schrodinger.application.matsci.elasticity.stress module

This module provides the Stress class used to create, manipulate, and calculate relevant properties of the stress tensor.

Copyright Schrodinger, LLC. All rights reserved.

class schrodinger.application.matsci.elasticity.stress.Stress(stress_matrix)

Bases: schrodinger.application.matsci.elasticity.tensors.SquareTensor

This class extends SquareTensor as a representation of the stress

property dev_principal_invariants

returns the principal invariants of the deviatoric stress tensor, which is calculated by finding the coefficients of the characteristic polynomial of the stress tensor minus the identity times the mean stress

property von_mises

returns the von mises stress

property mean_stress

returns the mean stress

property deviator_stress

returns the deviatoric component of the stress


calculates the first Piola-Kirchoff stress


def_grad (3x3 array-like): deformation gradient tensor


calculates the second Piola-Kirchoff stress


def_grad (3x3 array-like): rate of deformation tensor


View of the transposed array.

Same as self.transpose().

>>> a = np.array([[1, 2], [3, 4]])
>>> a
array([[1, 2],
       [3, 4]])
>>> a.T
array([[1, 3],
       [2, 4]])
>>> a = np.array([1, 2, 3, 4])
>>> a
array([1, 2, 3, 4])
>>> a.T
array([1, 2, 3, 4])


__contains__(key, /)

Return key in self.


Return len(self).

all(axis=None, out=None, keepdims=False, *, where=True)

Returns True if all elements evaluate to True.

Refer to numpy.all for full documentation.

numpy.all : equivalent function

any(axis=None, out=None, keepdims=False, *, where=True)

Returns True if any of the elements of a evaluate to True.

Refer to numpy.any for full documentation.

numpy.any : equivalent function

argmax(axis=None, out=None, *, keepdims=False)

Return indices of the maximum values along the given axis.

Refer to numpy.argmax for full documentation.

numpy.argmax : equivalent function

argmin(axis=None, out=None, *, keepdims=False)

Return indices of the minimum values along the given axis.

Refer to numpy.argmin for detailed documentation.

numpy.argmin : equivalent function

argpartition(kth, axis=- 1, kind='introselect', order=None)

Returns the indices that would partition this array.

Refer to numpy.argpartition for full documentation.

New in version 1.8.0.

numpy.argpartition : equivalent function

argsort(axis=- 1, kind=None, order=None)

Returns the indices that would sort this array.

Refer to numpy.argsort for full documentation.

numpy.argsort : equivalent function

astype(dtype, order='K', casting='unsafe', subok=True, copy=True)

Copy of the array, cast to a specified type.

dtypestr or dtype

Typecode or data-type to which the array is cast.

order{‘C’, ‘F’, ‘A’, ‘K’}, optional

Controls the memory layout order of the result. ‘C’ means C order, ‘F’ means Fortran order, ‘A’ means ‘F’ order if all the arrays are Fortran contiguous, ‘C’ order otherwise, and ‘K’ means as close to the order the array elements appear in memory as possible. Default is ‘K’.

casting{‘no’, ‘equiv’, ‘safe’, ‘same_kind’, ‘unsafe’}, optional

Controls what kind of data casting may occur. Defaults to ‘unsafe’ for backwards compatibility.

  • ‘no’ means the data types should not be cast at all.

  • ‘equiv’ means only byte-order changes are allowed.

  • ‘safe’ means only casts which can preserve values are allowed.

  • ‘same_kind’ means only safe casts or casts within a kind, like float64 to float32, are allowed.

  • ‘unsafe’ means any data conversions may be done.

subokbool, optional

If True, then sub-classes will be passed-through (default), otherwise the returned array will be forced to be a base-class array.

copybool, optional

By default, astype always returns a newly allocated array. If this is set to false, and the dtype, order, and subok requirements are satisfied, the input array is returned instead of a copy.


Unless copy is False and the other conditions for returning the input array are satisfied (see description for copy input parameter), arr_t is a new array of the same shape as the input array, with dtype, order given by dtype, order.

Changed in version 1.17.0: Casting between a simple data type and a structured one is possible only for “unsafe” casting. Casting to multiple fields is allowed, but casting from multiple fields is not.

Changed in version 1.9.0: Casting from numeric to string types in ‘safe’ casting mode requires that the string dtype length is long enough to store the max integer/float value converted.


When casting from complex to float or int. To avoid this, one should use a.real.astype(t).

>>> x = np.array([1, 2, 2.5])
>>> x
array([1. ,  2. ,  2.5])
>>> x.astype(int)
array([1, 2, 2])

Base object if memory is from some other object.

The base of an array that owns its memory is None:

>>> x = np.array([1,2,3,4])
>>> x.base is None

Slicing creates a view, whose memory is shared with x:

>>> y = x[2:]
>>> y.base is x

Swap the bytes of the array elements

Toggle between low-endian and big-endian data representation by returning a byteswapped array, optionally swapped in-place. Arrays of byte-strings are not swapped. The real and imaginary parts of a complex number are swapped individually.

inplacebool, optional

If True, swap bytes in-place, default is False.


The byteswapped array. If inplace is True, this is a view to self.

>>> A = np.array([1, 256, 8755], dtype=np.int16)
>>> list(map(hex, A))
['0x1', '0x100', '0x2233']
>>> A.byteswap(inplace=True)
array([  256,     1, 13090], dtype=int16)
>>> list(map(hex, A))
['0x100', '0x1', '0x3322']

Arrays of byte-strings are not swapped

>>> A = np.array([b'ceg', b'fac'])
>>> A.byteswap()
array([b'ceg', b'fac'], dtype='|S3')
A.newbyteorder().byteswap() produces an array with the same values

but different representation in memory

>>> A = np.array([1, 2, 3])
>>> A.view(np.uint8)
array([1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0,
       0, 0], dtype=uint8)
>>> A.newbyteorder().byteswap(inplace=True)
array([1, 2, 3])
>>> A.view(np.uint8)
array([0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0,
       0, 3], dtype=uint8)
choose(choices, out=None, mode='raise')

Use an index array to construct a new array from a set of choices.

