Optimal orthonormalization of the strapdown matrix by using singular value decomposition. (English) Zbl 0606.65027
The mathematical problem considered in this paper is computing for a given real quadratic matrix A the nearest orthonormal matrix with respect to the Frobenius norm. The well-known fact that this matrix is given by \(UV^ T\), where \(A=U\Sigma V^ T\) is the singular value decomposition (SVD) of A is rediscovered and the computation of \(UV^ T\) via the usual SVD computation is proposed as a solution to the problem. An application and numerical experiments are given.
Reviewer: A.Bunse-Gerstner
MSC:
| 65F30 | Other matrix algorithms (MSC2010) |
| 65F25 | Orthogonalization in numerical linear algebra |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
Keywords:
nearest orthonormal matrix; Frobenius norm; singular value decomposition; numerical experimentsSoftware:
LINPACKReferences:
| [1] | Bar-Itzhack, Y., Iterative optimal orthogonalization of the strapdown matrix, IEEE Trans. Aerosp Electron Syst, AES-11, 30-37 (1975) |
| [2] | Bar-Itzhack, I. Y., Optimum normalization of a computed quaternion of rotation, IEEE Trans. Aerosp Electron Syst, AES-7, 401-402 (1971) |
| [3] | Bar-Itzhack, I. Y.; Fegley, K. A., Orthogonalization techniques for a direction cosine matrix, IEEE Trans. Aerosp Electron Syst, AES-5, 798-804 (1969) |
| [4] | Bar-Itzhack, I. Y.; Meyer, J.; Fuhrmann, P. A., Strapdown matrix orthogonalization: The dual iterative algorithm, IEEE Trans. Aerosp Electron Syst, AES-12, 32-37 (1976) |
| [5] | Bar-Itzhack, I. Y.; Meyer, J., On the convergence of iterative orthogonalization processes, IEEE Trans. Aerosp Electron Syst, AES-12, 146-151 (1976) |
| [6] | Bar-Itzhack, I. Y., Optimal orthonormalization of strapdown guidance systems, IEEE PLANS, 76, 122-127 (1976) |
| [7] | Bar-Itzhack, I. Y., A unidimensional convergence test for matrix iterative processes applied to strapdown navigation, Int J Numer Methods Eng, II, 115-130 (1977) · Zbl 0344.65021 |
| [8] | Giardina, C. R.; Bronson, R.; Wallen, L., An optimal normalization scheme, IEEE Trans. Aerosp Electron Syst, AES-11, 443-446 (1975) |
| [9] | Björk, A.; Bowie, C., An iterative algorithm for computing the best estimate of an orthogonal matrix, J. SIAM. Numerical Analysis, 8, 358-364 (1971) · Zbl 0221.65075 |
| [10] | Stewart, G. W., Introduction To Matrix Computations (1973), Academic Press: Academic Press New York · Zbl 0302.65021 |
| [11] | Klema, V. C.; Laub, Alan J., The singular value decomposition: Its computation and some applications, IEEE Trans. Aerosp Electron Syst, AC-25, 164-176 (1980) · Zbl 0433.93018 |
| [12] | Dongarra, J. J., LINPACK User’s Guide (1979), SIAM: SIAM PA |
| [13] | Bar-Itzhack, I. Y., Practical comparison of iterative matrix orthogonalization algorithms, IEEE Trans. Aerosp Electron Syst, AES-13, 230-235 (1977) |
| [14] | Golub, G. H.; Reinsch, C., Singular value decomposition and least square solutions, Numer. Math., 14, 403-420 (1970) · Zbl 0181.17602 |
| [15] | Jianqin Mao, The perturbation analysis of the product of singular vector matrices \(UV^τ\); Jianqin Mao, The perturbation analysis of the product of singular vector matrices \(UV^τ\) · Zbl 0626.65039 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
