Line data Source code
1 : /* +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
2 : Copyright (c) 2011-2023 The plumed team
3 : (see the PEOPLE file at the root of the distribution for a list of names)
4 :
5 : See http://www.plumed.org for more information.
6 :
7 : This file is part of plumed, version 2.
8 :
9 : plumed is free software: you can redistribute it and/or modify
10 : it under the terms of the GNU Lesser General Public License as published by
11 : the Free Software Foundation, either version 3 of the License, or
12 : (at your option) any later version.
13 :
14 : plumed is distributed in the hope that it will be useful,
15 : but WITHOUT ANY WARRANTY; without even the implied warranty of
16 : MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
17 : GNU Lesser General Public License for more details.
18 :
19 : You should have received a copy of the GNU Lesser General Public License
20 : along with plumed. If not, see <http://www.gnu.org/licenses/>.
21 : +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
22 : #ifndef __PLUMED_tools_Matrix_h
23 : #define __PLUMED_tools_Matrix_h
24 : #include <vector>
25 : #include <string>
26 : #include <set>
27 : #include <cmath>
28 : #include "Exception.h"
29 : #include "MatrixSquareBracketsAccess.h"
30 : #include "Tools.h"
31 : #include "Log.h"
32 : #include "lapack/lapack.h"
33 :
34 : namespace PLMD {
35 :
36 : /// Calculate the dot product between two vectors
37 : template <typename T> T dotProduct( const std::vector<T>& A, const std::vector<T>& B ) {
38 : plumed_assert( A.size()==B.size() );
39 : T val; for(unsigned i=0; i<A.size(); ++i) { val+=A[i]*B[i]; }
40 : return val;
41 : }
42 :
43 : /// Calculate the dot product between a vector and itself
44 : template <typename T> T norm( const std::vector<T>& A ) {
45 : T val; for(unsigned i=0; i<A.size(); ++i) { val+=A[i]*A[i]; }
46 : return val;
47 : }
48 :
49 : /// This class stores a full matrix and allows one to do some simple matrix operations
50 : template <typename T>
51 36868233 : class Matrix:
52 : public MatrixSquareBracketsAccess<Matrix<T>,T>
53 : {
54 : /// Multiply matrix by scalar
55 : template <typename U> friend Matrix<U> operator*(U&, const Matrix<U>& );
56 : /// Matrix matrix multiply
57 : template <typename U> friend void mult( const Matrix<U>&, const Matrix<U>&, Matrix<U>& );
58 : /// Matrix times a std::vector
59 : template <typename U> friend void mult( const Matrix<U>&, const std::vector<U>&, std::vector<U>& );
60 : /// std::vector times a Matrix
61 : template <typename U> friend void mult( const std::vector<U>&, const Matrix<U>&, std::vector<U>& );
62 : /// Matrix transpose
63 : template <typename U> friend void transpose( const Matrix<U>&, Matrix<U>& );
64 : /// Output the entire matrix on a single line
65 : template <typename U> friend Log& operator<<(Log&, const Matrix<U>& );
66 : /// Output the Matrix in matrix form
67 : template <typename U> friend void matrixOut( Log&, const Matrix<U>& );
68 : /// Diagonalize a symmetric matrix - returns zero if diagonalization worked
69 : template <typename U> friend int diagMat( const Matrix<U>&, std::vector<double>&, Matrix<double>& );
