = 0.'''
>>> _classify_variables(constraints, nvars=5)
{'dependent': ['x0','x2'], 'independent': [], 'unconstrained': ['x1','x3','x4']}
""" #XXX: should simplify first?
dep.sort()
indep.sort()
# return the actual variable names (not the indicies)
if varname == variables: # then was single variable
variables = [varname+str(i) for i in range(ndim)]
dep = [variables[i] for i in dep]
indep = [variables[i] for i in indep]
indices = [variables[i] for i in indices]
d = {'dependent':dep, 'independent':indep, 'unconstrained':indices}
return d
def _prepare_sympy(constraints, variables='x', nvars=None):
"""Parse an equation string and prepare input for sympy. Returns a tuple
of sympy-specific input: (code for variable declaration, left side of equation
string, right side of equation string, list of variables, and the number of
sympy equations).
Inputs:
constraints -- a string of symbolic constraints, with one constraint
equation per line. Constraints must be equality constraints only.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> constraints = '''
... x0 = x4**2
... x4 - x3 = 0.
... x4 - x0 = x2'''
>>> code, lhs, rhs, vars, neqn = _prepare_sympy(constraints, nvars=5)
>>> print code
x0=Symbol('x0')
x1=Symbol('x1')
x2=Symbol('x2')
x3=Symbol('x3')
x4=Symbol('x4')
rand = Symbol('rand')
>>> print lhs, rhs
['x0 ', 'x4 - x3 ', 'x4 - x0 '] [' x4**2', ' 0.', ' x2']
print "%s in %s eqns" % (vars, neqn)
x0,x1,x2,x3,x4, in 3 eqns
Additional Inputs:
nvars -- number of variables. Includes variables not explicitly
given by the constraint equations (e.g. 'x1' in the example above).
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
"""
if ">" in constraints or "<" in constraints:
raise NotImplementedError, "cannot simplify inequalities"
from mystic.symbolic import replace_variables, get_variables
#XXX: if constraints contain x0,x1,x3 for 'x', should x2 be in code,xlist?
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
constraints = replace_variables(constraints, variables, markers='_')
varname = '_'
ndim = len(variables)
else:
varname = variables # varname used below instead of variables
myvar = get_variables(constraints, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
# split constraints_str into lists of left hand sides and right hand sides
eacheqn = constraints.splitlines()
neqns = 0
left = []
right = []
for eq in eacheqn: #XXX: Le/Ge instead of Eq; Max/Min... (NotImplemented ?)
splitlist = eq.replace('==','=').split('=') #FIXME: no inequalities
if len(splitlist) == 2: #FIXME: first convert >/< to min/max ?
# If equation is blank on one side, raise error.
if len(splitlist[0].strip()) == 0 or len(splitlist[1].strip()) == 0:
print eq, "is not an equation!" # Raise exception?
else:
left.append(splitlist[0])
right.append(splitlist[1])
neqns += 1
# If equation doesn't have one equal sign, raise error.
if len(splitlist) != 2 and len(splitlist) != 1:
print eq, "is not an equation!" # Raise exception?
# First create list of x variables
xlist = ""
for i in range(ndim):
xn = varname + str(i)
xlist += xn + ","
# Start constructing the code string
code = ""
for i in range(ndim):
xn = varname + str(i)
code += xn + '=' + "Symbol('" + xn + "')\n"
code += "rand = Symbol('rand')\n"
return code, left, right, xlist, neqns
def _solve_nonlinear(constraints, variables='x', target=None, **kwds):
"""Build a constraints function given a string of nonlinear constraints.
Returns a constraints function.
Inputs:
constraints -- a string of symbolic constraints, with one constraint
equation per line. Constraints must be equality constraints only.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> constraints = '''x1 = x3*3. + x0*x2'''
>>> print(_solve_nonlinear(constraints))
x0 = (x1 - 3.0*x3)/x2
>>> constraints = '''
... spread([x0,x1]) - 1.0 = mean([x0,x1])
... mean([x0,x1,x2]) = x2'''
>>> print(_solve_nonlinear(constraints))
x0 = -0.5 + 0.5*x2
x1 = 0.5 + 1.5*x2
Additional Inputs:
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
target -- list providing the order for which the variables will be solved.
