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# This will be our main interest when developing

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Unformatted text preview: his, let us de ne the rst forward and backward di erence operators as fi = fi+1 ; fi (5.2.13a) rfi = fi ; fi;1 : (5.2.13b) Although both operators are used, backward di erences better suit our current needs (because multistep methods will be interpolating with data at prior times). Thus, using (5.2.6a) and (5.2.13b), let us write the rst divided di erence with uniform spacing as f ti;1 ti] = fi;1 ; fi = rfi : t ;t h i;1 i (5.2.14) Higher-order operators follow by iteration thus, using (5.2.7a) ;f i f ti;2 ti;1 ti] = f ti;2 tti;1] ; t ti;1 ti] = rfi ;hrfi;1 = fi ; 2f2;12 + fi;2 : 22 h i;2 i In a similar manner, we de ne the higher-order backward di erences recursively as rm fi = rm;1 fi ; rm;1 fi ; 1 (5.2.15a) r 0 fi = f i : (5.2.15b) with the understanding that 11 Thus, f ti;2 ti;1 r2 fi : t]= i (5.2.15c) 2h2 The k th divided di erence adn k th backward di erence operator are related by (Problem 1) f ti;k ti;k+1 : : : ti] = r!hfki : k k (5.2.15d) We can easily write the divided-di erence polynomial (5.2.9) in terms of backward di erences however, our intended use with ODEs will call for polynomials proceeding from an advanced point (tn;1 or tn) into the past. Hence, it will be convenient to re-index the points in (5.2.9) to account for this. For the present, We'll illustrate this by reversing the indexing thus, let t0 become tk , t1 become tk;1 etc. Then, (5.2.9) becomes Pk (t) = k X i=0 f tk tk;1 : : : tk;i] i;1 Y j ;1 (t ; tk;j ) or Pk (t) = f tk ] + f tk tk;1](t ; tk ) + : : : + f tk tk;1 : : : t0 ](t ; tk )(t ; tk;1) : : : (t ; t1 ): Now, for uniform spacing, we use (5.2.15) to obtain i;1 k X rifk Y Pk (t) = i!hi j=0(t ; tk;j ) i=0 (5.2.16a) or kf f Pk (t) = fk + rh k (t ; tk ) + : : : + r!hkk (t ; tk )(t ; tk;1) : : : (t ; t1): k (5.2.16b) Since our primary purpose is the development of multistep methods for ODEs, we won't illustrate any interpolation examples at this point but will illustrate use of (5.2.16) in subsequent sections. Problems 1. Problem 1. Derive the identity (5.2.15d) using, e.g., an induction argument. 12 5.3 Explicit Methods: Adams-Bashforth Methods Th...
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