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errors & uncertainties in computations 371.00.8l = 00.6l = 10.4j (x) l0.2l = 30.00.0 2.0 4.0 6.0 8.0 10.0 12.0xFigure 2.1 The first four spherical Bessel functions j l (x) as functions of x. Notice that for small x,the values for increasing l become progressively smaller.TABLE 2.1Approximate Values for Spherical Bessel Functions of Orders 3, 5, and 8(from Maple)x j 3(x) j 5(x) j 8(x)0.1 +9.518519719 10 −6 +9.616310231 10 −10 +2.901200102 10 −161 +9.006581118 10 −3 +9.256115862 10 −05 +2.826498802 10 −0810 −3.949584498 10 −1 −5.553451162 10 −01 +1.255780236 10 +00Equations (2.23) and (2.24) are the same relation, one written for upward recurrencefrom small to large l values, and the other for downward recurrence to small lvalues. With just a few additions and multiplications, recurrence relations permitrapid, simple computation of the entire set of j l values for fixed x and all l.To recur upward in l for fixed x, we start with the known forms for j 0 and j 1(2.21) and use (2.23). As you will prove for yourself, this upward recurrence usuallyseems to work at first but then fails. The reason for the failure can be seen fromthe plots of j l (x) and n l (x) versus x (Figure 2.1). If we start at x ≃ 2 and l =0,wewill see that as we recur j l up to larger l values with (2.23), we are essentiallytaking the difference of two “large” functions to produce a “small” value for j l .This process suffers from subtractive cancellation and always reduces the precision.As we continue recurring, we take the difference of two small functions with largeerrors and produce a smaller function with yet a larger error. After a while, we areleft with only round-off error (garbage).−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 37