• DOI:10.1145/365170.365195
• Corpus ID: 8876563

Algorithm 277: Computation of Chebyshev series coefficients

@article{Smith1966Algorithm2C,
  title={Algorithm 277: Computation of Chebyshev series coefficients},
  author={Lyle B. Smith},
  journal={Commun. ACM},
  year={1966},
  volume={9},
  pages={86-87}
}

• L. B. Smith
• Published 1 February 1966
• Mathematics
• Commun. ACM

modification of the classical least squares method is utilized to approximate a solution to the system of nonlinear equations of condition. After every iteration, the statistic E squared is computed as a measure of the goodness of fit. Commencing with the second iteration, the successive values of E squared are differenced, and when the difference in absolute value becomes less than epsilon, the calculations cease. If the number of iterations necessary to achieve this result exceeds 1 max, aflag is set to a nonzero value and the procedure is terminated. In normal usage, the jump parameter is brought in as a ZERO. With certain data sets, convergence difficulties will be experienced. In these cases it is sometimes helpful to first utilize the procedure EXPCRVFT [Algorithm 275, Comm. ACM 9 (Feb. 1966), 85] to obtain initial values for b and c, and then bring the jump parameter in as a ONE in order to bypass the following starting value computations for b and c.; begin integer i, l, m; real exp factor; if jump = 1 then go to entry; comment Computation of initial estimates follows; b := 2 X ln(abs(((y[n]-y[n-1]) X (x[2]-x[ll))/ ((y[21-y[l]) X (x[n]-x[n-ll))))/ (x[n] + xtn-1]-x[2]-~[11); m := (n+l) + 2; exp factor := exp(b X (x[m]-x[k])); c := (y[m]-z X exp factor)l(1-exp factor); a := (z-c) X exp(-b X x[k]); E squared := 0; for i := 1 step 1 until n do E squared := E squared + (y[i]-c-a X exp(b X x[i])) 1" 2; eonunent Computation of corrections follows; q := x[i]-z[k]; exl := exp(b X q); ex2 := exl ~ 2; qexl := q X exl; qex2 := q X ex2; qexllsex2 := qexl-qex2; q2ex2 := qex2 X q; sumexl := sumexl + exl; sumex2 := sumex2-~ ez2; sumqexl := sumqexl ~ qexl; sumqez2 := sumqex2 ~ qex2; sumqexllsex2 := sumqexllsex2-4-qexllsex2; sumq2ex2 := sumq2ex2 + q2ex2; sumyi := sumyi-4-y[i]; sumyiexl := sumyiexl-4-exl X y[i]; sumqyiexl := sumqyiexl ~ qexl X y[i]; end computation of sum terms in normal equations; zlsc := z-c; dll := sumq2ex2 X zlsc ~ 2; d12 := sumqexllsex2 X zlsc; d22 := n-2 X sumexl-~ sumex2; el := sumqyiexl X zlsc-sumqex2 X zlsc ~ 2-sumqexl X zlsc X c; e2 := sumyi-sumyiexl + sumexl X (2 X c-z) + sumex2 X zlsc-n X c; delta := dll X d22-d12 ~'2; v := (el X d22-e2 X dl2)/delta; w := (e2 …