Abstract
In this paper we present an algorithm ASVD for the computation of the singular value decomposition (SVD). The method presented is a power method for calculating the largest triplets of the SVD of a matrix A when multiplication is "cheap". The method used bears a lot of similitude with the power method for finding the eigenvalues of a symmetric matrix M. The triplets are found one after another and also some deflation techniques (orthogonalization) are used. The algorithm can take profit of the SVD of slightly different matrices and it is based on the geometric properties of the SVD. Tests have shown that it is more efficient than Golub's algorithm if only the dominant part of the SVD of a long sequence of slowly varying matrices is needed. Also storage efficiency is obtained whenever the matrices are structured.
The paper describes the basic ASVD algorithm, its numerical properties using the shift mechanism, an acceleration method, a computer implementation and its use in adaptive state space realization of noisy impulse responses. It is expected that the new ASVD algorithm and its many strategies will be useful in the domain of signal processing, system theory and automatic control, where SVD is becoming more and more an important concept.
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References
Special Issue on Multivariable Control Systems, IEEE Trans. Aut. Contr., Vol AC-26, No. 1, 1981.
V.C. Klema and A.J. Laub, "The singular Value Decomposition: Its computation and some applications." IEEE Trans. Aut. Contr., Vol AC-25,No 2, p.164–176, 1980.
J. Staar, "Concepts for reliable modelling of linear systems with application to on-line identification of multivariable state space descriptions", doct. diss., EAST laboratory, Kath. Univ. Leuven, June 1982.
J. Staar, J. Vandewalle & M. Wemans, "A Comparison of multivariable MBH-algorithms in the presence of multiple poles, and noise disturbing the Markov sequence." Proc. 4-th Int. Conf. on analysis and optimisation of systems (Lecture notes in Control and information Sciences), p. 141–160, Springer Verlag, Versailles, December 1980.
J. Staar & J. Vandewalle, "On the numerical implication of the choice of a reference node in the nodal analysis of large circuits", Int. Jrnl. Circuit Theory and Applications, Vol 9, p.488–492, 1981.
J. Vandewalle and J. Staar, "On the recursive identification of multivariable state space models", Proc. Symposium on applications of multivariable system theory, Plymouth, October 1982.
J. Vandewalle & J. Staar, "Modelling of linear systems: critical examples and numerically reliable approaches", Proc. ISCAS, p.915–918,Rome May 1982.
H.P. Zeiger and A.J. Mc. Ewen, "Approximate linear realizations of given dimension via Ho's algorithm", IEEE Trans. Aut. Contr., Vol AC-19 p.153,1974.
S.Kung, "A new identification and model reduction algorithm via singular value decomposition", Proc. 12-th Asilomar Conf. Circ. Syst. Nov 1978
L.S. De Jong. "Numerical aspects of the recursive realization algorithm", SIAM Journal on Control and optimization, Vol.16, p.646–659, 1978.
C.L. Lawson and R.J. Hanson, "Solving least squares problems", Prentice Hall Series in Automatic Computation, Englewood Cliffs 1974.
J. Vanderschoot, J. Vandewalle, J. Janssen, W. Sansen, H. Van Trappen: "Extraction of weak bioelectrical signals by means of singular value decomposition", sub. to this conference.
G.H. Golub and W.Kahan, "Calculating the Singular values and Pseudo-inverse of a matrix", SIAM J. Num. Anal. No.3, p.205–224, 1965.
G.H. Golub and C. Reinsch, "Singular Value Decomposition and Least squares solutions", Num. Math., Vol.14,p.403–420, 1970.
J.Staar and J. Vandewalle, "Oriented energy and signal to signal ratios concepts and use" in preparation for sub. to Systems and Control letters.
J.Lauwers, "Singuliere waardenontbinding in signaalverwerking; ontwerp en analyse van een efficient algoritme", Master's thesis, ESAT lab., Kath. Univ. Leuven, July 1983.
B.De Moor, "Een betrouwbaar adaptief realisatiealgoritme gebaseerd op recursieve singuliere waardenontbinding.", Master's thesis, ESAT Lab., Kath. Univ. Leuven, July 1983.
B. Garbow et al., "Matrix eigensystem routines-EISPACK guide extension", (Lecture Notes in Computing Sciences, Vol 51) New York:Springer 1977.
J. Dongerra et al., "Linpack User's guide", Philadelphia, SIAM 1979.
Y.S. Hung and A.G.J. Mac Farlane, "Multivariable feedback: A quasi-classical approach", Springer Lecture notes in Control and Information Sciences, Berlin 1982.
J.H. Wilkinson, "The algebraic eigenvalue problem", Clarendon Press, Oxford 1965.
Beresford N. Parlett, "The symmetric eigenvalue problem." Prentice Hall Series in computational mathematics, 1980.
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Vandewalle, J., Staar, J., De Moor, B., Lauwers, J. (1984). An adaptive singular value decomposition algorithm and its application to adaptive realization. In: Bensoussan, A., Lions, J.L. (eds) Analysis and Optimization of Systems. Lecture Notes in Control and Information Sciences, vol 63. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0006275
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DOI: https://doi.org/10.1007/BFb0006275
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