def get_linear_system(eqs, variables):
"""
Convert equations into a linear system using sympy.
Parameters
----------
eqs : `Equations`
The model equations.
Returns
-------
(diff_eq_names, coefficients, constants) : (list of str, `sympy.Matrix`, `sympy.Matrix`)
A tuple containing the variable names (`diff_eq_names`) corresponding
to the rows of the matrix `coefficients` and the vector `constants`,
representing the system of equations in the form M * X + B
Raises
------
ValueError
If the equations cannot be converted into an M * X + B form.
"""
diff_eqs = eqs.get_substituted_expressions(variables)
diff_eq_names = [name for name, _ in diff_eqs]
symbols = [Symbol(name, real=True) for name in diff_eq_names]
coefficients = sp.zeros(len(diff_eq_names))
constants = sp.zeros(len(diff_eq_names), 1)
for row_idx, (name, expr) in enumerate(diff_eqs):
s_expr = str_to_sympy(expr.code, variables).expand()
current_s_expr = s_expr
for col_idx, symbol in enumerate(symbols):
current_s_expr = current_s_expr.collect(symbol)
constant_wildcard = Wild('c', exclude=[symbol])
factor_wildcard = Wild(f"c_{name}", exclude=symbols)
one_pattern = factor_wildcard * symbol + constant_wildcard
matches = current_s_expr.match(one_pattern)
if matches is None:
raise UnsupportedEquationsException(
f"The expression '{expr}', "
f"defining the variable "
f"'{name}', could not be "
f"separated into linear "
f"components.")
coefficients[row_idx, col_idx] = matches[factor_wildcard]
current_s_expr = matches[constant_wildcard]
# The remaining constant should be a true constant
constants[row_idx] = current_s_expr
return (diff_eq_names, coefficients, constants)
def _check_for_locally_constant(expression, variables, dt_value, t_symbol):
for arg in expression.args:
if arg is t_symbol:
# We found "t" -- if it is not the only argument of a locally
# constant function we bail out
func_name = str(expression.func)
if not (func_name in variables
and variables[func_name].is_locally_constant(dt_value)):
raise UnsupportedEquationsException(
('t is used in a context '
'where we cannot '
'guarantee that it can be '
'considered locally '
'constant.'))
else:
_check_for_locally_constant(arg, variables, dt_value, t_symbol)
def translate(
self, code, dtype
): # TODO: it's not so nice we have to copy the contents of this function..
'''
Translates an abstract code block into the target language.
'''
# first check if user code is not using variables that are also used by GSL
reserved_variables = [
'_dataholder', '_fill_y_vector', '_empty_y_vector',
'_GSL_dataholder', '_GSL_y', '_GSL_func'
]
if any([var in self.variables for var in reserved_variables]):
# import here to avoid circular import
raise ValueError(("The variables %s are reserved for the GSL "
"internal code." % (str(reserved_variables))))
# if the following statements are not added, Brian translates the
# differential expressions in the abstract code for GSL to scalar statements
# in the case no non-scalar variables are used in the expression
diff_vars = self.find_differential_variables(code.values())
self.add_gsl_variables_as_non_scalar(diff_vars)
# add arrays we want to use in generated code before self.generator.translate() so
# brian does namespace unpacking for us
pointer_names = self.add_meta_variables(self.method_options)
scalar_statements = {}
vector_statements = {}
for ac_name, ac_code in code.iteritems():
statements = make_statements(ac_code,
self.variables,
dtype,
optimise=True,
blockname=ac_name)
scalar_statements[ac_name], vector_statements[ac_name] = statements
for vs in vector_statements.itervalues():
# Check that the statements are meaningful independent on the order of
# execution (e.g. for synapses)
try:
if self.has_repeated_indices(
vs
): # only do order dependence if there are repeated indices
check_for_order_independence(
vs, self.generator.variables,
self.generator.variable_indices)
except OrderDependenceError:
# If the abstract code is only one line, display it in ful l
if len(vs) <= 1:
error_msg = 'Abstract code: "%s"\n' % vs[0]
else:
error_msg = (
'%_GSL_driver lines of abstract code, first line is: '
'"%s"\n') % (len(vs), vs[0])
# save function names because self.generator.translate_statement_sequence
# deletes these from self.variables but we need to know which identifiers
# we can safely ignore (i.e. we can ignore the functions because they are
# handled by the original generator)
self.function_names = self.find_function_names()
scalar_code, vector_code, kwds = self.generator.translate_statement_sequence(
scalar_statements, vector_statements)
############ translate code for GSL
# first check if any indexing other than '_idx' is used (currently not supported)
for code_list in scalar_code.values() + vector_code.values():
for code in code_list:
m = re.search('\[(\w+)\]', code)
if m is not None:
if m.group(1) != '0' and m.group(1) != '_idx':
from brian2.stateupdaters.base import UnsupportedEquationsException
raise UnsupportedEquationsException(
("Equations result in state "
"updater code with indexing "
"other than '_idx', which "
"is currently not supported "
"in combination with the "
"GSL stateupdater."))
