GSVDGeneralized Singular Value Decomposition
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It formulates GSVD as an expansion or usual simplification of singular value decomposition (SVD) [20, 57, 58].
As shown in figure 2, the inputs to the trained ANN for GSVD are the dimensions or the total number of educational and noneducational websites accessed by each user.
By the simultaneous factorization (3.6) it is possible to define a truncated GSVD (TGSVD) solution [s.sub.l]; see [9] for details.
For a fixed value of the regularization parameter l, we substitute the truncated SVD or GSVD solution of (3.2) [s.sub.l]) to the step size [s.sub.k] in (3.4), obtaining the following iterative method
Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], then the GSVD of the matrix pair (N1, N2) is given by
Our approach is based upon the Projection Theorem in Hilbert products spaces, as well as GSVD and CCD of matrix pairs, and can be essentially divided into three parts:
part 3: Find the symmetric solutions of this consistent matrix equation by using GSVD.
We underline that the superscripts S and G have been introduced to better distinguish the SVD of A and the GSVD of (A, L), respectively, from the SVD and GSVD of the matrices associated to the projected problems that we will consider in the following sections.
Now let [[bar.D].sub.m] = [[bar.U].sub.m] [[bar.S].sub.m] [[bar.X].sup.-1.sub.m] and [L.sub.m] = [[bar.V].sub.m] [[bar.C].sub.m] [[bar.X].sup.-1.sub.m] be the GSVD decomposition of the matrix pair ([[bar.D].sub.m], [L.sub.m]), where [[bar.U].sub.m] [member of] [R.sup.(m+1)x(m+1)] and [[bar.V].sub.m] [member of] [R.sup.mxm] are orthogonal, [[bar.X].sub.m] [member of] [R.sup.mxm] is nonsingular, and
In these literatures, the generalized inverses or some complicated matrix decompositions such as canonical correlation decomposition (CCD) [23] and GSVD are employed.
In this paper, we are interested in developing solution methods that can be applied when the matrices A and B are too large to compute the GSVD or a related decomposition of the matrix pair {A, B}.
This method is based on the partial GSVD method described by Zha [24].