391 lines
14 KiB
C++
391 lines
14 KiB
C++
/*
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LA: linear algebra C++ interface library
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Copyright (C) 2008 Jiri Pittner <jiri.pittner@jh-inst.cas.cz> or <jiri@pittnerovi.com>
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This program is free software: you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation, either version 3 of the License, or
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(at your option) any later version.
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This program is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with this program. If not, see <http://www.gnu.org/licenses/>.
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*/
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#ifndef _LA_NONCLASS_H_
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#define _LA_NONCLASS_H_
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#include "vec.h"
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#include "smat.h"
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#include "mat.h"
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#include "la_traits.h"
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namespace LA {
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//MISC
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template <class T>
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const NRSMat<T> twoside_transform(const NRSMat<T> &S, const NRMat<T> &C, bool transp=0) //calculate C^dagger S C
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{
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if(transp)
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{
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NRMat<T> tmp = C * S;
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NRMat<T> result(C.nrows(),C.nrows());
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result.gemm((T)0,tmp,'n',C,'t',(T)1);
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return NRSMat<T>(result);
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}
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NRMat<T> tmp = S * C;
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NRMat<T> result(C.ncols(),C.ncols());
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result.gemm((T)0,C,'t',tmp,'n',(T)1);
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return NRSMat<T>(result);
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}
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template <class T>
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const NRMat<T> diagonalmatrix(const NRVec<T> &x)
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{
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int n=x.size();
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NRMat<T> result((T)0,n,n);
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T *p = result[0];
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for(int j=0; j<n; j++) {*p = x[j]; p+=(n+1);}
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return result;
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}
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//more efficient commutator for a special case of full matrices
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template<class T>
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inline const NRMat<T> commutator ( const NRMat<T> &x, const NRMat<T> &y, const bool trx=0, const bool tryy=0)
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{
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NRMat<T> r(trx?x.ncols():x.nrows(), tryy?y.nrows():y.ncols());
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r.gemm((T)0,x,trx?'t':'n',y,tryy?'t':'n',(T)1);
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r.gemm((T)1,y,tryy?'t':'n',x,trx?'t':'n',(T)-1);
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return r;
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}
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//more efficient commutator for a special case of full matrices
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template<class T>
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inline const NRMat<T> anticommutator ( const NRMat<T> &x, const NRMat<T> &y, const bool trx=0, const bool tryy=0)
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{
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NRMat<T> r(trx?x.ncols():x.nrows(), tryy?y.nrows():y.ncols());
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r.gemm((T)0,x,trx?'t':'n',y,tryy?'t':'n',(T)1);
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r.gemm((T)1,y,tryy?'t':'n',x,trx?'t':'n',(T)1);
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return r;
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}
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//////////////////////
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// LAPACK interface //
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//////////////////////
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#define declare_la(T) \
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extern const NRVec<T> diagofproduct(const NRMat<T> &a, const NRMat<T> &b,\
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bool trb=0, bool conjb=0); \
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extern T trace2(const NRMat<T> &a, const NRMat<T> &b, bool trb=0); \
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extern T trace2(const NRSMat<T> &a, const NRSMat<T> &b, const bool diagscaled=0);\
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extern T trace2(const NRSMat<T> &a, const NRMat<T> &b, const bool diagscaled=0);\
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extern T trace2(const NRMat<T> &a, const NRSMat<T> &b, const bool diagscaled=0);\
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extern void linear_solve(NRMat<T> &a, NRMat<T> *b, T *det=0,int n=0); /*solve Ax^T=b^T (b is nrhs x n) */ \
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extern void linear_solve(NRSMat<T> &a, NRMat<T> *b, T *det=0, int n=0); /*solve Ax^T=b^T (b is nrhs x n) */\
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extern void linear_solve(NRMat<T> &a, NRVec<T> &b, double *det=0, int n=0); \
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extern void linear_solve(NRSMat<T> &a, NRVec<T> &b, double *det=0, int n=0); \
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extern void diagonalize(NRMat<T> &a, NRVec<LA_traits<T>::normtype> &w, const bool eivec=1, const bool corder=1, int n=0, NRMat<T> *b=NULL, const int itype=1); \
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extern void diagonalize(NRSMat<T> &a, NRVec<LA_traits<T>::normtype> &w, NRMat<T> *v, const bool corder=1, int n=0, NRSMat<T> *b=NULL, const int itype=1);\
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extern void singular_decomposition(NRMat<T> &a, NRMat<T> *u, NRVec<LA_traits<T>::normtype> &s, NRMat<T> *v, const bool vnotdagger=0, int m=0, int n=0);
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/*NOTE!!! all versions of diagonalize DESTROY A and generalized diagonalize also B matrix */
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declare_la(double)
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declare_la(complex<double>)
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// Separate declarations
