def get_rhs(self, row_indices=None):
'''get the rhs vector of the constraints
row_indices - a list of requested indices (None=all)
this returns an np.array of rhs values for the requested indices
'''
rv = []
if row_indices is None:
lp_rows = glpk.glp_get_num_rows(self.lp)
row_indices = range(lp_rows)
for row in row_indices:
row_type = glpk.glp_get_row_type(self.lp, row + 1)
if row_type in [glpk.GLP_FX, glpk.GLP_UP]:
limit = glpk.glp_get_row_ub(self.lp, row + 1)
elif row_type == glpk.GLP_LO:
limit = glpk.glp_get_row_ub(self.lp, row + 1)
else:
raise RuntimeError(
"Error: Unsupported type ({}) in getRhs() in row {}".
format(row_type, row))
rv.append(limit)
return np.array(rv, dtype=float)
def test_change_constraint_bounds(self):
inner_prob = self.model.problem
inner_problem_bounds = [(glp_get_row_lb(inner_prob, i), glp_get_row_ub(inner_prob, i)) for i in
range(1, glp_get_num_rows(inner_prob) + 1)]
bounds = [(constr.lb, constr.ub) for constr in self.model.constraints]
self.assertEqual(bounds, inner_problem_bounds)
for constr in self.model.constraints:
constr.lb = random.uniform(-1000, constr.ub)
constr.ub = random.uniform(constr.lb, 1000)
inner_problem_bounds_new = [(glp_get_row_lb(inner_prob, i), glp_get_row_ub(inner_prob, i)) for i in
range(1, glp_get_num_rows(inner_prob) + 1)]
bounds_new = [(constr.lb, constr.ub) for constr in self.model.constraints]
self.assertNotEqual(bounds, bounds_new)
self.assertNotEqual(inner_problem_bounds, inner_problem_bounds_new)
self.assertEqual(bounds_new, inner_problem_bounds_new)
def test_change_constraint_bounds(self):
inner_prob = self.model.problem
inner_problem_bounds = [
(glp_get_row_lb(inner_prob, i), glp_get_row_ub(inner_prob, i))
for i in range(1,
glp_get_num_rows(inner_prob) + 1)
]
bounds = [(constr.lb, constr.ub) for constr in self.model.constraints]
self.assertEqual(bounds, inner_problem_bounds)
for constr in self.model.constraints:
constr.lb = random.uniform(-1000, constr.ub)
constr.ub = random.uniform(constr.lb, 1000)
inner_problem_bounds_new = [
(glp_get_row_lb(inner_prob, i), glp_get_row_ub(inner_prob, i))
for i in range(1,
glp_get_num_rows(inner_prob) + 1)
]
bounds_new = [(constr.lb, constr.ub)
for constr in self.model.constraints]
self.assertNotEqual(bounds, bounds_new)
self.assertNotEqual(inner_problem_bounds, inner_problem_bounds_new)
self.assertEqual(bounds_new, inner_problem_bounds_new)
def _constraints_str(self, zero_print):
'get the constraints matrix lines for __str__'
rv = ""
lp = self.lp
rows = self.get_num_rows()
cols = self.get_num_cols()
stat_labels = ["?(0)?", "BS", "NL", "NU", "NF", "NS"]
inds = SwigArray.get_int_array(cols + 1)
vals = SwigArray.get_double_array(cols + 1)
for row in range(1, rows + 1):
stat = glpk.glp_get_row_stat(lp, row)
assert 0 <= stat <= len(stat_labels)
rv += "{:2}: {} ".format(row, stat_labels[stat])
num_inds = glpk.glp_get_mat_row(lp, row, inds, vals)
for col in range(1, cols + 1):
val = 0
for index in range(1, num_inds + 1):
if inds[index] == col:
val = vals[index]
break
num = str(val)
num = num.rjust(6)[:6] # fix width to exactly 6
rv += (zero_print(num) if val == 0 else num) + " "
row_type = glpk.glp_get_row_type(lp, row)
assert row_type == glpk.GLP_UP
val = glpk.glp_get_row_ub(lp, row)
rv += " <= "
num = str(val)
num = num.rjust(6)[:6] # fix width to exactly 6
rv += (zero_print(num) if val == 0 else num) + " "
rv += "\n"
return rv
def serialize(self):
'serialize self.lp from a glpk instance into a tuple'
Timers.tic('serialize')
# get constraints as csr matrix
lp_rows = self.get_num_rows()
lp_cols = self.get_num_cols()
inds_row = SwigArray.get_int_array(lp_cols + 1)
vals_row = SwigArray.get_double_array(lp_cols + 1)
data = []
glpk_indices = []
indptr = [0]
for row in range(lp_rows):
got_len = glpk.glp_get_mat_row(self.lp, row + 1, inds_row,
vals_row)
for i in range(1, got_len + 1):
data.append(vals_row[i])
glpk_indices.append(inds_row[i])
indptr.append(len(data))
# rhs
rhs = []
for row in range(lp_rows):
assert glpk.glp_get_row_type(self.lp, row + 1) == glpk.GLP_UP
rhs.append(glpk.glp_get_row_ub(self.lp, row + 1))
col_bounds = self.get_col_bounds()
# remember to free lp object before overwriting with tuple
glpk.glp_delete_prob(self.lp)
self.lp = (data, glpk_indices, indptr, rhs, col_bounds)
Timers.toc('serialize')
def _initialize_model_from_problem(self, problem):
try:
self.problem = problem
glp_create_index(self.problem)
except TypeError:
raise TypeError("Provided problem is not a valid GLPK model.")
