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1"""Cholesky decomposition functions."""
3from numpy import asarray_chkfinite, asarray, atleast_2d
5# Local imports
6from .misc import LinAlgError, _datacopied
7from .lapack import get_lapack_funcs
9__all__ = ['cholesky', 'cho_factor', 'cho_solve', 'cholesky_banded',
10 'cho_solve_banded']
13def _cholesky(a, lower=False, overwrite_a=False, clean=True,
14 check_finite=True):
15 """Common code for cholesky() and cho_factor()."""
17 a1 = asarray_chkfinite(a) if check_finite else asarray(a)
18 a1 = atleast_2d(a1)
20 # Dimension check
21 if a1.ndim != 2:
22 raise ValueError('Input array needs to be 2D but received '
23 'a {}d-array.'.format(a1.ndim))
24 # Squareness check
25 if a1.shape[0] != a1.shape[1]:
26 raise ValueError('Input array is expected to be square but has '
27 'the shape: {}.'.format(a1.shape))
29 # Quick return for square empty array
30 if a1.size == 0:
31 return a1.copy(), lower
33 overwrite_a = overwrite_a or _datacopied(a1, a)
34 potrf, = get_lapack_funcs(('potrf',), (a1,))
35 c, info = potrf(a1, lower=lower, overwrite_a=overwrite_a, clean=clean)
36 if info > 0:
37 raise LinAlgError("%d-th leading minor of the array is not positive "
38 "definite" % info)
39 if info < 0:
40 raise ValueError('LAPACK reported an illegal value in {}-th argument'
41 'on entry to "POTRF".'.format(-info))
42 return c, lower
45def cholesky(a, lower=False, overwrite_a=False, check_finite=True):
46 """
47 Compute the Cholesky decomposition of a matrix.
49 Returns the Cholesky decomposition, :math:`A = L L^*` or
50 :math:`A = U^* U` of a Hermitian positive-definite matrix A.
52 Parameters
53 ----------
54 a : (M, M) array_like
55 Matrix to be decomposed
56 lower : bool, optional
57 Whether to compute the upper- or lower-triangular Cholesky
58 factorization. Default is upper-triangular.
59 overwrite_a : bool, optional
60 Whether to overwrite data in `a` (may improve performance).
61 check_finite : bool, optional
62 Whether to check that the input matrix contains only finite numbers.
63 Disabling may give a performance gain, but may result in problems
64 (crashes, non-termination) if the inputs do contain infinities or NaNs.
66 Returns
67 -------
68 c : (M, M) ndarray
69 Upper- or lower-triangular Cholesky factor of `a`.
71 Raises
72 ------
73 LinAlgError : if decomposition fails.
75 Examples
76 --------
77 >>> from scipy.linalg import cholesky
78 >>> a = np.array([[1,-2j],[2j,5]])
79 >>> L = cholesky(a, lower=True)
80 >>> L
81 array([[ 1.+0.j, 0.+0.j],
82 [ 0.+2.j, 1.+0.j]])
83 >>> L @ L.T.conj()
84 array([[ 1.+0.j, 0.-2.j],
85 [ 0.+2.j, 5.+0.j]])
87 """
88 c, lower = _cholesky(a, lower=lower, overwrite_a=overwrite_a, clean=True,
89 check_finite=check_finite)
90 return c
93def cho_factor(a, lower=False, overwrite_a=False, check_finite=True):
94 """
95 Compute the Cholesky decomposition of a matrix, to use in cho_solve
97 Returns a matrix containing the Cholesky decomposition,
98 ``A = L L*`` or ``A = U* U`` of a Hermitian positive-definite matrix `a`.
99 The return value can be directly used as the first parameter to cho_solve.
101 .. warning::
102 The returned matrix also contains random data in the entries not
103 used by the Cholesky decomposition. If you need to zero these
104 entries, use the function `cholesky` instead.
106 Parameters
107 ----------
108 a : (M, M) array_like
109 Matrix to be decomposed
110 lower : bool, optional
111 Whether to compute the upper or lower triangular Cholesky factorization
112 (Default: upper-triangular)
113 overwrite_a : bool, optional
114 Whether to overwrite data in a (may improve performance)
115 check_finite : bool, optional
116 Whether to check that the input matrix contains only finite numbers.
117 Disabling may give a performance gain, but may result in problems
118 (crashes, non-termination) if the inputs do contain infinities or NaNs.