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//general nonsymmetric matrix and generalized diagonalization
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//corder =0 ... C rows are eigenvectors, =1 ... C columns are eigenvectors
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extern void gdiagonalize(NRMat<double> &a, NRVec<double> &wr, NRVec<double> &wi,
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NRMat<double> *vl, NRMat<double> *vr, const bool corder=1, int n=0, const int sorttype=0, const int biorthonormalize=0,
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NRMat<double> *b=NULL, NRVec<double> *beta=NULL); //this used real storage of eigenvectors like dgeev
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template<typename T>
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extern void gdiagonalize(NRMat<T> &a, NRVec< complex<double> > &w,
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NRMat< complex<double> >*vl, NRMat< complex<double> > *vr,
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const bool corder=1, int n=0, const int sorttype=0, const int biorthonormalize=0,
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NRMat<T> *b=NULL, NRVec<T> *beta=NULL); //eigenvectors are stored in complex matrices for T both double and complex
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//complex,real,imaginary parts of various entities
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template<typename T>
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extern const typename LA_traits<T>::realtype realpart(const T&);
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template<typename T>
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extern const typename LA_traits<T>::realtype imagpart(const T&);
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template<typename T>
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extern const typename LA_traits<T>::complextype realmatrix (const T&);
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template<typename T>
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extern const typename LA_traits<T>::complextype imagmatrix (const T&);
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template<typename T>
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extern const typename LA_traits<T>::complextype complexmatrix (const T&, const T&);
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//Cholesky decomposition
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extern void cholesky(NRMat<double> &a, bool upper=1);
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extern void cholesky(NRMat<complex<double> > &a, bool upper=1);
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//inverse by means of linear solve, preserving rhs intact
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template<typename T>
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const NRMat<T> inverse(NRMat<T> a, T *det=0)
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{
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#ifdef DEBUG
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if(a.nrows()!=a.ncols()) laerror("inverse() for non-square matrix");
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#endif
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NRMat<T> result(a.nrows(),a.nrows());
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result = (T)1.;
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a.copyonwrite();
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linear_solve(a, &result, det);
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result.transposeme(); //tested with noncblas
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return result;
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}
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//several matrix norms
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template<class MAT>
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typename LA_traits<MAT>::normtype MatrixNorm(const MAT &A, const char norm);
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//condition number
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template<class MAT>
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typename LA_traits<MAT>::normtype CondNumber(const MAT &A, const char norm);
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//general determinant
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template<class MAT>
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const typename LA_traits<MAT>::elementtype determinant(MAT a)//passed by value
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{
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typename LA_traits<MAT>::elementtype det;
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if(a.nrows()!=a.ncols()) laerror("determinant of non-square matrix");
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linear_solve(a,NULL,&det);
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return det;
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}
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//general determinant destructive on input
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template<class MAT>
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const typename LA_traits<MAT>::elementtype determinant_destroy(MAT &a) //passed by reference
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{
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typename LA_traits<MAT>::elementtype det;
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if(a.nrows()!=a.ncols()) laerror("determinant of non-square matrix");
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linear_solve(a,NULL,&det);
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return det;
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}
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//------------------------------------------------------------------------------
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// solves set of linear equations using gesvx
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// input:
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// A double precision matrix of dimension nn x mm, where min(nn, mm) >= n
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// B double prec. array dimensioned as nrhs x n
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// rhsCount nrhs - count of right hand sides
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// eqCount n - count of equations
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// eq use equilibration of matrix A before solving
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// saveA if set, do no overwrite A if equilibration in effect
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// rcond if not NULL, store the returned rcond value from dgesvx
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// output:
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// solution is stored in B
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// the info parameter of gesvx is returned (see man dgesvx)
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//------------------------------------------------------------------------------
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template<class T>
