Refactor is_linear_expression() in expr/expr.py

sympde/expr/expr.py

@@ -127,7 +127,7 @@ def __new__(cls, arguments, expr, **options):
 
         args = _sanitize_arguments(arguments, is_linear=True)
 
-        if not is_linear_expression(expr, args, integral=False):
+        if not is_linear_expression(expr, args):
             msg = '> Expression is not linear'
             raise UnconsistentLinearExpressionError(msg)
 
@@ -739,71 +739,86 @@ def linearize(form, fields, trials=None):
     return BilinearForm((trials, tests), bilinear_expr)
 
 #==============================================================================
-def is_linear_expression(expr, args, integral=True, debug=True):
-    """checks if an expression is linear with respect to the given arguments."""
-    # ...
-    left_args  = []
-    right_args = []
+def is_linear_expression(expr, args, debug=True):
+    """
+    Checks if expression is linear with respect to ``args``:
+
+        1. Additivity:   f(x + y) = f(x) + f(y)
+        2. Homogeneity:  f(alpha * x) = alpha * f(x)
+
+    Parameters
+    ----------
+    expr
+        Symbolic expression to test.
+    args : iterable
+        Each argument must be ScalarFunction or VectorFunction.
+    debug : bool, optional
+        Print diagnostic info if a check fails.
+
+    Returns
+    -------
+    bool
+        True if the expression is linear with respect to all given arguments, False otherwise.
+    """
+
+    x_args = []
+    y_args = []
 
+    # create 2 independent copies (x and y) of every original argument
     for arg in args:
-        tag    = random_string( 4 )
+        tag = random_string(4)
 
         if isinstance(arg, ScalarFunction):
-            left  = ScalarFunction(arg.space, name='l_' + tag)
-            right = ScalarFunction(arg.space, name='r_' + tag)
+            x  = ScalarFunction(arg.space, name='x_' + tag)
+            y = ScalarFunction(arg.space, name='y_' + tag)
 
         elif isinstance(arg, VectorFunction):
-            left  = VectorFunction(arg.space, name='l_' + tag)
-            right = VectorFunction(arg.space, name='r_' + tag)
+            x  = VectorFunction(arg.space, name='x_' + tag)
+            y = VectorFunction(arg.space, name='y_' + tag)
         else:
             raise TypeError('argument must be a {Scalar|Vector}Function')
 
-        left_args  += [left]
-        right_args += [right]
-    # ...
-
-    # ... check addition
-    newargs = [left + right for left, right in zip(left_args, right_args)]
-
-    newexpr    = expr.subs(zip(args, newargs))
-    left_expr  = expr.subs(zip(args, left_args))
-    right_expr = expr.subs(zip(args, right_args))
-
-    a = newexpr
-    b = left_expr + right_expr
-
-    if not( (a-b).expand() == 0 or a.expand() == b.expand()):
-        # TODO use a warning or exception?
-        if debug:
-            print('Failed to assert addition property')
-            print('{} != {}'.format(a.expand(), b.expand()))
-        return False
-
-    # ...
-
-    # ... check multiplication
-    tag   = random_string( 4 )
-    coeff = Constant('alpha_' + tag)
-
-    newexpr = expr
-    for arg, left in zip(args, left_args):
-        newarg  = coeff * left
-        newexpr = newexpr.subs(arg, newarg)
-
-    atoms     = list(newexpr.atoms(BasicOperator))
+        x_args.append(x)
+        y_args.append(y)
+
+    # ---------------------------------------------------------------------------
+    # check addition property: f(x + y) = f(x) + f(y)
+    summed_args = [x + y for x, y in zip(x_args, y_args)]
+    expr_at_x  = expr.subs(zip(args, x_args)) # f(x)
+    expr_at_y = expr.subs(zip(args, y_args)) # f(y)
+    expr_at_sum    = expr.subs(zip(args, summed_args)) # f(x + y)
+    expected_sum = expr_at_x + expr_at_y # = f(x) + f (y)
+
+    if (expr_at_sum - expected_sum).expand() != 0:
+        expr1 = expr_at_sum.expand()
+        expr2 = expected_sum.expand()
+        if expr1 != expr2:
+            if debug:
+                print('Failed to assert addition property')
+                print('{} != {}'.format(expr1, expr2))
+            return False
+
+    # ---------------------------------------------------------------------------
+    # check multiplication property: f(alpha * x) = alpha * f(x)
+    alpha = Constant(f"alpha_{random_string(4)}")
+
+    scaled_x_args = [alpha * x for x in x_args]
+    expr_at_scaled_x = expr.subs(zip(args, scaled_x_args))
+
+    atoms     = list(expr_at_scaled_x.atoms(BasicOperator))
     subs      = [e.func(*e.args, evaluate=True) for e in atoms]
-    newexpr   = newexpr.subs(zip(atoms, subs))
-
-
-    left_expr = expr.subs(list(zip(args, left_args)))
-    left_expr = coeff * left_expr
-    if not( (newexpr-left_expr).expand() == 0 or newexpr.expand()==left_expr.expand()):
-        # TODO use a warning or exception?
-        if debug:
-            print('Failed to assert multiplication property')
-            print('{} != {}'.format(newexpr, left_expr))
-        return False
-    # ...
+    expr_at_scaled_x   = expr_at_scaled_x.subs(zip(atoms, subs))
+
+    scaled_expr = alpha * expr_at_x
+
+    if (expr_at_scaled_x - scaled_expr).expand() != 0:
+        expr1 = expr_at_scaled_x.expand()
+        expr2 = scaled_expr.expand()
+        if expr1 != expr2:
+            if debug:
+                print('Failed to assert multiplication property')
+                print('{} != {}'.format(expr1, expr2))
+            return False
 
     return True