evaluable¶
The function module defines the Evaluable
class and derived objects,
commonly referred to as nutils functions. They represent mappings from a
nutils.topology
onto Python space. The notabe class of Array
objects map onto the space of Numpy arrays of predefined dimension and shape.
Most functions used in nutils applicatons are of this latter type, including the
geometry and function bases for analysis.
Nutils functions are essentially postponed python functions, stored in a tree
structure of input/output dependencies. Many Array
objects have
directly recognizable numpy equivalents, such as Sin
or
Inverse
. By not evaluating directly but merely stacking operations,
complex operations can be defined prior to entering a quadrature loop, allowing
for a higher level style programming. It also allows for automatic
differentiation and code optimization.
It is important to realize that nutils functions do not map for a physical xy-domain but from a topology, where a point is characterized by the combination of an element and its local coordinate. This is a natural fit for typical finite element operations such as quadrature. Evaluation from physical coordinates is possible only via inverting of the geometry function, which is a fundamentally expensive and currently unsupported operation.
- nutils.evaluable.equalindex(n, m)¶
Compare two array indices.
Returns True if the two indices are certainly equal, False if they are certainly not equal, or None if equality cannot be determined at compile time.
- nutils.evaluable.equalshape(N, M)¶
Compare two array shapes.
Returns True if all indices are certainly equal, False if any indices are certainly not equal, or None if equality cannot be determined at compile time.
- nutils.evaluable.replace(func=None, depthfirst=False, recursive=False, lru=4)¶
decorator for deep object replacement
Generates a deep replacement method for general objects based on a callable that is applied (recursively) on individual constructor arguments.
- Parameters
func – Callable which maps an object onto a new object, or None if no replacement is made. It must have one positional argument for the object, and may have any number of additional positional and/or keyword arguments.
depthfirst (
bool
) – If True, decompose each object as far a possible, then apply func to all arguments as the objects are reconstructed. Otherwise apply func directly on each new object that is encountered in the decomposition, proceding only if the return value is None.recursive (
bool
) – If True, repeat replacement for any object returned by func until it returns None. Otherwise perform a single, non-recursive sweep.lru (
int
) – Maximum size of the least-recently-used cache. A persistent weak-key dictionary is maintained for every unique set of function arguments. When the size of lru is reached, the least recently used cache is dropped.
- Returns
The method that searches the object to perform the replacements.
- Return type
- class nutils.evaluable.Evaluable(args)¶
Bases:
nutils.types.Singleton
Base class
- dependencies¶
collection of all function arguments
- arguments¶
a frozenset of all arguments of this evaluable
- ordereddeps¶
collection of all function arguments such that the arguments to dependencies[i] can be found in dependencies[:i]
- dependencytree¶
lookup table of function arguments into ordereddeps, such that ordereddeps[i].__args[j] == ordereddeps[dependencytree[i][j]], and self.__args[j] == ordereddeps[dependencytree[-1][j]]
- asciitree(self, richoutput=False)¶
string representation
- eval(self, **evalargs)¶
Evaluate function on a specified element, point set.
- eval_withtimes(self, times, **evalargs)¶
Evaluate function on a specified element, point set while measure time of each step.
- class nutils.evaluable.EvaluableConstant(value)¶
Bases:
nutils.evaluable.Evaluable
Evaluate to the given constant value.
- Parameters
value – The return value of
eval
.
- class nutils.evaluable.SparseArray(chunks, shape, dtype)¶
Bases:
nutils.evaluable.Evaluable
sparse array
- nutils.evaluable.sum(arg, axis=None)¶
Sum array elements over a given axis.
- nutils.evaluable.dot(a, b, axes)¶
Contract
a
andb
alongaxes
.
- nutils.evaluable.align(arg, where, shape)¶
Align array to target shape.
The align operation can be considered the opposite of transpose: instead of specifying for each axis of the return value the original position in the argument, align specifies for each axis of the argument the new position in the return value. In addition, the return value may be of higher dimension, with new axes being inserted according to the
shape
argument.
- nutils.evaluable.unalign(*args)¶
Remove (joint) inserted axes.
