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**Additional resources for Algebraic Theory of the Bianchi Groups**

**Example text**

37), the solutions x corresponding to only a few of the Singular-value decomposition, and use in least-squares problems 41 dominant principal components can be computed. Furthermore, at this stage in the calculation UT b should already have been computed and saved, so that only a simple matrix-vector multiplication is involved in finding each of the solutions. 6). 39) BT Bw = S2 w = BT b. 40) w = S - 1UT b. 6) gives Should the problem be singular, then w = S+ U T b. 42) can be used. 44) x = Vw .

It may be required that in the non-unique case, x shall be that member of the set of vectors which minimises rT r which has x T x a minimum also. In this case x is the unique minimum-length least-squares solution. 22) as the set of conditions which x must satisfy. These are simply n simultaneous linear equations in n unknowns x and are called the normal equations. Solution of the least-squares problem via the normal equations is the most common method by which such problems are solved. Unfortunately, there are several objections to such an approach if it is not carefully executed, since the special structure of ATA and the numerical instabilities which attend its formation are ignored at the peril of meaningless computed values for the parameters x.

2. A SINGULAR-VALUE DECOMPOSITION ALGORITHM It may seem odd that the first algorithm to be described in this work is designed to compute the singular-value decomposition (svd) of a matrix. Such computations are topics well to the back of most books on numerical linear algebra. However, it was the algorithm below which first interested the author in the capabilities of small computers. Moreover, while the svd is somewhat of a sledgehammer method for many nutshell problems, its versatility in finding the eigensolutions of a real symmetric matrix, in solving sets of simultaneous linear equations or in computing minimum-length solutions to least-squares problems makes it a valuable building block in programs used to tackle a variety of real problems.