If there are "N" constraint equations, the first "N" variables given
will be selected as the dependent variables. By default, increasing
order is used.
For example:
>>> constraints = '''
... spread([x0,x1]) - 1.0 = mean([x0,x1])
... mean([x0,x1,x2]) = x2'''
>>> print(_solve_nonlinear(constraints, target=['x1']))
x1 = -0.833333333333333 + 0.166666666666667*x2
x0 = -0.5 + 0.5*x2
Further Inputs:
locals -- a dictionary of additional variables used in the symbolic
constraints equations, and their desired values.
"""
nvars = None
permute = False # if True, return all permutations
warn = True # if True, don't suppress warnings
verbose = False # if True, print details from _classify_variables
#-----------------------undocumented-------------------------------
permute = kwds['permute'] if 'permute' in kwds else permute
warn = kwds['warn'] if 'warn' in kwds else warn
verbose = kwds['verbose'] if 'verbose' in kwds else verbose
#------------------------------------------------------------------
if target in [None, False]:
target = []
elif isinstance(target, str):
target = target.split(',')
else:
target = list(target) # not the best for ndarray, but should work
from mystic.symbolic import replace_variables, get_variables
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
constraints = replace_variables(constraints, variables, '_')
varname = '_'
ndim = len(variables)
else:
varname = variables # varname used below instead of variables
myvar = get_variables(constraints, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
# create function to replace "_" with original variables
def restore(variables, mystring):
if list_or_tuple_or_ndarray(variables):
vars = get_variables(mystring,'_')
indices = [int(v.strip('_')) for v in vars]
for i in range(len(vars)):
mystring = mystring.replace(vars[i],variables[indices[i]])
return mystring
locals = kwds['locals'] if 'locals' in kwds else None
if locals is None: locals = {}
eqns = constraints.splitlines()
# Remove empty strings:
actual_eqns = []
for j in range(len(eqns)):
if eqns[j].strip():
actual_eqns.append(eqns[j].strip())
orig_eqns = actual_eqns[:]
neqns = len(actual_eqns)
xperms = [varname+str(i) for i in range(ndim)]
if target:
[target.remove(i) for i in target if i not in xperms]
[target.append(i) for i in xperms if i not in target]
_target = []
[_target.append(i) for i in target if i not in _target]
target = _target
target = tuple(target)
xperms = list(permutations(xperms)) #XXX: takes a while if nvars is ~10
if target: # Try the suggested order first.
xperms.remove(target)
xperms.insert(0, target)
complete_list = []
constraints_function_list = []
# Some of the permutations will give the same answer;
# look into reducing the number of repeats?
for perm in xperms:
# Sort the list actual_eqns so any equation containing x0 is first, etc.
sorted_eqns = []
actual_eqns_copy = orig_eqns[:]
usedvars = []
for variable in perm: # range(ndim):
for eqn in actual_eqns_copy:
if eqn.find(variable) != -1:
sorted_eqns.append(eqn)
actual_eqns_copy.remove(eqn)
usedvars.append(variable)
break
if actual_eqns_copy: # Append the remaining equations
for item in actual_eqns_copy:
sorted_eqns.append(item)
actual_eqns = sorted_eqns
# Append the remaining variables to usedvars
tempusedvar = usedvars[:]
tempusedvar.sort()
nmissing = ndim - len(tempusedvar)
for m in range(nmissing):
usedvars.append(varname + str(len(tempusedvar) + m))
#FIXME: not sure if the code below should be totally trusted...