row_num = glp_get_num_rows(self.problem)
col_num = glp_get_num_cols(self.problem)
for i in range(1, col_num + 1):
var = Variable(
glp_get_col_name(self.problem, i),
lb=glp_get_col_lb(self.problem, i),
ub=glp_get_col_ub(self.problem, i),
problem=self,
type=_GLPK_VTYPE_TO_VTYPE[
glp_get_col_kind(self.problem, i)]
)
# This avoids adding the variable to the glpk problem
super(Model, self)._add_variables([var])
variables = self.variables
for j in range(1, row_num + 1):
ia = intArray(col_num + 1)
da = doubleArray(col_num + 1)
nnz = glp_get_mat_row(self.problem, j, ia, da)
constraint_variables = [variables[ia[i] - 1] for i in range(1, nnz + 1)]
# Since constraint expressions are lazily retrieved from the solver they don't have to be built here
# lhs = _unevaluated_Add(*[da[i] * constraint_variables[i - 1]
# for i in range(1, nnz + 1)])
lhs = 0
glpk_row_type = glp_get_row_type(self.problem, j)
if glpk_row_type == GLP_FX:
row_lb = glp_get_row_lb(self.problem, j)
row_ub = row_lb
elif glpk_row_type == GLP_LO:
row_lb = glp_get_row_lb(self.problem, j)
row_ub = None
elif glpk_row_type == GLP_UP:
row_lb = None
row_ub = glp_get_row_ub(self.problem, j)
elif glpk_row_type == GLP_DB:
row_lb = glp_get_row_lb(self.problem, j)
row_ub = glp_get_row_ub(self.problem, j)
elif glpk_row_type == GLP_FR:
row_lb = None
row_ub = None
else:
raise Exception(
"Currently, optlang does not support glpk row type %s"
% str(glpk_row_type)
)
log.exception()
if isinstance(lhs, int):
lhs = symbolics.Integer(lhs)
elif isinstance(lhs, float):
lhs = symbolics.Real(lhs)
constraint_id = glp_get_row_name(self.problem, j)
for variable in constraint_variables:
try:
self._variables_to_constraints_mapping[variable.name].add(constraint_id)
except KeyError:
self._variables_to_constraints_mapping[variable.name] = set([constraint_id])
super(Model, self)._add_constraints(
[Constraint(lhs, lb=row_lb, ub=row_ub, name=constraint_id, problem=self, sloppy=True)],
sloppy=True
)
term_generator = (
(glp_get_obj_coef(self.problem, index), variables[index - 1])
for index in range(1, glp_get_num_cols(problem) + 1)
)
self._objective = Objective(
symbolics.add(
[symbolics.mul((symbolics.Real(term[0]), term[1])) for term in term_generator if
term[0] != 0.]