70 : /// Calculate the Moore-Penrose Pseudoinverse of a matrix
71 : template <typename U> friend int pseudoInvert( const Matrix<U>&, Matrix<double>& );
72 : /// Calculate the logarithm of the determinant of a symmetric matrix - returns zero if succesfull
73 : template <typename U> friend int logdet( const Matrix<U>&, double& );
74 : /// Invert a matrix (works for both symmetric and asymmetric matrices) - returns zero if sucesfull
75 : template <typename U> friend int Invert( const Matrix<U>&, Matrix<double>& );
76 : /// Do a cholesky decomposition of a matrix
77 : template <typename U> friend void cholesky( const Matrix<U>&, Matrix<U>& );
78 : /// Solve a system of equations using the cholesky decomposition
79 : template <typename U> friend void chol_elsolve( const Matrix<U>&, const std::vector<U>&, std::vector<U>& );
80 : private:
81 : /// Number of elements in matrix (nrows*ncols)
82 : unsigned sz;
83 : /// Number of rows in matrix
84 : unsigned rw;
85 : /// Number of columns in matrix
86 : unsigned cl;
87 : /// The data in the matrix
88 : std::vector<T> data;
89 : public:
90 36864898 : explicit Matrix(const unsigned nr=0, const unsigned nc=0 ) : sz(nr*nc), rw(nr), cl(nc), data(nr*nc) {}
91 272 : Matrix(const Matrix<T>& t) : sz(t.sz), rw(t.rw), cl(t.cl), data(t.data) {}
92 : /// Resize the matrix
93 899 : void resize( const unsigned nr, const unsigned nc ) { rw=nr; cl=nc; sz=nr*nc; data.resize(sz); }
94 : /// Return the number of rows
95 3019033 : inline unsigned nrows() const { return rw; }
96 : /// Return the number of columns
97 11458232 : inline unsigned ncols() const { return cl; }
98 : /// Return the contents of the matrix as a vector of length rw*cl
99 : inline std::vector<T>& getVector() { return data; }
100 : /// Set the matrix from a vector input
101 : inline void setFromVector( const std::vector<T>& vecin ) { plumed_assert( vecin.size()==sz ); for(unsigned i=0; i<sz; ++i) data[i]=vecin[i]; }
102 : /// Return element i,j of the matrix
103 2015393449 : inline const T& operator () (const unsigned& i, const unsigned& j) const { return data[j+i*cl]; }
104 : /// Return a referenre to element i,j of the matrix
105 601302118 : inline T& operator () (const unsigned& i, const unsigned& j) { return data[j+i*cl]; }
106 : /// Set all elements of the matrix equal to the value of v
107 : Matrix<T>& operator=(const T& v) {
108 4829318 : for(unsigned i=0; i<sz; ++i) { data[i]=v; }
109 : return *this;
110 : }
111 : /// Set the Matrix equal to another Matrix
112 : Matrix<T>& operator=(const Matrix<T>& m) {
113 56904 : sz=m.sz;
114 56904 : rw=m.rw;
115 56904 : cl=m.cl;
116 56904 : data=m.data;
117 56904 : return *this;
118 : }
119 : /// Set the Matrix equal to the value of a standard vector - used for readin
120 : Matrix<T>& operator=(const std::vector<T>& v) {
121 : plumed_dbg_assert( v.size()==sz );
122 : for(unsigned i=0; i<sz; ++i) { data[i]=v[i]; }
123 : return *this;
124 : }
125 : /// Add v to all elements of the Matrix
126 : Matrix<T> operator+=(const T& v) {
127 : for(unsigned i=0; i<sz; ++i) { data[i]+=v; }
128 : return *this;
129 : }
130 : /// Multiply all elements by v
