The Fornberg Algorithm

Equation (3.25) holds for equal intervals but also for arbitrary (unequal) intervals, if the coefficients are computed accordingly. For unequal intervals, the coefficients must be computed (probably precomputed in a given program), as they cannot be tabulated, and this is best done using the Fornberg algorithm [10], to be described in the later Chap. 7. It is implemented in the routine GOFORN, also described in Appendix E. 

Theoretically, there is no limit on the number n of points used in the approximations. In practice, however, a limit is set by roundoff in the computations, making an increase in n useless, and the factorials in the matrix H will increase to impractical levels. In any case, there seems little point in n values greater than about 8, although for the usual 32-bit computers in use today, up to 12-point formulas can be accommodated, and up to 15 if the Fornberg algorithm (Sect. 3.9) is used. 

For chronopotentiometry, where we have a known gradient and wish to compute the value of the concentration at the electrode that produces this value, the same formula as (5.10) on page 93 applies, again with appropriate coefficients. Again the Fornberg algorithm is most convenient and this is implemented in the Examples routine COFORN. 

Spatial discretisations on unequally spaced points are best done using multi-point stencils, for example five-point. However, in order to keep things simple, three-point approximations are used in what follows. The symbols like i)k refer to three fi coefficients k = 1,2,3 pertaining to point i, used to approximate a first derivative at point i, and similarly for the a coefficients. These are all precomputed using the Fornberg algorithm [63], implemented in the subroutines forn and fornberg, described in Appendix E. 

For unequal intervals, there are equivalent functions GOFORN and COFORN, both using the Fornberg algorithm [1]. These two need theX array to operate. 

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