diff --git a/Eigen/src/IterativeLinearSolvers/MINRES.h b/Eigen/src/IterativeLinearSolvers/MINRES.h index a87b7cb28..5bc4773d7 100644 --- a/Eigen/src/IterativeLinearSolvers/MINRES.h +++ b/Eigen/src/IterativeLinearSolvers/MINRES.h @@ -35,73 +35,133 @@ namespace Eigen { typedef typename Dest::RealScalar RealScalar; typedef typename Dest::Scalar Scalar; typedef Matrix VectorType; + + // initialize const int maxIters(iters); // initialize maxIters to iters const int N(mat.cols()); // the size of the matrix const RealScalar rhsNorm2(rhs.squaredNorm()); // const RealScalar threshold(tol_error); // threshold for original convergence criterion, see below const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold - VectorType v(VectorType::Zero(N)); - VectorType v_hat(rhs-mat*x); - RealScalar beta(v_hat.norm()); + + +// VectorType v(VectorType::Zero(N)); +// VectorType v_hat(rhs-mat*x); + + // Compute initial residual + VectorType residual(rhs-mat*x); + + + // Initialize preconditioned Lanczos + VectorType v_old(N); // will be initialized inside loop + VectorType v = VectorType::Zero(N); //initialize v + VectorType v_new = residual; //initialize v_new + VectorType w(N); // will be initialized inside loop + VectorType w_new = precond.solve(v_new); // initialize w_new + RealScalar beta; // will be initialized inside loop + RealScalar beta_new = sqrt(v_new.dot(w_new)); + v_new /= beta_new; + w_new /= beta_new; + + + + // RealScalar beta(v_hat.norm()); RealScalar c(1.0); // the cosine of the Givens rotation RealScalar c_old(1.0); RealScalar s(0.0); // the sine of the Givens rotation RealScalar s_old(0.0); // the sine of the Givens rotation - VectorType w(VectorType::Zero(N)); - VectorType w_old(w); - RealScalar eta(beta); + VectorType p_oold(VectorType::Zero(N)); // initialize p_oold=0 + VectorType p_old(p_oold); // initialize p_old=0 + VectorType p(N); // will be initialized in loop + + //RealScalar eta(beta); // CHANGE THIS RealScalar norm_rMR=beta; const RealScalar norm_r0(beta); - VectorType v_old(N), Av(N), w_oold(N); // preallocate temporaty vectors used in iteration + RealScalar eta(1.0); + + // VectorType v_old(N), Av(N), w_oold(N); // preallocate temporaty vectors used in iteration RealScalar residualNorm2; // not needed for original convergnce criterion int n = 0; while ( n < maxIters ){ - // Lanczos - // VectorType v_old(v); // now pre-allocated - v_old = v; - v=v_hat/beta; -// VectorType Av(mat*v); // now pre-allocated - Av = mat*v; - RealScalar alpha(v.transpose()*Av); - v_hat=Av-alpha*v-beta*v_old; - RealScalar beta_old(beta); - beta=v_hat.norm(); + // Preconditioned Lanczos + /* Note that there are 4 variants on the Lanczos algorithm. These are + * described in Paige, C. C. (1972). Computational variants of + * the Lanczos method for the eigenproblem. IMA Journal of Applied + * Mathematics, 10(3), 373–381. The current implementation corresonds + * to the case A(2,7) in the paper. It also corresponds to + * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear + * Systems, 2003 p.173. For the preconditioned version see + * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987). + */ + beta = beta_new; + v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter + v = v_new; // update + w = w_new; // update + v_new.noalias() = mat*w - beta*v_old; // compute v_new + const RealScalar alpha = v_new.dot(w); + v_new -= alpha*v; // overwrite v_new + w_new = precond.solve(v_new); // overwrite w_new + beta_new = sqrt(v_new.dot(w_new)); // compute beta_new + v_new /= beta_new; // overwrite v_new + w_new /= beta_new; // overwrite w_new - // QR - RealScalar c_oold(c_old); - c_old=c; - RealScalar s_oold(s_old); - s_old=s; - RealScalar r1_hat=c_old *alpha-c_oold*s_old *beta_old; - RealScalar r1 =std::pow(std::pow(r1_hat,2)+std::pow(beta,2),0.5); - RealScalar r2 =s_old *alpha+c_oold*c_old*beta_old; - RealScalar r3 =s_oold*beta_old; +// +// +// +// +// +// +// +// +// // VectorType v_old(v); // now pre-allocated +// v_old = v; +// v=v_hat/beta; +//// VectorType Av(mat*v); // now pre-allocated +// Av = mat*v; +// RealScalar alpha(v.transpose()*Av); +// v_hat=Av-alpha*v-beta*v_old; +// RealScalar beta_old(beta); +// beta=v_hat.norm(); - // Givens rotation - c=r1_hat/r1; - s=beta/r1; + // Apply QR +// RealScalar c_oold(c_old); // store old-old cosine +// c_old=c; // store old cosine +// RealScalar s_oold(s_old); // store old-old sine +// s_old=s; // store old sine +// const RealScalar r1_hat=c_old *alpha-c_oold*s_old *beta_old; +// const RealScalar r1 =std::pow(std::pow(r1_hat,2)+std::pow(beta,2),0.5); + const RealScalar r2 =s*alpha+c*c_old*beta; // s, s_old, c and c_old are still from previous iteration + const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration - // update + // Compute new Givens rotation + const RealScalar r1_hat=c*alpha-c_old*s*beta; + const RealScalar r1 =std::pow(std::pow(r1_hat,2)+std::pow(beta_new,2),0.5); + c_old = c; // store for next iteration + s_old = s; // store for next iteration + c=r1_hat/r1; // new cosine + s=beta/r1; // new sine + + // update w // VectorType w_oold(w_old); // now pre-allocated - w_oold = w_old; - - w_old=w; - w=(v-r3*w_oold-r2*w_old) /r1; - x += c*eta*w; + p_oold = p_old; + p_old = p; + p=(w-r2*p_old-r3*p_oold) /r1; + // update x + x += c*eta*p; norm_rMR *= std::fabs(s); - eta=-s*eta; residualNorm2 = (mat*x-rhs).squaredNorm(); // DOES mat*x NEED TO BE RECOMPUTED ???? //if(norm_rMR/norm_r0 < threshold){ // original convergence criterion, does not require "mat*x" if ( residualNorm2 < threshold2){ break; } - n++; + + eta=-s*eta; // update eta + n++; // increment iteration } tol_error = std::sqrt(residualNorm2 / rhsNorm2); // return error iters = n; // return number of iterations