# differential variable specific operations
to_replace = self.diff_var_to_replace(diff_vars)
GSL_support_code = self.get_dimension_code(len(diff_vars))
GSL_support_code += self.yvector_code(diff_vars)
# analyze all needed variables; if not in self.variables: put in separate dic.
# also keep track of variables needed for scalar statements and vector statements
other_variables = self.find_undefined_variables(
scalar_statements[None] + vector_statements[None])
variables_in_scalar = self.find_used_variables(scalar_statements[None],
other_variables)
variables_in_vector = self.find_used_variables(vector_statements[None],
other_variables)
# so that _dataholder holds diff_vars as well, even if they don't occur
# in the actual statements
for var in diff_vars.keys():
if not var in variables_in_vector:
variables_in_vector[var] = self.variables[var]
# lets keep track of the variables that eventually need to be added to
# the _GSL_dataholder somehow
self.variables_to_be_processed = variables_in_vector.keys()
# add code for _dataholder struct
GSL_support_code = self.write_dataholder(
variables_in_vector) + GSL_support_code
# add e.g. _lio_1 --> _GSL_dataholder._lio_1 to replacer
to_replace.update(
self.to_replace_vector_vars(variables_in_vector,
ignore=diff_vars.keys()))
# write statements that unpack (python) namespace to _dataholder struct
# or local namespace
GSL_main_code = self.unpack_namespace(variables_in_vector,
variables_in_scalar, ['t'])
# rewrite actual calculations described by vector_code and put them in _GSL_func
func_code = self.translate_one_statement_sequence(
vector_statements[None], scalar=False)
GSL_support_code += self.make_function_code(
self.translate_vector_code(func_code, to_replace))
scalar_func_code = self.translate_one_statement_sequence(
scalar_statements[None], scalar=True)
# rewrite scalar code, keep variables that are needed in scalar code normal
# and add variables to _dataholder for vector_code
GSL_main_code += '\n' + self.translate_scalar_code(
scalar_func_code, variables_in_scalar, variables_in_vector)
if len(self.variables_to_be_processed) > 0:
raise AssertionError(
("Not all variables that will be used in the vector "
"code have been added to the _GSL_dataholder. This "
"might mean that the _GSL_func is using unitialized "
"variables."
"\nThe unprocessed variables "
"are: %s" % (str(self.variables_to_be_processed))))
scalar_code['GSL'] = GSL_main_code
kwds['define_GSL_scale_array'] = self.scale_array_code(
diff_vars, self.method_options)
kwds['n_diff_vars'] = len(diff_vars)
kwds['GSL_settings'] = dict(self.method_options)
kwds['GSL_settings']['integrator'] = self.integrator
kwds['support_code_lines'] += GSL_support_code.split('\n')
kwds['t_array'] = self.get_array_name(self.variables['t']) + '[0]'
kwds['dt_array'] = self.get_array_name(self.variables['dt']) + '[0]'
kwds['define_dt'] = 'dt' not in variables_in_scalar
kwds['cpp_standalone'] = self.is_cpp_standalone()
for key, value in pointer_names.items():
kwds[key] = value
return scalar_code, vector_code, kwds
def __call__(self, equations, variables=None, method_options=None):
logger.warn(
"The 'independent' state updater is deprecated and might be "
"removed in future versions of Brian.",
'deprecated_independent',
once=True)
extract_method_options(method_options, {})
if equations.is_stochastic:
raise UnsupportedEquationsException("Cannot solve stochastic "
"equations with this state "
"updater")
if variables is None:
variables = {}
diff_eqs = equations.get_substituted_expressions(variables)
t = Symbol('t', real=True, positive=True)
dt = Symbol('dt', real=True, positive=True)
t0 = Symbol('t0', real=True, positive=True)
code = []
for name, expression in diff_eqs:
rhs = str_to_sympy(expression.code, variables)
# We have to be careful and use the real=True assumption as well,
# otherwise sympy doesn't consider the symbol a match to the content
# of the equation
var = Symbol(name, real=True)
f = sp.Function(name)
rhs = rhs.subs(var, f(t))
derivative = sp.Derivative(f(t), t)
diff_eq = sp.Eq(derivative, rhs)
# TODO: simplify=True sometimes fails with 0.7.4, see:
# https://github.com/sympy/sympy/issues/2666
try:
general_solution = sp.dsolve(diff_eq, f(t), simplify=True)
except RuntimeError:
general_solution = sp.dsolve(diff_eq, f(t), simplify=False)