Refer to numpy.choose for full documentation.

numpy.choose : equivalent function

clip(min=None, max=None, out=None, **kwargs)

Return an array whose values are limited to [min, max]. One of max or min must be given.

Refer to numpy.clip for full documentation.

numpy.clip : equivalent function

compress(condition, axis=None, out=None)

Return selected slices of this array along given axis.

Refer to numpy.compress for full documentation.

numpy.compress : equivalent function


Complex-conjugate all elements.

Refer to numpy.conjugate for full documentation.

numpy.conjugate : equivalent function


Return the complex conjugate, element-wise.

Refer to numpy.conjugate for full documentation.

numpy.conjugate : equivalent function


Return a copy of the array.

order{‘C’, ‘F’, ‘A’, ‘K’}, optional

Controls the memory layout of the copy. ‘C’ means C-order, ‘F’ means F-order, ‘A’ means ‘F’ if a is Fortran contiguous, ‘C’ otherwise. ‘K’ means match the layout of a as closely as possible. (Note that this function and numpy.copy() are very similar but have different default values for their order= arguments, and this function always passes sub-classes through.)

numpy.copy : Similar function with different default behavior numpy.copyto

This function is the preferred method for creating an array copy. The function numpy.copy() is similar, but it defaults to using order ‘K’, and will not pass sub-classes through by default.

>>> x = np.array([[1,2,3],[4,5,6]], order='F')
>>> y = x.copy()
>>> x.fill(0)
>>> x
array([[0, 0, 0],
       [0, 0, 0]])
>>> y
array([[1, 2, 3],
       [4, 5, 6]])
>>> y.flags['C_CONTIGUOUS']

An object to simplify the interaction of the array with the ctypes module.

This attribute creates an object that makes it easier to use arrays when calling shared libraries with the ctypes module. The returned object has, among others, data, shape, and strides attributes (see Notes below) which themselves return ctypes objects that can be used as arguments to a shared library.


cPython object

Possessing attributes data, shape, strides, etc.


Below are the public attributes of this object which were documented in “Guide to NumPy” (we have omitted undocumented public attributes, as well as documented private attributes):

A pointer to the memory area of the array as a Python integer. This memory area may contain data that is not aligned, or not in correct byte-order. The memory area may not even be writeable. The array flags and data-type of this array should be respected when passing this attribute to arbitrary C-code to avoid trouble that can include Python crashing. User Beware! The value of this attribute is exactly the same as self._array_interface_['data'][0].

Note that unlike data_as, a reference will not be kept to the array: code like ctypes.c_void_p((a + b) will result in a pointer to a deallocated array, and should be spelt (a + b).ctypes.data_as(ctypes.c_void_p)


(c_intp*self.ndim): A ctypes array of length self.ndim where the basetype is the C-integer corresponding to dtype('p') on this platform (see c_intp). This base-type could be ctypes.c_int, ctypes.c_long, or ctypes.c_longlong depending on the platform. The ctypes array contains the shape of the underlying array.


(c_intp*self.ndim): A ctypes array of length self.ndim where the basetype is the same as for the shape attribute. This ctypes array contains the strides information from the underlying array. This strides information is important for showing how many bytes must be jumped to get to the next element in the array.


Return the data pointer cast to a particular c-types object. For example, calling self._as_parameter_ is equivalent to self.data_as(ctypes.c_void_p). Perhaps you want to use the data as a pointer to a ctypes array of floating-point data: self.data_as(ctypes.POINTER(ctypes.c_double)).

The returned pointer will keep a reference to the array.


Return the shape tuple as an array of some other c-types type. For example: self.shape_as(ctypes.c_short).


Return the strides tuple as an array of some other c-types type. For example: self.strides_as(ctypes.c_longlong).

If the ctypes module is not available, then the ctypes attribute of array objects still returns something useful, but ctypes objects are not returned and errors may be raised instead. In particular, the object will still have the as_parameter attribute which will return an integer equal to the data attribute.

>>> import ctypes
>>> x = np.array([[0, 1], [2, 3]], dtype=np.int32)
>>> x
array([[0, 1],
       [2, 3]], dtype=int32)
31962608 # may vary
>>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_uint32))
<__main__.LP_c_uint object at 0x7ff2fc1fc200> # may vary
>>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_uint32)).contents
>>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_uint64)).contents
>>> x.ctypes.shape
<numpy.core._internal.c_long_Array_2 object at 0x7ff2fc1fce60> # may vary
>>> x.ctypes.strides
<numpy.core._internal.c_long_Array_2 object at 0x7ff2fc1ff320> # may vary
cumprod(axis=None, dtype=None, out=None)

Return the cumulative product of the elements along the given axis.

Refer to numpy.cumprod for full documentation.

numpy.cumprod : equivalent function

cumsum(axis=None, dtype=None, out=None)

Return the cumulative sum of the elements along the given axis.

Refer to numpy.cumsum for full documentation.

numpy.cumsum : equivalent function


Python buffer object pointing to the start of the array’s data.

property det

shorthand for the determinant of the SquareTensor

diagonal(offset=0, axis1=0, axis2=1)

Return specified diagonals. In NumPy 1.9 the returned array is a read-only view instead of a copy as in previous NumPy versions. In a future version the read-only restriction will be removed.

Refer to numpy.diagonal() for full documentation.

numpy.diagonal : equivalent function


Data-type of the array’s elements.


Setting arr.dtype is discouraged and may be deprecated in the future. Setting will replace the dtype without modifying the memory (see also ndarray.view and ndarray.astype).


d : numpy dtype object

ndarray.astype : Cast the values contained in the array to a new data-type. ndarray.view : Create a view of the same data but a different data-type. numpy.dtype

>>> x
array([[0, 1],
       [2, 3]])
>>> x.dtype
>>> type(x.dtype)
<type 'numpy.dtype'>

Dump a pickle of the array to the specified file. The array can be read back with pickle.load or numpy.load.

filestr or Path

A string naming the dump file.

