# Equation

User API to specify equations.

class devito.types.equation.Eq(lhs, rhs=0, subdomain=None, coefficients=None, implicit_dims=None, **kwargs)[source]

Bases: `Equality`, `Evaluable`

An equal relation between two objects, the left-hand side and the right-hand side.

The left-hand side may be a Function or a SparseFunction. The right-hand side may be any arbitrary expressions with numbers, Dimensions, Constants, Functions and SparseFunctions as operands.

Parameters:
• lhs (Function or SparseFunction) – The left-hand side.

• rhs (expr-like, optional) – The right-hand side. Defaults to 0.

• subdomain (SubDomain, optional) – To restrict the computation of the Eq to a particular sub-region in the computational domain.

• coefficients (Substitutions, optional) – Can be used to replace symbolic finite difference weights with user defined weights.

• implicit_dims (Dimension or list of Dimension, optional) – An ordered list of Dimensions that do not explicitly appear in either the left-hand side or in the right-hand side, but that should be honored when constructing an Operator.

Examples

```>>> from devito import Grid, Function, Eq
>>> grid = Grid(shape=(4, 4))
>>> f = Function(name='f', grid=grid)
>>> Eq(f, f + 1)
Eq(f(x, y), f(x, y) + 1)
```

Any SymPy expressions may be used in the right-hand side.

```>>> from devito import sin
>>> Eq(f, sin(f.dx)**2)
Eq(f(x, y), sin(Derivative(f(x, y), x))**2)
```

Notes

An Eq can be thought of as an assignment in an imperative programming language (e.g., `a[i] = b[i]*c`).

func(*args, **kwargs)

Reconstruct self via self.__class__(*args, **kwargs) using self’s __rargs__ and __rkwargs__ if and where *args and **kwargs lack entries.

Examples

Given

class Foo(object):

__rargs__ = (‘a’, ‘b’) __rkwargs__ = (‘c’,) def __init__(self, a, b, c=4):

self.a = a self.b = b self.c = c

a = foo(3, 5)`

Then:

• a._rebuild() -> x(3, 5, 4) (i.e., copy of a).

• a._rebuild(4) -> x(4, 5, 4)

• a._rebuild(4, 7) -> x(4, 7, 4)

• a._rebuild(c=5) -> x(3, 5, 5)

• a._rebuild(1, c=7) -> x(1, 5, 7)

property subdomain

The SubDomain in which the Eq is defined.

xreplace(rules)[source]

Replace occurrences of objects within the expression.

Parameters:

rule (dict-like) – Expresses a replacement rule

Returns:

xreplace

Return type:

the result of the replacement

Examples

```>>> from sympy import symbols, pi, exp
>>> x, y, z = symbols('x y z')
>>> (1 + x*y).xreplace({x: pi})
pi*y + 1
>>> (1 + x*y).xreplace({x: pi, y: 2})
1 + 2*pi
```

Replacements occur only if an entire node in the expression tree is matched:

```>>> (x*y + z).xreplace({x*y: pi})
z + pi
>>> (x*y*z).xreplace({x*y: pi})
x*y*z
>>> (2*x).xreplace({2*x: y, x: z})
y
>>> (2*2*x).xreplace({2*x: y, x: z})
4*z
>>> (x + y + 2).xreplace({x + y: 2})
x + y + 2
>>> (x + 2 + exp(x + 2)).xreplace({x + 2: y})
x + exp(y) + 2
```

xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does:

```>>> from sympy import Integral
>>> Integral(x, (x, 1, 2*x)).xreplace({x: y})
Integral(y, (y, 1, 2*y))
```

Trying to replace x with an expression raises an error:

```>>> Integral(x, (x, 1, 2*x)).xreplace({x: 2*y})
ValueError: Invalid limits given: ((2*y, 1, 4*y),)
```

`replace`

replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements

`subs`

substitution of subexpressions as defined by the objects themselves.

class devito.types.equation.Inc(lhs, rhs=0, subdomain=None, coefficients=None, implicit_dims=None, **kwargs)[source]

Bases: `Reduction`

An increment Reduction.

Examples

Inc may be used to express tensor contractions. Below, a summation along the user-defined Dimension i.

```>>> from devito import Grid, Dimension, Function, Inc
>>> grid = Grid(shape=(4, 4))
>>> x, y = grid.dimensions
>>> i = Dimension(name='i')
>>> f = Function(name='f', grid=grid)
>>> g = Function(name='g', shape=(10, 4, 4), dimensions=(i, x, y))
>>> Inc(f, g)
Inc(f(x, y), g(i, x, y))
```

Notes

An Inc can be thought of as the augmented assignment ‘+=’ in an imperative programming language (e.g., `a[i] += c`).

class devito.types.equation.ReduceMax(lhs, rhs=0, subdomain=None, coefficients=None, implicit_dims=None, **kwargs)[source]

Bases: `Reduction`

class devito.types.equation.ReduceMin(lhs, rhs=0, subdomain=None, coefficients=None, implicit_dims=None, **kwargs)[source]

Bases: `Reduction`