Merge pull request #19805 from pearu/gh-8062-dimdecs

ENH: Symbolic solver for dimension specifications.

2 files changed, 561 insertions, 0 deletions

diff --git a/numpy/f2py/tests/test_crackfortran.py b/numpy/f2py/tests/test_crackfortran.py
index 140f42cbc..da7974d1a 100644
--- a/numpy/f2py/tests/test_crackfortran.py
+++ b/numpy/f2py/tests/test_crackfortran.py
@@ -1,3 +1,4 @@
+import pytest
 import numpy as np
 from numpy.testing import assert_array_equal, assert_equal
 from numpy.f2py.crackfortran import markinnerspaces
@@ -39,6 +40,7 @@ class TestNoSpace(util.F2PyTest):
 
 
 class TestPublicPrivate():
+
     def test_defaultPrivate(self, tmp_path):
         f_path = tmp_path / "mod.f90"
         with f_path.open('w') as ff:
@@ -165,3 +167,100 @@ class TestMarkinnerspaces():
     def test_multiple_relevant_spaces(self):
         assert_equal(markinnerspaces("a 'b c' 'd e'"), "a 'b@_@c' 'd@_@e'")
         assert_equal(markinnerspaces(r'a "b c" "d e"'), r'a "b@_@c" "d@_@e"')
+
+
+class TestDimSpec(util.F2PyTest):
+    """This test site tests various expressions that are used as dimension
+    specifications.
+
+    There exists two usage cases where analyzing dimensions
+    specifications are important.
+
+    In the first case, the size of output arrays must be defined based
+    on the inputs to a Fortran function. Because Fortran supports
+    arbitrary bases for indexing, for instance, `arr(lower:upper)`,
+    f2py has to evaluate an expression `upper - lower + 1` where
+    `lower` and `upper` are arbitrary expressions of input parameters.
+    The evaluation is performed in C, so f2py has to translate Fortran
+    expressions to valid C expressions (an alternative approach is
+    that a developer specifies the corresponing C expressions in a
+    .pyf file).
+
+    In the second case, when user provides an input array with a given
+    size but some hidden parameters used in dimensions specifications
+    need to be determined based on the input array size. This is a
+    harder problem because f2py has to solve the inverse problem: find
+    a parameter `p` such that `upper(p) - lower(p) + 1` equals to the
+    size of input array. In the case when this equation cannot be
+    solved (e.g. because the input array size is wrong), raise an
+    error before calling the Fortran function (that otherwise would
+    likely crash Python process when the size of input arrays is
+    wrong). f2py currently supports this case only when the equation
+    is linear with respect to unknown parameter.
+
+    """
+
+    suffix = '.f90'
+
+    code_template = textwrap.dedent("""
+      function get_arr_size_{count}(a, n) result (length)
+        integer, intent(in) :: n
+        integer, dimension({dimspec}), intent(out) :: a
+        integer length
+        length = size(a)
+      end function
+
+      subroutine get_inv_arr_size_{count}(a, n)
+        integer :: n
+        ! the value of n is computed in f2py wrapper
+        !f2py intent(out) n
+        integer, dimension({dimspec}), intent(in) :: a
+        if (a({first}).gt.0) then
+          print*, "a=", a
+        endif
+      end subroutine
+    """)
+
+    linear_dimspecs = ['n', '2*n', '2:n', 'n/2', '5 - n/2', '3*n:20',
+                       'n*(n+1):n*(n+5)']
+    nonlinear_dimspecs = ['2*n:3*n*n+2*n']
+    all_dimspecs = linear_dimspecs + nonlinear_dimspecs