Changed in version 1.17.0: pathlib.Path objects are now accepted.


Returns the pickle of the array as a string. pickle.loads will convert the string back to an array.


einsum_sequence(other_arrays, einsum_string=None)

Calculates the result of an einstein summation expression


Fill the array with a scalar value.


All elements of a will be assigned this value.

>>> a = np.array([1, 2])
>>> a.fill(0)
>>> a
array([0, 0])
>>> a = np.empty(2)
>>> a.fill(1)
>>> a
array([1.,  1.])

Fill expects a scalar value and always behaves the same as assigning to a single array element. The following is a rare example where this distinction is important:

>>> a = np.array([None, None], dtype=object)
>>> a[0] = np.array(3)
>>> a
array([array(3), None], dtype=object)
>>> a.fill(np.array(3))
>>> a
array([array(3), array(3)], dtype=object)

Where other forms of assignments will unpack the array being assigned:

>>> a[...] = np.array(3)
>>> a
array([3, 3], dtype=object)

Information about the memory layout of the array.


The data is in a single, C-style contiguous segment.


The data is in a single, Fortran-style contiguous segment.


The array owns the memory it uses or borrows it from another object.


The data area can be written to. Setting this to False locks the data, making it read-only. A view (slice, etc.) inherits WRITEABLE from its base array at creation time, but a view of a writeable array may be subsequently locked while the base array remains writeable. (The opposite is not true, in that a view of a locked array may not be made writeable. However, currently, locking a base object does not lock any views that already reference it, so under that circumstance it is possible to alter the contents of a locked array via a previously created writeable view onto it.) Attempting to change a non-writeable array raises a RuntimeError exception.


The data and all elements are aligned appropriately for the hardware.


This array is a copy of some other array. The C-API function PyArray_ResolveWritebackIfCopy must be called before deallocating to the base array will be updated with the contents of this array.




F_CONTIGUOUS or C_CONTIGUOUS (one-segment test).







The flags object can be accessed dictionary-like (as in a.flags['WRITEABLE']), or by using lowercased attribute names (as in a.flags.writeable). Short flag names are only supported in dictionary access.

Only the WRITEBACKIFCOPY, WRITEABLE, and ALIGNED flags can be changed by the user, via direct assignment to the attribute or dictionary entry, or by calling ndarray.setflags.

The array flags cannot be set arbitrarily:

  • WRITEBACKIFCOPY can only be set False.

  • ALIGNED can only be set True if the data is truly aligned.

  • WRITEABLE can only be set True if the array owns its own memory or the ultimate owner of the memory exposes a writeable buffer interface or is a string.

Arrays can be both C-style and Fortran-style contiguous simultaneously. This is clear for 1-dimensional arrays, but can also be true for higher dimensional arrays.

Even for contiguous arrays a stride for a given dimension arr.strides[dim] may be arbitrary if arr.shape[dim] == 1 or the array has no elements. It does not generally hold that self.strides[-1] == self.itemsize for C-style contiguous arrays or self.strides[0] == self.itemsize for Fortran-style contiguous arrays is true.


A 1-D iterator over the array.

This is a numpy.flatiter instance, which acts similarly to, but is not a subclass of, Python’s built-in iterator object.

flatten : Return a copy of the array collapsed into one dimension.


>>> x = np.arange(1, 7).reshape(2, 3)
>>> x
array([[1, 2, 3],
       [4, 5, 6]])
>>> x.flat[3]
>>> x.T
array([[1, 4],
       [2, 5],
       [3, 6]])
>>> x.T.flat[3]
>>> type(x.flat)
<class 'numpy.flatiter'>

An assignment example:

>>> x.flat = 3; x
array([[3, 3, 3],
       [3, 3, 3]])
>>> x.flat[[1,4]] = 1; x
array([[3, 1, 3],
       [3, 1, 3]])

Return a copy of the array collapsed into one dimension.

order{‘C’, ‘F’, ‘A’, ‘K’}, optional

‘C’ means to flatten in row-major (C-style) order. ‘F’ means to flatten in column-major (Fortran- style) order. ‘A’ means to flatten in column-major order if a is Fortran contiguous in memory, row-major order otherwise. ‘K’ means to flatten a in the order the elements occur in memory. The default is ‘C’.


A copy of the input array, flattened to one dimension.

ravel : Return a flattened array. flat : A 1-D flat iterator over the array.

>>> a = np.array([[1,2], [3,4]])
>>> a.flatten()
array([1, 2, 3, 4])
>>> a.flatten('F')
array([1, 3, 2, 4])
classmethod from_values_indices(values, indices, populate=False, structure=None, voigt_rank=None, vsym=True, verbose=False)

Creates a tensor from values and indices, with options for populating the remainder of the tensor.

  • values (list[float]) – numbers to place at indices

  • indices – array-like collection of indices to place values at

  • populate (bool) – whether to populate the tensor

  • structure (Structure) – structure to base population or fit_to_structure on

  • voigt_rank (int) – full tensor rank to indicate the shape of the resulting tensor. This is necessary if one provides a set of indices more minimal than the shape of the tensor they want, e.g. Tensor.from_values_indices((0, 0), 100)

  • vsym (bool) – whether to voigt symmetrize during the optimization procedure

  • verbose (bool) – whether to populate verbosely

classmethod from_voigt(voigt_input)

Constructor based on the voigt notation vector or matrix.


voigt_input (array-like): voigt input for a given tensor


Scales the tensor by a certain multiplicative scale factor

scale_factor (float): scalar multiplier to be applied to the

SquareTensor object

static get_voigt_dict(rank)

Returns a dictionary that maps indices in the tensor to those in a voigt representation based on input rank


rank (int): Tensor rank to generate the voigt map

getfield(dtype, offset=0)

Returns a field of the given array as a certain type.