131 125 : Matrix<T> operator*=(const T& v) {
132 55038 : for(unsigned i=0; i<sz; ++i) { data[i]*=v; }
133 125 : return *this;
134 : }
135 : /// Matrix addition
136 : Matrix<T>& operator+=(const Matrix<T>& m) {
137 : plumed_dbg_assert( m.rw==rw && m.cl==cl );
138 : data+=m.data;
139 : return *this;
140 : }
141 : /// Subtract v from all elements of the Matrix
142 : Matrix<T> operator-=(const T& v) {
143 : for(unsigned i=0; i<sz; ++i) { data-=v; }
144 : return *this;
145 : }
146 : /// Matrix subtraction
147 : Matrix<T>& operator-=(const Matrix<T>& m) {
148 : plumed_dbg_assert( m.rw==rw && m.cl==cl );
149 : data-=m.data;
150 : return *this;
151 : }
152 : /// Test if the matrix is symmetric or not
153 40203 : unsigned isSymmetric() const {
154 40203 : if (rw!=cl) { return 0; }
155 : unsigned sym=1;
156 291278 : for(unsigned i=1; i<rw; ++i) for(unsigned j=0; j<i; ++j) if( std::fabs(data[i+j*cl]-data[j+i*cl])>1.e-10 ) { sym=0; break; }
157 : return sym;
158 : }
159 : };
160 :
161 : /// Multiply matrix by scalar
162 : template <typename T> Matrix<T> operator*(T& v, const Matrix<T>& m ) {
163 : Matrix<T> new_m(m);
164 : new_m*=v;
165 : return new_m;
166 : }
167 :
168 16490 : template <typename T> void mult( const Matrix<T>& A, const Matrix<T>& B, Matrix<T>& C ) {
169 16490 : plumed_assert(A.cl==B.rw);
170 16490 : if( A.rw !=C.rw || B.cl !=C.cl ) { C.resize( A.rw, B.cl ); } C=static_cast<T>( 0 );
171 2012861479 : for(unsigned i=0; i<A.rw; ++i) for(unsigned j=0; j<B.cl; ++j) for (unsigned k=0; k<A.cl; ++k) C(i,j)+=A(i,k)*B(k,j);
172 16490 : }
173 :
174 18594 : template <typename T> void mult( const Matrix<T>& A, const std::vector<T>& B, std::vector<T>& C) {
175 18594 : plumed_assert( A.cl==B.size() );
176 18594 : if( C.size()!=A.rw ) { C.resize(A.rw); }
177 86682 : for(unsigned i=0; i<A.rw; ++i) { C[i]= static_cast<T>( 0 ); }
178 796630 : for(unsigned i=0; i<A.rw; ++i) for(unsigned k=0; k<A.cl; ++k) C[i]+=A(i,k)*B[k] ;
179 18594 : }
180 :
181 : template <typename T> void mult( const std::vector<T>& A, const Matrix<T>& B, std::vector<T>& C) {
182 : plumed_assert( B.rw==A.size() );
183 : if( C.size()!=B.cl ) {C.resize( B.cl );}
184 : for(unsigned i=0; i<B.cl; ++i) { C[i]=static_cast<T>( 0 ); }
185 : for(unsigned i=0; i<B.cl; ++i) for(unsigned k=0; k<B.rw; ++k) C[i]+=A[k]*B(k,i);
186 : }
187 :
188 730 : template <typename T> void transpose( const Matrix<T>& A, Matrix<T>& AT ) {
189 730 : if( A.rw!=AT.cl || A.cl!=AT.rw ) AT.resize( A.cl, A.rw );
190 52990 : for(unsigned i=0; i<A.cl; ++i) for(unsigned j=0; j<A.rw; ++j) AT(i,j)=A(j,i);
191 730 : }
192 :
193 : template <typename T> Log& operator<<(Log& ostr, const Matrix<T>& mat) {
194 : for(unsigned i=0; i<mat.sz; ++i) ostr<<mat.data[i]<<" ";
195 : return ostr;
196 : }
197 :
198 : template <typename T> void matrixOut( Log& ostr, const Matrix<T>& mat) {
199 : for(unsigned i=0; i<mat.rw; ++i) {
200 : for(unsigned j=0; j<mat.cl; ++j) { ostr<<mat(i,j)<<" "; }
201 : ostr<<"\n";
202 : }
203 : return;
204 : }
205 :
206 14055 : template <typename T> int diagMat( const Matrix<T>& A, std::vector<double>& eigenvals, Matrix<double>& eigenvecs ) {
