Coverage for /builds/ase/ase/ase/geometry/minkowski_reduction.py: 96.95%
131 statements
« prev ^ index » next coverage.py v7.5.3, created at 2025-08-02 00:12 +0000
1# fmt: off
3import itertools
5import numpy as np
7from ase.cell import Cell
8from ase.utils import pbc2pbc
10TOL = 1E-12
11MAX_IT = 100000 # in practice this is not exceeded
14class CycleChecker:
16 def __init__(self, d):
17 assert d in [2, 3]
19 # worst case is the hexagonal cell in 2D and the fcc cell in 3D
20 n = {2: 6, 3: 12}[d]
22 # max cycle length is total number of primtive cell descriptions
23 max_cycle_length = np.prod([n - i for i in range(d)]) * np.prod(d)
24 self.visited = np.zeros((max_cycle_length, 3 * d), dtype=int)
26 def add_site(self, H):
27 # flatten array for simplicity
28 H = H.ravel()
30 # check if site exists
31 found = (self.visited == H).all(axis=1).any()
33 # shift all visited sites down and place current site at the top
34 self.visited = np.roll(self.visited, 1, axis=0)
35 self.visited[0] = H
36 return found
39def reduction_gauss(B, hu, hv):
40 """Calculate a Gauss-reduced lattice basis (2D reduction)."""
41 cycle_checker = CycleChecker(d=2)
42 u = hu @ B
43 v = hv @ B
45 for _ in range(MAX_IT):
46 x = int(round(np.dot(u, v) / np.dot(u, u)))
47 hu, hv = hv - x * hu, hu
48 u = hu @ B
49 v = hv @ B
50 site = np.array([hu, hv])
51 if np.dot(u, u) >= np.dot(v, v) or cycle_checker.add_site(site):
52 return hv, hu
54 raise RuntimeError(f"Gaussian basis not found after {MAX_IT} iterations")
57def relevant_vectors_2D(u, v):
58 cs = np.array([e for e in itertools.product([-1, 0, 1], repeat=2)])
59 vs = cs @ [u, v]
60 indices = np.argsort(np.linalg.norm(vs, axis=1))[:7]
61 return vs[indices], cs[indices]
64def closest_vector(t0, u, v):
65 t = t0
66 a = np.zeros(2, dtype=int)
67 rs, cs = relevant_vectors_2D(u, v)
69 dprev = float("inf")
70 for _ in range(MAX_IT):
71 ds = np.linalg.norm(rs + t, axis=1)
72 index = np.argmin(ds)
73 if index == 0 or ds[index] >= dprev:
74 return a
76 dprev = ds[index]
77 r = rs[index]
78 kopt = int(round(-np.dot(t, r) / np.dot(r, r)))
79 a += kopt * cs[index]
80 t = t0 + a[0] * u + a[1] * v
82 raise RuntimeError(f"Closest vector not found after {MAX_IT} iterations")
85def reduction_full(B):
86 """Calculate a Minkowski-reduced lattice basis (3D reduction)."""
87 cycle_checker = CycleChecker(d=3)
88 H = np.eye(3, dtype=int)
89 norms = np.linalg.norm(B, axis=1)
91 for _ in range(MAX_IT):
92 # Sort vectors by norm
93 H = H[np.argsort(norms, kind='merge')]
95 # Gauss-reduce smallest two vectors
96 hw = H[2]
97 hu, hv = reduction_gauss(B, H[0], H[1])
98 H = np.array([hu, hv, hw])
99 R = H @ B
101 # Orthogonalize vectors using Gram-Schmidt
102 u, v, _ = R
103 X = u / np.linalg.norm(u)
104 Y = v - X * np.dot(v, X)
105 Y /= np.linalg.norm(Y)
107 # Find closest vector to last element of R
108 pu, pv, pw = R @ np.array([X, Y]).T
109 nb = closest_vector(pw, pu, pv)
111 # Update basis
112 H[2] = [nb[0], nb[1], 1] @ H
113 R = H @ B
115 norms = np.linalg.norm(R, axis=1)
116 if norms[2] >= norms[1] or cycle_checker.add_site(H):
117 return R, H
119 raise RuntimeError(f"Reduced basis not found after {MAX_IT} iterations")
122def is_minkowski_reduced(cell, pbc=True):
123 """Tests if a cell is Minkowski-reduced.
