For which δ s 2 s 2 s 2 min s 2 min m 1 102 11 using

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for which Δ s 2 = s 2 s 2 min = s 2 min M 1 (10.2) 11. USING SINGULAR VALUE DECOMPOSITION (SVD) Occasionally, a normal-equation matrix [ α ] = X T · X is degenerate, or at least sufficiently ill-posed that inverting it using standard matrix inversion doesn’t work. In its invert function, IDL even provides a keyword called status to check on this (although I find that it is not perfectly reliable; the best indicator of reliability is to check that the matrix product [ α - 1 ] · [ α ] = I , the unitary matrix). In these cases, Singular Value Decomposition (SVD) comes to the rescue. First, we reiterate the least-squares problem. Least squares begins with equations of condition (equation 2.2), which are expressed in matrix form as X · a = y (11.1) In our treatments above we premultiply both sides by X T , on the left generating the curvature matrix [ α ] to obtain the normal equations (equation 2.3 or its equivalent derived from equations 8.8) [ α ] · a = X T · y , (11.2)
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– 39 – for which the solution is, of course, a = ( [ α ] - 1 · X T ) · y (11.3) We need to find the inverse matrix [ α ] - 1 . Above in this document we did this using simple matrix inversion, which doesn’t work if [ α ] is degenerate. SVD provides a bombproof and interestingly informative way do least squares. Perhaps sur- prising, with SVD you don’t form normal equations. Rather, you solve for the coefficients directly. In essence, SVD provides the combination ( [ α ] - 1 · X T ) without taking any inverses. By itself, this doesn’t prevent blowup for degenerate cases; however, SVD provides a straightforward way to eliminate the blowup and get reasonable solutions. For a discussion of the details of SVD, see NR § 2.6; for SVD applied to least squares, see NR § 15.4; if you are rusty on matrix algebra, look at NR § 11.0. Below, we provide a brief description of SVD. Implementing SVD in our least-squares solutions is trivially easy, and we provide the IDL prescription below in § 11.5 and § 11.5.2. Be sure to look at § 11.3!! 11.1. Phenomenological description of SVD The cornerstone of SVD is that our (or any) M × N ( M rows, N columns) matrix X , where M N , can be expressed as a product of three matrices: X = U · [ w ] · V T , (11.4) where 1. U is M × N , [ w ] is N × N and diagonal, and V is N × N ; and 2. the columns of U and V are unit vectors that are orthonormal. Because V is square, its rows are also orthonormal so that V · V T = I . Recall that, for square orthonormal vectors, the transpose equals the inverse so V T = V - 1 . Similarly, because U has columns that are orthonormal, the matrix product U T · U = I N × N (the N × N unitary matrix). 3. The columns of V T , which are orthonormal vectors, are the eigenvectors of [ X T · X ]. That is, in our case, they are the eigenvectors of the curvature matrix of X . That is, they define the principal axes of the error ellipsoid Δ χ 2 ; see NR § 14.5.
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