+
+    code = ''
+    for count, dimspec in enumerate(all_dimspecs):
+        code += code_template.format(
+            count=count, dimspec=dimspec,
+            first=dimspec.split(':')[0] if ':' in dimspec else '1')
+
+    @pytest.mark.parametrize('dimspec', all_dimspecs)
+    def test_array_size(self, dimspec):
+
+        count = self.all_dimspecs.index(dimspec)
+        get_arr_size = getattr(self.module, f'get_arr_size_{count}')
+
+        for n in [1, 2, 3, 4, 5]:
+            sz, a = get_arr_size(n)
+            assert len(a) == sz
+
+    @pytest.mark.parametrize('dimspec', all_dimspecs)
+    def test_inv_array_size(self, dimspec):
+
+        count = self.all_dimspecs.index(dimspec)
+        get_arr_size = getattr(self.module, f'get_arr_size_{count}')
+        get_inv_arr_size = getattr(self.module, f'get_inv_arr_size_{count}')
+
+        for n in [1, 2, 3, 4, 5]:
+            sz, a = get_arr_size(n)
+            if dimspec in self.nonlinear_dimspecs:
+                # one must specify n as input, the call we'll ensure
+                # that a and n are compatible:
+                n1 = get_inv_arr_size(a, n)
+            else:
+                # in case of linear dependence, n can be determined
+                # from the shape of a:
+                n1 = get_inv_arr_size(a)
+            # n1 may be different from n (for instance, when `a` size
+            # is a function of some `n` fraction) but it must produce
+            # the same sized array
+            sz1, _ = get_arr_size(n1)
+            assert sz == sz1, (n, n1, sz, sz1)
diff --git a/numpy/f2py/tests/test_symbolic.py b/numpy/f2py/tests/test_symbolic.py
new file mode 100644
index 000000000..52cabac53
--- /dev/null
+++ b/numpy/f2py/tests/test_symbolic.py
@@ -0,0 +1,462 @@
+from numpy.testing import assert_raises
+from numpy.f2py.symbolic import (
+    Expr, Op, ArithOp, Language,
+    as_symbol, as_number, as_string, as_array, as_complex,
+    as_terms, as_factors, eliminate_quotes, insert_quotes,
+    fromstring, as_expr, as_apply,
+    as_numer_denom, as_ternary, as_ref, as_deref,
+    normalize, as_eq, as_ne, as_lt, as_gt, as_le, as_ge
+    )
+from . import util
+
+
+class TestSymbolic(util.F2PyTest):
+
+    def test_eliminate_quotes(self):
+        def worker(s):
+            r, d = eliminate_quotes(s)
+            s1 = insert_quotes(r, d)
+            assert s1 == s
+
+        for kind in ['', 'mykind_']:
+            worker(kind + '"1234" // "ABCD"')
+            worker(kind + '"1234" // ' + kind + '"ABCD"')
+            worker(kind + '"1234" // \'ABCD\'')
+            worker(kind + '"1234" // ' + kind + '\'ABCD\'')
+            worker(kind + '"1\\"2\'AB\'34"')
+            worker('a = ' + kind + "'1\\'2\"AB\"34'")
+
+    def test_sanity(self):
+        x = as_symbol('x')
+        y = as_symbol('y')
+        z = as_symbol('z')
+
+        assert x.op == Op.SYMBOL
+        assert repr(x) == "Expr(Op.SYMBOL, 'x')"
+        assert x == x
+        assert x != y
+        assert hash(x) is not None
+
+        n = as_number(123)
+        m = as_number(456)
+        assert n.op == Op.INTEGER
+        assert repr(n) == "Expr(Op.INTEGER, (123, 4))"
+        assert n == n
+        assert n != m
+        assert hash(n) is not None
+
+        fn = as_number(12.3)
+        fm = as_number(45.6)
+        assert fn.op == Op.REAL
