Abstract
Many problems in physics and especially computational physics involve systems of linear equations which arise e.g. from linearization of a general nonlinear problem or from discretization of differential equations. If the dimension of the system is not too large standard methods like Gaussian elimination or QR decomposition are sufficient. Systems with a tridiagonal matrix are important for cubic spline interpolation and numerical second derivatives. They can be solved very efficiently with a specialized Gaussian elimination method. Practical applications often involve very large dimensions and require iterative methods. Convergence of Jacobi and Gauss-Seidel methods is slow and can be improved by relaxation or over-relaxation. An alternative for large systems is the method of conjugate gradients.
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Notes
- 1.
Alternatively Givens rotations [108] can be employed which need slightly more floating point operations.
- 2.
This algorithm is only well behaved if the matrix is diagonal dominant |b i |>|a i |+|c i |.
- 3.
Here uv T is the outer or matrix product of the two vectors.
- 4.
This is also known as the method of successive over-relaxation (SOR).
- 5.
The vector norm used here is not necessarily the Euclidian norm.
- 6.
Using the Frobenius norm \(\|A\|=\sqrt{\sum_{ij}A_{ij}^{2}}\).
References
B. Beckermann, Numer. Math. 85, 553 (2000)
A. Bermann, S. Gueron, Math. Gaz. 86, 274 (2002)
I.N. Bronshtein, K.A. Semendyayev, Handbook of Mathematics, 3rd edn. (Springer, New York, 1997), p. 892
J.W. Daniel, W.B. Gragg, L. Kaufmann, G.W. Stewart, Math. Comput. 30, 772 (1976)
G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore, 1976). ISBN 978-0-8018-5414-9
M.R. Hestenes, E. Stiefel, J. Res. Natl. Bur. Stand. 49, 435 (1952)
D. Hilbert, L. Nordheim, J. von Neumann, Math. Ann. 98, 1 (1927)
H. Jeffreys, B.S. Jeffreys, Methods of Mathematical Physics, 3rd edn. (Cambridge University Press, Cambridge, 1988), pp. 305–306
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, LU decomposition and its applications, in Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007), pp. 48–55
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Cholesky decomposition, in Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007), pp. 100–101
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Cyclic tridiagonal systems, in Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007), p. 79
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Relaxation mehods for boundary value problems, in Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007), pp. 1059–1065
I.R. Savage, E. Lukacs, in Contributions to the Solution of Systems of Linear Equations and the Determination of Eigenvalues, ed. by O. Taussky. National Bureau of Standards, Applied Mathematics Series, vol. 39 (1954)
J. Sherman, W.J. Morrison, Ann. Math. Stat. 20, 621 (1949)
J. Stör, R. Bulirsch, Introduction to Numerical Analysis, 3rd revised edn. (Springer, New York, 2010). ISBN 978-1441930064
L.N. Trefethen, D. Bau III, Numerical Linear Algebra (Society for Industrial and Applied Mathematics, Philadelphia, 1997), p. 1
J.H. Wilkinson, J. ACM 8, 281 (1961)
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Scherer, P.O.J. (2013). Systems of Inhomogeneous Linear Equations. In: Computational Physics. Graduate Texts in Physics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00401-3_5
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