A field is a view of the array data with a given data-type. The values in the view are determined by the given type and the offset into the current array in bytes. The offset needs to be such that the view dtype fits in the array dtype; for example an array of dtype complex128 has 16-byte elements. If taking a view with a 32-bit integer (4 bytes), the offset needs to be between 0 and 12 bytes.

dtypestr or dtype

The data type of the view. The dtype size of the view can not be larger than that of the array itself.


Number of bytes to skip before beginning the element view.

>>> x = np.diag([1.+1.j]*2)
>>> x[1, 1] = 2 + 4.j
>>> x
array([[1.+1.j,  0.+0.j],
       [0.+0.j,  2.+4.j]])
>>> x.getfield(np.float64)
array([[1.,  0.],
       [0.,  2.]])

By choosing an offset of 8 bytes we can select the complex part of the array for our view:

>>> x.getfield(np.float64, offset=8)
array([[1.,  0.],
       [0.,  4.]])

The imaginary part of the array.

>>> x = np.sqrt([1+0j, 0+1j])
>>> x.imag
array([ 0.        ,  0.70710678])
>>> x.imag.dtype
property inv

shorthand for matrix inverse on SquareTensor

is_fit_to_structure(structure, tol=0.01)

Tests whether a tensor is invariant with respect to the symmetry operations of a particular structure by testing whether the residual of the symmetric portion is below a tolerance


structure (Structure): structure to be fit to tol (float): tolerance for symmetry testing

is_rotation(tol=0.001, include_improper=True)

Test to see if tensor is a valid rotation matrix, performs a test to check whether the inverse is equal to the transpose and if the determinant is equal to one within the specified tolerance

tol (float): tolerance to both tests of whether the

the determinant is one and the inverse is equal to the transpose

include_improper (bool): whether to include improper

rotations in the determination of validity


Tests whether a tensor is symmetric or not based on the residual with its symmetric part, from self.symmetrized


tol (float): tolerance to test for symmetry


Tests symmetry of tensor to that necessary for voigt-conversion by grouping indices into pairs and constructing a sequence of possible permutations to be used in a tensor transpose


Copy an element of an array to a standard Python scalar and return it.

*args : Arguments (variable number and type)

  • none: in this case, the method only works for arrays with one element (a.size == 1), which element is copied into a standard Python scalar object and returned.

  • int_type: this argument is interpreted as a flat index into the array, specifying which element to copy and return.

  • tuple of int_types: functions as does a single int_type argument, except that the argument is interpreted as an nd-index into the array.

zStandard Python scalar object

A copy of the specified element of the array as a suitable Python scalar

When the data type of a is longdouble or clongdouble, item() returns a scalar array object because there is no available Python scalar that would not lose information. Void arrays return a buffer object for item(), unless fields are defined, in which case a tuple is returned.

item is very similar to a[args], except, instead of an array scalar, a standard Python scalar is returned. This can be useful for speeding up access to elements of the array and doing arithmetic on elements of the array using Python’s optimized math.

>>> np.random.seed(123)
>>> x = np.random.randint(9, size=(3, 3))
>>> x
array([[2, 2, 6],
       [1, 3, 6],
       [1, 0, 1]])
>>> x.item(3)
>>> x.item(7)
>>> x.item((0, 1))
>>> x.item((2, 2))

Insert scalar into an array (scalar is cast to array’s dtype, if possible)

There must be at least 1 argument, and define the last argument as item. Then, a.itemset(*args) is equivalent to but faster than a[args] = item. The item should be a scalar value and args must select a single item in the array a.


If one argument: a scalar, only used in case a is of size 1. If two arguments: the last argument is the value to be set and must be a scalar, the first argument specifies a single array element location. It is either an int or a tuple.

Compared to indexing syntax, itemset provides some speed increase for placing a scalar into a particular location in an ndarray, if you must do this. However, generally this is discouraged: among other problems, it complicates the appearance of the code. Also, when using itemset (and item) inside a loop, be sure to assign the methods to a local variable to avoid the attribute look-up at each loop iteration.

>>> np.random.seed(123)
>>> x = np.random.randint(9, size=(3, 3))
>>> x
array([[2, 2, 6],
       [1, 3, 6],
       [1, 0, 1]])
>>> x.itemset(4, 0)
>>> x.itemset((2, 2), 9)
>>> x
array([[2, 2, 6],
       [1, 0, 6],
       [1, 0, 9]])

Length of one array element in bytes.

>>> x = np.array([1,2,3], dtype=np.float64)
>>> x.itemsize
>>> x = np.array([1,2,3], dtype=np.complex128)
>>> x.itemsize
max(axis=None, out=None, keepdims=False, initial=<no value>, where=True)

Return the maximum along a given axis.

Refer to numpy.amax for full documentation.

numpy.amax : equivalent function

mean(axis=None, dtype=None, out=None, keepdims=False, *, where=True)

Returns the average of the array elements along given axis.

Refer to numpy.mean for full documentation.

numpy.mean : equivalent function

min(axis=None, out=None, keepdims=False, initial=<no value>, where=True)

Return the minimum along a given axis.

Refer to numpy.amin for full documentation.

numpy.amin : equivalent function


Total bytes consumed by the elements of the array.

Does not include memory consumed by non-element attributes of the array object.

>>> x = np.zeros((3,5,2), dtype=np.complex128)
>>> x.nbytes
>>> * x.itemsize

Number of array dimensions.

>>> x = np.array([1, 2, 3])
>>> x.ndim
>>> y = np.zeros((2, 3, 4))
>>> y.ndim
newbyteorder(new_order='S', /)

Return the array with the same data viewed with a different byte order.

