esys.escript.symbolic Package — esys.escript 5.6 documentation

# esys.escript.symbolic Package

## Classes

- `Evaluator`
- `Symbol`

class `esys.escript.symbolic.` `Evaluator`(*expressions)

  Bases: `object`

  `__init__`(*expressions)

    Returns a symbolic evaluator.

    Parameters: | expressions – optional expressions to initialise with

  `addExpression`(expression)

    Adds an expression to this evaluator.

    Returns: | the modified Evaluator object

  `evaluate`(evalf=False, **args)

    Evaluates all expressions in this evaluator and returns the result as a tuple.

    Returns: | the evaluated expressions in the order they were added to this Evaluator.

  `subs`(**args)

    Symbol substitution.

    Returns: | the modified Evaluator object

class `esys.escript.symbolic.` `Symbol`(*args, **kwargs)

  Bases: `object`

  `Symbol` objects are placeholders for a single mathematical symbol, such as ‘x’, or for arbitrarily complex mathematical expressions such as ‘c*x**4+alpha*exp(x)-2*sin(beta*x)’, where ‘alpha’, ‘beta’, ‘c’, and ‘x’ are also Symbols (the symbolic ‘atoms’ of the expression).

  With the help of the ‘Evaluator’ class these symbols and expressions can be resolved by substituting numeric values and/or escript `Data` objects for the atoms. To facilitate the use of `Data` objects a `Symbol` has a shape (and thus a rank) as well as a dimension (see constructor). Symbols are useful to perform mathematical simplifications, compute derivatives and as coefficients for nonlinear PDEs which can be solved by the `NonlinearPDE` class.

  `__init__`(*args, **kwargs)

    Initialises a new `Symbol` object in one of three ways:

        u=Symbol('u')

    returns a scalar symbol by the name ‘u’.

    > alpha=Symbol(‘alpha’, (4,3))

    returns a rank 2 symbol with the shape (4,3), whose elements are named ‘[alpha]_i_j’ (with i=0..3, j=0..2).

    > a,b,c=symbols(‘a,b,c’) x=Symbol([[a+b,0,0],[0,b-c,0],[0,0,c-a]])

    returns a rank 2 symbol with the shape (3,3) whose elements are explicitly specified by numeric values and other symbols/expressions within a list or numpy array.

    The dimensionality of the symbol can be specified through the `dim` keyword. All other keywords are passed to the underlying symbolic library (currently sympy).

    Parameters: | args – initialisation arguments as described above dim (`int`) – dimensionality of the new Symbol (default: 2)

  `applyfunc`(f, on_type=None)

    Applies the function `f` to all elements (if on_type is None) or to all elements of type `on_type`.

  `atoms`(*types)

    Returns the atoms that form the current Symbol.

    By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however.

    Note that if this symbol contains components such as [x]_i_j then only their main symbol ‘x’ is returned.

    Parameters: | types – types to restrict result to
    Returns: | list of atoms of specified type
    Return type: | `set`

  `coeff`(x, expand=True)

    Returns the coefficient of the term “x” or 0 if there is no “x”.

    If “x” is a scalar symbol then “x” is searched in all components of this symbol. Otherwise the shapes must match and the coefficients are checked component by component.

    Example:

        x=Symbol('x', (2,2))
        y=3*x
        print y.coeff(x)
        print y.coeff(x[1,1])

    will print:

        [[3 3]
         [3 3]]

        [[0 0]
         [0 3]]

    Parameters: | x (`Symbol`, `numpy.ndarray`, `list`) – the term whose coefficients are to be found
    Returns: | the coefficient(s) of the term
    Return type: | `Symbol`

  `diff`(*symbols, **assumptions)
  `evalf`()

    Applies the sympy.evalf operation on all elements in this symbol

  `expand`()

    Applies the sympy.expand operation on all elements in this symbol

  `getDataSubstitutions`()

    Returns a dictionary of symbol names and the escript `Data` objects they represent within this Symbol.

    Returns: | the dictionary of substituted `Data` objects
    Return type: | `dict`

  `getDim`()

    Returns the spatial dimensionality of this symbol.

    Returns: | the symbol’s spatial dimensionality, or -1 if undefined
    Return type: | `int`

  `getRank`()

    Returns the rank of this symbol.

    Returns: | the symbol’s rank which is equal to the length of the shape.
    Return type: | `int`

  `getShape`()

    Returns the shape of this symbol.

    Returns: | the symbol’s shape
    Return type: | `tuple` of `int`

  `grad`(where=None)

    Returns a symbol which represents the gradient of this symbol. :type where: `Symbol`, `FunctionSpace`

  `inverse`()
  `is_Add` = False
  `is_Float` = False
  `item`(*args)

    Returns an element of this symbol. This method behaves like the item() method of numpy.ndarray. If this is a scalar Symbol, no arguments are allowed and the only element in this Symbol is returned. Otherwise, ‘args’ specifies a flat or nd-index and the element at that index is returned.

    Parameters: | args – index of item to be returned
    Returns: | the requested element
    Return type: | `sympy.Symbol`, `int`, or `float`

  `lambdarepr`()
  `simplify`()

    Applies the sympy.simplify operation on all elements in this symbol

  `subs`(old, new)

    Substitutes an expression.

  `swap_axes`(axis0, axis1)
  `tensorProduct`(other, axis_offset)
  `tensorTransposedProduct`(other, axis_offset)
  `trace`(axis_offset)

    Returns the trace of this Symbol.

  `transpose`(axis_offset)

    Returns the transpose of this Symbol.

  `transposedTensorProduct`(other, axis_offset)

## Functions

`esys.escript.symbolic.` `combineData`(array, shape)

`esys.escript.symbolic.` `getTotalDifferential`(f, x, order=0)

  This function computes:

      | Df/Dx = del_f/del_x + del_f/del_grad(x)*del_grad(x)/del_x + ...
      |            \   /         \   /
      |              a             b

`esys.escript.symbolic.` `isSymbol`(arg)

  Returns True if the argument `arg` is an escript `Symbol` or `sympy.Basic` object, False otherwise.

`esys.escript.symbolic.` `pprint`(expr, use_unicode=None)

  Prints expr in pretty form.

  pprint is just a shortcut for this function

`esys.escript.symbolic.` `pretty_print`(expr, use_unicode=None)

  Prints expr in pretty form.

  pprint is just a shortcut for this function

`esys.escript.symbolic.` `removeFsFromGrad`(sym)

  Returns `sym` with all occurrences grad_n(a,b,c) replaced by grad_n(a,b). That is, all functionspace parameters are removed.

`esys.escript.symbolic.` `symbols`(*names, **kwargs)

  Emulates the behaviour of sympy.symbols.

## Others

- HAVE_SYMBOLS

## Packages