+        assert repr(fn) == "Expr(Op.REAL, (12.3, 4))"
+        assert fn == fn
+        assert fn != fm
+        assert hash(fn) is not None
+
+        c = as_complex(1, 2)
+        c2 = as_complex(3, 4)
+        assert c.op == Op.COMPLEX
+        assert repr(c) == ("Expr(Op.COMPLEX, (Expr(Op.INTEGER, (1, 4)),"
+                           " Expr(Op.INTEGER, (2, 4))))")
+        assert c == c
+        assert c != c2
+        assert hash(c) is not None
+
+        s = as_string("'123'")
+        s2 = as_string('"ABC"')
+        assert s.op == Op.STRING
+        assert repr(s) == "Expr(Op.STRING, (\"'123'\", 1))", repr(s)
+        assert s == s
+        assert s != s2
+
+        a = as_array((n, m))
+        b = as_array((n,))
+        assert a.op == Op.ARRAY
+        assert repr(a) == ("Expr(Op.ARRAY, (Expr(Op.INTEGER, (123, 4)),"
+                           " Expr(Op.INTEGER, (456, 4))))")
+        assert a == a
+        assert a != b
+
+        t = as_terms(x)
+        u = as_terms(y)
+        assert t.op == Op.TERMS
+        assert repr(t) == "Expr(Op.TERMS, {Expr(Op.SYMBOL, 'x'): 1})"
+        assert t == t
+        assert t != u
+        assert hash(t) is not None
+
+        v = as_factors(x)
+        w = as_factors(y)
+        assert v.op == Op.FACTORS
+        assert repr(v) == "Expr(Op.FACTORS, {Expr(Op.SYMBOL, 'x'): 1})"
+        assert v == v
+        assert w != v
+        assert hash(v) is not None
+
+        t = as_ternary(x, y, z)
+        u = as_ternary(x, z, y)
+        assert t.op == Op.TERNARY
+        assert t == t
+        assert t != u
+        assert hash(t) is not None
+
+        e = as_eq(x, y)
+        f = as_lt(x, y)
+        assert e.op == Op.RELATIONAL
+        assert e == e
+        assert e != f
+        assert hash(e) is not None
+
+    def test_tostring_fortran(self):
+        x = as_symbol('x')
+        y = as_symbol('y')
+        z = as_symbol('z')
+        n = as_number(123)
+        m = as_number(456)
+        a = as_array((n, m))
+        c = as_complex(n, m)
+
+        assert str(x) == 'x'
+        assert str(n) == '123'
+        assert str(a) == '[123, 456]'
+        assert str(c) == '(123, 456)'
+
+        assert str(Expr(Op.TERMS, {x: 1})) == 'x'
+        assert str(Expr(Op.TERMS, {x: 2})) == '2 * x'
+        assert str(Expr(Op.TERMS, {x: -1})) == '-x'
+        assert str(Expr(Op.TERMS, {x: -2})) == '-2 * x'
+        assert str(Expr(Op.TERMS, {x: 1, y: 1})) == 'x + y'
+        assert str(Expr(Op.TERMS, {x: -1, y: -1})) == '-x - y'
+        assert str(Expr(Op.TERMS, {x: 2, y: 3})) == '2 * x + 3 * y'
+        assert str(Expr(Op.TERMS, {x: -2, y: 3})) == '-2 * x + 3 * y'
+        assert str(Expr(Op.TERMS, {x: 2, y: -3})) == '2 * x - 3 * y'
+
+        assert str(Expr(Op.FACTORS, {x: 1})) == 'x'
+        assert str(Expr(Op.FACTORS, {x: 2})) == 'x ** 2'
+        assert str(Expr(Op.FACTORS, {x: -1})) == 'x ** -1'
+        assert str(Expr(Op.FACTORS, {x: -2})) == 'x ** -2'
+        assert str(Expr(Op.FACTORS, {x: 1, y: 1})) == 'x * y'
+        assert str(Expr(Op.FACTORS, {x: 2, y: 3})) == 'x ** 2 * y ** 3'
+
+        v = Expr(Op.FACTORS, {x: 2, Expr(Op.TERMS, {x: 1, y: 1}): 3})
+        assert str(v) == 'x ** 2 * (x + y) ** 3', str(v)
+        v = Expr(Op.FACTORS, {x: 2, Expr(Op.FACTORS, {x: 1, y: 1}): 3})
+        assert str(v) == 'x ** 2 * (x * y) ** 3', str(v)
+
+        assert str(Expr(Op.APPLY, ('f', (), {}))) == 'f()'
