Peter Benner, Ralph Byers, Enrique S. Quintana-Ort'i, and Gregorio Quintana-Ort'i. Solving algebraic Riccati equations on parallel computers using Newton's method with exact line search. Report 98-05, Zentrum fur Technomathematik, Universitat Bremen, Bremen, Germany, August 1998. 33 pp. Home/Search Document Details and Download
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Efficient Numerical Algorithms for Balanced.. - Benner.. Self-citation (Benner Quintana-ort) (Correct)
....(21) to solve the Lyapunov equations in each iteration step in order to determine the Newton step N j . As this is the major computational step in Newton s method for solving (5) this algorithm can be eciently implemented on parallel computers; see the discussion at the end of Subsection 3. 1 and [9]. Possible stopping criteria for this algorithm are discussed in [9] Roughly speaking, the iteration can be stopped once kR (X j ) k kX j k or kN j k kX j k for some user de ned tolerance threshold . The line search parameter t j is determined as a local minimizer of the one dimensional ....
....to determine the Newton step N j . As this is the major computational step in Newton s method for solving (5) this algorithm can be eciently implemented on parallel computers; see the discussion at the end of Subsection 3. 1 and [9] Possible stopping criteria for this algorithm are discussed in [9]. Roughly speaking, the iteration can be stopped once kR (X j ) k kX j k or kN j k kX j k for some user de ned tolerance threshold . The line search parameter t j is determined as a local minimizer of the one dimensional optimization problem to minimize the residual of the next step with ....
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P. Benner, R. Byers, E.S. Quintana-Ort, and G. Quintana-Ort. Solving algebraic Riccati equations on parallel computers using Newton's method with exact line search. Parallel Comput., 26(10):1345-1368, 2000.
Solving Algebraic Riccati Equations on Parallel.. - Benner, Byers.. (1999) (1 citation) Self-citation (Benner Byers) (Correct)
....Tables 1 and 2. Note that the performance model only refers to a single iteration of Newton s method. In the tables, iter stands for the number of iterations necessary for the convergence of the corresponding matrix sign function iteration for the Lyapunov equation. Further details are given in [10]. Iteration for Computation cost Lyapunov eq. Theta n 3 p Newton 14 6 Theta iter Halley 14 50 3 Theta iter Newton Schulz 14 12 Theta iter Generalized Newton 64 3 26 3 Theta iter Table 1 Computation cost of a single iteration of Newton s method for the CARE. We have ....
P. Benner, R. Byers, E.S. Quintana-Ort'i, and G. Quintana-Ort'i. Solving algebraic Riccati equations on parallel computers using Newton's method with exact line search. Berichte aus der Technomathematik, Report 98--05, Universitat Bremen, 28334 Bremen (Germany), August 1998.
Solving Linear and Quadratic Matrix Equations on.. - Benner.. Self-citation (Benner) (Correct)
....(3) ffl PDGECRNW. Newton s method for the ARE (4) with E = I n using PDGECLNW for solving the Lyapunov equations. ffl PDGGCRNW. Newton s method for the ARE (4) using PDGGCLNW for solving the generalized Lyapunov equations. For more details on accuracy, algorithms, and implementation see [4, 5, 6, 7]. For the naming convention and a survey on other subroutines implemented in this area see [8] Example 1 We generated the coefficient matrices A = Vn diag (ff 1 ; ff n ) Wn ; E = VnWn ; where the scalars ff 1 ; ff n are uniformly distributed in [ Gamma10; 0) Wn is an n Theta n ....
P. Benner, R. Byers, E.S. Quintana-Ort'i, and G. Quintana-Ort'i. Solving algebraic Riccati equations on parallel computers using Newton 's method with exact line search. Berichte aus der Technomathematik, Report 98--05, Universit at Bremen, August 1998. Available from www.math.uni-bremen.de/zetem/berichte.html.
Numerical Analysis of a Quadratic Matrix Equation - Higham, Kim (1999) (1 citation) (Correct)
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Peter Benner, Ralph Byers, Enrique S. Quintana-Ort'i, and Gregorio Quintana-Ort'i. Solving algebraic Riccati equations on parallel computers using Newton's method with exact line search. Report 98-05, Zentrum fur Technomathematik, Universitat Bremen, Bremen, Germany, August 1998. 33 pp.
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