By J. S. Rollett
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Extra info for Computing Methods in Crystallography
15. The output of the program described in Sects. 12 and 14 gives standard deviation estimates for the individual parameters, but does not provide the maximum information available for calculation of standard deviation estimates of functions of parameters. This cannot be obtained unless all elements of the inverse matrices calculated are available, but this is such a volume of information that if the matrices are large, output is very expensive. It is desirable that, if the computer system concerned has sufficient capacity and speed, the least-squares output parameters and inverse matrices should be left in a form suitable for use by a separate routine designed to calculate functions of parameters and their standard deviations.
15. 16. 17. 18. Advance ι, go to 13 until i = nu+1 ; Go to 21 if right-hand sides only to be computed; S e t / = 1; Set y = i; Accumulate Σ*(Α*Αλ(ΜΙ) 19. Advance j , go to 18 until y = nu+1 ; 20. Advance i, go to 17 until i = nu+1 ; 21. A d d a t o * ; ; 22. Advance w, go to 11 until last matrix done; 23. Exit. If this scheme is used it is clear that the right-hand sides accumulated at step 13, and the coefficients at step 18, will form lists which vary in length greatly from one case to another.
CHAPTER 7 LATENT ROOTS AND VECTORS 1. Chapters 1 and 2 dealt with the transformation Ax = y and the simultaneous equations Ax = b respectively. In this chapter we shall consider the homogeneous equations Ax = λχ (1) These can be put in the alternative form (Α-Ιλ)χ = 0 (2) Here A is a square matrix of order n, x a vector of order n and λ a scalar. There are n values of A for which (1) has a solution with x#0, and these values are called the latent roots, the characteristic roots, the eigenvalues, or the proper values of the matrix A.