+        assert str(Expr(Op.APPLY, ('f', (x,), {}))) == 'f(x)'
+        assert str(Expr(Op.APPLY, ('f', (x, y), {}))) == 'f(x, y)'
+        assert str(Expr(Op.INDEXING, ('f', x))) == 'f[x]'
+
+        assert str(as_ternary(x, y, z)) == 'merge(y, z, x)'
+        assert str(as_eq(x, y)) == 'x .eq. y'
+        assert str(as_ne(x, y)) == 'x .ne. y'
+        assert str(as_lt(x, y)) == 'x .lt. y'
+        assert str(as_le(x, y)) == 'x .le. y'
+        assert str(as_gt(x, y)) == 'x .gt. y'
+        assert str(as_ge(x, y)) == 'x .ge. y'
+
+    def test_tostring_c(self):
+        language = Language.C
+        x = as_symbol('x')
+        y = as_symbol('y')
+        z = as_symbol('z')
+        n = as_number(123)
+
+        assert Expr(Op.FACTORS, {x: 2}).tostring(language=language) == 'x * x'
+        assert Expr(Op.FACTORS, {x + y: 2}).tostring(
+            language=language) == '(x + y) * (x + y)'
+        assert Expr(Op.FACTORS, {x: 12}).tostring(
+            language=language) == 'pow(x, 12)'
+
+        assert as_apply(ArithOp.DIV, x, y).tostring(
+            language=language) == 'x / y'
+        assert as_apply(ArithOp.DIV, x, x + y).tostring(
+            language=language) == 'x / (x + y)'
+        assert as_apply(ArithOp.DIV, x - y, x + y).tostring(
+            language=language) == '(x - y) / (x + y)'
+        assert (x + (x - y) / (x + y) + n).tostring(
+            language=language) == '123 + x + (x - y) / (x + y)'
+
+        assert as_ternary(x, y, z).tostring(language=language) == '(x ? y : z)'
+        assert as_eq(x, y).tostring(language=language) == 'x == y'
+        assert as_ne(x, y).tostring(language=language) == 'x != y'
+        assert as_lt(x, y).tostring(language=language) == 'x < y'
+        assert as_le(x, y).tostring(language=language) == 'x <= y'
+        assert as_gt(x, y).tostring(language=language) == 'x > y'
+        assert as_ge(x, y).tostring(language=language) == 'x >= y'
+
+    def test_operations(self):
+        x = as_symbol('x')
+        y = as_symbol('y')
+        z = as_symbol('z')
+
+        assert x + x == Expr(Op.TERMS, {x: 2})
+        assert x - x == Expr(Op.INTEGER, (0, 4))
+        assert x + y == Expr(Op.TERMS, {x: 1, y: 1})
+        assert x - y == Expr(Op.TERMS, {x: 1, y: -1})
+        assert x * x == Expr(Op.FACTORS, {x: 2})
+        assert x * y == Expr(Op.FACTORS, {x: 1, y: 1})
+
+        assert +x == x
+        assert -x == Expr(Op.TERMS, {x: -1}), repr(-x)
+        assert 2 * x == Expr(Op.TERMS, {x: 2})
+        assert 2 + x == Expr(Op.TERMS, {x: 1, as_number(1): 2})
+        assert 2 * x + 3 * y == Expr(Op.TERMS, {x: 2, y: 3})
+        assert (x + y) * 2 == Expr(Op.TERMS, {x: 2, y: 2})
+
+        assert x ** 2 == Expr(Op.FACTORS, {x: 2})
+        assert (x + y) ** 2 == Expr(Op.TERMS,
+                                    {Expr(Op.FACTORS, {x: 2}): 1,
+                                     Expr(Op.FACTORS, {y: 2}): 1,
+                                     Expr(Op.FACTORS, {x: 1, y: 1}): 2})
+        assert (x + y) * x == x ** 2 + x * y
+        assert (x + y) ** 2 == x ** 2 + 2 * x * y + y ** 2
+        assert (x + y) ** 2 + (x - y) ** 2 == 2 * x ** 2 + 2 * y ** 2
+        assert (x + y) * z == x * z + y * z
+        assert z * (x + y) == x * z + y * z
+
+        assert (x / 2) == as_apply(ArithOp.DIV, x, as_number(2))
+        assert (2 * x / 2) == x