207 :
208 : // Check matrix is square and symmetric
209 14055 : plumed_assert( A.rw==A.cl ); plumed_assert( A.isSymmetric()==1 );
210 14055 : std::vector<double> da(A.sz);
211 : unsigned k=0;
212 14055 : std::vector<double> evals(A.cl);
213 : // Transfer the matrix to the local array
214 475053 : for (unsigned i=0; i<A.cl; ++i) for (unsigned j=0; j<A.rw; ++j) da[k++]=static_cast<double>( A(j,i) );
215 :
216 14055 : int n=A.cl; int lwork=-1, liwork=-1, m, info, one=1;
217 14055 : std::vector<double> work(A.cl);
218 14055 : std::vector<int> iwork(A.cl);
219 14055 : double vl, vu, abstol=0.0;
220 14055 : std::vector<int> isup(2*A.cl);
221 14055 : std::vector<double> evecs(A.sz);
222 :
223 14055 : plumed_lapack_dsyevr("V", "I", "U", &n, da.data(), &n, &vl, &vu, &one, &n,
224 : &abstol, &m, evals.data(), evecs.data(), &n,
225 : isup.data(), work.data(), &lwork, iwork.data(), &liwork, &info);
226 14055 : if (info!=0) return info;
227 :
228 : // Retrieve correct sizes for work and iwork then reallocate
229 14055 : liwork=iwork[0]; iwork.resize(liwork);
230 14055 : lwork=static_cast<int>( work[0] ); work.resize(lwork);
231 :
232 14055 : plumed_lapack_dsyevr("V", "I", "U", &n, da.data(), &n, &vl, &vu, &one, &n,
233 : &abstol, &m, evals.data(), evecs.data(), &n,
234 : isup.data(), work.data(), &lwork, iwork.data(), &liwork, &info);
235 14055 : if (info!=0) return info;
236 :
237 14055 : if( eigenvals.size()!=A.cl ) { eigenvals.resize( A.cl ); }
238 14055 : if( eigenvecs.rw!=A.rw || eigenvecs.cl!=A.cl ) { eigenvecs.resize( A.rw, A.cl ); }
239 : k=0;
240 43814 : for(unsigned i=0; i<A.cl; ++i) {
241 29759 : eigenvals[i]=evals[i];
242 : // N.B. For ease of producing projectors we store the eigenvectors
243 : // ROW-WISE in the eigenvectors matrix. The first index is the
244 : // eigenvector number and the second the component
245 460998 : for(unsigned j=0; j<A.rw; ++j) { eigenvecs(i,j)=evecs[k++]; }
246 : }
247 :
248 : // This changes eigenvectors so that the first non-null element
249 : // of each of them is positive
250 : // We can do it because the phase is arbitrary, and helps making
251 : // the result reproducible
252 43814 : for(int i=0; i<n; ++i) {
253 : int j;
254 30984 : for(j=0; j<n; j++) if(eigenvecs(i,j)*eigenvecs(i,j)>1e-14) break;
255 217054 : if(j<n) if(eigenvecs(i,j)<0.0) for(j=0; j<n; j++) eigenvecs(i,j)*=-1;
256 : }
257 : return 0;
258 : }
259 :
260 1 : template <typename T> int pseudoInvert( const Matrix<T>& A, Matrix<double>& pseudoinverse ) {
261 1 : std::vector<double> da(A.sz);
262 : unsigned k=0;
263 : // Transfer the matrix to the local array
264 250501 : for (unsigned i=0; i<A.cl; ++i) for (unsigned j=0; j<A.rw; ++j) da[k++]=static_cast<double>( A(j,i) );
265 :
266 1 : int nsv, info, nrows=A.rw, ncols=A.cl;
267 1 : if(A.rw>A.cl) {nsv=A.cl;} else {nsv=A.rw;}
268 :
269 : // Create some containers for stuff from single value decomposition
270 1 : std::vector<double> S(nsv);
271 1 : std::vector<double> U(nrows*nrows);
272 1 : std::vector<double> VT(ncols*ncols);
273 1 : std::vector<int> iwork(8*nsv);
274 :
275 : // This optimizes the size of the work array used in lapack singular value decomposition
