1 | /// \ingroup newmat |
---|
2 | ///@{ |
---|
3 | |
---|
4 | /// \file cholesky.cpp |
---|
5 | /// Cholesky decomposition. |
---|
6 | /// Cholesky decomposition of symmetric and band symmetric matrices, |
---|
7 | /// update, downdate, manipulate a Cholesky decomposition |
---|
8 | |
---|
9 | |
---|
10 | // Copyright (C) 1991,2,3,4: R B Davies |
---|
11 | |
---|
12 | #define WANT_MATH |
---|
13 | //#define WANT_STREAM |
---|
14 | |
---|
15 | #include "include.h" |
---|
16 | |
---|
17 | #include "newmat.h" |
---|
18 | #include "newmatrm.h" |
---|
19 | |
---|
20 | #ifdef use_namespace |
---|
21 | namespace NEWMAT { |
---|
22 | #endif |
---|
23 | |
---|
24 | #ifdef DO_REPORT |
---|
25 | #define REPORT { static ExeCounter ExeCount(__LINE__,14); ++ExeCount; } |
---|
26 | #else |
---|
27 | #define REPORT {} |
---|
28 | #endif |
---|
29 | |
---|
30 | /********* Cholesky decomposition of a positive definite matrix *************/ |
---|
31 | |
---|
32 | // Suppose S is symmetrix and positive definite. Then there exists a unique |
---|
33 | // lower triangular matrix L such that L L.t() = S; |
---|
34 | |
---|
35 | |
---|
36 | ReturnMatrix Cholesky(const SymmetricMatrix& S) |
---|
37 | { |
---|
38 | REPORT |
---|
39 | Tracer trace("Cholesky"); |
---|
40 | int nr = S.Nrows(); |
---|
41 | LowerTriangularMatrix T(nr); |
---|
42 | Real* s = S.Store(); Real* t = T.Store(); Real* ti = t; |
---|
43 | for (int i=0; i<nr; i++) |
---|
44 | { |
---|
45 | Real* tj = t; Real sum; int k; |
---|
46 | for (int j=0; j<i; j++) |
---|
47 | { |
---|
48 | Real* tk = ti; sum = 0.0; k = j; |
---|
49 | while (k--) { sum += *tj++ * *tk++; } |
---|
50 | *tk = (*s++ - sum) / *tj++; |
---|
51 | } |
---|
52 | sum = 0.0; k = i; |
---|
53 | while (k--) { sum += square(*ti++); } |
---|
54 | Real d = *s++ - sum; |
---|
55 | if (d<=0.0) Throw(NPDException(S)); |
---|
56 | *ti++ = sqrt(d); |
---|
57 | } |
---|
58 | T.release(); return T.for_return(); |
---|
59 | } |
---|
60 | |
---|
61 | ReturnMatrix Cholesky(const SymmetricBandMatrix& S) |
---|
62 | { |
---|
63 | REPORT |
---|
64 | Tracer trace("Band-Cholesky"); |
---|
65 | int nr = S.Nrows(); int m = S.lower_val; |
---|
66 | LowerBandMatrix T(nr,m); |
---|
67 | Real* s = S.Store(); Real* t = T.Store(); Real* ti = t; |
---|
68 | |
---|
69 | for (int i=0; i<nr; i++) |
---|
70 | { |
---|
71 | Real* tj = t; Real sum; int l; |
---|
72 | if (i<m) { REPORT l = m-i; s += l; ti += l; l = i; } |
---|
73 | else { REPORT t += (m+1); l = m; } |
---|
74 | |
---|
75 | for (int j=0; j<l; j++) |
---|
76 | { |
---|
77 | Real* tk = ti; sum = 0.0; int k = j; tj += (m-j); |
---|
78 | while (k--) { sum += *tj++ * *tk++; } |
---|
79 | *tk = (*s++ - sum) / *tj++; |
---|
80 | } |
---|
81 | sum = 0.0; |
---|
82 | while (l--) { sum += square(*ti++); } |
---|
83 | Real d = *s++ - sum; |
---|
84 | if (d<=0.0) Throw(NPDException(S)); |
---|
85 | *ti++ = sqrt(d); |
---|
86 | } |
---|
87 | |
---|
88 | T.release(); return T.for_return(); |
---|
89 | } |
---|
90 | |
---|
91 | |
---|
92 | |
---|
93 | |
---|
94 | // Contributed by Nick Bennett of Schlumberger-Doll Research; modified by RBD |
---|
95 | |
---|
96 | // The enclosed routines can be used to update the Cholesky decomposition of |
---|
97 | // a positive definite symmetric matrix. A good reference for this routines |
---|
98 | // can be found in |
---|
99 | // LINPACK User's Guide, Chapter 10, Dongarra et. al., SIAM, Philadelphia, 1979 |
---|
100 | |
---|
101 | // produces the Cholesky decomposition of A + x.t() * x where A = chol.t() * chol |
---|
102 | void update_Cholesky(UpperTriangularMatrix &chol, RowVector x) |
---|
103 | { |
