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.

  • 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.


The method that searches the object to perform the replacements.

Return type:


class nutils.evaluable.Evaluable(args)

Bases: Singleton

Base class


collection of all function arguments


a frozenset of all arguments of this evaluable


collection of all function arguments such that the arguments to dependencies[i] can be found in dependencies[:i]


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: Evaluable

Evaluate to the given constant value.


value – The return value of eval.

class nutils.evaluable.SparseArray(chunks, shape, dtype)

Bases: Evaluable

sparse array

nutils.evaluable.sum(arg, axis=None)

Sum array elements over a given axis.

nutils.evaluable.dot(a, b, axes)

Contract a and b along axes.

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.

  • arg (Array) – Original array.

  • where (tuple of integers) – New axis positions.

  • shape (tuple) – Shape of the aligned array.


The aligned array.

Return type:



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: Array

Lower this object to a nutils.evaluable.Array.


list of weak references to the object (if defined)

class nutils.evaluable.Array(args, shape, dtype)

Bases: Evaluable

Base class for array valued functions.


The shape of this array function.


tuple of ints


The number of dimensions of this array array function. Equal to len(shape).




The dtype of the array elements.


int, float

sum(arg, axis=None)

Sum array elements over a given axis.

dot(a, b, axes)

Contract a and b along axes.


Convert to a SparseArray.

property as_evaluable_array

return self

class nutils.evaluable.NPoints

Bases: Array

The length of the points axis.

class nutils.evaluable.Normal(lgrad)

Bases: Array


class nutils.evaluable.Inverse(func)

Bases: 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: Array

interpolate uniformly spaced data; stepwise for now

class nutils.evaluable.Pointwise(*args)

Bases: 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: Pointwise

Cosine, element-wise.

evalf = <ufunc 'cos'>
class nutils.evaluable.Sin(*args)

Bases: Pointwise

Sine, element-wise.

evalf = <ufunc 'sin'>
class nutils.evaluable.Tan(*args)

Bases: Pointwise

Tangent, element-wise.

evalf = <ufunc 'tan'>
class nutils.evaluable.ArcSin(*args)

Bases: Pointwise

Inverse sine, element-wise.

evalf = <ufunc 'arcsin'>
class nutils.evaluable.ArcCos(*args)

Bases: Pointwise

Inverse cosine, element-wise.

evalf = <ufunc 'arccos'>
class nutils.evaluable.ArcTan(*args)

Bases: Pointwise

Inverse tangent, element-wise.

evalf = <ufunc 'arctan'>
class nutils.evaluable.CosH(*args)

Bases: Pointwise

Hyperbolic cosine, element-wise.

evalf = <ufunc 'cosh'>
class nutils.evaluable.SinH(*args)

Bases: Pointwise

Hyperbolic sine, element-wise.

evalf = <ufunc 'sinh'>
class nutils.evaluable.TanH(*args)

Bases: Pointwise

Hyperbolic tangent, element-wise.

evalf = <ufunc 'tanh'>
class nutils.evaluable.ArcTanH(*args)

Bases: Pointwise

Inverse tangent, element-wise.

evalf = <ufunc 'arctanh'>
class nutils.evaluable.Sampled(points, expect)

Bases: 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.

  • points (1d Array) – Present point coordinates.

  • expect (2d Array) – Elementwise constant that evaluates to the predefined point coordinates; used for error checking and to inherit the shape.

class nutils.evaluable.Zeros(shape, dtype)

Bases: Array


class nutils.evaluable.Guard(fun)

Bases: Array

bar all simplifications

class nutils.evaluable.Find(where)

Bases: Array

indices of boolean index vector

class nutils.evaluable.DerivativeTargetBase(args, shape, dtype)

Bases: Array

base class for derivative targets

class nutils.evaluable.WithDerivative(func, var, derivative)

Bases: 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.

  • func (Array) – The function to wrap.

  • var (DerivativeTargetBase) – The derivative target.

  • derivative (Array) – The derivative with shape func.shape + var.shape.

See also


a virtual derivative target

class nutils.evaluable.Argument(name, shape, dtype=<class 'float'>)

Bases: DerivativeTargetBase

Array argument, to be substituted before evaluation.

The Argument is an Array with a known shape, but whose values are to be defined later, before evaluation, e.g. using replace_arguments().

It is possible to take the derivative of an Array to an Argument:

>>> 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
  • name (str) – The Identifier of this argument.

  • shape (tuple of ints) – The shape of this argument.

class nutils.evaluable.IdentifierDerivativeTarget(identifier, shape)

Bases: DerivativeTargetBase

Virtual derivative target distinguished by an identifier.

  • identifier (hashable object) – The identifier for this derivative target.

  • shape (tuple of Array or int) – The shape of this derivative target.

See also


Array wrapper with additional derivative

class nutils.evaluable.Polyval(coeffs, points, ngrad=0)

Bases: 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 of coeffs.


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: Array

Series of Legendre polynomial up to and including the given degree.

  • x (Array) – The coordinates to evaluate the series at.

  • degree (int) – The degree of the last polynomial of the series.

class nutils.evaluable.Choose(index, choices)

Bases: 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 in value.

Replace Argument objects in value according to the arguments map, taking into account derivatives to the local coordinates.


The edited value.

Return type:


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]))

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]))

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]))

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.eval_sparse(funcs, **arguments)

Evaluate one or several Array objects as sparse data.

  • funcs (tuple of Array objects) – Arrays to be evaluated.

  • arguments (dict (default: None)) – Optional arguments for function evaluation.



Return type:

tuple of sparse data arrays