Matrix Decomposition with Eigen: QR, Cholesky Decomposition LU, UL
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void qRDecomposition(Eigen::MatrixXd &A,Eigen::MatrixXd &Q, Eigen::MatrixXd &R) { /* A=QR Q: is orthogonal matrix-> columns of Q are orthonormal R: is upper triangulate matrix this is possible when columns of A are linearly indipendent */ Eigen::MatrixXd thinQ(A.rows(),A.cols() ), q(A.rows(),A.rows()); Eigen::HouseholderQR<Eigen::MatrixXd> householderQR(A); q = householderQR.householderQ(); thinQ.setIdentity(); Q = householderQR.householderQ() * thinQ; R=Q.transpose()*A; } void qRExample() { Eigen::MatrixXd A; A.setRandom(3,4); std::cout<<"A" <<std::endl; std::cout<<A <<std::endl; Eigen::MatrixXd Q(A.rows(),A.rows()); Eigen::MatrixXd R(A.rows(),A.cols()); /////////////////////////////////HouseholderQR//////////////////////// Eigen::MatrixXd thinQ(A.rows(),A.cols() ), q(A.rows(),A.rows()); Eigen::HouseholderQR<Eigen::MatrixXd> householderQR(A); q = householderQR.householderQ(); thinQ.setIdentity(); Q = householderQR.householderQ() * thinQ; std::cout << "HouseholderQR" <<std::endl; std::cout << "Q" <<std::endl; std::cout << Q <<std::endl; R = householderQR.matrixQR().template triangularView<Eigen::Upper>(); std::cout << R<<std::endl; std::cout << R.rows()<<std::endl; std::cout << R.cols()<<std::endl; R=Q.transpose()*A; // std::cout << "R" <<std::endl; // std::cout << R<<std::endl; std::cout << "A-Q*R" <<std::endl; std::cout << A-Q*R <<std::endl; /////////////////////////////////ColPivHouseholderQR//////////////////////// Eigen::ColPivHouseholderQR<Eigen::MatrixXd> colPivHouseholderQR(A.rows(), A.cols()); colPivHouseholderQR.compute(A); //R = colPivHouseholderQR.matrixR().template triangularView<Upper>(); R = colPivHouseholderQR.matrixR(); Q = colPivHouseholderQR.matrixQ(); std::cout << "ColPivHouseholderQR" <<std::endl; std::cout << "Q" <<std::endl; std::cout << Q <<std::endl; std::cout << "R" <<std::endl; std::cout << R <<std::endl; std::cout << "A-Q*R" <<std::endl; std::cout << A-Q*R <<std::endl; /////////////////////////////////FullPivHouseholderQR//////////////////////// std::cout << "FullPivHouseholderQR" <<std::endl; Eigen::FullPivHouseholderQR<Eigen::MatrixXd> fullPivHouseholderQR(A.rows(), A.cols()); fullPivHouseholderQR.compute(A); Q=fullPivHouseholderQR.matrixQ(); R=fullPivHouseholderQR.matrixQR().template triangularView<Eigen::Upper>(); std::cout << "Q" <<std::endl; std::cout << Q <<std::endl; std::cout << "R" <<std::endl; std::cout << R <<std::endl; std::cout << "A-Q*R" <<std::endl; std::cout << A-Q*R <<std::endl; } void lDUDecomposition() { /* L: lower triangular matrix L U: upper triangular matrix U D: is a diagonal matrix A=LDU */ } void lUDecomposition() { /* L: lower triangular matrix L U: upper triangular matrix U A=LU */ } double exp(double x) // the functor we want to apply { std::setprecision(5); return std::trunc(x); } void gramSchmidtOrthogonalization(Eigen::MatrixXd &matrix,Eigen::MatrixXd &orthonormalMatrix) { /* In this method you make every column perpendicular to it's previous columns, here if a and b are representation