13.2.7. Mathematical helper functions — `MDAnalysis.lib.mdamath`

Helper functions for common mathematical operations. Some of these functions are written in C/cython for higher performance.

13.2.7.1. Linear algebra

MDAnalysis.lib.mdamath.normal(vec1, vec2)[source]

Returns the unit vector normal to two vectors.

\[\hat{\mathbf{n}} = \frac{\mathbf{v}_1 \times \mathbf{v}_2}{|\mathbf{v}_1 \times \mathbf{v}_2|}\]

If the two vectors are collinear, the vector \(\mathbf{0}\) is returned.

Changed in version 0.11.0: Moved into lib.mdamath

MDAnalysis.lib.mdamath.norm(v)[source]

Calculate the norm of a vector v.

\[v = \sqrt{\mathbf{v}\cdot\mathbf{v}}\]

This version is faster then numpy.linalg.norm because it only works for a single vector and therefore can skip a lot of the additional fuss linalg.norm does.

Parameters: v (array_like) – 1D array of shape (N) for a vector of length N
Returns: norm of the vector
Return type: float

MDAnalysis.lib.mdamath.pdot(a, b)[source]

Pairwise dot product.

a must be the same shape as b.

Parameters

a (numpy.ndarray of shape (N, M)) –
b (numpy.ndarray of shape (N, M)) –

Return type

numpy.ndarray of shape (N,)

MDAnalysis.lib.mdamath.pnorm(a)[source]

Euclidean norm of each vector in a matrix

Parameters: a (numpy.ndarray of shape (N, M)) –
Return type: numpy.ndarray of shape (N,)

MDAnalysis.lib.mdamath.angle(a, b)[source]: Returns the angle between two vectors in radians

Changed in version 0.11.0: Moved into lib.mdamath

Changed in version 1.0.0: Changed rounding-off code to use np.clip(). Values lower than -1.0 now return np.pi instead of -np.pi

MDAnalysis.lib.mdamath.dihedral(ab, bc, cd)[source]

Returns the dihedral angle in radians between vectors connecting A,B,C,D.

The dihedral measures the rotation around bc:

  ab
A---->B
       \ bc
       _\'
         C---->D
           cd

The dihedral angle is restricted to the range -π <= x <= π.

New in version 0.8.

Changed in version 0.11.0: Moved into lib.mdamath

MDAnalysis.lib.mdamath.stp(vec1, vec2, vec3)[source]

Takes the scalar triple product of three vectors.

Returns the volume V of the parallel epiped spanned by the three vectors

\[V = \mathbf{v}_3 \cdot (\mathbf{v}_1 \times \mathbf{v}_2)\]

Changed in version 0.11.0: Moved into lib.mdamath

MDAnalysis.lib.mdamath.sarrus_det(matrix)[source]

Computes the determinant of a 3x3 matrix according to the rule of Sarrus.

If an array of 3x3 matrices is supplied, determinants are computed per matrix and returned as an appropriately shaped numpy array.

Parameters: matrix (numpy.ndarray) – An array of shape (..., 3, 3) with the 3x3 matrices residing in the last two dimensions.
Returns: det – The determinant(s) of matrix. If matrix.shape == (3, 3), the determinant will be returned as a scalar. If matrix.shape == (..., 3, 3), the determinants will be returned as a numpy.ndarray of shape (...,) and dtype numpy.float64.
Return type: float or numpy.ndarray
Raises: ValueError: – If matrix has less than two dimensions or its last two dimensions are not of shape (3, 3).

New in version 0.20.0.

MDAnalysis.lib.mdamath.triclinic_box(x, y, z)[source]

Convert the three triclinic box vectors to [lx, ly, lz, alpha, beta, gamma].

If the resulting box is invalid, i.e., any box length is zero or negative, or any angle is outside the open interval (0, 180), a zero vector will be returned.

All angles are in degrees and defined as follows:

alpha = angle(y,z)
beta = angle(x,z)
gamma = angle(x,y)

Parameters

x (array_like) – Array of shape (3,) representing the first box vector
y (array_like) – Array of shape (3,) representing the second box vector
z (array_like) – Array of shape (3,) representing the third box vector

Returns

A numpy array of shape (6,) and dtype np.float32 providing the unitcell dimensions in the same format as returned by MDAnalysis.coordinates.base.Timestep.dimensions:

[lx, ly, lz, alpha, beta, gamma].

Invalid boxes are returned as a zero vector.

Return type

numpy.ndarray

Note

Definition of angles: http://en.wikipedia.org/wiki/Lattice_constant

13.2.7.2. Connectivity

MDAnalysis.lib.mdamath.make_whole()

Move all atoms in a single molecule so that bonds don’t split over images.

This function is most useful when atoms have been packed into the primary unit cell, causing breaks mid molecule, with the molecule then appearing on either side of the unit cell. This is problematic for operations such as calculating the center of mass of the molecule.

+-----------+     +-----------+
|           |     |           |
| 6       3 |     |         3 | 6
| !       ! |     |         ! | !
|-5-8   1-2-| ->  |       1-2-|-5-8
| !       ! |     |         ! | !
| 7       4 |     |         4 | 7
|           |     |           |
+-----------+     +-----------+

Parameters

atomgroup (AtomGroup) – The MDAnalysis.core.groups.AtomGroup to work with. The positions of this are modified in place. All these atoms must belong to the same molecule or fragment.
reference_atom (Atom) – The atom around which all other atoms will be moved. Defaults to atom 0 in the atomgroup.
inplace (bool, optional) – If True, coordinates are modified in place.

Returns

coords – The unwrapped atom coordinates.

Return type

numpy.ndarray

Raises

NoDataError – There are no bonds present. (See guess_bonds())
ValueError – The algorithm fails to work. This is usually caused by the atomgroup not being a single fragment. (ie the molecule can’t be traversed by following bonds)

Example

Make fragments whole:

from MDAnalysis.lib.mdamath import make_whole

# This algorithm requires bonds, these can be guessed!
u = mda.Universe(......, guess_bonds=True)

# MDAnalysis can split AtomGroups into their fragments
# based on bonding information.
# Note that this function will only handle a single fragment
# at a time, necessitating a loop.
for frag in u.atoms.fragments:
    make_whole(frag)

Alternatively, to keep a single atom in place as the anchor:

# This will mean that atomgroup[10] will NOT get moved,
# and all other atoms will move (if necessary).
make_whole(atomgroup, reference_atom=atomgroup[10])

13.2.7. Mathematical helper functions — MDAnalysis.lib.mdamath

13.2.7.1. Linear algebra

13.2.7.2. Connectivity

13.2.7. Mathematical helper functions — `MDAnalysis.lib.mdamath`