Matrix computations singular-value decomposition

Given any data matrix A of arbitrary size (as rows x columns) the matrix A can be written or defined using the computation of Singular Value Decomposition [6-8] as... 

The singular-value decomposition (SVD) is a computational method for simultaneously calculating the complete set of column-mode eigenvectors, row-mode eigenvectors, and singular values of any real data matrix. These eigenvectors and singular values can be used to build a principal component model of a data set. 

A final method for selecting a temperature control tray location is to use singular value decomposition (SVD) techniques. This approach was first presented by Downs and Moore and is summarized on p. 458 in Luyben and Luyben (1997). A steady-state rating program is used to obtain the gains between the two manipulated variables and the temperatures on all trays. The gain matrix is decomposed by using SVD to find the most sensitive tray locations. This method requires more computation than the others. 

The PCs can be computed by spectral decomposition [126], computation of eigenvalues and eigenvectors, or singular value decomposition. The covariance matrix S (S=X X/(m — 1)) of data matrix X can be decomposed by spectral decomposition as... 

At present, data projection is performed mainly by methods called PC A, FA, singular value decomposition (SVD), eigenvector projection, or rank annihilation. The different methods are linked to different science areas. They also differ mathematically in the way the projection is computed, that is, which dispersion matrix is the basis for data decomposition, which assumptions are valid, and whether the method is based on eigenvector analysis, SVD, or other iterative schemes. 

In this example, we have inverted a 2 X 2 matrix. Perhaps an inversion by head could also be performed in the case of a 3 X 3 matrix. For larger matrices, however, a computer algorithm is necessary. In addition, matrix inversion is a very sensitive procedure, so that powerful algorithms, such as singular value decomposition (cf. Section 5.2), are to be applied. 

The singular value decomposition (SVD) method, and the similar principal component analysis method, are powerful computational tools for parametric sensitivity analysis of the collective effects of a group of model parameters on a group of simulated properties. The SVD method is based on an elegant theorem of linear algebra. The theorem states that one can represent an w X n matrix M by a product of three matrices ... 

To summarize, we perform a singular value decomposition of the augmented formula matrix to obtain the matrices U, W, and V. With these, we use (11.2.10) to obtain a particular basis vector N for the range. From V, we form P and then use (11.2.7) to obtain all sets of stoichiometric coefficients Vy. Then we combine N and Vy into (11.2.5) to determine all sets of mole numbers that satisfy the elemental balances. Therefore, a singular value decomposition provides the number of independent reactions 91, all sets of 91 independent stoichiometric coefficients Vy, and all possible combinations of mole numbers N that satisfy the elemental balances. A computer program for performing the decomposition is contained in the book by Press et al. [9] routines for performing the decomposition are also available in MATLAB and in Mathematica . 

A little more expensive [n (m + I7nl3) flops and 2mn space versus n (m—nl3) and mn in the Householder transformation] but completely stable algorithm relies on computing the singular value decomposition (SVD) of A. Unlike Householder s transformation, that algorithm always computes the least-squares solution of the minimum 2-norm. The SVD of an m x n matrix A is the factorization A = ITEV, where U and V are two square orthogonal matrices (of sizes mxm and nxn, respectively), U U = Im, y V = In, and where the m x n matrix S... 

Regarding FDE-ET, both cases can be circumvented computationally by performing a singular value decomposition of the overlap matrix and then invert only those values which are larger than a threshold (i.e. Penrose inversion). For DNA presented in Sect. 4.2.1.3, the default inversion threshold of 10 was appropriate in most cases [67]. However, three systems stood out AG, GA and TT nucleobase pairs. All the systems above showed erratic behavior of the computed couplings for some specific donor-acceptor distances, speeifically 4.0 A for AG, 3.5 and 8.0 A for GA and 9.0 A for TT. We found that at those distances, the near singularity of the overlap matrix due to symmetry considerations (case 1 above) was the source of the erratic behavior. To circumvent these numerical issues, a threshold of 10 was adopted in these cases. 

In practice, however, matrix diagonalization is computationally expensive, and the calculation of R can take an intolerable long time even with a modern computer. The algorithm based on singular value decomposition is an... 

Singular value decomposition is a convenient way to compute the eigenvalues of a non-square matrix. Suppose the f matrix is ... 

The main algorithms used for eigenvectors/eigenvalues computation differ in two aspects the matrix to work on, either X X (eigenvalue decomposition (EVD) and the POWER method) or X (singular value decomposition (SVD) and non-linear iterative partial least squares (NIPALS)). However SVD may work as well on X X (giving the same results as eigenvalue decomposition). Another difference is whether PCs are obtained simultaneously (EVD and SVD) or sequentially (POWER and NIPALS) for details and comparison of efficiency see Wu et al. [38]. In all the cases for which rows dimension I is much smaller than columns dimension /, one can operate on XX instead (EVD, POWER, SVD), and on X (NIPALS). 

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