+        assert (3 * x / 2) == as_apply(ArithOp.DIV, 3*x, as_number(2))
+        assert (4 * x / 2) == 2 * x
+        assert (5 * x / 2) == as_apply(ArithOp.DIV, 5*x, as_number(2))
+        assert (6 * x / 2) == 3 * x
+        assert ((3*5) * x / 6) == as_apply(ArithOp.DIV, 5*x, as_number(2))
+        assert (30*x**2*y**4 / (24*x**3*y**3)) == as_apply(ArithOp.DIV,
+                                                           5*y, 4*x)
+        assert ((15 * x / 6) / 5) == as_apply(
+            ArithOp.DIV, x, as_number(2)), ((15 * x / 6) / 5)
+        assert (x / (5 / x)) == as_apply(ArithOp.DIV, x**2, as_number(5))
+
+        assert (x / 2.0) == Expr(Op.TERMS, {x: 0.5})
+
+        s = as_string('"ABC"')
+        t = as_string('"123"')
+
+        assert s // t == Expr(Op.STRING, ('"ABC123"', 1))
+        assert s // x == Expr(Op.CONCAT, (s, x))
+        assert x // s == Expr(Op.CONCAT, (x, s))
+
+        c = as_complex(1., 2.)
+        assert -c == as_complex(-1., -2.)
+        assert c + c == as_expr((1+2j)*2)
+        assert c * c == as_expr((1+2j)**2)
+
+    def test_substitute(self):
+        x = as_symbol('x')
+        y = as_symbol('y')
+        z = as_symbol('z')
+        a = as_array((x, y))
+
+        assert x.substitute({x: y}) == y
+        assert (x + y).substitute({x: z}) == y + z
+        assert (x * y).substitute({x: z}) == y * z
+        assert (x ** 4).substitute({x: z}) == z ** 4
+        assert (x / y).substitute({x: z}) == z / y
+        assert x.substitute({x: y + z}) == y + z
+        assert a.substitute({x: y + z}) == as_array((y + z, y))
+
+        assert as_ternary(x, y, z).substitute(
+            {x: y + z}) == as_ternary(y + z, y, z)
+        assert as_eq(x, y).substitute(
+            {x: y + z}) == as_eq(y + z, y)
+
+    def test_fromstring(self):
+
+        x = as_symbol('x')
+        y = as_symbol('y')
+        z = as_symbol('z')
+        f = as_symbol('f')
+        s = as_string('"ABC"')
+        t = as_string('"123"')
+        a = as_array((x, y))
+
+        assert fromstring('x') == x
+        assert fromstring('+ x') == x
+        assert fromstring('-  x') == -x
+        assert fromstring('x + y') == x + y
+        assert fromstring('x + 1') == x + 1
+        assert fromstring('x * y') == x * y
+        assert fromstring('x * 2') == x * 2
+        assert fromstring('x / y') == x / y
+        assert fromstring('x ** 2',
+                          language=Language.Python) == x ** 2
+        assert fromstring('x ** 2 ** 3',
+                          language=Language.Python) == x ** 2 ** 3
+        assert fromstring('(x + y) * z') == (x + y) * z
+
+        assert fromstring('f(x)') == f(x)
+        assert fromstring('f(x,y)') == f(x, y)
+        assert fromstring('f[x]') == f[x]
+        assert fromstring('f[x][y]') == f[x][y]
+
+        assert fromstring('"ABC"') == s
+        assert normalize(fromstring('"ABC" // "123" ',
+                                    language=Language.Fortran)) == s // t
+        assert fromstring('f("ABC")') == f(s)
+        assert fromstring('MYSTRKIND_"ABC"') == as_string('"ABC"', 'MYSTRKIND')
+
+        assert fromstring('(/x, y/)') == a, fromstring('(/x, y/)')
+        assert fromstring('f((/x, y/))') == f(a)
+        assert fromstring('(/(x+y)*z/)') == as_array(((x+y)*z,))
+
+        assert fromstring('123') == as_number(123)