Equivalent to:


Changes are also made in all fields and sub-arrays of the array data type.

new_orderstring, optional

Byte order to force; a value from the byte order specifications below. new_order codes can be any of:

  • ‘S’ - swap dtype from current to opposite endian

  • {‘<’, ‘little’} - little endian

  • {‘>’, ‘big’} - big endian

  • {‘=’, ‘native’} - native order, equivalent to sys.byteorder

  • {‘|’, ‘I’} - ignore (no change to byte order)

The default value (‘S’) results in swapping the current byte order.


New array object with the dtype reflecting given change to the byte order.


Return the indices of the elements that are non-zero.

Refer to numpy.nonzero for full documentation.

numpy.nonzero : equivalent function

partition(kth, axis=- 1, kind='introselect', order=None)

Rearranges the elements in the array in such a way that the value of the element in kth position is in the position it would be in a sorted array. All elements smaller than the kth element are moved before this element and all equal or greater are moved behind it. The ordering of the elements in the two partitions is undefined.

New in version 1.8.0.

kthint or sequence of ints

Element index to partition by. The kth element value will be in its final sorted position and all smaller elements will be moved before it and all equal or greater elements behind it. The order of all elements in the partitions is undefined. If provided with a sequence of kth it will partition all elements indexed by kth of them into their sorted position at once.

Deprecated since version 1.22.0: Passing booleans as index is deprecated.

axisint, optional

Axis along which to sort. Default is -1, which means sort along the last axis.

kind{‘introselect’}, optional

Selection algorithm. Default is ‘introselect’.

orderstr or list of str, optional

When a is an array with fields defined, this argument specifies which fields to compare first, second, etc. A single field can be specified as a string, and not all fields need to be specified, but unspecified fields will still be used, in the order in which they come up in the dtype, to break ties.

numpy.partition : Return a partitioned copy of an array. argpartition : Indirect partition. sort : Full sort.

See np.partition for notes on the different algorithms.

>>> a = np.array([3, 4, 2, 1])
>>> a.partition(3)
>>> a
array([2, 1, 3, 4])
>>> a.partition((1, 3))
>>> a
array([1, 2, 3, 4])

calculates matrices for polar decomposition

property principal_invariants

Returns a list of principal invariants for the tensor, which are the values of the coefficients of the characteristic polynomial for the matrix

prod(axis=None, dtype=None, out=None, keepdims=False, initial=1, where=True)

Return the product of the array elements over the given axis

Refer to for full documentation. : equivalent function

ptp(axis=None, out=None, keepdims=False)

Peak to peak (maximum - minimum) value along a given axis.

Refer to numpy.ptp for full documentation.

numpy.ptp : equivalent function

put(indices, values, mode='raise')

Set a.flat[n] = values[n] for all n in indices.

Refer to numpy.put for full documentation.

numpy.put : equivalent function


Return a flattened array.

Refer to numpy.ravel for full documentation.

numpy.ravel : equivalent function

ndarray.flat : a flat iterator on the array.


The real part of the array.

>>> x = np.sqrt([1+0j, 0+1j])
>>> x.real
array([ 1.        ,  0.70710678])
>>> x.real.dtype

numpy.real : equivalent function


Helper method for refining rotation matrix by ensuring that second and third rows are perpindicular to the first. Gets new y vector from an orthogonal projection of x onto y and the new z vector from a cross product of the new x and y


tol to test for rotation


new rotation matrix

repeat(repeats, axis=None)

Repeat elements of an array.

Refer to numpy.repeat for full documentation.

numpy.repeat : equivalent function

reshape(shape, order='C')

Returns an array containing the same data with a new shape.

Refer to numpy.reshape for full documentation.

numpy.reshape : equivalent function

Unlike the free function numpy.reshape, this method on ndarray allows the elements of the shape parameter to be passed in as separate arguments. For example, a.reshape(10, 11) is equivalent to a.reshape((10, 11)).

resize(new_shape, refcheck=True)

Change shape and size of array in-place.

new_shapetuple of ints, or n ints

Shape of resized array.

refcheckbool, optional

If False, reference count will not be checked. Default is True.



If a does not own its own data or references or views to it exist, and the data memory must be changed. PyPy only: will always raise if the data memory must be changed, since there is no reliable way to determine if references or views to it exist.


If the order keyword argument is specified. This behaviour is a bug in NumPy.

resize : Return a new array with the specified shape.

This reallocates space for the data area if necessary.

Only contiguous arrays (data elements consecutive in memory) can be resized.

The purpose of the reference count check is to make sure you do not use this array as a buffer for another Python object and then reallocate the memory. However, reference counts can increase in other ways so if you are sure that you have not shared the memory for this array with another Python object, then you may safely set refcheck to False.

Shrinking an array: array is flattened (in the order that the data are stored in memory), resized, and reshaped:

>>> a = np.array([[0, 1], [2, 3]], order='C')
>>> a.resize((2, 1))
>>> a
>>> a = np.array([[0, 1], [2, 3]], order='F')
>>> a.resize((2, 1))
>>> a

Enlarging an array: as above, but missing entries are filled with zeros:

>>> b = np.array([[0, 1], [2, 3]])
>>> b.resize(2, 3) # new_shape parameter doesn't have to be a tuple
>>> b
array([[0, 1, 2],
       [3, 0, 0]])

Referencing an array prevents resizing…

>>> c = a
>>> a.resize((1, 1))
Traceback (most recent call last):
ValueError: cannot resize an array that references or is referenced ...

Unless refcheck is False:

>>> a.resize((1, 1), refcheck=False)
>>> a
>>> c
rotate(matrix, tol=0.001)

Applies a rotation directly, and tests input matrix to ensure a valid rotation.


matrix (3x3 array-like): rotation matrix to be applied to tensor tol (float): tolerance for testing rotation matrix validity

round(decimals=0, out=None)

Return a with each element rounded to the given number of decimals.

Refer to numpy.around for full documentation.

numpy.around : equivalent function

searchsorted(v, side='left', sorter=None)

Find indices where elements of v should be inserted in a to maintain order.