---|
104 | int nc = chol.Nrows(); |
---|
105 | ColumnVector cGivens(nc); cGivens = 0.0; |
---|
106 | ColumnVector sGivens(nc); sGivens = 0.0; |
---|
107 | |
---|
108 | for(int j = 1; j <= nc; ++j) // process the jth column of chol |
---|
109 | { |
---|
110 | // apply the previous Givens rotations k = 1,...,j-1 to column j |
---|
111 | for(int k = 1; k < j; ++k) |
---|
112 | GivensRotation(cGivens(k), sGivens(k), chol(k,j), x(j)); |
---|
113 | |
---|
114 | // determine the jth Given's rotation |
---|
115 | pythag(chol(j,j), x(j), cGivens(j), sGivens(j)); |
---|
116 | |
---|
117 | // apply the jth Given's rotation |
---|
118 | { |
---|
119 | Real tmp0 = cGivens(j) * chol(j,j) + sGivens(j) * x(j); |
---|
120 | chol(j,j) = tmp0; x(j) = 0.0; |
---|
121 | } |
---|
122 | |
---|
123 | } |
---|
124 | |
---|
125 | } |
---|
126 | |
---|
127 | |
---|
128 | // produces the Cholesky decomposition of A - x.t() * x where A = chol.t() * chol |
---|
129 | void downdate_Cholesky(UpperTriangularMatrix &chol, RowVector x) |
---|
130 | { |
---|
131 | int nRC = chol.Nrows(); |
---|
132 | |
---|
133 | // solve R^T a = x |
---|
134 | LowerTriangularMatrix L = chol.t(); |
---|
135 | ColumnVector a(nRC); a = 0.0; |
---|
136 | int i, j; |
---|
137 | |
---|
138 | for (i = 1; i <= nRC; ++i) |
---|
139 | { |
---|
140 | // accumulate subtr sum |
---|
141 | Real subtrsum = 0.0; |
---|
142 | for(int k = 1; k < i; ++k) subtrsum += a(k) * L(i,k); |
---|
143 | |
---|
144 | a(i) = (x(i) - subtrsum) / L(i,i); |
---|
145 | } |
---|
146 | |
---|
147 | // test that l2 norm of a is < 1 |
---|
148 | Real squareNormA = a.SumSquare(); |
---|
149 | if (squareNormA >= 1.0) |
---|
150 | Throw(ProgramException("downdate_Cholesky() fails", chol)); |
---|
151 | |
---|
152 | Real alpha = sqrt(1.0 - squareNormA); |
---|
153 | |
---|
154 | // compute and apply Givens rotations to the vector a |
---|
155 | ColumnVector cGivens(nRC); cGivens = 0.0; |
---|
156 | ColumnVector sGivens(nRC); sGivens = 0.0; |
---|
157 | for(i = nRC; i >= 1; i--) |
---|
158 | alpha = pythag(alpha, a(i), cGivens(i), sGivens(i)); |
---|
159 | |
---|
160 | // apply Givens rotations to the jth column of chol |
---|
161 | ColumnVector xtilde(nRC); xtilde = 0.0; |
---|
162 | for(j = nRC; j >= 1; j--) |
---|
163 | { |
---|
164 | // only the first j rotations have an affect on chol,0 |
---|
165 | for(int k = j; k >= 1; k--) |
---|
166 | GivensRotation(cGivens(k), -sGivens(k), chol(k,j), xtilde(j)); |
---|
167 | } |
---|
168 | } |
---|
169 | |
---|
170 | |
---|
171 | |
---|
172 | // produces the Cholesky decomposition of EAE where A = chol.t() * chol |
---|
173 | // and E produces a RIGHT circular shift of the rows and columns from |
---|
174 | // 1,...,k-1,k,k+1,...l,l+1,...,p to |
---|
175 | // 1,...,k-1,l,k,k+1,...l-1,l+1,...p |
---|
176 | void right_circular_update_Cholesky(UpperTriangularMatrix &chol, int k, int l) |
---|
177 | { |
---|
178 | int nRC = chol.Nrows(); |
---|
179 | int i, j; |
---|
180 | |
---|
181 | // I. compute shift of column l to the kth position |
---|
182 | Matrix cholCopy = chol; |
---|
183 | // a. grab column l |
---|
184 | ColumnVector columnL = cholCopy.Column(l); |
---|
185 | // b. shift columns k,...l-1 to the RIGHT |
---|
186 | for(j = l-1; j >= k; --j) |
---|
187 | cholCopy.Column(j+1) = cholCopy.Column(j); |
---|
188 | // c. copy the top k-1 elements of columnL into the kth column of cholCopy |
---|
189 | cholCopy.Column(k) = 0.0; |
---|
190 | for(i = 1; i < k; ++i) cholCopy(i,k) = columnL(i); |
---|
191 | |
---|
192 | // II. determine the l-k Given's rotations |
---|
193 | int nGivens = l-k; |
---|
