formulas for the values of the solution of difference equation and its derivatives of the given order and at the given point. - Maple Help

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LREtools[ValuesAtPoint] - formulas for the values of the solution of difference equation and its derivatives of the given order and at the given point.

Calling Sequence

ValuesAtPoint(L, E, fun, HalfInt_opt, Point_opt, Order_opt)

Parameters

L

-

linear difference operator in E with coefficients which are polynomials in x

E

-

name of the shift operator acting on x

fun

-

function f(x) that is a solution of Lfx=0

HalfInt_opt

-

(optional) 'HalfInterval'= A, A is a rational number, 0 by default

Point_opt

-

(optional) 'Point'=p, p is a rational number or an algebraic number in the indexed RootOf representation (see,RootOf,indexed), 0 by default

Order_opt

-

(optional) 'OrderDer'=m, m is non-negative integer, 0 by default.

Description

• 

The ValuesAtPoint command returns formulas for the values of the function and its derivatives of the given order and at the given point in Point_opt. It also computes conditions for the analyticity of the function at the given point.

• 

The input includes a difference operator

L := sum(a[i](x)* E^i,i=1..d);

L:=i=1daixEi

(1)
  

and the point A. Specify the point 'Point'=p to compute the value f(x) and its derivatives at x=p, and non-negative integer via the option Order_opt to specify the highest order of required derivatives of f(x) at x=p. 

• 

The procedure returns 2 sets:

1. 

The set of conditions. f(x) is assumed to be analytic on some open set which contains a set A<=Rex<A+d. Elements of the set give the conditions of the analyticity of f(x) at x&equals;p. They are relations between the values of the function and, possibly several of its derivatives at the points into A<=Rex<A+d.

2. 

The set of formulas for computing fp&comma;&DifferentialD;&DifferentialD;pfp,...,&DifferentialD;m&DifferentialD;pmfp. (f(x) must satisfy the conditions in the first set.) These formulas give the values of fp&comma;&DifferentialD;&DifferentialD;pfp,...,&DifferentialD;m&DifferentialD;pmfp as linear combinations of f(x) and several of its derivatives in A<=Rex<A+d. For m&equals;0, we have one unique formula for fp.

Examples

withLREtools&colon;

L1:=xE23x3E&plus;2x312x&plus;4

L1:=xE23x3E&plus;2x312x&plus;4

(2)

ValuesAtPointL1&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;2&comma;&apos;Point&apos;&equals;13

f113&equals;185f83&comma;f13&equals;275f83&plus;1440&DifferentialD;&DifferentialD;xfxx&equals;83|&DifferentialD;&DifferentialD;xfxx&equals;83&plus;11584&DifferentialD;&DifferentialD;xfxx&equals;113|&DifferentialD;&DifferentialD;xfxx&equals;113

(3)

ValuesAtPointL1&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;2&comma;&apos;Point&apos;&equals;RootOfx2&plus;1&comma;x&comma;index&equals;1&comma;&apos;OrderDer&apos;&equals;5

&comma;&DifferentialD;&DifferentialD;xfxx&equals;I|&DifferentialD;&DifferentialD;xfxx&equals;I&equals;1071422500If2&plus;I&plus;1647422500If3&plus;I&plus;9650IDf2&plus;I35200IDf3&plus;I&plus;1587422500f2&plus;I&plus;17311690000f3&plus;I&plus;435200Df2&plus;I&plus;31040Df3&plus;I&comma;&DifferentialD;2&DifferentialD;x2fxx&equals;I|&DifferentialD;2&DifferentialD;x2fxx&equals;I&equals;91300ID2f2&plus;I310400ID2f3&plus;I&plus;1669431274625000If2&plus;I&plus;657849549250000If3&plus;I&plus;1071422500IDf2&plus;I&plus;1647422500IDf3&plus;I&plus;4310400D2f2&plus;I&plus;32080D2f3&plus;I&plus;987417274625000f2&plus;I1106091274625000f3&plus;I&plus;1587422500Df2&plus;I&plus;17311690000Df3&plus;I&comma;&DifferentialD;3&DifferentialD;x3fxx&equals;I|&DifferentialD;3&DifferentialD;x3fxx&equals;I&equals;1647845000ID2f3&plus;I&plus;657849549250000IDf3&plus;I&plus;6041699111156640625If2&plus;I&plus;1071845000ID2f2&plus;I110400ID3f3&plus;I4226896811156640625If3&plus;I&plus;31300ID3f2&plus;I&plus;1669431274625000IDf2&plus;I&plus;1587845000D2f2&plus;I&plus;4331200D3f2&plus;I&plus;17313380000D2f3&plus;I&plus;12080D3f3&plus;I17845797989253125000f2&plus;I204172941178506250000f3&plus;I&plus;987417274625000Df2&plus;I1106091274625000Df3&plus;I&comma;&DifferentialD;4&DifferentialD;x4fxx&equals;I|&DifferentialD;4&DifferentialD;x4fxx&equals;I&equals;4226896811156640625IDf3&plus;I&plus;1669431549250000ID2f2&plus;I&plus;549845000ID3f3&plus;I141600ID4f3&plus;I&plus;357845000ID3f2&plus;I&plus;6578491098500000ID2f3&plus;I&plus;35200ID4f2&plus;I&plus;6881034150358014531250000If2&plus;I2578072904729007265625000If3&plus;I&plus;6041699111156640625IDf2&plus;I&plus;987417549250000D2f2&plus;I&plus;529845000D3f2&plus;I&plus;43124800D4f2&plus;I1106091549250000D2f3&plus;I&plus;5773380000D3f3&plus;I&plus;18320D4f3&plus;I25020203832958014531250000f2&plus;I&plus;19002130751758014531250000f3&plus;I17845797989253125000Df2&plus;I204172941178506250000Df3&plus;I&comma;&DifferentialD;5&DifferentialD;x5fxx&equals;I|&DifferentialD;5&DifferentialD;x5fxx&equals;I&equals;2578072904729007265625000IDf3&plus;I&plus;6041699122313281250ID2f2&plus;I&plus;2192831098500000ID3f3&plus;I&plus;5493380000ID4f3&plus;I&plus;556477549250000ID3f2&plus;I1208000ID5f3&plus;I2113448411156640625ID2f3&plus;I&plus;3573380000ID4f2&plus;I4883949953396118854722656250000If2&plus;I&plus;4696284071715318854722656250000If3&plus;I&plus;6881034150358014531250000IDf2&plus;I&plus;326000ID5f2&plus;I178457979178506250000D2f2&plus;I&plus;329139549250000D3f2&plus;I&plus;5293380000D4f2&plus;I&plus;43624000D5f2&plus;I204172941357012500000D2f3&plus;I368697549250000D3f3&plus;I&plus;57713520000D4f3&plus;I&plus;141600D5f3&plus;I38537180240192356840332031250f2&plus;I&plus;831981883997118854722656250000f3&plus;I25020203832958014531250000Df2&plus;I&plus;19002130751758014531250000Df3&plus;I&comma;fI&equals;9650If2&plus;I35200If3&plus;I&plus;435200f2&plus;I&plus;31040f3&plus;I