+        assert fromstring('123_2') == as_number(123, 2)
+        assert fromstring('123_myintkind') == as_number(123, 'myintkind')
+
+        assert fromstring('123.0') == as_number(123.0, 4)
+        assert fromstring('123.0_4') == as_number(123.0, 4)
+        assert fromstring('123.0_8') == as_number(123.0, 8)
+        assert fromstring('123.0e0') == as_number(123.0, 4)
+        assert fromstring('123.0d0') == as_number(123.0, 8)
+        assert fromstring('123d0') == as_number(123.0, 8)
+        assert fromstring('123e-0') == as_number(123.0, 4)
+        assert fromstring('123d+0') == as_number(123.0, 8)
+        assert fromstring('123.0_myrealkind') == as_number(123.0, 'myrealkind')
+        assert fromstring('3E4') == as_number(30000.0, 4)
+
+        assert fromstring('(1, 2)') == as_complex(1, 2)
+        assert fromstring('(1e2, PI)') == as_complex(
+            as_number(100.0), as_symbol('PI'))
+
+        assert fromstring('[1, 2]') == as_array((as_number(1), as_number(2)))
+
+        assert fromstring('POINT(x, y=1)') == as_apply(
+            as_symbol('POINT'), x, y=as_number(1))
+        assert (fromstring('PERSON(name="John", age=50, shape=(/34, 23/))')
+                == as_apply(as_symbol('PERSON'),
+                            name=as_string('"John"'),
+                            age=as_number(50),
+                            shape=as_array((as_number(34), as_number(23)))))
+
+        assert fromstring('x?y:z') == as_ternary(x, y, z)
+
+        assert fromstring('*x') == as_deref(x)
+        assert fromstring('**x') == as_deref(as_deref(x))
+        assert fromstring('&x') == as_ref(x)
+        assert fromstring('(*x) * (*y)') == as_deref(x) * as_deref(y)
+        assert fromstring('(*x) * *y') == as_deref(x) * as_deref(y)
+        assert fromstring('*x * *y') == as_deref(x) * as_deref(y)
+        assert fromstring('*x**y') == as_deref(x) * as_deref(y)
+
+        assert fromstring('x == y') == as_eq(x, y)
+        assert fromstring('x != y') == as_ne(x, y)
+        assert fromstring('x < y') == as_lt(x, y)
+        assert fromstring('x > y') == as_gt(x, y)
+        assert fromstring('x <= y') == as_le(x, y)
+        assert fromstring('x >= y') == as_ge(x, y)
+
+        assert fromstring('x .eq. y', language=Language.Fortran) == as_eq(x, y)
+        assert fromstring('x .ne. y', language=Language.Fortran) == as_ne(x, y)
+        assert fromstring('x .lt. y', language=Language.Fortran) == as_lt(x, y)
+        assert fromstring('x .gt. y', language=Language.Fortran) == as_gt(x, y)
+        assert fromstring('x .le. y', language=Language.Fortran) == as_le(x, y)
+        assert fromstring('x .ge. y', language=Language.Fortran) == as_ge(x, y)
+
+    def test_traverse(self):
+        x = as_symbol('x')
+        y = as_symbol('y')
+        z = as_symbol('z')
+        f = as_symbol('f')
+
+        # Use traverse to substitute a symbol
+        def replace_visit(s, r=z):
+            if s == x:
+                return r
+
+        assert x.traverse(replace_visit) == z
+        assert y.traverse(replace_visit) == y
+        assert z.traverse(replace_visit) == z
+        assert (f(y)).traverse(replace_visit) == f(y)
+        assert (f(x)).traverse(replace_visit) == f(z)
+        assert (f[y]).traverse(replace_visit) == f[y]
+        assert (f[z]).traverse(replace_visit) == f[z]
+        assert (x + y + z).traverse(replace_visit) == (2 * z + y)