for i in range(neqns):
# Trying to use xi as a pivot. Loop through the equations
# looking for one containing xi.
_target = usedvars[i%len(usedvars)] #XXX: ...to make it len of neqns
for eqn in actual_eqns[i:]:
invertedstring = _solve_single(eqn, variables=varname, target=_target, warn=warn)
if invertedstring:
warn = False
break
if invertedstring is None: continue #XXX: ...when _solve_single fails
# substitute into the remaining equations. the equations' order
# in the list newsystem is like in a linear coefficient matrix.
newsystem = ['']*neqns
j = actual_eqns.index(eqn)
newsystem[j] = invertedstring #XXX: ...was eqn. I think correct now
othereqns = actual_eqns[:j] + actual_eqns[j+1:]
for othereqn in othereqns:
expression = invertedstring.split("=")[1]
fixed = othereqn.replace(_target, '(' + expression + ')')
k = actual_eqns.index(othereqn)
newsystem[k] = fixed
actual_eqns = newsystem #XXX: potentially carrying too many eqns
# Invert so that it can be fed properly to generate_constraint
simplified = []
for eqn in actual_eqns[:len(usedvars)]: #XXX: ...needs to be same len
_target = usedvars[actual_eqns.index(eqn)]
mysoln = _solve_single(eqn, variables=varname, target=_target, warn=warn)
if mysoln: simplified.append(mysoln)
simplified = restore(variables, '\n'.join(simplified).rstrip())
if permute:
complete_list.append(simplified)
continue
if verbose:
print(_classify_variables(simplified, variables, ndim))
return simplified
warning='Warning: an error occurred in building the constraints.'
if warn: print(warning)
if verbose:
print(_classify_variables(simplified, variables, ndim))
if permute: #FIXME: target='x3,x1' may order correct, while 'x1,x3' doesn't
filter = []; results = []
for i in complete_list:
_eqs = '\n'.join(sorted(i.split('\n')))
if _eqs and (_eqs not in filter):
filter.append(_eqs)
results.append(i)
return tuple(results) #FIXME: somehow 'rhs = xi' can be in results
return simplified
def _classify_variables(constraints, variables='x', nvars=None):
"""Takes a string of constraint equations and determines which variables
are dependent, independent, and unconstrained. Assumes there are no duplicate
equations. Returns a dictionary with keys: 'dependent', 'independent', and
'unconstrained', and with values that enumerate the variables that match
each variable type.
Inputs:
constraints -- a string of symbolic constraints, with one constraint
equation per line. Constraints must be equality constraints only.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> constraints = '''
... x0 = x4**2
... x2 = x3 + x4'''
>>> _classify_variables(constraints, nvars=5)
{'dependent':['x0','x2'], 'independent':['x3','x4'], 'unconstrained':['x1']}
>>> constraints = '''
... x0 = x4**2
... x4 - x3 = 0.
... x4 - x0 = x2'''
>>> _classify_variables(constraints, nvars=5)
{'dependent': ['x0','x2','x4'], 'independent': ['x3'], 'unconstrained': ['x1']}
Additional Inputs:
nvars -- number of variables. Includes variables not explicitly
given by the constraint equations (e.g. 'x1' in the example above).
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
"""
if ">" in constraints or "<" in constraints:
raise NotImplementedError("cannot classify inequalities")
from mystic.symbolic import replace_variables, get_variables
#XXX: use solve? or first if not in form xi = ... ?
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
constraints = replace_variables(constraints, variables)
varname = '$'
ndim = len(variables)
else:
varname = variables # varname used below instead of variables
myvar = get_variables(constraints, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
eqns = constraints.splitlines()
indices = list(range(ndim))
dep = []
indep = []
for eqn in eqns: # find which variables are used
if eqn:
for var in range(ndim):
if indices.count(var) != 0:
if eqn.find(varname + str(var)) != -1:
indep.append(var)
indices.remove(var)
indep.sort()
_dep = []
for eqn in eqns: # find which variables are on the LHS
if eqn:
split = eqn.split('=')
for var in indep:
if split[0].find(varname + str(var)) != -1:
_dep.append(var)
indep.remove(var)
break
_dep.sort()
indep = _dep + indep # prefer variables found on LHS
for eqn in eqns: # find one dependent variable per equation
_dep = []
_indep = indep[:]
if eqn:
for var in _indep:
if eqn.find(varname + str(var)) != -1:
_dep.append(var)
_indep.remove(var)
if _dep:
dep.append(_dep[0])
indep.remove(_dep[0])
#FIXME: 'equivalent' equations not ignored (e.g. x2=x2; or x2=1, 2*x2=2)
"""These are good:
>>> constraints = '''
... x0 = x4**2
... x2 - x4 - x3 = 0.'''