125 Parameters:
127 cell: array
128 The lattice basis to test (in row-vector format).
129 pbc: array, optional
130 The periodic boundary conditions of the cell (Default `True`).
131 If `pbc` is provided, only periodic cell vectors are tested.
133 Returns:
135 is_reduced: bool
136 True if cell is Minkowski-reduced, False otherwise.
137 """
139 """These conditions are due to Minkowski, but a nice description in English
140 can be found in the thesis of Carine Jaber: "Algorithmic approaches to
141 Siegel's fundamental domain", https://www.theses.fr/2017UBFCK006.pdf
142 This is also good background reading for Minkowski reduction.
144 0D and 1D cells are trivially reduced. For 2D cells, the conditions which
145 an already-reduced basis fulfil are:
146 |b1| ≤ |b2|
147 |b2| ≤ |b1 - b2|
148 |b2| ≤ |b1 + b2|
150 For 3D cells, the conditions which an already-reduced basis fulfil are:
151 |b1| ≤ |b2| ≤ |b3|
153 |b1 + b2| ≥ |b2|
154 |b1 + b3| ≥ |b3|
155 |b2 + b3| ≥ |b3|
156 |b1 - b2| ≥ |b2|
157 |b1 - b3| ≥ |b3|
158 |b2 - b3| ≥ |b3|
159 |b1 + b2 + b3| ≥ |b3|
160 |b1 - b2 + b3| ≥ |b3|
161 |b1 + b2 - b3| ≥ |b3|
162 |b1 - b2 - b3| ≥ |b3|
163 """
164 pbc = pbc2pbc(pbc)
165 dim = pbc.sum()
166 if dim <= 1:
167 return True
169 if dim == 2:
170 # reorder cell vectors to [shortest, longest, aperiodic]
171 cell = cell.copy()
172 cell[np.argmin(pbc)] = 0
173 norms = np.linalg.norm(cell, axis=1)
174 cell = cell[np.argsort(norms)[[1, 2, 0]]]
176 A = [[0, 1, 0],
177 [1, -1, 0],
178 [1, 1, 0]]
179 lhs = np.linalg.norm(A @ cell, axis=1)
180 norms = np.linalg.norm(cell, axis=1)
181 rhs = norms[[0, 1, 1]]
182 else:
183 A = [[0, 1, 0],
184 [0, 0, 1],
185 [1, 1, 0],
186 [1, 0, 1],
187 [0, 1, 1],
188 [1, -1, 0],
189 [1, 0, -1],
190 [0, 1, -1],
191 [1, 1, 1],
192 [1, -1, 1],
193 [1, 1, -1],
194 [1, -1, -1]]
195 lhs = np.linalg.norm(A @ cell, axis=1)
196 norms = np.linalg.norm(cell, axis=1)
197 rhs = norms[[0, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2]]
198 return (lhs >= rhs - TOL).all()
201def minkowski_reduce(cell, pbc=True):
202 """Calculate a Minkowski-reduced lattice basis. The reduced basis
203 has the shortest possible vector lengths and has
204 norm(a) <= norm(b) <= norm(c).
206 Implements the method described in:
208 Low-dimensional Lattice Basis Reduction Revisited
209 Nguyen, Phong Q. and Stehlé, Damien,
210 ACM Trans. Algorithms 5(4) 46:1--46:48, 2009
211 :doi:`10.1145/1597036.1597050`
213 Parameters:
215 cell: array
216 The lattice basis to reduce (in row-vector format).
217 pbc: array, optional
218 The periodic boundary conditions of the cell (Default `True`).
219 If `pbc` is provided, only periodic cell vectors are reduced.
221 Returns:
223 rcell: array
224 The reduced lattice basis.
225 op: array
226 The unimodular matrix transformation (rcell = op @ cell).
227 """
228 cell = Cell(cell)
229 pbc = pbc2pbc(pbc)
230 dim = pbc.sum()
231 op = np.eye(3, dtype=int)
232 if is_minkowski_reduced(cell, pbc):
233 return cell, op
235 if dim == 2:
236 # permute cell so that first two vectors are the periodic ones
237 perm = np.argsort(pbc, kind='merge')[::-1] # stable sort
238 pcell = cell[perm][:, perm]
240 # perform gauss reduction
241 norms = np.linalg.norm(pcell, axis=1)
242 norms[2] = float("inf")
243 indices = np.argsort(norms)
244 op = op[indices]
245 hu, hv = reduction_gauss(pcell, op[0], op[1])
246 op[0] = hu
247 op[1] = hv
249 # undo above permutation
250 invperm = np.argsort(perm)
251 op = op[invperm][:, invperm]
253 # maintain cell handedness
254 index = np.argmin(pbc)
255 normal = np.cross(cell[index - 2], cell[index - 1])
256 normal /= np.linalg.norm(normal)
258 _cell = cell.copy()
259 _cell[index] = normal
260 _rcell = op @ cell
261 _rcell[index] = normal
262 if _cell.handedness != Cell(_rcell).handedness:
263 op[index - 1] *= -1
265 elif dim == 3:
266 _, op = reduction_full(cell)
267 # maintain cell handedness
268 if cell.handedness != Cell(op @ cell).handedness:
269 op = -op
271 norms1 = np.sort(np.linalg.norm(cell, axis=1))
272 norms2 = np.sort(np.linalg.norm(op @ cell, axis=1))
273 if (norms2 > norms1 + TOL).any():
274 raise RuntimeError("Minkowski reduction failed")
275 return op @ cell, op