Given one or more equally shaped array arguments, return the shortest common axis vector along with function arguments such that the original arrays can be recovered by
align()
.
- class nutils.evaluable.AsEvaluableArray¶
Bases:
object
Protocol for conversion into an
Array
.- property as_evaluable_array: nutils.evaluable.Array¶
Lower this object to a
nutils.evaluable.Array
.
- __weakref__¶
list of weak references to the object (if defined)
- class nutils.evaluable.Array(args, shape, dtype)¶
Bases:
nutils.evaluable.Evaluable
Base class for array valued functions.
- sum(arg, axis=None)¶
Sum array elements over a given axis.
- dot(a, b, axes)¶
Contract
a
andb
alongaxes
.
- assparse¶
Convert to a
SparseArray
.
- property as_evaluable_array¶
return self
- class nutils.evaluable.NPoints¶
Bases:
nutils.evaluable.Array
The length of the points axis.
- class nutils.evaluable.Normal(lgrad)¶
Bases:
nutils.evaluable.Array
normal
- class nutils.evaluable.Inverse(func)¶
Bases:
nutils.evaluable.Array
Matrix inverse of
func
over the last two axes. All other axes are treated element-wise.
- class nutils.evaluable.Interpolate(x, xp, fp, left=None, right=None)¶
Bases:
nutils.evaluable.Array
interpolate uniformly spaced data; stepwise for now
- class nutils.evaluable.Pointwise(*args)¶
Bases:
nutils.evaluable.Array
Abstract base class for pointwise array functions.
- classmethod outer(*args)¶
Alternative constructor that outer-aligns the arguments.
The output shape of this pointwise function is the sum of all shapes of its arguments. When called with multiple arguments, the first argument will be appended with singleton axes to match the output shape, the second argument will be prepended with as many singleton axes as the dimension of the original first argument and appended to match the output shape, and so forth and so on.
- class nutils.evaluable.Cos(*args)¶
Bases:
nutils.evaluable.Pointwise
Cosine, element-wise.
- evalf = <ufunc 'cos'>¶
- class nutils.evaluable.Sin(*args)¶
Bases:
nutils.evaluable.Pointwise
Sine, element-wise.
- evalf = <ufunc 'sin'>¶
- class nutils.evaluable.Tan(*args)¶
Bases:
nutils.evaluable.Pointwise
Tangent, element-wise.
- evalf = <ufunc 'tan'>¶
- class nutils.evaluable.ArcSin(*args)¶
Bases:
nutils.evaluable.Pointwise
Inverse sine, element-wise.
- evalf = <ufunc 'arcsin'>¶
- class nutils.evaluable.ArcCos(*args)¶
Bases:
nutils.evaluable.Pointwise
Inverse cosine, element-wise.
- evalf = <ufunc 'arccos'>¶
- class nutils.evaluable.ArcTan(*args)¶
Bases:
nutils.evaluable.Pointwise
Inverse tangent, element-wise.
- evalf = <ufunc 'arctan'>¶
- class nutils.evaluable.Sampled(points, expect)¶
Bases:
nutils.evaluable.Array
Basis-like identity operator.
Basis-like function that for every point in a predefined set evaluates to the unit vector corresponding to its index.
- class nutils.evaluable.Zeros(shape, dtype)¶
Bases:
nutils.evaluable.Array
zero
- class nutils.evaluable.Guard(fun)¶
Bases:
nutils.evaluable.Array
bar all simplifications
- class nutils.evaluable.TrigNormal(angle)¶
Bases:
nutils.evaluable.Array
cos, sin
- class nutils.evaluable.TrigTangent(angle)¶
Bases:
nutils.evaluable.Array
-sin, cos
- class nutils.evaluable.Find(where)¶
Bases:
nutils.evaluable.Array
indices of boolean index vector
- class nutils.evaluable.DerivativeTargetBase(args, shape, dtype)¶
Bases:
nutils.evaluable.Array
base class for derivative targets
- class nutils.evaluable.WithDerivative(func, var, derivative)¶
Bases:
nutils.evaluable.Array
Wrap the given function and define the derivative to a target.