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int linear_solve_x(NRMat<T> &A, T *B, const int rhsCount, const int eqCount, const bool eq, const bool saveA, double *rcond);
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//------------------------------------------------------------------------------
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// for given square matrices A, B computes X = AB^{-1} as follows
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// XB = A => B^TX^T = A^T
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// input:
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// _A double precision matrix of dimension nn x nn
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// _B double prec. matrix of dimension nn x nn
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// _useEq use equilibration suitable for badly conditioned matrices
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// _rcond if not NULL, store the returned value of rcond fromd dgesvx
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// output:
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// solution is stored in _B
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// the info parameter of dgesvx is returned (see man dgesvx)
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//------------------------------------------------------------------------------
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template<class T>
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int multiply_by_inverse(NRMat<T> &A, NRMat<T> &B, bool useEq, double *rcond);
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//general submatrix, INDEX will typically be NRVec<int> or even int*
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//NOTE: in order to check consistency between nrows and rows in rows is a NRVec
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//some advanced metaprogramming would be necessary
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//application: e.g. ignoresign=true, equalsigns=true, indexshift= -1 ... elements of Slater overlaps for RHF
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template<class MAT, class INDEX>
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const NRMat<typename LA_traits<MAT>::elementtype> submatrix(const MAT a, const int nrows, const INDEX rows, const int ncols, const INDEX cols, int indexshift=0, bool ignoresign=false, bool equalsigns=false)
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{
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NRMat<typename LA_traits<MAT>::elementtype> r(nrows,ncols);
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if(equalsigns) //make the element zero if signs of both indices are opposite
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{
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if(ignoresign)
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{
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for(int i=0; i<nrows; ++i)
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for(int j=0; j<ncols; ++j)
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r(i,j) = rows[i]*cols[j]<0?0.:a(std::abs(rows[i])+indexshift,std::abs(cols[j])+indexshift);
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}
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else
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{
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for(int i=0; i<nrows; ++i)
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for(int j=0; j<ncols; ++j)
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r(i,j) = rows[i]*cols[j]<0?0.:a(rows[i]+indexshift,cols[j]+indexshift);
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}
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}
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else
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{
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if(ignoresign)
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{
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for(int i=0; i<nrows; ++i)
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for(int j=0; j<ncols; ++j)
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r(i,j) = a(std::abs(rows[i])+indexshift,std::abs(cols[j])+indexshift);
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}
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else
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{
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for(int i=0; i<nrows; ++i)
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for(int j=0; j<ncols; ++j)
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r(i,j) = a(rows[i]+indexshift,cols[j]+indexshift);
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}
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}
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return r;
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}
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//auxiliary routine to adjust eigenvectors to guarantee real logarithm
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extern void adjustphases(NRMat<double> &v);
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//declaration of template interface to cblas routines with full options available
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//just to facilitate easy change between float, double, complex in a user code
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//very incomplete, add new ones as needed
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template<class T> inline void xcopy(int n, const T *x, int incx, T *y, int incy);
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template<class T> inline void xaxpy(int n, const T &a, const T *x, int incx, T *y, int incy);
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template<class T> inline T xdot(int n, const T *x, int incx, const T *y, int incy);
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//specialized definitions have to be in the header file to be inlineable, eliminating any runtime overhead
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template<>
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inline void xcopy<double> (int n, const double *x, int incx, double *y, int incy)
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{
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cblas_dcopy(n, x, incx, y, incy);
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}
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template<>
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inline void xaxpy<double>(int n, const double &a, const double *x, int incx, double *y, int incy)
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{
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cblas_daxpy(n, a, x, incx, y, incy);
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}
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template<>
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inline double xdot<double>(int n, const double *x, int incx, const double *y, int incy)
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{
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return cblas_ddot(n,x,incx,y,incy);
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}
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//debugging aid: reconstruct an explicit matrix from the implicit version
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//which provides gemv only