For full documentation, see numpy.searchsorted

numpy.searchsorted : equivalent function

setfield(val, dtype, offset=0)

Put a value into a specified place in a field defined by a data-type.

Place val into a’s field defined by dtype and beginning offset bytes into the field.


Value to be placed in field.

dtypedtype object

Data-type of the field in which to place val.

offsetint, optional

The number of bytes into the field at which to place val.



>>> x = np.eye(3)
>>> x.getfield(np.float64)
array([[1.,  0.,  0.],
       [0.,  1.,  0.],
       [0.,  0.,  1.]])
>>> x.setfield(3, np.int32)
>>> x.getfield(np.int32)
array([[3, 3, 3],
       [3, 3, 3],
       [3, 3, 3]], dtype=int32)
>>> x
array([[1.0e+000, 1.5e-323, 1.5e-323],
       [1.5e-323, 1.0e+000, 1.5e-323],
       [1.5e-323, 1.5e-323, 1.0e+000]])
>>> x.setfield(np.eye(3), np.int32)
>>> x
array([[1.,  0.,  0.],
       [0.,  1.,  0.],
       [0.,  0.,  1.]])
setflags(write=None, align=None, uic=None)

Set array flags WRITEABLE, ALIGNED, WRITEBACKIFCOPY, respectively.

These Boolean-valued flags affect how numpy interprets the memory area used by a (see Notes below). The ALIGNED flag can only be set to True if the data is actually aligned according to the type. The WRITEBACKIFCOPY and flag can never be set to True. The flag WRITEABLE can only be set to True if the array owns its own memory, or the ultimate owner of the memory exposes a writeable buffer interface, or is a string. (The exception for string is made so that unpickling can be done without copying memory.)

writebool, optional

Describes whether or not a can be written to.

alignbool, optional

Describes whether or not a is aligned properly for its type.

uicbool, optional

Describes whether or not a is a copy of another “base” array.

Array flags provide information about how the memory area used for the array is to be interpreted. There are 7 Boolean flags in use, only four of which can be changed by the user: WRITEBACKIFCOPY, WRITEABLE, and ALIGNED.

WRITEABLE (W) the data area can be written to;

ALIGNED (A) the data and strides are aligned appropriately for the hardware (as determined by the compiler);

WRITEBACKIFCOPY (X) this array is a copy of some other array (referenced by .base). When the C-API function PyArray_ResolveWritebackIfCopy is called, the base array will be updated with the contents of this array.

All flags can be accessed using the single (upper case) letter as well as the full name.

>>> y = np.array([[3, 1, 7],
...               [2, 0, 0],
...               [8, 5, 9]])
>>> y
array([[3, 1, 7],
       [2, 0, 0],
       [8, 5, 9]])
>>> y.flags
  OWNDATA : True
  ALIGNED : True
>>> y.setflags(write=0, align=0)
>>> y.flags
  OWNDATA : True
  ALIGNED : False
>>> y.setflags(uic=1)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: cannot set WRITEBACKIFCOPY flag to True

Tuple of array dimensions.

The shape property is usually used to get the current shape of an array, but may also be used to reshape the array in-place by assigning a tuple of array dimensions to it. As with numpy.reshape, one of the new shape dimensions can be -1, in which case its value is inferred from the size of the array and the remaining dimensions. Reshaping an array in-place will fail if a copy is required.


Setting arr.shape is discouraged and may be deprecated in the future. Using ndarray.reshape is the preferred approach.

>>> x = np.array([1, 2, 3, 4])
>>> x.shape
>>> y = np.zeros((2, 3, 4))
>>> y.shape
(2, 3, 4)
>>> y.shape = (3, 8)
>>> y
array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.]])
>>> y.shape = (3, 6)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: total size of new array must be unchanged
>>> np.zeros((4,2))[::2].shape = (-1,)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: Incompatible shape for in-place modification. Use
`.reshape()` to make a copy with the desired shape.

numpy.shape : Equivalent getter function. numpy.reshape : Function similar to setting shape. ndarray.reshape : Method similar to setting shape.


Number of elements in the array.

Equal to, i.e., the product of the array’s dimensions.

a.size returns a standard arbitrary precision Python integer. This may not be the case with other methods of obtaining the same value (like the suggested, which returns an instance of np.int_), and may be relevant if the value is used further in calculations that may overflow a fixed size integer type.

>>> x = np.zeros((3, 5, 2), dtype=np.complex128)
>>> x.size
sort(axis=- 1, kind=None, order=None)

Sort an array in-place. Refer to numpy.sort for full documentation.

axisint, optional

Axis along which to sort. Default is -1, which means sort along the last axis.

kind{‘quicksort’, ‘mergesort’, ‘heapsort’, ‘stable’}, optional

Sorting algorithm. The default is ‘quicksort’. Note that both ‘stable’ and ‘mergesort’ use timsort under the covers and, in general, the actual implementation will vary with datatype. The ‘mergesort’ option is retained for backwards compatibility.

Changed in version 1.15.0: The ‘stable’ option was added.

orderstr or list of str, optional

When a is an array with fields defined, this argument specifies which fields to compare first, second, etc. A single field can be specified as a string, and not all fields need be specified, but unspecified fields will still be used, in the order in which they come up in the dtype, to break ties.

numpy.sort : Return a sorted copy of an array. numpy.argsort : Indirect sort. numpy.lexsort : Indirect stable sort on multiple keys. numpy.searchsorted : Find elements in sorted array. numpy.partition: Partial sort.

See numpy.sort for notes on the different sorting algorithms.

>>> a = np.array([[1,4], [3,1]])
>>> a.sort(axis=1)
>>> a
array([[1, 4],
       [1, 3]])
>>> a.sort(axis=0)
>>> a
array([[1, 3],
       [1, 4]])

Use the order keyword to specify a field to use when sorting a structured array:

>>> a = np.array([('a', 2), ('c', 1)], dtype=[('x', 'S1'), ('y', int)])
>>> a.sort(order='y')
>>> a
array([(b'c', 1), (b'a', 2)],
      dtype=[('x', 'S1'), ('y', '<i8')])

Remove axes of length one from a.