276 1 : int lwork=-1;
277 1 : std::vector<double> work(1);
278 1 : plumed_lapack_dgesdd( "A", &nrows, &ncols, da.data(), &nrows, S.data(), U.data(), &nrows, VT.data(), &ncols, work.data(), &lwork, iwork.data(), &info );
279 1 : if(info!=0) return info;
280 :
281 : // Retrieve correct sizes for work and rellocate
282 1 : lwork=(int) work[0]; work.resize(lwork);
283 :
284 : // This does the singular value decomposition
285 1 : plumed_lapack_dgesdd( "A", &nrows, &ncols, da.data(), &nrows, S.data(), U.data(), &nrows, VT.data(), &ncols, work.data(), &lwork, iwork.data(), &info );
286 1 : if(info!=0) return info;
287 :
288 : // Compute the tolerance on the singular values ( machine epsilon * number of singular values * maximum singular value )
289 500 : double tol; tol=S[0]; for(int i=1; i<nsv; ++i) { if( S[i]>tol ) { tol=S[i]; } } tol*=nsv*epsilon;
290 :
291 : // Get the inverses of the singlular values
292 1 : Matrix<double> Si( ncols, nrows ); Si=0.0;
293 501 : for(int i=0; i<nsv; ++i) { if( S[i]>tol ) { Si(i,i)=1./S[i]; } else { Si(i,i)=0.0; } }
294 :
295 : // Now extract matrices for pseudoinverse
296 1 : Matrix<double> V( ncols, ncols ), UT( nrows, nrows ), tmp( ncols, nrows );
297 250501 : k=0; for(int i=0; i<nrows; ++i) { for(int j=0; j<nrows; ++j) { UT(i,j)=U[k++]; } }
298 250501 : k=0; for(int i=0; i<ncols; ++i) { for(int j=0; j<ncols; ++j) { V(i,j)=VT[k++]; } }
299 :
300 : // And do matrix algebra to construct the pseudoinverse
301 1 : if( pseudoinverse.rw!=ncols || pseudoinverse.cl!=nrows ) pseudoinverse.resize( ncols, nrows );
302 1 : mult( V, Si, tmp ); mult( tmp, UT, pseudoinverse );
303 :
304 : return 0;
305 : }
306 :
307 25702 : template <typename T> int Invert( const Matrix<T>& A, Matrix<double>& inverse ) {
308 :
309 25702 : if( A.isSymmetric()==1 ) {
310 : // GAT -- I only ever use symmetric matrices so I can invert them like this.
311 : // I choose to do this as I have had problems with the more general way of doing this that
312 : // is implemented below.
313 9191 : std::vector<double> eval(A.rw); Matrix<double> evec(A.rw,A.cl), tevec(A.rw,A.cl);
314 9191 : int err; err=diagMat( A, eval, evec );
315 9191 : if(err!=0) return err;
316 63179 : for (unsigned i=0; i<A.rw; ++i) for (unsigned j=0; j<A.cl; ++j) tevec(i,j)=evec(j,i)/eval[j];
317 9191 : mult(tevec,evec,inverse);
318 : } else {
319 16511 : std::vector<double> da(A.sz);
320 16511 : std::vector<int> ipiv(A.cl);
321 16511 : unsigned k=0; int n=A.rw, info;
322 115601 : for(unsigned i=0; i<A.cl; ++i) for(unsigned j=0; j<A.rw; ++j) da[k++]=static_cast<double>( A(j,i) );
323 :
324 16511 : plumed_lapack_dgetrf(&n,&n,da.data(),&n,ipiv.data(),&info);
325 16511 : if(info!=0) return info;
326 :
327 16511 : int lwork=-1;
328 16511 : std::vector<double> work(A.cl);
329 16511 : plumed_lapack_dgetri(&n,da.data(),&n,ipiv.data(),work.data(),&lwork,&info);
330 16511 : if(info!=0) return info;
331 :
332 16511 : lwork=static_cast<int>( work[0] ); work.resize(lwork);
333 16511 : plumed_lapack_dgetri(&n,da.data(),&n,ipiv.data(),work.data(),&lwork,&info);
334 16511 : if(info!=0) return info;
335 :
