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1: 16.25 Methods of Computation
In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …
2: 5.21 Methods of Computation
Or we can use forward recurrence, with an initial value obtained e. …
3: 11.13 Methods of Computation
Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that H ν ( x ) and L ν ( x ) can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. The solution K ν ( x ) needs to be integrated backwards for small x , and either forwards or backwards for large x depending whether or not ν exceeds 1 2 . For M ν ( x ) both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …
4: 3.6 Linear Difference Equations
where Δ w n - 1 = w n - w n - 1 , Δ 2 w n - 1 = Δ w n - Δ w n - 1 , and n . … If, as n , the wanted solution w n grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. … Then computation of w n by forward recursion is unstable. … (This part of the process is equivalent to forward elimination.) … Within this framework forward and backward recursion may be regarded as the special cases = 0 and = k , respectively. …
5: 3.9 Acceleration of Convergence
3.9.2 S = k = 0 ( - 1 ) k 2 - k - 1 Δ k a 0 ,
Here Δ is the forward difference operator:
3.9.3 Δ k a 0 = Δ k - 1 a 1 - Δ k - 1 a 0 , k = 1 , 2 , .
3.9.4 Δ k a 0 = m = 0 k ( - 1 ) m ( k m ) a k - m .
6: 3.10 Continued Fractions
Forward Recurrence Algorithm
In general this algorithm is more stable than the forward algorithm; see Jones and Thron (1974).
Forward Series Recurrence Algorithm
In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). … This forward algorithm achieves efficiency and stability in the computation of the convergents C n = A n / B n , and is related to the forward series recurrence algorithm. …
7: 7.22 Methods of Computation
See Gautschi (1977a), where forward and backward recursions are used; see also Gautschi (1961). …
8: 18.1 Notation
Forward differences:
Δ x ( f ( x ) ) = f ( x + 1 ) - f ( x ) ,
Δ x n + 1 ( f ( x ) ) = Δ x ( Δ x n ( f ( x ) ) ) .
9: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.19 Δ x Q n ( x ; α , β , N ) = - n ( n + α + β + 1 ) ( α + 1 ) N Q n - 1 ( x ; α + 1 , β + 1 , N - 1 ) ,
18.22.21 Δ x K n ( x ; p , N ) = - n p N K n - 1 ( x ; p , N - 1 ) ,
18.22.23 Δ x M n ( x ; β , c ) = - n ( 1 - c ) β c M n - 1 ( x ; β + 1 , c ) ,
18.22.25 Δ x C n ( x ; a ) = - n a C n - 1 ( x ; a ) ,
10: 10.74 Methods of Computation
In the interval 0 < x < ν , J ν ( x ) needs to be integrated in the forward direction and Y ν ( x ) in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). … Similarly, to maintain stability in the interval 0 < x < the integration direction has to be forwards in the case of I ν ( x ) and backwards in the case of K ν ( x ) , with initial values obtained in an analogous manner to those for J ν ( x ) and Y ν ( x ) . … Then J n ( x ) and Y n ( x ) can be generated by either forward or backward recurrence on n when n < x , but if n > x then to maintain stability J n ( x ) has to be generated by backward recurrence on n , and Y n ( x ) has to be generated by forward recurrence on n . …