# Check whether this is an explicit solution
if not getattr(general_solution, 'lhs', None) == f(t):
raise UnsupportedEquationsException(
f"Cannot explicitly solve: {str(diff_eq)}")
# Solve for C1 (assuming "var" as the initial value and "t0" as time)
if general_solution.has(Symbol('C1')):
if general_solution.has(Symbol('C2')):
raise UnsupportedEquationsException(
f'Too many constants in solution: {str(general_solution)}'
)
constant_solution = sp.solve(general_solution, Symbol('C1'))
if len(constant_solution) != 1:
raise UnsupportedEquationsException(
("Couldn't solve for the constant "
"C1 in : %s ") % str(general_solution))
constant = constant_solution[0].subs(t, t0).subs(f(t0), var)
solution = general_solution.rhs.subs('C1', constant)
else:
solution = general_solution.rhs.subs(t, t0).subs(f(t0), var)
# Evaluate the expression for one timestep
solution = solution.subs(t, t + dt).subs(t0, t)
# only try symplifying it -- it sometimes raises an error
try:
solution = solution.simplify()
except ValueError:
pass
code.append(f"{name} = {sympy_to_str(solution)}")
return '\n'.join(code)
def __call__(self, equations, variables=None, method_options=None):
method_options = extract_method_options(method_options,
{'simplify': True})
if equations.is_stochastic:
raise UnsupportedEquationsException("Cannot solve stochastic "
"equations with this state "
"updater.")
if variables is None:
variables = {}
# Get a representation of the ODE system in the form of
# dX/dt = M*X + B
varnames, matrix, constants = get_linear_system(equations, variables)
# No differential equations, nothing to do (this occurs sometimes in the
# test suite where the whole model is nothing more than something like
# 'v : 1')
if matrix.shape == (0, 0):
return ''
# Make sure that the matrix M is constant, i.e. it only contains
# external variables or constant variables
# Check for time dependence
dt_value = variables['dt'].get_value(
)[0] if 'dt' in variables else None
# This will raise an error if we meet the symbol "t" anywhere
# except as an argument of a locally constant function
for entry in itertools.chain(matrix, constants):
if not is_constant_over_dt(entry, variables, dt_value):
raise UnsupportedEquationsException(
f"Expression '{sympy_to_str(entry)}' is not guaranteed to be "
f"constant over a time step.")
symbols = [Symbol(variable, real=True) for variable in varnames]
solution = sp.solve_linear_system(matrix.row_join(constants), *symbols)
if solution is None or set(symbols) != set(solution.keys()):
raise UnsupportedEquationsException("Cannot solve the given "
"equations with this "
"stateupdater.")
b = sp.ImmutableMatrix([solution[symbol] for symbol in symbols])
# Solve the system
dt = Symbol('dt', real=True, positive=True)
try:
A = (matrix * dt).exp()
except NotImplementedError:
raise UnsupportedEquationsException("Cannot solve the given "
"equations with this "
"stateupdater.")
if method_options['simplify']:
A = A.applyfunc(
lambda x: sp.factor_terms(sp.cancel(sp.signsimp(x))))
C = sp.ImmutableMatrix(A * b) - b
_S = sp.MatrixSymbol('_S', len(varnames), 1)
updates = A * _S + C
updates = updates.as_explicit()
# The solution contains _S[0, 0], _S[1, 0] etc. for the state variables,
# replace them with the state variable names
abstract_code = []
for idx, (variable, update) in enumerate(zip(varnames, updates)):
rhs = update
if rhs.has(I, re, im):
raise UnsupportedEquationsException(
"The solution to the linear system "
"contains complex values "
"which is currently not implemented.")
for row_idx, varname in enumerate(varnames):
rhs = rhs.subs(_S[row_idx, 0], varname)
# Do not overwrite the real state variables yet, the update step
# of other state variables might still need the original values
abstract_code.append(f"_{variable} = {sympy_to_str(rhs)}")
# Update the state variables
for variable in varnames:
abstract_code.append(f"{variable} = _{variable}")
return '\n'.join(abstract_code)
def __call__(self, equations, variables=None):
if equations.is_stochastic:
raise UnsupportedEquationsException('Cannot solve stochastic '
'equations with this state '
'updater')
if variables is None:
variables = {}
diff_eqs = equations.get_substituted_expressions(variables)
t = Symbol('t', real=True, positive=True)
dt = Symbol('dt', real=True, positive=True)
t0 = Symbol('t0', real=True, positive=True)
f0 = Symbol('f0', real=True)