>>> _classify_variables(constraints, nvars=5)
{'dependent': ['x0','x2'], 'independent': ['x3','x4'], 'unconstrained': ['x1']}
>>> constraints = '''
... x0 + x2 = 0.
... x0 + 2*x2 = 0.'''
>>> _classify_variables(constraints, nvars=5)
{'dependent': ['x0','x2'], 'independent': [], 'unconstrained': ['x1','x3','x4']}
This is a bug:
>>> constraints = '''
... x0 + x2 = 0.
... 2*x0 + 2*x2 = 0.'''
>>> _classify_variables(constraints, nvars=5)
{'dependent': ['x0','x2'], 'independent': [], 'unconstrained': ['x1','x3','x4']}
""" #XXX: should simplify first?
dep.sort()
indep.sort()
# return the actual variable names (not the indices)
if varname == variables: # then was single variable
variables = [varname+str(i) for i in range(ndim)]
dep = [variables[i] for i in dep]
indep = [variables[i] for i in indep]
indices = [variables[i] for i in indices]
d = {'dependent':dep, 'independent':indep, 'unconstrained':indices}
return d
def _prepare_sympy(constraints, variables='x', nvars=None):
"""Parse an equation string and prepare input for sympy. Returns a tuple
of sympy-specific input: (code for variable declaration, left side of equation
string, right side of equation string, list of variables, and the number of
sympy equations).
Inputs:
constraints -- a string of symbolic constraints, with one constraint
equation per line. Constraints must be equality constraints only.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> constraints = '''
... x0 = x4**2
... x4 - x3 = 0.
... x4 - x0 = x2'''
>>> code, lhs, rhs, vars, neqn = _prepare_sympy(constraints, nvars=5)
>>> print(code)
x0=Symbol('x0')
x1=Symbol('x1')
x2=Symbol('x2')
x3=Symbol('x3')
x4=Symbol('x4')
rand = Symbol('rand')
>>> print("%s %s" % (lhs, rhs))
['x0 ', 'x4 - x3 ', 'x4 - x0 '] [' x4**2', ' 0.', ' x2']
>>> print("%s in %s eqns" % (vars, neqn))
x0,x1,x2,x3,x4, in 3 eqns
Additional Inputs:
nvars -- number of variables. Includes variables not explicitly
given by the constraint equations (e.g. 'x1' in the example above).
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
"""
if ">" in constraints or "<" in constraints:
raise NotImplementedError("cannot simplify inequalities")
from mystic.symbolic import replace_variables, get_variables
#XXX: if constraints contain x0,x1,x3 for 'x', should x2 be in code,xlist?
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
constraints = replace_variables(constraints, variables, markers='_')
varname = '_'
ndim = len(variables)
else:
varname = variables # varname used below instead of variables
myvar = get_variables(constraints, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
# split constraints_str into lists of left hand sides and right hand sides
eacheqn = constraints.splitlines()
neqns = 0
left = []
right = []
for eq in eacheqn: #XXX: Le/Ge instead of Eq; Max/Min... (NotImplemented ?)
splitlist = eq.replace('==','=').split('=') #FIXME: no inequalities
if len(splitlist) == 2: #FIXME: first convert >/< to min/max ?