120 Returns
121 -------
122 c : (M, M) ndarray
123 Matrix whose upper or lower triangle contains the Cholesky factor
124 of `a`. Other parts of the matrix contain random data.
125 lower : bool
126 Flag indicating whether the factor is in the lower or upper triangle
128 Raises
129 ------
130 LinAlgError
131 Raised if decomposition fails.
133 See also
134 --------
135 cho_solve : Solve a linear set equations using the Cholesky factorization
136 of a matrix.
138 Examples
139 --------
140 >>> from scipy.linalg import cho_factor
141 >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]])
142 >>> c, low = cho_factor(A)
143 >>> c
144 array([[3. , 1. , 0.33333333, 1.66666667],
145 [3. , 2.44948974, 1.90515869, -0.27216553],
146 [1. , 5. , 2.29330749, 0.8559528 ],
147 [5. , 1. , 2. , 1.55418563]])
148 >>> np.allclose(np.triu(c).T @ np. triu(c) - A, np.zeros((4, 4)))
149 True
151 """
152 c, lower = _cholesky(a, lower=lower, overwrite_a=overwrite_a, clean=False,
153 check_finite=check_finite)
154 return c, lower
157def cho_solve(c_and_lower, b, overwrite_b=False, check_finite=True):
158 """Solve the linear equations A x = b, given the Cholesky factorization of A.
160 Parameters
161 ----------
162 (c, lower) : tuple, (array, bool)
163 Cholesky factorization of a, as given by cho_factor
164 b : array
165 Right-hand side
166 overwrite_b : bool, optional
167 Whether to overwrite data in b (may improve performance)
168 check_finite : bool, optional
169 Whether to check that the input matrices contain only finite numbers.
170 Disabling may give a performance gain, but may result in problems
171 (crashes, non-termination) if the inputs do contain infinities or NaNs.
173 Returns
174 -------
175 x : array
176 The solution to the system A x = b
178 See also
179 --------
180 cho_factor : Cholesky factorization of a matrix
182 Examples
183 --------
184 >>> from scipy.linalg import cho_factor, cho_solve
185 >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]])
186 >>> c, low = cho_factor(A)
187 >>> x = cho_solve((c, low), [1, 1, 1, 1])
188 >>> np.allclose(A @ x - [1, 1, 1, 1], np.zeros(4))
189 True
191 """
192 (c, lower) = c_and_lower
193 if check_finite:
194 b1 = asarray_chkfinite(b)
195 c = asarray_chkfinite(c)
196 else:
197 b1 = asarray(b)
198 c = asarray(c)
199 if c.ndim != 2 or c.shape[0] != c.shape[1]:
200 raise ValueError("The factored matrix c is not square.")
201 if c.shape[1] != b1.shape[0]:
202 raise ValueError("incompatible dimensions.")
204 overwrite_b = overwrite_b or _datacopied(b1, b)
206 potrs, = get_lapack_funcs(('potrs',), (c, b1))
207 x, info = potrs(c, b1, lower=lower, overwrite_b=overwrite_b)
208 if info != 0:
209 raise ValueError('illegal value in %dth argument of internal potrs'
210 % -info)
211 return x
214def cholesky_banded(ab, overwrite_ab=False, lower=False, check_finite=True):
215 """
216 Cholesky decompose a banded Hermitian positive-definite matrix
218 The matrix a is stored in ab either in lower-diagonal or upper-
219 diagonal ordered form::
221 ab[u + i - j, j] == a[i,j] (if upper form; i <= j)
222 ab[ i - j, j] == a[i,j] (if lower form; i >= j)
224 Example of ab (shape of a is (6,6), u=2)::
226 upper form:
227 * * a02 a13 a24 a35
228 * a01 a12 a23 a34 a45
229 a00 a11 a22 a33 a44 a55
231 lower form:
232 a00 a11 a22 a33 a44 a55
233 a10 a21 a32 a43 a54 *
234 a20 a31 a42 a53 * *
236 Parameters
237 ----------
238 ab : (u + 1, M) array_like
239 Banded matrix
240 overwrite_ab : bool, optional
241 Discard data in ab (may enhance performance)
242 lower : bool, optional
243 Is the matrix in the lower form. (Default is upper form)
244 check_finite : bool, optional
245 Whether to check that the input matrix contains only finite numbers.
246 Disabling may give a performance gain, but may result in problems
247 (crashes, non-termination) if the inputs do contain infinities or NaNs.