# TODO: Shortcut for simple linear equations? Is all this effort really
# worth it?
code = []
for name, expression in diff_eqs:
rhs = str_to_sympy(expression.code, variables)
# We have to be careful and use the real=True assumption as well,
# otherwise sympy doesn't consider the symbol a match to the content
# of the equation
var = Symbol(name, real=True)
f = sp.Function(name)
rhs = rhs.subs(var, f(t))
derivative = sp.Derivative(f(t), t)
diff_eq = sp.Eq(derivative, rhs)
# TODO: simplify=True sometimes fails with 0.7.4, see:
# https://github.com/sympy/sympy/issues/2666
try:
general_solution = sp.dsolve(diff_eq, f(t), simplify=True)
except RuntimeError:
general_solution = sp.dsolve(diff_eq, f(t), simplify=False)
# Check whether this is an explicit solution
if not getattr(general_solution, 'lhs', None) == f(t):
raise UnsupportedEquationsException(
'Cannot explicitly solve: ' + str(diff_eq))
# Solve for C1 (assuming "var" as the initial value and "t0" as time)
if general_solution.has(Symbol('C1')):
if general_solution.has(Symbol('C2')):
raise UnsupportedEquationsException(
'Too many constants in solution: %s' %
str(general_solution))
constant_solution = sp.solve(general_solution, Symbol('C1'))
if len(constant_solution) != 1:
raise UnsupportedEquationsException(
("Couldn't solve for the constant "
"C1 in : %s ") % str(general_solution))
constant = constant_solution[0].subs(t, t0).subs(f(t0), var)
solution = general_solution.rhs.subs('C1', constant)
else:
solution = general_solution.rhs.subs(t, t0).subs(f(t0), var)
# Evaluate the expression for one timestep
solution = solution.subs(t, t + dt).subs(t0, t)
# only try symplifying it -- it sometimes raises an error
try:
solution = solution.simplify()
except ValueError:
pass
code.append(name + ' = ' + sympy_to_str(solution))
return '\n'.join(code)
def __call__(self, equations, variables=None, simplify=True):
if equations.is_stochastic:
raise UnsupportedEquationsException('Cannot solve stochastic '
'equations with this state '
'updater.')
if variables is None:
variables = {}
# Get a representation of the ODE system in the form of
# dX/dt = M*X + B
varnames, matrix, constants = get_linear_system(equations, variables)
# No differential equations, nothing to do (this occurs sometimes in the
# test suite where the whole model is nothing more than something like
# 'v : 1')
if matrix.shape == (0, 0):
return ''
# Make sure that the matrix M is constant, i.e. it only contains
# external variables or constant variables
t = Symbol('t', real=True, positive=True)
# Check for time dependence
if 'dt' in variables:
dt_value = variables['dt'].get_value()[0]
# This will raise an error if we meet the symbol "t" anywhere
# except as an argument of a locally constant function
t = Symbol('t', real=True, positive=True)
for entry in itertools.chain(matrix, constants):
_check_for_locally_constant(entry, variables, dt_value, t)
symbols = [Symbol(variable, real=True) for variable in varnames]
solution = sp.solve_linear_system(matrix.row_join(constants), *symbols)
if solution is None or set(symbols) != set(solution.keys()):
raise UnsupportedEquationsException('Cannot solve the given '
'equations with this '
'stateupdater.')
b = sp.ImmutableMatrix([solution[symbol]
for symbol in symbols]).transpose()
# Solve the system
dt = Symbol('dt', real=True, positive=True)
try:
A = (matrix * dt).exp()
except NotImplementedError:
raise UnsupportedEquationsException('Cannot solve the given '
'equations with this '
'stateupdater.')
if simplify:
A = A.applyfunc(
lambda x: sp.factor_terms(sp.cancel(sp.signsimp(x))))
C = sp.ImmutableMatrix([A.dot(b)]) - b
_S = sp.MatrixSymbol('_S', len(varnames), 1)
updates = A * _S + C.transpose()
updates = updates.as_explicit()
# The solution contains _S[0, 0], _S[1, 0] etc. for the state variables,
# replace them with the state variable names
abstract_code = []
for idx, (variable, update) in enumerate(zip(varnames, updates)):
rhs = update
for row_idx, varname in enumerate(varnames):
rhs = rhs.subs(_S[row_idx, 0], varname)
# Do not overwrite the real state variables yet, the update step
# of other state variables might still need the original values
abstract_code.append('_' + variable + ' = ' + sympy_to_str(rhs))
# Update the state variables
for variable in varnames:
abstract_code.append(
'{variable} = _{variable}'.format(variable=variable))
return '\n'.join(abstract_code)
