forward differences
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11—18 of 18 matching pages
11: 18.20 Hahn Class: Explicit Representations
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βΊFor comments on the use of the forward-difference operator , the backward-difference operator , and the central-difference operator , see §18.2(ii).
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12: 10.21 Zeros
13: 18.2 General Orthogonal Polynomials
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βΊIf the orthogonality discrete set is or , then the role of the differentiation operator in the case of classical OP’s (§18.3) is played by , the forward-difference operator, or by , the backward-difference operator; compare §18.1(i).
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14: 2.11 Remainder Terms; Stokes Phenomenon
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βΊWe now compute the forward differences
, , of the moduli of the rounded values of the first 6 neglected terms:
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15: 11.13 Methods of Computation
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βΊThen 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 and can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity.
The solution needs to be integrated backwards for small , and either forwards or backwards for large depending whether or not exceeds .
For both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)).
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§11.13(v) Difference Equations
… βΊIn consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …16: 16.25 Methods of Computation
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βΊThere is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
…In these cases integration, or recurrence, in either a forward or a backward direction is unstable.
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17: 3.10 Continued Fractions
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βΊ
Quotient-Difference Algorithm
… βΊ … βΊForward Recurrence Algorithm
… βΊIn general this algorithm is more stable than the forward algorithm; see Jones and Thron (1974). βΊForward Series Recurrence Algorithm
…18: 30.8 Expansions in Series of Ferrers Functions
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βΊ
30.8.1
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βΊThen the set of coefficients , is the solution of the difference equation
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βΊ
30.8.9
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βΊFor they are determined from (30.8.4) by forward recursion using .
…It should be noted that if the forward recursion (30.8.4) beginning with , leads to , then is undefined for and does not exist.
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