194 | ColumnVector cGivens(nGivens); cGivens = 0.0; |
---|
195 | ColumnVector sGivens(nGivens); sGivens = 0.0; |
---|
196 | for(i = l; i > k; i--) |
---|
197 | { |
---|
198 | int givensIndex = l-i+1; |
---|
199 | columnL(i-1) = pythag(columnL(i-1), columnL(i), |
---|
200 | cGivens(givensIndex), sGivens(givensIndex)); |
---|
201 | columnL(i) = 0.0; |
---|
202 | } |
---|
203 | // the kth entry of columnL is the new diagonal element in column k of cholCopy |
---|
204 | cholCopy(k,k) = columnL(k); |
---|
205 | |
---|
206 | // III. apply these Given's rotations to subsequent columns |
---|
207 | // for columns k+1,...,l-1 we only need to apply the last nGivens-(j-k) rotations |
---|
208 | for(j = k+1; j <= nRC; ++j) |
---|
209 | { |
---|
210 | ColumnVector columnJ = cholCopy.Column(j); |
---|
211 | int imin = nGivens - (j-k) + 1; if (imin < 1) imin = 1; |
---|
212 | for(int gIndex = imin; gIndex <= nGivens; ++gIndex) |
---|
213 | { |
---|
214 | // apply gIndex Given's rotation |
---|
215 | int topRowIndex = k + nGivens - gIndex; |
---|
216 | GivensRotationR(cGivens(gIndex), sGivens(gIndex), |
---|
217 | columnJ(topRowIndex), columnJ(topRowIndex+1)); |
---|
218 | } |
---|
219 | cholCopy.Column(j) = columnJ; |
---|
220 | } |
---|
221 | |
---|
222 | chol << cholCopy; |
---|
223 | } |
---|
224 | |
---|
225 | |
---|
226 | |
---|
227 | // produces the Cholesky decomposition of EAE where A = chol.t() * chol |
---|
228 | // and E produces a LEFT circular shift of the rows and columns from |
---|
229 | // 1,...,k-1,k,k+1,...l,l+1,...,p to |
---|
230 | // 1,...,k-1,k+1,...l,k,l+1,...,p to |
---|
231 | void left_circular_update_Cholesky(UpperTriangularMatrix &chol, int k, int l) |
---|
232 | { |
---|
233 | int nRC = chol.Nrows(); |
---|
234 | int i, j; |
---|
235 | |
---|
236 | // I. compute shift of column k to the lth position |
---|
237 | Matrix cholCopy = chol; |
---|
238 | // a. grab column k |
---|
239 | ColumnVector columnK = cholCopy.Column(k); |
---|
240 | // b. shift columns k+1,...l to the LEFT |
---|
241 | for(j = k+1; j <= l; ++j) |
---|
242 | cholCopy.Column(j-1) = cholCopy.Column(j); |
---|
243 | // c. copy the elements of columnK into the lth column of cholCopy |
---|
244 | cholCopy.Column(l) = 0.0; |
---|
245 | for(i = 1; i <= k; ++i) |
---|
246 | cholCopy(i,l) = columnK(i); |
---|
247 | |
---|
248 | // II. apply and compute Given's rotations |
---|
249 | int nGivens = l-k; |
---|
250 | ColumnVector cGivens(nGivens); cGivens = 0.0; |
---|
251 | ColumnVector sGivens(nGivens); sGivens = 0.0; |
---|
252 | for(j = k; j <= nRC; ++j) |
---|
253 | { |
---|
254 | ColumnVector columnJ = cholCopy.Column(j); |
---|
255 | |
---|
256 | // apply the previous Givens rotations to columnJ |
---|
257 | int imax = j - k; if (imax > nGivens) imax = nGivens; |
---|
258 | for(int i = 1; i <= imax; ++i) |
---|
259 | { |
---|
260 | int gIndex = i; |
---|
261 | int topRowIndex = k + i - 1; |
---|
262 | GivensRotationR(cGivens(gIndex), sGivens(gIndex), |
---|
263 | columnJ(topRowIndex), columnJ(topRowIndex+1)); |
---|
264 | } |
---|
265 | |
---|
266 | // compute a new Given's rotation when j < l |
---|
267 | if(j < l) |
---|
268 | { |
---|
269 | int gIndex = j-k+1; |
---|
270 | columnJ(j) = pythag(columnJ(j), columnJ(j+1), cGivens(gIndex), |
---|
271 | sGivens(gIndex)); |
---|
272 | columnJ(j+1) = 0.0; |
---|
273 | } |
---|
274 | |
---|
275 | cholCopy.Column(j) = columnJ; |
---|
276 | } |
---|
277 | |
---|
278 | chol << cholCopy; |
---|
279 | |
---|
280 | } |
---|
281 | |
---|
282 | |
---|
283 | |
---|
284 | |
---|
285 | #ifdef use_namespace |
---|
286 | } |
---|
287 | #endif |
---|
288 | |
---|
289 | ///@} |
---|