),
problem=self,
direction={GLP_MIN: 'min', GLP_MAX: 'max'}[glp_get_obj_dir(self.problem)])
glp_scale_prob(self.problem, GLP_SF_AUTO)
def test_add_constraints(self):
x = Variable('x', lb=0, ub=1, type='binary')
y = Variable('y', lb=-181133.3, ub=12000., type='continuous')
z = Variable('z', lb=0., ub=10., type='integer')
constr1 = Constraint(0.3 * x + 0.4 * y + 66. * z,
lb=-100,
ub=0.,
name='test')
constr2 = Constraint(2.333 * x + y + 3.333, ub=100.33, name='test2')
constr3 = Constraint(2.333 * x + y + z, lb=-300)
constr4 = Constraint(x, lb=-300, ub=-300)
constr5 = Constraint(3 * x)
self.model.add(constr1)
self.model.add(constr2)
self.model.add(constr3)
self.model.add([constr4, constr5])
self.assertIn(constr1.name, self.model.constraints)
self.assertIn(constr2.name, self.model.constraints)
self.assertIn(constr3.name, self.model.constraints)
self.assertIn(constr4.name, self.model.constraints)
self.assertIn(constr5.name, self.model.constraints)
# constr1
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr1._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 0.3, 'y': 0.4, 'z': 66.})
self.assertEqual(glp_get_row_type(self.model.problem, constr1._index),
GLP_DB)
self.assertEqual(glp_get_row_lb(self.model.problem, constr1._index),
-100)
self.assertEqual(glp_get_row_ub(self.model.problem, constr1._index), 0)
# constr2
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr2._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 2.333, 'y': 1.})
self.assertEqual(glp_get_row_type(self.model.problem, constr2._index),
GLP_UP)
self.assertEqual(glp_get_row_lb(self.model.problem, constr2._index),
-1.7976931348623157e+308)
self.assertEqual(glp_get_row_ub(self.model.problem, constr2._index),
96.997)
# constr3
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr3._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 2.333, 'y': 1., 'z': 1.})
self.assertEqual(glp_get_row_type(self.model.problem, constr3._index),
GLP_LO)
self.assertEqual(glp_get_row_lb(self.model.problem, constr3._index),
-300)
self.assertEqual(glp_get_row_ub(self.model.problem, constr3._index),
1.7976931348623157e+308)
# constr4
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr4._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 1})
self.assertEqual(glp_get_row_type(self.model.problem, constr4._index),
GLP_FX)
self.assertEqual(glp_get_row_lb(self.model.problem, constr4._index),
-300)
self.assertEqual(glp_get_row_ub(self.model.problem, constr4._index),
-300)
# constr5
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr5._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 3})
self.assertEqual(glp_get_row_type(self.model.problem, constr5._index),
GLP_FR)
self.assertLess(glp_get_row_lb(self.model.problem, constr5._index),
-1e30)
self.assertGreater(glp_get_row_ub(self.model.problem, constr5._index),
1e30)
def test_add_constraints(self):
x = self.interface.Variable('x', lb=0, ub=1, type='binary')
y = self.interface.Variable('y', lb=-181133.3, ub=12000., type='continuous')
z = self.interface.Variable('z', lb=0., ub=10., type='integer')
constr1 = self.interface.Constraint(0.3 * x + 0.4 * y + 66. * z, lb=-100, ub=0., name='test')
constr2 = self.interface.Constraint(2.333 * x + y + 3.333, ub=100.33, name='test2')
constr3 = self.interface.Constraint(2.333 * x + y + z, lb=-300)
constr4 = self.interface.Constraint(x, lb=-300, ub=-300)
constr5 = self.interface.Constraint(3 * x)
self.model.add(constr1)
self.model.add(constr2)
self.model.add(constr3)
self.model.add([constr4, constr5])
self.assertIn(constr1.name, self.model.constraints)
self.assertIn(constr2.name, self.model.constraints)
self.assertIn(constr3.name, self.model.constraints)
self.assertIn(constr4.name, self.model.constraints)
self.assertIn(constr5.name, self.model.constraints)
# constr1
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr1._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 0.3, 'y': 0.4, 'z': 66.})