# If equation is blank on one side, raise error.
if len(splitlist[0].strip()) == 0 or len(splitlist[1].strip()) == 0:
print("%r is not an equation!" % eq) # Raise exception?
else:
left.append(splitlist[0])
right.append(splitlist[1])
neqns += 1
# If equation doesn't have one equal sign, raise error.
if len(splitlist) != 2 and len(splitlist) != 1:
print("%r is not an equation!" % eq) # Raise exception?
# First create list of x variables
xlist = ""
for i in range(ndim):
xn = varname + str(i)
xlist += xn + ","
# Start constructing the code string
code = ""
for i in range(ndim):
xn = varname + str(i)
code += xn + '=' + "Symbol('" + xn + "')\n"
code += "rand = Symbol('rand')\n"
return code, left, right, xlist, neqns
def _solve_linear(constraints, variables='x', target=None, **kwds):
"""Solve a system of symbolic linear constraints equations.
Inputs:
constraints -- a string of symbolic constraints, with one constraint
equation per line. Constraints must be equality constraints only.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> constraints = '''
... x0 - x2 = 2.
... x2 = x3*2.'''
>>> print _solve_linear(constraints)
x2 = 2.0*x3
x0 = 2.0 + 2.0*x3
Additional Inputs:
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
target -- list providing the order for which the variables will be solved.
If there are "N" constraint equations, the first "N" variables given
will be selected as the dependent variables. By default, increasing
order is used.
For example:
>>> constraints = '''
... x0 - x2 = 2.
... x2 = x3*2.'''
>>> print _solve_linear(constraints, target=['x3','x2'])
x3 = -1.0 + 0.5*x0
x2 = -2.0 + x0
Further Inputs:
locals -- a dictionary of additional variables used in the symbolic
constraints equations, and their desired values.
"""
nvars = None
permute = False # if True, return all permutations
warn = True # if True, don't supress warnings
verbose = False # if True, print debug info
#-----------------------undocumented-------------------------------
permute = kwds['permute'] if 'permute' in kwds else permute
warn = kwds['warn'] if 'warn' in kwds else warn
verbose = kwds['verbose'] if 'verbose' in kwds else verbose
#------------------------------------------------------------------
if target in [None, False]:
target = []
elif isinstance(target, str):
target = target.split(',')
else:
target = list(target) # not the best for ndarray, but should work
from mystic.symbolic import replace_variables, get_variables
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
_constraints = replace_variables(constraints, variables, '_')
varname = '_'
ndim = len(variables)
for i in range(len(target)):
if variables.count(target[i]):
target[i] = replace_variables(target[i],
variables,
markers='_')
else:
_constraints = constraints
varname = variables # varname used below instead of variables
myvar = get_variables(constraints, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
# create function to replace "_" with original variables
def restore(variables, mystring):
if list_or_tuple_or_ndarray(variables):
vars = get_variables(mystring, '_')
indices = [int(v.strip('_')) for v in vars]
for i in range(len(vars)):
mystring = mystring.replace(vars[i], variables[indices[i]])
return mystring
# default is _locals with sympy imported
_locals = {}
locals = kwds['locals'] if 'locals' in kwds else None
if locals is None: locals = {}
# if sympy not installed, return original constraints
try:
code = """from sympy import Eq, Symbol;"""
code += """from sympy import solve as symsol;"""
code = compile(code, '<string>', 'exec')
exec code in _locals
except ImportError: # Equation will not be simplified."
if warn: print "Warning: sympy not installed."
return constraints
# default is _locals with numpy and math imported
# numpy throws an 'AttributeError', but math passes error to sympy
code = """from numpy import *; from math import *;""" # prefer math
code += """from numpy import mean as average;""" # use np.mean not average
code += """from numpy import var as variance;""" # look like mystic.math
code += """from numpy import ptp as spread;""" # look like mystic.math
code = compile(code, '<string>', 'exec')
exec code in _locals
_locals.update(locals) #XXX: allow this?
