1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
|
/* MIT License
*
* Copyright (c) 2016--2017 Felix Lenders
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in all
* copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
* SOFTWARE.
*
*/
#ifndef TRLIB_EIGEN_INVERSE_H
#define TRLIB_EIGEN_INVERSE_H
#define TRLIB_EIR_CONV (0)
#define TRLIB_EIR_ITMAX (-1)
#define TRLIB_EIR_FAIL_FACTOR (-2)
#define TRLIB_EIR_FAIL_LINSOLVE (-3)
#define TRLIB_EIR_N_STARTVEC (5)
/** Computes eigenvector to provided eigenvalue of symmetric tridiagonal matrix :math:`T \in \mathbb R^{n\times n}`,
* using inverse iteration.
*
* For a description of the method see https://en.wikipedia.org/wiki/Inverse_iteration.
*
* **Convergence**
*
* Convergence is reported if :math:`\vert \frac{1}{\Vert w_{i+1} \Vert} - \texttt{pert} \vert \le \texttt{tol}\_\texttt{abs}`, where :math:`(T-\lambda I) w_{i+1} = v_i`, :math:`v_i` the current normalized iterate and :math:`\texttt{pert}` is the perturbation applied to the provided eigenvalue.
*
* :param n: dimension, ensure :math:`n > 0`
* :type n: trlib_int_t, input
* :param diag: pointer to array holding diagonal of :math:`T`, length :c:data:`n`
* :type diag: trlib_flt_t, input
* :param offdiag: pointer to array holding offdiagonal of :math:`T`, length :c:data:`n-1`
* :type offdiag: trlib_flt_t, input
* :param lam_init: estimation of eigenvalue corresponding to eigenvector to compute
* :type lam_init: trlib_flt_t, input
* :param itmax: maximum number of iterations
* :type itmax: trlib_int_t, input
* :param tol_abs: absolute stopping tolerance in inverse iteration, good default may be :math:`\sqrt{\texttt{macheps}}` (:c:macro:`TRLIB_EPS_POW_5`)
* :type tol_abs: trlib_flt_t, input
* :param ones: array with every value ``1.0``, length :c:data:`n`
* :type ones: trlib_flt_t, input
* :param diag_fac: pointer to array holding diagonal of Cholesky factorization of :math:`T - \lambda I`, length :c:data:`n`
*
* - on entry: allocated memory
* - on exit: factorization corresponding to computed eigenvalue @p lam
*
* :type diag_fac: trlib_flt_t, input/output
* :param offdiag_fac: pointer to array holding offdiagonal of Cholesky factorization of :math:`T - \lambda I`, length :c:data:`n-1`
*
* - on entry: allocated memory
* - on exit: factorization corresponding to computed eigenvalue @p lam
*
* :type offidag_fac: trlib_flt_t, input/output
* :param eig: pointer to array holding eigenvector, length :c:data:`n`
* :type eig: trlib_flt_t, input/output
* :param verbose: determines the verbosity level of output that is written to :c:data:`fout`
* :type verbose: trlib_int_t, input
* :param unicode: set to ``1`` if :c:data:`fout` can handle unicode, otherwise to ``0``
* :type unicode: trlib_int_t, input
* :param prefix: string that is printed before iteration output
* :type prefix: char, input
* :param fout: output stream
* :type fout: FILE, input
* :param timing: gives timing details, provide allocated zero initialized memory of length :c:func:`trlib_eigen_timing_size`
*
* ====== ================================
* block description
* ====== ================================
* 0 total duration
* 1 timing of linear algebra calls
* ====== ================================
*
* :type timing: trlib_int_t, input/output
* :param lam_pert: eigenvalue corresponding to eigenvector
* :type lam_pert: trlib_flt_t, output
* :param pert: perturbation applied to provided eigenvalue
* :type pert: trlib_flt_t, output
* :param iter_inv: number of inverse iterations
* :type iter_inv: trlib_int_t, output
*
* :returns: status
*
* - :c:macro:`TRLIB_EIR_CONV` success
* - :c:macro:`TRLIB_EIR_ITMAX` iteration limit exceeded
* - :c:macro:`TRLIB_EIR_FAIL_FACTOR` failure on matrix factorization
* - :c:macro:`TRLIB_EIR_FAIL_LINSOLVE` failure on backsolve
*
* :rtype: trlib_int_t
*/
trlib_int_t trlib_eigen_inverse(
trlib_int_t n, trlib_flt_t *diag, trlib_flt_t *offdiag,
trlib_flt_t lam_init, trlib_int_t itmax, trlib_flt_t tol_abs,
trlib_flt_t *ones, trlib_flt_t *diag_fac, trlib_flt_t *offdiag_fac,
trlib_flt_t *eig, trlib_int_t verbose, trlib_int_t unicode, char *prefix, FILE *fout,
trlib_int_t *timing, trlib_flt_t *lam_pert, trlib_flt_t *pert, trlib_int_t *iter_inv);
/** size that has to be allocated for :c:data:`timing` in :c:func:`trlib_eigen_inverse`
*/
trlib_int_t trlib_eigen_timing_size(void);
#endif
|