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template<typename M, typename T>
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NRMat<T> reconstructmatrix(const M &implicitmat)
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{
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NRMat<T> r(implicitmat.nrows(),implicitmat.ncols());
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NRVec<T> rhs(0.,implicitmat.ncols());
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NRVec<T> tmp(implicitmat.nrows());
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for(int i=0; i<implicitmat.ncols(); ++i)
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{
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rhs[i]=1.;
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implicitmat.gemv(0.,tmp,'n',1.,rhs);
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for(int j=0; j<implicitmat.nrows(); ++j) r(j,i)=tmp[j];
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rhs[i]=0.;
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}
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return r;
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}
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//matrix functions via diagonalization
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extern NRMat<double> realmatrixfunction(NRMat<double> a, double (*f) (double)); //a has to by in fact symmetric
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extern NRMat<complex<double> > complexmatrixfunction(NRMat<double> a, double (*fre) (double), double (*fim) (double)); //a has to by in fact symmetric
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template<typename T>
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NRMat<T> matrixfunction(NRSMat<T> a, double (*f) (double)) //of symmetric/hermitian matrix
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{
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int n = a.nrows();
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NRVec<double> w(n);
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NRMat<T> v(n, n);
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diagonalize(a, w, &v, 0);
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for (int i=0; i<a.nrows(); i++) w[i] = (*f)(w[i]);
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NRMat<T> u = v;
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NRVec<T> ww=w; //diagmultl needs same type
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v.diagmultl(ww);
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NRMat<T> r(n, n);
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r.gemm(0.0, u, 't', v, 'n', 1.0); //gemm will use 'c' for complex ones
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return r;
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}
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template<typename T>
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extern NRMat<T> matrixfunction(NRMat<T> a, complex<double> (*f)(const complex<double> &)) //of a general real/complex matrix
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{
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int n = a.nrows();
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NRVec<complex<double> > w(n);
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NRMat<complex<double> > u(n,n),v(n,n);
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#ifdef debugmf
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NRMat<complex<double> > a0=a;
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#endif
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gdiagonalize<T>(a, w, &u, &v, false,n,0,false,NULL,NULL);//a gets destroyed, eigenvectors are rows
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NRVec< complex<double> > z = diagofproduct(u, v, 1, 1);
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#ifdef debugmf
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std::cout <<"TEST matrixfunction\n"<<w<<u<<v<<z;
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std::cout <<"TEST matrixfunction1 "<< u*a0 - diagonalmatrix(w)*u<<std::endl;
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std::cout <<"TEST matrixfunction2 "<< a0*v.transpose(1) - v.transpose(1)*diagonalmatrix(w)<<std::endl;
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std::cout <<"TEST matrixfunction3 "<< u*v.transpose(1)<<diagonalmatrix(z)<<std::endl;
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#endif
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NRVec< complex<double> > wz(n);
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for (int i=0; i<a.nrows(); i++) wz[i] = w[i]/z[i];
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#ifdef debugmf
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std::cout <<"TEST matrixfunction4 "<< a0<< v.transpose(true)*diagonalmatrix(wz)*u<<std::endl;
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#endif
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for (int i=0; i<a.nrows(); i++) w[i] = (*f)(w[i])/z[i];
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u.diagmultl(w);
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NRMat< complex<double> > r(n, n);
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r.gemm(0.0, v, 'c', u, 'n', 1.0);
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return (NRMat<T>) r; //convert back to real if applicable by the explicit decomplexifying constructor; it is NOT checked to which accuracy the imaginary part is actually zero
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}
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extern complex<double> sqrtinv(const complex<double> &);
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extern double sqrtinv(const double);
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//functions on matrices
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inline NRMat<double> sqrt(const NRSMat<double> &a) { return matrixfunction(a,&std::sqrt); }
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inline NRMat<double> sqrtinv(const NRSMat<double> &a) { return matrixfunction(a,&sqrtinv); }
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inline NRMat<double> realsqrt(const NRMat<double> &a) { return realmatrixfunction(a,&std::sqrt); }
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inline NRMat<double> realsqrtinv(const NRMat<double> &a) { return realmatrixfunction(a,&sqrtinv); }
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inline NRMat<double> log(const NRSMat<double> &a) { return matrixfunction(a,&std::log); }
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extern NRMat<double> log(const NRMat<double> &a);
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extern NRMat<complex<double> > log(const NRMat<complex<double> > &a);
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extern NRMat<complex<double> > exp0(const NRMat<complex<double> > &a);
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extern NRMat<complex<double> > copytest(const NRMat<complex<double> > &a);
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extern NRMat<double> copytest(const NRMat<double> &a);
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extern NRMat<double> exp0(const NRMat<double> &a);
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}//namespace
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#endif
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