code, left, right, xlist, neqns = _prepare_sympy(_constraints, varname,
ndim)
eqlist = ""
for i in range(1, neqns + 1):
eqn = 'eq' + str(i)
eqlist += eqn + ","
code += eqn + '= Eq(' + left[i - 1] + ',' + right[i - 1] + ')\n'
eqlist = eqlist.rstrip(',')
# get full list of variables in 'targeted' order
xperms = xlist.split(',')[:-1]
targeted = target[:]
[targeted.remove(i) for i in targeted if i not in xperms]
[targeted.append(i) for i in xperms if i not in targeted]
_target = []
[_target.append(i) for i in targeted if i not in _target]
targeted = _target
targeted = tuple(targeted)
if permute:
# Form constraints equations for each permutation.
# This will change the order of the x variables passed to symsol()
# to get different variables solved for.
xperms = list(
permutations(xperms)) #XXX: takes a while if nvars is ~10
if target: # put the tuple with the 'targeted' order first
xperms.remove(targeted)
xperms.insert(0, targeted)
else:
xperms = [tuple(targeted)]
solns = []
for perm in xperms:
_code = code
xlist = ','.join(perm).rstrip(',') #XXX: if not all, use target ?
# solve dependent xi: symsol([linear_system], [x0,x1,...,xi,...,xn])
# returns: {x0: f(xn,...), x1: f(xn,...), ...}
_code += 'soln = symsol([' + eqlist + '], [' + xlist + '])'
#XXX: need to convert/check soln similarly as in _solve_single ?
if verbose: print _code
_code = compile(_code, '<string>', 'exec')
try:
exec _code in globals(), _locals
soln = _locals['soln']
if not soln:
if warn: print "Warning: could not simplify equation."
soln = {}
except NotImplementedError: # catch 'multivariate' error
if warn: print "Warning: could not simplify equation."
soln = {}
except NameError, error: # catch when variable is not defined
if warn: print "Warning:", error
soln = {}
if verbose: print soln
solved = ""
for key, value in soln.iteritems():
solved += str(key) + ' = ' + str(value) + '\n'
if solved: solns.append(restore(variables, solved.rstrip()))
def _solve_single(constraint, variables='x', target=None, **kwds):
"""Solve a symbolic constraints equation for a single variable.
Inputs:
constraint -- a string of symbolic constraints. Only a single constraint
equation should be provided, and must be an equality constraint.
Standard python syntax should be followed (with the math and numpy
modules already imported).
For example:
>>> equation = "x1 - 3. = x0*x2"
>>> print _solve_single(equation)
x0 = -(3.0 - x1)/x2
Additional Inputs:
variables -- desired variable name. Default is 'x'. A list of variable
name strings is also accepted for when desired variable names
don't have the same base, and can include variables that are not
found in the constraints equation string.
target -- list providing the order for which the variables will be solved.
By default, increasing order is used.
For example:
>>> equation = "x1 - 3. = x0*x2"
>>> print _solve_single(equation, target='x1')
x1 = 3.0 + x0*x2
Further Inputs:
locals -- a dictionary of additional variables used in the symbolic
constraints equations, and their desired values.
