By Sabine Van Huffel, Joos Vandewalle
This can be the 1st ebook dedicated totally to overall least squares. The authors supply a unified presentation of the TLS challenge. an outline of its easy rules are given, some of the algebraic, statistical and sensitivity houses of the matter are mentioned, and generalizations are provided. functions are surveyed to facilitate makes use of in a fair wider variety of functions. at any time when attainable, comparability is made with the well known least squares equipment.
A simple wisdom of numerical linear algebra, matrix computations, and a few suggestion of effortless records is needed of the reader; in spite of the fact that, a few history fabric is incorporated to make the booklet quite self-contained.
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Example text
22) and with p < n and Vjj the jth component of v\. Hence, for TLS problem d — 1 : one- dimensional d > 1 : multidimensional TLS solution is generic and unique. 2. TLS solution is computed. 15 describe the conditions. 4) TLS solution is generic and not unique. Minimum norm TLS solution is computed. 3. If we allow more than one observation vector, the TLS problem definition can be generalized as follows. 1 (Multidimensional TLS problem). mxd, in n X d unknowns X.
If these sets Ax w b are overdetermined, then generally only an approximate solution can be computed, for instance by using a LS or TLS approach. The basic principle of TLS is that the noisy data [A] 6], while not satisfying a linear relation, are modified with minimal effort, as measured by the Frobenius norm, in a "nearby" matrix [A; 6] that is rank-deficient so that the set Ax = b is compatible. This matrix [A; b] is a rank one modification of the data matrix [A; 6]. 3). A simple algorithm outlines the computations of the solution of the basic TLS problem.
To preserve uniqueness, TLS seeks for the minimum norm solution. 4, TLS problems that fail to have a solution altogether are investigated. The properties of these so-called nongeneric problems are described and an extension of the generic TLS problem, called nongeneric TLS, is proposed to make these problems solvable. It is proved that the solution to these nongeneric TLS problems is still optimal with respect to the TLS criteria for any number of observation vectors in B if additional constraints are imposed on the TLS solution space.