+        assert (x + f(y, x - z)).traverse(
+            replace_visit) == (z + f(y, as_number(0)))
+        assert as_eq(x, y).traverse(replace_visit) == as_eq(z, y)
+
+        # Use traverse to collect symbols, method 1
+        function_symbols = set()
+        symbols = set()
+
+        def collect_symbols(s):
+            if s.op is Op.APPLY:
+                oper = s.data[0]
+                function_symbols.add(oper)
+                if oper in symbols:
+                    symbols.remove(oper)
+            elif s.op is Op.SYMBOL and s not in function_symbols:
+                symbols.add(s)
+
+        (x + f(y, x - z)).traverse(collect_symbols)
+        assert function_symbols == {f}
+        assert symbols == {x, y, z}
+
+        # Use traverse to collect symbols, method 2
+        def collect_symbols2(expr, symbols):
+            if expr.op is Op.SYMBOL:
+                symbols.add(expr)
+
+        symbols = set()
+        (x + f(y, x - z)).traverse(collect_symbols2, symbols)
+        assert symbols == {x, y, z, f}
+
+        # Use traverse to partially collect symbols
+        def collect_symbols3(expr, symbols):
+            if expr.op is Op.APPLY:
+                # skip traversing function calls
+                return expr
+            if expr.op is Op.SYMBOL:
+                symbols.add(expr)
+
+        symbols = set()
+        (x + f(y, x - z)).traverse(collect_symbols3, symbols)
+        assert symbols == {x}
+
+    def test_linear_solve(self):
+        x = as_symbol('x')
+        y = as_symbol('y')
+        z = as_symbol('z')
+
+        assert x.linear_solve(x) == (as_number(1), as_number(0))
+        assert (x+1).linear_solve(x) == (as_number(1), as_number(1))
+        assert (2*x).linear_solve(x) == (as_number(2), as_number(0))
+        assert (2*x+3).linear_solve(x) == (as_number(2), as_number(3))
+        assert as_number(3).linear_solve(x) == (as_number(0), as_number(3))
+        assert y.linear_solve(x) == (as_number(0), y)
+        assert (y*z).linear_solve(x) == (as_number(0), y * z)
+
+        assert (x+y).linear_solve(x) == (as_number(1), y)
+        assert (z*x+y).linear_solve(x) == (z, y)
+        assert ((z+y)*x+y).linear_solve(x) == (z + y, y)
+        assert (z*y*x+y).linear_solve(x) == (z * y, y)
+
+        assert_raises(RuntimeError, lambda: (x*x).linear_solve(x))
+
+    def test_as_numer_denom(self):
+        x = as_symbol('x')
+        y = as_symbol('y')
+        n = as_number(123)
+
+        assert as_numer_denom(x) == (x, as_number(1))
+        assert as_numer_denom(x / n) == (x, n)
+        assert as_numer_denom(n / x) == (n, x)
+        assert as_numer_denom(x / y) == (x, y)
+        assert as_numer_denom(x * y) == (x * y, as_number(1))
+        assert as_numer_denom(n + x / y) == (x + n * y, y)
+        assert as_numer_denom(n + x / (y - x / n)) == (y * n ** 2, y * n - x)
+
+    def test_polynomial_atoms(self):
+        x = as_symbol('x')
+        y = as_symbol('y')
+        n = as_number(123)
+
+        assert x.polynomial_atoms() == {x}
+        assert n.polynomial_atoms() == set()
+        assert (y[x]).polynomial_atoms() == {y[x]}
+        assert (y(x)).polynomial_atoms() == {y(x)}
+        assert (y(x) + x).polynomial_atoms() == {y(x), x}
+        assert (y(x) * x[y]).polynomial_atoms() == {y(x), x[y]}
+        assert (y(x) ** x).polynomial_atoms() == {y(x)}