Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems
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Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems by
M.ENGELI·TH.GINSBURG H.RUTISHAUSER· E.STIEFEL
BIRKHAUSER VERLAG· BASEL/STUTTGART 1959
ISBN 978-3-0348-7226-3 ISBN 978-3-0348-7224-9 (eBook) DOl 10.1007/978-3-0348-7224-9
© Birkhauser Verlag Basel 1959 Softcover reprint of the hardcover 1st edition 1959
3
In 1957, I gathered a team of four scientists in my Institute of Applied Mathematics in order to study relaxation methods for solving elliptic partial differential equations from the theoretical and practical point of view as well. The basic idea was to test different methods by applying them to a specific problem of elasticity and to collect the theoretical fruits of this activity. During many years we have been interested in relaxation and I wanted to give a final report enabling myself to turn the light red for further investigations along this line and to look around for new flowers in the field of numerical analysis. HEINZ RUTISHAUSER was our indefatigable theoretical expert; he discovered the beautiful link between gradient methods and Q,D-algorithm; THEO GINSBURG was manager of the project and carried out most of the experiments; MAX ENGEL! took care of overrelaxation. The fourth member of the group was very happy to collaborate with the remaining three during two years of satisfying and adventurous research. We are very indebted to IBM and in particular to Dr. SPEISER for sponsoring this work and also to the staff of the IBM-center at Paris for assistance during our computations on their IBM 704. November 1959
E.
STIEFEL
5 TABLE OF CONTENTS
Chapter
I: The Self-Adjoint Boundary Value Problem (E. STIEFEL)
1. Problems of Dirichlet's and Poisson's type ..................... Energy on the boundary ....................................
9 13 15
Eigenvalue problems ....................................... Biharmonic problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Adaption for practical purposes; the test example . . . . . . . . . . . . . ..
15 16 19
Modes of oscillation of the plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
23
2. Better a pproxima tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 4. 5. 6. 7.
Chapter II: Theory
rif Gradient
Methods (H. RUTISHAUSER)
1. Introduction ..............................................
24
2. The residual polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Methods with two-term recursive formulae. . . . . . . . . . . . . . . . . . . .. Methods with three-term recursive formulae. . . . . . . . . . . . . . . . . . .. Combined methods ........................................
25 27 30 35
The cgT -method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Determination of eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
38 43
3. 4. 5. 6. 7.
Chapter III: Experiments on Gradient Methods (TH. GINSBURG)
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