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266 CHAPTER 14. NUMERICAL EXERCISES • G is a model of concept glas::VectorExpression. Next, we write the code for the Richardson iteration. We store the variables ui in u and ri in r. #ifndef athens richardson hpp #define athens richardson hpp #include namespace athens { template double richardson( Op const& op, F const& f, U& u, double const& tol, int max it ) { double resid norm ; // Create residual vector glas::dense vector< typename glas::value type::type > r( glas::size(u) ) ; for ( int iter =0; iter
14.1. COMPUTING AN EIGENFUNCTION OF THE POISSON EQUATION 267 glas::random( f, seed ) ; v type x( 10 ) ; x = 0.0 ; // Richardson iteration double res nrm = athens::richardson( poisson scaled(), 0.5∗f, x, 1.e−4, 1000 ) ; { glas::dense vector r( size(x) ) ; athens::poisson 1d( x, r ) ; std::cout ≪ ”res nrm = ” ≪ norm 2( f −r ) ≪ std::endl ; std::cout ≪ ”f = ” ≪ f ≪ std::endl ; std::cout ≪ ”x = ” ≪ x ≪ std::endl ; } return 0 ; } We multiply right-hand side and matrix vector product by 0.5 to make sure the Richardson method converges. The output looks like res_nrm = 0.000195164 f = (10)[0.0484811,0.822283,0.102721,0.436631,0.46112,0.0475317,0.864644,0.0772845,0.920099,0.105434] x = (10)[1.85463,3.66081,4.64473,5.52601,5.97071,5.9544,5.8906,4.96226,3.95668,2.03105] Note that the Richardson method converges very slowly. For the Poisson equation, there exist much faster methods. 14.1.3 LAPACK tridiagonal solver The LAPACK [?] software package contains routines for solving linear systems with a symmetric positive definite tridiagonal matrix. This package is written in FORTRAN 77. The corresponding functions are • Cholesky factorization: A = LL T by SUBROUTINE DPTTRF( N, D, E, INFO ) • Linear solve: Ax = b using LL T x = b by SUBROUTINE DPTTRS( N, NRHS, D, E, B, LDB, INFO ) In order to solve Au = f, first A is factorized by the Cholesky factorization into A = LDL T where L is a matrix consisting of a main diagonal of ones and a diagonal below the main diagonal and D is a diagonal matrix. Once the factorization is performed, the solution is computed as u = L −T D(L −1 f). Note the inversions of L and L T are not computed explicitly. For example L −1 f is computed as a linear solve with L. Linear solves for triangular matrices are easy to program. This is what DPTTRS does for us. A C++ interface to DPTTRF and DPTTRS is available from the BoostSandbox.Bindings. For our application, we can solve a linear system as follows.
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Technische Universität Dresden Fak
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Contents I Understanding C++ 7 Intr
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CONTENTS 5 10.5 Unix and Linux . .
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Part I Understanding C ++ 7
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10 C ++ was not a reliable computer
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12 CHAPTER 1. GOOD AND BAD SCIENTIF
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20 CHAPTER 2. C++ BASICS • std::c
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22 CHAPTER 2. C++ BASICS In the fir
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24 CHAPTER 2. C++ BASICS int main (
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26 CHAPTER 2. C++ BASICS Operator A
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36 CHAPTER 2. C++ BASICS eps/= 2.0;
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40 CHAPTER 2. C++ BASICS 2.6.1 Inli
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48 CHAPTER 2. C++ BASICS #ifndef at
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56 CHAPTER 2. C++ BASICS A function
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66 CHAPTER 3. CLASSES apply symm bl
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68 CHAPTER 3. CLASSES int main() {
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74 CHAPTER 3. CLASSES a real number
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76 CHAPTER 3. CLASSES } return ∗t
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Page 293: Bibliography [AG04] David Abrahams
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