Refer to numpy.squeeze for full documentation.

numpy.squeeze : equivalent function

std(axis=None, dtype=None, out=None, ddof=0, keepdims=False, *, where=True)

Returns the standard deviation of the array elements along given axis.

Refer to numpy.std for full documentation.

numpy.std : equivalent function


Tuple of bytes to step in each dimension when traversing an array.

The byte offset of element (i[0], i[1], ..., i[n]) in an array a is:

offset = sum(np.array(i) * a.strides)

A more detailed explanation of strides can be found in the “ndarray.rst” file in the NumPy reference guide.


Setting arr.strides is discouraged and may be deprecated in the future. numpy.lib.stride_tricks.as_strided should be preferred to create a new view of the same data in a safer way.

Imagine an array of 32-bit integers (each 4 bytes):

x = np.array([[0, 1, 2, 3, 4],
              [5, 6, 7, 8, 9]], dtype=np.int32)

This array is stored in memory as 40 bytes, one after the other (known as a contiguous block of memory). The strides of an array tell us how many bytes we have to skip in memory to move to the next position along a certain axis. For example, we have to skip 4 bytes (1 value) to move to the next column, but 20 bytes (5 values) to get to the same position in the next row. As such, the strides for the array x will be (20, 4).


>>> y = np.reshape(np.arange(2*3*4), (2,3,4))
>>> y
array([[[ 0,  1,  2,  3],
        [ 4,  5,  6,  7],
        [ 8,  9, 10, 11]],
       [[12, 13, 14, 15],
        [16, 17, 18, 19],
        [20, 21, 22, 23]]])
>>> y.strides
(48, 16, 4)
>>> y[1,1,1]
>>> offset=sum(y.strides * np.array((1,1,1)))
>>> offset/y.itemsize
>>> x = np.reshape(np.arange(5*6*7*8), (5,6,7,8)).transpose(2,3,1,0)
>>> x.strides
(32, 4, 224, 1344)
>>> i = np.array([3,5,2,2])
>>> offset = sum(i * x.strides)
>>> x[3,5,2,2]
>>> offset / x.itemsize
sum(axis=None, dtype=None, out=None, keepdims=False, initial=0, where=True)

Return the sum of the array elements over the given axis.

Refer to numpy.sum for full documentation.

numpy.sum : equivalent function

swapaxes(axis1, axis2)

Return a view of the array with axis1 and axis2 interchanged.

Refer to numpy.swapaxes for full documentation.

numpy.swapaxes : equivalent function

property symmetrized

Returns a generally symmetrized tensor, calculated by taking the sum of the tensor and its transpose with respect to all possible permutations of indices

take(indices, axis=None, out=None, mode='raise')

Return an array formed from the elements of a at the given indices.

Refer to numpy.take for full documentation.

numpy.take : equivalent function


Construct Python bytes containing the raw data bytes in the array.

Constructs Python bytes showing a copy of the raw contents of data memory. The bytes object is produced in C-order by default. This behavior is controlled by the order parameter.

New in version 1.9.0.

order{‘C’, ‘F’, ‘A’}, optional

Controls the memory layout of the bytes object. ‘C’ means C-order, ‘F’ means F-order, ‘A’ (short for Any) means ‘F’ if a is Fortran contiguous, ‘C’ otherwise. Default is ‘C’.


Python bytes exhibiting a copy of a’s raw data.


Inverse of this operation, construct a 1-dimensional array from Python bytes.

>>> x = np.array([[0, 1], [2, 3]], dtype='<u2')
>>> x.tobytes()
>>> x.tobytes('C') == x.tobytes()
>>> x.tobytes('F')
tofile(fid, sep='', format='%s')

Write array to a file as text or binary (default).

Data is always written in ‘C’ order, independent of the order of a. The data produced by this method can be recovered using the function fromfile().

fidfile or str or Path

An open file object, or a string containing a filename.

Changed in version 1.17.0: pathlib.Path objects are now accepted.


Separator between array items for text output. If “” (empty), a binary file is written, equivalent to file.write(a.tobytes()).


Format string for text file output. Each entry in the array is formatted to text by first converting it to the closest Python type, and then using “format” % item.

This is a convenience function for quick storage of array data. Information on endianness and precision is lost, so this method is not a good choice for files intended to archive data or transport data between machines with different endianness. Some of these problems can be overcome by outputting the data as text files, at the expense of speed and file size.

When fid is a file object, array contents are directly written to the file, bypassing the file object’s write method. As a result, tofile cannot be used with files objects supporting compression (e.g., GzipFile) or file-like objects that do not support fileno() (e.g., BytesIO).


Return the array as an a.ndim-levels deep nested list of Python scalars.

Return a copy of the array data as a (nested) Python list. Data items are converted to the nearest compatible builtin Python type, via the item function.

If a.ndim is 0, then since the depth of the nested list is 0, it will not be a list at all, but a simple Python scalar.


yobject, or list of object, or list of list of object, or …

The possibly nested list of array elements.

The array may be recreated via a = np.array(a.tolist()), although this may sometimes lose precision.

For a 1D array, a.tolist() is almost the same as list(a), except that tolist changes numpy scalars to Python scalars:

>>> a = np.uint32([1, 2])
>>> a_list = list(a)
>>> a_list
[1, 2]
>>> type(a_list[0])
<class 'numpy.uint32'>
>>> a_tolist = a.tolist()
>>> a_tolist
[1, 2]
>>> type(a_tolist[0])
<class 'int'>

Additionally, for a 2D array, tolist applies recursively:

>>> a = np.array([[1, 2], [3, 4]])
>>> list(a)
[array([1, 2]), array([3, 4])]
>>> a.tolist()
[[1, 2], [3, 4]]

The base case for this recursion is a 0D array:

>>> a = np.array(1)
>>> list(a)
Traceback (most recent call last):
TypeError: iteration over a 0-d array
>>> a.tolist()

A compatibility alias for tobytes, with exactly the same behavior.