""" #XXX: an very similar version of this code is found in _solve_linear XXX#
# for now, we abort on multi-line equations or inequalities
if len(constraint.replace('==', '=').split('=')) != 2:
raise NotImplementedError, "requires a single valid equation"
if ">" in constraint or "<" in constraint:
raise NotImplementedError, "cannot simplify inequalities"
nvars = None
permute = False # if True, return all permutations
warn = True # if True, don't supress warnings
verbose = False # if True, print debug info
#-----------------------undocumented-------------------------------
permute = kwds['permute'] if 'permute' in kwds else permute
warn = kwds['warn'] if 'warn' in kwds else warn
verbose = kwds['verbose'] if 'verbose' in kwds else verbose
#------------------------------------------------------------------
if target in [None, False]:
target = []
elif isinstance(target, str):
target = target.split(',')
else:
target = list(target) # not the best for ndarray, but should work
from mystic.symbolic import replace_variables, get_variables
if list_or_tuple_or_ndarray(variables):
if nvars is not None: variables = variables[:nvars]
constraints = replace_variables(constraint, variables, markers='_')
varname = '_'
ndim = len(variables)
for i in range(len(target)):
if variables.count(target[i]):
target[i] = replace_variables(target[i],
variables,
markers='_')
else:
constraints = constraint # constraints used below
varname = variables # varname used below instead of variables
myvar = get_variables(constraint, variables)
if myvar: ndim = max([int(v.strip(varname)) for v in myvar]) + 1
else: ndim = 0
if nvars is not None: ndim = nvars
# create function to replace "_" with original variables
def restore(variables, mystring):
if list_or_tuple_or_ndarray(variables):
vars = get_variables(mystring, '_')
indices = [int(v.strip('_')) for v in vars]
for i in range(len(vars)):
mystring = mystring.replace(vars[i], variables[indices[i]])
return mystring
# default is _locals with sympy imported
_locals = {}
locals = kwds['locals'] if 'locals' in kwds else None
if locals is None: locals = {}
try:
code = """from sympy import Eq, Symbol;"""
code += """from sympy import solve as symsol;"""
code = compile(code, '<string>', 'exec')
exec code in _locals
except ImportError: # Equation will not be simplified."
if warn: print "Warning: sympy not installed."
return constraint
# default is _locals with numpy and math imported
# numpy throws an 'AttributeError', but math passes error to sympy
code = """from numpy import *; from math import *;""" # prefer math
code += """from numpy import mean as average;""" # use np.mean not average
code += """from numpy import var as variance;""" # look like mystic.math
code += """from numpy import ptp as spread;""" # look like mystic.math
code = compile(code, '<string>', 'exec')
exec code in _locals
_locals.update(locals) #XXX: allow this?
code, left, right, xlist, neqns = _prepare_sympy(constraints, varname,
ndim)
eqlist = ""
for i in range(1, neqns + 1):
eqn = 'eq' + str(i)
eqlist += eqn + ","
code += eqn + '= Eq(' + left[i - 1] + ',' + right[i - 1] + ')\n'
eqlist = eqlist.rstrip(',')
# get full list of variables in 'targeted' order
xperms = xlist.split(',')[:-1]
targeted = target[:]
[targeted.remove(i) for i in targeted if i not in xperms]
[targeted.append(i) for i in xperms if i not in targeted]
_target = []
[_target.append(i) for i in targeted if i not in _target]
targeted = _target
targeted = tuple(targeted)
########################################################################
# solve each xi: symsol(single_equation, [x0,x1,...,xi,...,xn])
# returns: {x0: f(xn,...), x1: f(xn,...), ..., xn: f(...,x0)}
if permute or not target: #XXX: the goal is solving *only one* equation
code += '_xlist = %s\n' % ','.join(targeted)
code += '_elist = [symsol([' + eqlist + '], [i]) for i in _xlist]\n'
code += '_elist = [i if isinstance(i, dict) else {j:i[-1][-1]} for j,i in zip(_xlist,_elist)]\n'
code += 'soln = {}\n'
code += '[soln.update(i) for i in _elist if i]\n'
else:
code += 'soln = symsol([' + eqlist + '], [' + target[0] + '])\n'
#code += 'soln = symsol([' + eqlist + '], [' + targeted[0] + '])\n'
code += 'soln = soln if isinstance(soln, dict) else {' + target[
0] + ': soln[-1][-1]}\n'
########################################################################
if verbose: print code
code = compile(code, '<string>', 'exec')
try:
exec code in globals(), _locals
soln = _locals['soln']
if not soln:
if warn: print "Warning: target variable is not valid"
soln = {}
except NotImplementedError: # catch 'multivariate' error for older sympy
if warn: print "Warning: could not simplify equation."
return constraint #FIXME: resolve diff with _solve_linear
except NameError, error: # catch when variable is not defined
if warn: print "Warning:", error
soln = {}