336 16511 : if( inverse.cl!=A.cl || inverse.rw!=A.rw ) { inverse.resize(A.rw,A.cl); }
337 115601 : k=0; for(unsigned i=0; i<A.rw; ++i) for(unsigned j=0; j<A.cl; ++j) inverse(j,i)=da[k++];
338 : }
339 :
340 : return 0;
341 : }
342 :
343 446 : template <typename T> void cholesky( const Matrix<T>& A, Matrix<T>& B ) {
344 :
345 446 : plumed_assert( A.rw==A.cl && A.isSymmetric() );
346 : Matrix<T> L(A.rw,A.cl); L=0.;
347 446 : std::vector<T> D(A.rw,0.);
348 1047 : for(unsigned i=0; i<A.rw; ++i) {
349 601 : L(i,i)=static_cast<T>( 1 );
350 756 : for (unsigned j=0; j<i; ++j) {
351 155 : L(i,j)=A(i,j);
352 155 : for (unsigned k=0; k<j; ++k) L(i,j)-=L(i,k)*L(j,k)*D[k];
353 155 : if (D[j]!=0.) L(i,j)/=D[j]; else L(i,j)=static_cast<T>( 0 );
354 : }
355 601 : D[i]=A(i,i);
356 756 : for (unsigned k=0; k<i; ++k) D[i]-=L(i,k)*L(i,k)*D[k];
357 : }
358 :
359 1047 : for(unsigned i=0; i<A.rw; ++i) D[i]=(D[i]>0.?std::sqrt(D[i]):0.);
360 446 : if( B.rw!=A.rw || B.cl!=A.cl ) { B.resize( A.rw, A.cl); }
361 1803 : B=0.; for(unsigned i=0; i<A.rw; ++i) for(unsigned j=0; j<=i; ++j) B(i,j)+=L(i,j)*D[j];
362 446 : }
363 :
364 : template <typename T> void chol_elsolve( const Matrix<T>& M, const std::vector<T>& b, std::vector<T>& y ) {
365 :
366 : plumed_assert( M.rw==M.cl && M(0,1)==0.0 && b.size()==M.rw );
367 : if( y.size()!=M.rw ) { y.resize( M.rw ); }
368 : for(unsigned i=0; i<M.rw; ++i) {
369 : y[i]=b[i];
370 : for(unsigned j=0; j<i; ++j) y[i]-=M(i,j)*y[j];
371 : y[i]*=1.0/M(i,i);
372 : }
373 : }
374 :
375 0 : template <typename T> int logdet( const Matrix<T>& M, double& ldet ) {
376 : // Check matrix is square and symmetric
377 0 : plumed_assert( M.rw==M.cl || M.isSymmetric() );
378 :
379 0 : std::vector<double> da(M.sz);
380 : unsigned k=0;
381 0 : std::vector<double> evals(M.cl);
382 : // Transfer the matrix to the local array
383 0 : for (unsigned i=0; i<M.rw; ++i) for (unsigned j=0; j<M.cl; ++j) da[k++]=static_cast<double>( M(j,i) );
384 :
385 0 : int n=M.cl; int lwork=-1, liwork=-1, info, m, one=1;
386 0 : std::vector<double> work(M.rw);
387 0 : std::vector<int> iwork(M.rw);
388 0 : double vl, vu, abstol=0.0;
389 0 : std::vector<int> isup(2*M.rw);
390 0 : std::vector<double> evecs(M.sz);
391 0 : plumed_lapack_dsyevr("N", "I", "U", &n, da.data(), &n, &vl, &vu, &one, &n,
392 : &abstol, &m, evals.data(), evecs.data(), &n,
393 : isup.data(), work.data(), &lwork, iwork.data(), &liwork, &info);
394 0 : if (info!=0) return info;
395 :
396 : // Retrieve correct sizes for work and iwork then reallocate
397 0 : lwork=static_cast<int>( work[0] ); work.resize(lwork);
398 0 : liwork=iwork[0]; iwork.resize(liwork);
399 :
400 0 : plumed_lapack_dsyevr("N", "I", "U", &n, da.data(), &n, &vl, &vu, &one, &n,
401 : &abstol, &m, evals.data(), evecs.data(), &n,
402 : isup.data(), work.data(), &lwork, iwork.data(), &liwork, &info);
403 0 : if (info!=0) return info;
404 :
405 : // Transfer the eigenvalues and eigenvectors to the output
406 0 : ldet=0; for(unsigned i=0; i<M.cl; i++) { ldet+=log(evals[i]); }
407 :
408 : return 0;
409 : }
410 :
411 :
412 :
413 : }
414 : #endif
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