vector of two columns, c=b-((b.a)/|a|).a ^ / b / / / ----------> a ^ /| b / | / | c / | ----------> a you just have to normilze every vector after make it perpendicular to previous columns so: q1=a.normalized(); q2=b-(b.q1).q1 q2=q2.normalized(); q3=c-(c.q1).q1 - (c.q2).q2 q3=q3.normalized(); Now we have Q, but we want A=QR so we just multiply both side by Q.transpose(), since Q is orthonormal, Q*Q.transpose() is I A=QR; Q.transpose()*A=R; */ Eigen::VectorXd col; for(int i=0;i<matrix.cols();i++) { col=matrix.col(i); col=col.normalized(); for(int j=0;j<i-1;j++) { //orthonormalMatrix.col(i) } orthonormalMatrix.col(i)=col; } Eigen::MatrixXd A(4,3); A<<1,2,3,-1,1,1,1,1,1,1,1,1; Eigen::Vector4d a=A.col(0); Eigen::Vector4d b=A.col(1); Eigen::Vector4d c=A.col(2); Eigen::Vector4d q1= a.normalized(); Eigen::Vector4d q2=b-(b.dot(q1))*q1; q2=q2.normalized(); Eigen::Vector4d q3=c-(c.dot(q1))*q1 - (c.dot(q2))*q2; q3=q3.normalized(); std::cout<< "q1:"<<std::endl; std::cout<< q1<<std::endl; std::cout<< "q2"<<std::endl; std::cout<< q2<<std::endl; std::cout<< "q3:"<<std::endl; std::cout<< q3<<std::endl; Eigen::MatrixXd Q(4,3); Q.col(0)=q1; Q.col(1)=q2; Q.col(2)=q3; Eigen::MatrixXd R(3,3); R=Q.transpose()*(A); std::cout<<"Q"<<std::endl; std::cout<< Q<<std::endl; std::cout<<"R"<<std::endl; std::cout<< R.unaryExpr(std::ptr_fun(exp))<<std::endl; //MatrixXd A(4,3), thinQ(4,3), Q(4,4); Eigen::MatrixXd thinQ(4,3), q(4,4); //A.setRandom(); Eigen::HouseholderQR<Eigen::MatrixXd> qr(A); q = qr.householderQ(); thinQ.setIdentity(); thinQ = qr.householderQ() * thinQ; std::cout << "Q computed by Eigen" << "\n\n" << thinQ << "\n\n"; std::cout << q << "\n\n" << thinQ << "\n\n"; } void gramSchmidtOrthogonalizationExample() { Eigen::MatrixXd matrix(3,4),orthonormalMatrix(3,4) ; matrix=Eigen::MatrixXd::Random(3,4);////A.setRandom(); gramSchmidtOrthogonalization(matrix,orthonormalMatrix); } void choleskyDecompositionExample() { /* Positive-definite matrix: Matrix Mnxn is said to be positive definite if the scalar zTMz is strictly positive for every non-zero column vector z n real numbers. zTMz>0 Hermitian matrix: Matrix Mnxn is said to be positive definite if the scalar z*Mz is strictly positive for every non-zero column vector z n real numbers. z*Mz>0 z* is the conjugate transpose of z. Positive semi-definite same as above except zTMz>=0 or z*Mz>=0 Example: ┌2 -1 0┐ M= |-1 2 -1| |0 - 1 2| └ ┘ ┌ a ┐ z=| b | | c | └ ┘ zTMz=a^2 +c^2+ (a-b)^2+ (b-c)^2 Cholesky decomposition: Cholesky decomposition of a Hermitian positive-definite matrix A is: A=LL* L is a lower triangular matrix with real and positive diagonal entries L* is the conjugate transpose of L */ Eigen::MatrixXd A(3,3); A << 6, 0, 0, 0, 4, 0, 0, 0, 7; Eigen::MatrixXd L( A.llt().matrixL() ); Eigen::MatrixXd L_T=L.adjoint();//conjugate transpose std::cout << "L" << std::endl; std::cout << L << std::endl; std::cout << "L_T" << std::endl; std::cout << L_T << std::endl; std::cout << "A" << std::endl; std::cout << A << std::endl; std::cout << "L*L_T" << std::endl; std::cout << L*L_T << std::endl; } |
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