Despite its name, it returns bytes not strs.

Deprecated since version 1.19.0.

trace(offset=0, axis1=0, axis2=1, dtype=None, out=None)

Return the sum along diagonals of the array.

Refer to numpy.trace for full documentation.

numpy.trace : equivalent function

property trans

shorthand for transpose on SquareTensor


Applies a transformation (via a symmetry operation) to a tensor.


symm_op (SymmOp): a symmetry operation to apply to the tensor


Returns a view of the array with axes transposed.

Refer to numpy.transpose for full documentation.

axes : None, tuple of ints, or n ints

  • None or no argument: reverses the order of the axes.

  • tuple of ints: i in the j-th place in the tuple means that the array’s i-th axis becomes the transposed array’s j-th axis.

  • n ints: same as an n-tuple of the same ints (this form is intended simply as a “convenience” alternative to the tuple form).


View of the array with its axes suitably permuted.

transpose : Equivalent function. ndarray.T : Array property returning the array transposed. ndarray.reshape : Give a new shape to an array without changing its data.

>>> a = np.array([[1, 2], [3, 4]])
>>> a
array([[1, 2],
       [3, 4]])
>>> a.transpose()
array([[1, 3],
       [2, 4]])
>>> a.transpose((1, 0))
array([[1, 3],
       [2, 4]])
>>> a.transpose(1, 0)
array([[1, 3],
       [2, 4]])
>>> a = np.array([1, 2, 3, 4])
>>> a
array([1, 2, 3, 4])
>>> a.transpose()
array([1, 2, 3, 4])
var(axis=None, dtype=None, out=None, ddof=0, keepdims=False, *, where=True)

Returns the variance of the array elements, along given axis.

Refer to numpy.var for full documentation.

numpy.var : equivalent function

view([dtype][, type])

New view of array with the same data.


Passing None for dtype is different from omitting the parameter, since the former invokes dtype(None) which is an alias for dtype('float_').

dtypedata-type or ndarray sub-class, optional

Data-type descriptor of the returned view, e.g., float32 or int16. Omitting it results in the view having the same data-type as a. This argument can also be specified as an ndarray sub-class, which then specifies the type of the returned object (this is equivalent to setting the type parameter).

typePython type, optional

Type of the returned view, e.g., ndarray or matrix. Again, omission of the parameter results in type preservation.

a.view() is used two different ways:

a.view(some_dtype) or a.view(dtype=some_dtype) constructs a view of the array’s memory with a different data-type. This can cause a reinterpretation of the bytes of memory.

a.view(ndarray_subclass) or a.view(type=ndarray_subclass) just returns an instance of ndarray_subclass that looks at the same array (same shape, dtype, etc.) This does not cause a reinterpretation of the memory.

For a.view(some_dtype), if some_dtype has a different number of bytes per entry than the previous dtype (for example, converting a regular array to a structured array), then the last axis of a must be contiguous. This axis will be resized in the result.

Changed in version 1.23.0: Only the last axis needs to be contiguous. Previously, the entire array had to be C-contiguous.

>>> x = np.array([(1, 2)], dtype=[('a', np.int8), ('b', np.int8)])

Viewing array data using a different type and dtype:

>>> y = x.view(dtype=np.int16, type=np.matrix)
>>> y
matrix([[513]], dtype=int16)
>>> print(type(y))
<class 'numpy.matrix'>

Creating a view on a structured array so it can be used in calculations

>>> x = np.array([(1, 2),(3,4)], dtype=[('a', np.int8), ('b', np.int8)])
>>> xv = x.view(dtype=np.int8).reshape(-1,2)
>>> xv
array([[1, 2],
       [3, 4]], dtype=int8)
>>> xv.mean(0)
array([2.,  3.])

Making changes to the view changes the underlying array

>>> xv[0,1] = 20
>>> x
array([(1, 20), (3,  4)], dtype=[('a', 'i1'), ('b', 'i1')])

Using a view to convert an array to a recarray:

>>> z = x.view(np.recarray)
>>> z.a
array([1, 3], dtype=int8)

Views share data:

>>> x[0] = (9, 10)
>>> z[0]
(9, 10)

Views that change the dtype size (bytes per entry) should normally be avoided on arrays defined by slices, transposes, fortran-ordering, etc.:

>>> x = np.array([[1, 2, 3], [4, 5, 6]], dtype=np.int16)
>>> y = x[:, ::2]
>>> y
array([[1, 3],
       [4, 6]], dtype=int16)
>>> y.view(dtype=[('width', np.int16), ('length', np.int16)])
Traceback (most recent call last):
ValueError: To change to a dtype of a different size, the last axis must be contiguous
>>> z = y.copy()
>>> z.view(dtype=[('width', np.int16), ('length', np.int16)])
array([[(1, 3)],
       [(4, 6)]], dtype=[('width', '<i2'), ('length', '<i2')])

However, views that change dtype are totally fine for arrays with a contiguous last axis, even if the rest of the axes are not C-contiguous:

>>> x = np.arange(2 * 3 * 4, dtype=np.int8).reshape(2, 3, 4)
>>> x.transpose(1, 0, 2).view(np.int16)
array([[[ 256,  770],
        [3340, 3854]],

       [[1284, 1798],
        [4368, 4882]],

       [[2312, 2826],
        [5396, 5910]]], dtype=int16)
property voigt

Returns the tensor in Voigt notation

property voigt_symmetrized

Returns a “voigt”-symmetrized tensor, i. e. a voigt-notation tensor such that it is invariant wrt permutation of indices


returns the matrix with all entries below a certain threshold (i.e. tol) set to zero