The wrapper is typically used together with a virtual derivative target like
IdentifierDerivativeTarget
. The wrapper is removed in the simplified form.- Parameters
func (
Array
) – The function to wrap.var (
DerivativeTargetBase
) – The derivative target.derivative (
Array
) – The derivative with shapefunc.shape + var.shape
.
See also
IdentifierDerivativeTarget
a virtual derivative target
- class nutils.evaluable.Argument(name, shape, dtype=<class 'float'>)¶
Bases:
nutils.evaluable.DerivativeTargetBase
Array argument, to be substituted before evaluation.
The
Argument
is anArray
with a known shape, but whose values are to be defined later, before evaluation, e.g. usingreplace_arguments()
.It is possible to take the derivative of an
Array
to anArgument
:>>> from nutils import evaluable >>> a = evaluable.Argument('x', []) >>> b = evaluable.Argument('y', []) >>> f = a**3 + b**2 >>> evaluable.derivative(f, a).simplified == (3*a**2).simplified True
- class nutils.evaluable.IdentifierDerivativeTarget(identifier, shape)¶
Bases:
nutils.evaluable.DerivativeTargetBase
Virtual derivative target distinguished by an identifier.
- Parameters
See also
WithDerivative
Array
wrapper with additional derivative
- class nutils.evaluable.Polyval(coeffs, points, ngrad=0)¶
Bases:
nutils.evaluable.Array
Computes the \(k\)-dimensional array
\[\begin{split}j_0,\dots,j_{k-1} \mapsto \sum_{\substack{i_0,\dots,i_{n-1}\in\mathbb{N}\\i_0+\cdots+i_{n-1}\le d}} p_0^{i_0} \cdots p_{n-1}^{i_{n-1}} c_{j_0,\dots,j_{k-1},i_0,\dots,i_{n-1}},\end{split}\]where \(p\) are the \(n\)-dimensional local coordinates and \(c\) is the argument
coeffs
and \(d\) is the degree of the polynomial, where \(d\) is the length of the last \(n\) axes ofcoeffs
.Warning
All coefficients with a (combined) degree larger than \(d\) should be zero. Failing to do so won’t raise an
Exception
, but might give incorrect results.
- class nutils.evaluable.Legendre(x, degree)¶
Bases:
nutils.evaluable.Array
Series of Legendre polynomial up to and including the given degree.
- class nutils.evaluable.Choose(index, choices)¶
Bases:
nutils.evaluable.Array
Function equivalent of
numpy.choose()
.
- nutils.evaluable.derivative(func, var, seen=None)¶
- nutils.evaluable.prependaxes(func, shape)¶
Prepend axes with specified shape to func.
- nutils.evaluable.appendaxes(func, shape)¶
Append axes with specified shape to func.
- nutils.evaluable.replace_arguments(value, arguments)¶
Replace
Argument
objects invalue
.Replace
Argument
objects invalue
according to thearguments
map, taking into account derivatives to the local coordinates.
- nutils.evaluable.einsum(fmt, *args, **dims)¶
Multiply and/or contract arrays via format string.
The format string consists of a comma separated list of axis labels, followed by
->
and the axis labels of the return value. For example, the following swaps the axes of a matrix:>>> einsum('ij->ji', ones([2,3])) nutils.evaluable.Transpose<f:3,2>
Axis labels that do not occur in the return value are summed. For example, the following performs a dot product of three matrices:
>>> einsum('ij,jk,kl->il', ones([2,3]), ones([3,4]), ones([4,5])) nutils.evaluable.Sum<f:2,5>
In case the dimension of the input and output arrays may vary, a variable length axes group can be denoted by a capital. Its length is automatically established based on the dimension of the input arrays. The following example performs a tensor product of an array and a vector:
>>> einsum('A,i->Ai', ones([2,3,4]), ones([5])) nutils.evaluable.Multiply<f:2,3,4,5>
The format string may contain multiple variable length axes groups, but their lengths must be resolvable from left to right. In case this is not possible, lengths may be specified as keyword arguments.
>>> einsum('AjB,i->AijB', ones([2,3,4]), ones([5]), B=1) nutils.evaluable.Multiply<f:2,5,3,4>