self.assertEqual(glp_get_row_type(self.model.problem, constr1._index), GLP_DB)
self.assertEqual(glp_get_row_lb(self.model.problem, constr1._index), -100)
self.assertEqual(glp_get_row_ub(self.model.problem, constr1._index), 0)
# constr2
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr2._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 2.333, 'y': 1.})
self.assertEqual(glp_get_row_type(self.model.problem, constr2._index), GLP_UP)
self.assertEqual(glp_get_row_lb(self.model.problem, constr2._index), -1.7976931348623157e+308)
self.assertEqual(glp_get_row_ub(self.model.problem, constr2._index), 96.997)
# constr3
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr3._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 2.333, 'y': 1., 'z': 1.})
self.assertEqual(glp_get_row_type(self.model.problem, constr3._index), GLP_LO)
self.assertEqual(glp_get_row_lb(self.model.problem, constr3._index), -300)
self.assertEqual(glp_get_row_ub(self.model.problem, constr3._index), 1.7976931348623157e+308)
# constr4
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr4._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 1})
self.assertEqual(glp_get_row_type(self.model.problem, constr4._index), GLP_FX)
self.assertEqual(glp_get_row_lb(self.model.problem, constr4._index), -300)
self.assertEqual(glp_get_row_ub(self.model.problem, constr4._index), -300)
# constr5
ia = intArray(glp_get_num_rows(self.model.problem) + 1)
da = doubleArray(glp_get_num_rows(self.model.problem) + 1)
nnz = glp_get_mat_row(self.model.problem, constr5._index, ia, da)
coeff_dict = dict()
for i in range(1, nnz + 1):
coeff_dict[glp_get_col_name(self.model.problem, ia[i])] = da[i]
self.assertDictEqual(coeff_dict, {'x': 3})
self.assertEqual(glp_get_row_type(self.model.problem, constr5._index), GLP_FR)
self.assertLess(glp_get_row_lb(self.model.problem, constr5._index), -1e30)
self.assertGreater(glp_get_row_ub(self.model.problem, constr5._index), 1e30)
def __init__(self, problem=None, *args, **kwargs):
super(Model, self).__init__(*args, **kwargs)
self.configuration = Configuration()
if problem is None:
self.problem = glp_create_prob()
glp_create_index(self.problem)
if self.name is not None:
glp_set_prob_name(self.problem, str(self.name))
else:
try:
self.problem = problem
glp_create_index(self.problem)
except TypeError:
raise TypeError("Provided problem is not a valid GLPK model.")
row_num = glp_get_num_rows(self.problem)
col_num = glp_get_num_cols(self.problem)
for i in range(1, col_num + 1):
var = Variable(
glp_get_col_name(self.problem, i),
lb=glp_get_col_lb(self.problem, i),
ub=glp_get_col_ub(self.problem, i),
problem=self,
type=_GLPK_VTYPE_TO_VTYPE[
glp_get_col_kind(self.problem, i)]
)
# This avoids adding the variable to the glpk problem
super(Model, self)._add_variables([var])
variables = self.variables
for j in range(1, row_num + 1):
ia = intArray(col_num + 1)
da = doubleArray(col_num + 1)
nnz = glp_get_mat_row(self.problem, j, ia, da)
constraint_variables = [variables[ia[i] - 1] for i in range(1, nnz + 1)]
# Since constraint expressions are lazily retrieved from the solver they don't have to be built here
# lhs = _unevaluated_Add(*[da[i] * constraint_variables[i - 1]
# for i in range(1, nnz + 1)])
lhs = 0
glpk_row_type = glp_get_row_type(self.problem, j)
if glpk_row_type == GLP_FX:
row_lb = glp_get_row_lb(self.problem, j)
row_ub = row_lb
elif glpk_row_type == GLP_LO:
row_lb = glp_get_row_lb(self.problem, j)
row_ub = None
elif glpk_row_type == GLP_UP:
row_lb = None
row_ub = glp_get_row_ub(self.problem, j)
elif glpk_row_type == GLP_DB:
row_lb = glp_get_row_lb(self.problem, j)
row_ub = glp_get_row_ub(self.problem, j)
elif glpk_row_type == GLP_FR:
row_lb = None
row_ub = None
else:
raise Exception(
"Currently, optlang does not support glpk row type %s"
% str(glpk_row_type)
)
log.exception()
if isinstance(lhs, int):