249 Returns
250 -------
251 c : (u + 1, M) ndarray
252 Cholesky factorization of a, in the same banded format as ab
254 See also
255 --------
256 cho_solve_banded : Solve a linear set equations, given the Cholesky factorization
257 of a banded hermitian.
259 Examples
260 --------
261 >>> from scipy.linalg import cholesky_banded
262 >>> from numpy import allclose, zeros, diag
263 >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]])
264 >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1)
265 >>> A = A + A.conj().T + np.diag(Ab[2, :])
266 >>> c = cholesky_banded(Ab)
267 >>> C = np.diag(c[0, 2:], k=2) + np.diag(c[1, 1:], k=1) + np.diag(c[2, :])
268 >>> np.allclose(C.conj().T @ C - A, np.zeros((5, 5)))
269 True
271 """
272 if check_finite:
273 ab = asarray_chkfinite(ab)
274 else:
275 ab = asarray(ab)
277 pbtrf, = get_lapack_funcs(('pbtrf',), (ab,))
278 c, info = pbtrf(ab, lower=lower, overwrite_ab=overwrite_ab)
279 if info > 0:
280 raise LinAlgError("%d-th leading minor not positive definite" % info)
281 if info < 0:
282 raise ValueError('illegal value in %d-th argument of internal pbtrf'
283 % -info)
284 return c
287def cho_solve_banded(cb_and_lower, b, overwrite_b=False, check_finite=True):
288 """
289 Solve the linear equations ``A x = b``, given the Cholesky factorization of
290 the banded hermitian ``A``.
292 Parameters
293 ----------
294 (cb, lower) : tuple, (ndarray, bool)
295 `cb` is the Cholesky factorization of A, as given by cholesky_banded.
296 `lower` must be the same value that was given to cholesky_banded.
297 b : array_like
298 Right-hand side
299 overwrite_b : bool, optional
300 If True, the function will overwrite the values in `b`.
301 check_finite : bool, optional
302 Whether to check that the input matrices contain only finite numbers.
303 Disabling may give a performance gain, but may result in problems
304 (crashes, non-termination) if the inputs do contain infinities or NaNs.
306 Returns
307 -------
308 x : array
309 The solution to the system A x = b
311 See also
312 --------
313 cholesky_banded : Cholesky factorization of a banded matrix
315 Notes
316 -----
318 .. versionadded:: 0.8.0
320 Examples
321 --------
322 >>> from scipy.linalg import cholesky_banded, cho_solve_banded
323 >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]])
324 >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1)
325 >>> A = A + A.conj().T + np.diag(Ab[2, :])
326 >>> c = cholesky_banded(Ab)
327 >>> x = cho_solve_banded((c, False), np.ones(5))
328 >>> np.allclose(A @ x - np.ones(5), np.zeros(5))
329 True
331 """
332 (cb, lower) = cb_and_lower
333 if check_finite:
334 cb = asarray_chkfinite(cb)
335 b = asarray_chkfinite(b)
336 else:
337 cb = asarray(cb)
338 b = asarray(b)
340 # Validate shapes.
341 if cb.shape[-1] != b.shape[0]:
342 raise ValueError("shapes of cb and b are not compatible.")
344 pbtrs, = get_lapack_funcs(('pbtrs',), (cb, b))
345 x, info = pbtrs(cb, b, lower=lower, overwrite_b=overwrite_b)
346 if info > 0:
347 raise LinAlgError("%dth leading minor not positive definite" % info)
348 if info < 0:
349 raise ValueError('illegal value in %dth argument of internal pbtrs'
350 % -info)
351 return x