By J. S. Rollett
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A power series can be economised, for a specified range of the argument, by converting it into a Chebyshev series and truncating this (which normally converges rapidly) before converting back. This process is so widely applicable in numerical analysis that it is worth detailed description. It was devised by Lanczos who has given a full account of the polynomials concerned (Lanczos, 1952). If we write cos 0 = 2x-1, T*n(x) = cos ηθ, we get n rn(x) 1 =r0(x) 0 1 1 2x-l x =κη+τ\) (4) 2 8x2-8x+l x2 = (3r0+4T\+r2) 3 3 32x 3 -48x 2 + 18x-l x = (ioTo+i5r;+6r 2 +r 3 ) and so on.
S. parameter 0; Compute i IdA , , dB . s. parameter derivative. (Different routines for different parameter types); 9. Advance atomic parameter number, go to 5 until no more parameters this atom; 10. Advance r, go to 5 until r = w + l ; 54 J. S. ROLLETT [6, §12-13 11. Get order of present matrix, nu ; 12. S e t / = 1; 13. Accumulate 14. 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.
Clear all least-squares parameter derivatives 4. 5. 6. 7. 8. s. s. parameter 0; Compute i IdA , , dB . s. parameter derivative. (Different routines for different parameter types); 9. Advance atomic parameter number, go to 5 until no more parameters this atom; 10. Advance r, go to 5 until r = w + l ; 54 J. S. ROLLETT [6, §12-13 11. Get order of present matrix, nu ; 12. S e t / = 1; 13. Accumulate 14. 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.