lhs = sympy.Integer(lhs)
elif isinstance(lhs, float):
lhs = sympy.RealNumber(lhs)
constraint_id = glp_get_row_name(self.problem, j)
for variable in constraint_variables:
try:
self._variables_to_constraints_mapping[variable.name].add(constraint_id)
except KeyError:
self._variables_to_constraints_mapping[variable.name] = set([constraint_id])
super(Model, self)._add_constraints(
[Constraint(lhs, lb=row_lb, ub=row_ub, name=constraint_id, problem=self, sloppy=True)],
sloppy=True
)
term_generator = (
(glp_get_obj_coef(self.problem, index), variables[index - 1])
for index in range(1, glp_get_num_cols(problem) + 1)
)
self._objective = Objective(
_unevaluated_Add(
*[_unevaluated_Mul(sympy.RealNumber(term[0]), term[1]) for term in term_generator if
term[0] != 0.]),
problem=self,
direction={GLP_MIN: 'min', GLP_MAX: 'max'}[glp_get_obj_dir(self.problem)])
glp_scale_prob(self.problem, GLP_SF_AUTO)
def _constraints_str(self, bm_print, input_print, zero_print):
'get the constraints matrix lines for __str__'
rv = ""
lp = self.lp
rows = self.get_num_rows()
cols = self.get_num_cols()
stat_labels = ["?(0)?", "BS", "NL", "NU", "NF", "NS"]
inds = glpk.intArray(cols + 1)
vals = glpk.doubleArray(cols + 1)
for row in range(1, rows + 1):
stat = glpk.glp_get_row_stat(lp, row)
assert 0 <= stat <= len(stat_labels)
rv += "{:2}: {} ".format(row, stat_labels[stat])
num_inds = glpk.glp_get_mat_row(lp, row, inds, vals)
for col in range(1, cols + 1):
val = 0
for index in range(1, num_inds + 1):
if inds[index] == col:
val = vals[index]
break
num = str(val)
if len(num) < 6:
num = (" " * (6 - len(num))) + num
else:
num = num[0:6]
if self.basis_mat_rect[1] <= row - 1 < self.basis_mat_rect[0] + self.basis_mat_rect[3] and \
self.basis_mat_rect[0] <= col - 1 < self.basis_mat_rect[1] + self.basis_mat_rect[2]:
rv += bm_print(num) + " "
else:
rv += (zero_print(num) if val == 0 else num) + " "
row_type = glpk.glp_get_row_type(lp, row)
if row_type == glpk.GLP_FX:
val = glpk.glp_get_row_ub(lp, row)
rv += " == "
else:
assert row_type == glpk.GLP_UP
val = glpk.glp_get_row_ub(lp, row)
rv += " <= "
num = str(val)
if len(num) < 6:
num = (" " * (6 - len(num))) + num
else:
num = num[0:6]
rv += (zero_print(num) if val == 0 else num) + " "
rv += "\n"
return rv
def _constraints_str(self, bm_print, input_print, zero_print):
'get the constraints matrix lines for __str__'
rv = ""
lp = self.lp
rows = self.get_num_rows()
cols = self.get_num_cols()
stat_labels = ["?(0)?", "BS", "NL", "NU", "NF", "NS", "?(6)?"]
inds = glpk.intArray(cols + 1)
vals = glpk.doubleArray(cols + 1)
for row in range(1, rows + 1):
rv += "{:2}: {} ".format(row-1, stat_labels[glpk.glp_get_row_stat(lp, row)])
num_inds = glpk.glp_get_mat_row(lp, row, inds, vals)
for col in range(1, cols + 1):
val = 0
for index in range(1, num_inds+1):
if inds[index] == col:
val = vals[index]
break
num = str(val)
if len(num) < 6:
num = (" " * (6 - len(num))) + num
else:
num = num[0:6]
if self.basis_mat_pos[0] <= row - 1 < self.basis_mat_pos[0] + self.dims and \
self.basis_mat_pos[1] <= col - 1 < self.basis_mat_pos[1] + self.dims:
rv += bm_print(num) + " "
elif self.input_effects_offsets is not None and \
self.input_effects_offsets[0] <= row - 1 < self.input_effects_offsets[0] + self.dims and \
self.input_effects_offsets[1] <= col - 1 < self.input_effects_offsets[1] + self.dims:
rv += input_print(num) + " "
else:
rv += (zero_print(num) if val == 0 else num) + " "
row_type = glpk.glp_get_row_type(lp, row)
if row_type == glpk.GLP_FX:
val = glpk.glp_get_row_ub(lp, row)
rv += " == "
elif row_type == glpk.GLP_UP:
val = glpk.glp_get_row_ub(lp, row)
rv += " <= "
elif row_type == glpk.GLP_LO:
val = glpk.glp_get_row_lb(lp, row)
rv += " >= "
else:
rv += " <?> (unknown bounds)"
val = '?'
num = str(val)
if len(num) < 6:
num = (" " * (6 - len(num))) + num
else:
num = num[0:6]
rv += (zero_print(num) if val == 0 else num) + " "
rv += "\n"
return rv