(4)

ValuesAtPointL1&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;0&comma;&apos;Point&apos;&equals;2

f1&equals;4f0&comma;f2&equals;40f0&plus;12&DifferentialD;&DifferentialD;xfxx&equals;0|&DifferentialD;&DifferentialD;xfxx&equals;03&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;1

(5)

ValuesAtPointL1&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;0&comma;&apos;Point&apos;&equals;10&comma;&apos;OrderDer&apos;&equals;3

f1&equals;4f0&comma;&DifferentialD;&DifferentialD;xfxx&equals;10|&DifferentialD;&DifferentialD;xfxx&equals;10&equals;1807285457425f0&plus;1279142740325&DifferentialD;&DifferentialD;xfxx&equals;0|&DifferentialD;&DifferentialD;xfxx&equals;09773025123100&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;117432883310&DifferentialD;2&DifferentialD;x2fxx&equals;1|&DifferentialD;2&DifferentialD;x2fxx&equals;1&plus;3486576665&DifferentialD;2&DifferentialD;x2fxx&equals;0|&DifferentialD;2&DifferentialD;x2fxx&equals;0&comma;&DifferentialD;2&DifferentialD;x2fxx&equals;10|&DifferentialD;2&DifferentialD;x2fxx&equals;10&equals;402200989929500f0&plus;3674700025594000&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;1&plus;3554441804011000&DifferentialD;&DifferentialD;xfxx&equals;0|&DifferentialD;&DifferentialD;xfxx&equals;0&plus;1279142740350&DifferentialD;2&DifferentialD;x2fxx&equals;0|&DifferentialD;2&DifferentialD;x2fxx&equals;09773025123200&DifferentialD;2&DifferentialD;x2fxx&equals;1|&DifferentialD;2&DifferentialD;x2fxx&equals;15810961110&DifferentialD;3&DifferentialD;x3fxx&equals;1|&DifferentialD;3&DifferentialD;x3fxx&equals;1&plus;1162192225&DifferentialD;3&DifferentialD;x3fxx&equals;0|&DifferentialD;3&DifferentialD;x3fxx&equals;0&comma;&DifferentialD;3&DifferentialD;x3fxx&equals;10|&DifferentialD;3&DifferentialD;x3fxx&equals;10&equals;271315852855720000f0&plus;13102438497001120000&DifferentialD;&DifferentialD;xfxx&equals;0|&DifferentialD;&DifferentialD;xfxx&equals;0&plus;83425799085959480000&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;1&plus;3674700025598000&DifferentialD;2&DifferentialD;x2fxx&equals;1|&DifferentialD;2&DifferentialD;x2fxx&equals;1&plus;3554441804012000&DifferentialD;2&DifferentialD;x2fxx&equals;0|&DifferentialD;2&DifferentialD;x2fxx&equals;0&plus;12791427403150&DifferentialD;3&DifferentialD;x3fxx&equals;0|&DifferentialD;3&DifferentialD;x3fxx&equals;03257675041200&DifferentialD;3&DifferentialD;x3fxx&equals;1|&DifferentialD;3&DifferentialD;x3fxx&equals;15810961140&DifferentialD;4&DifferentialD;x4fxx&equals;1|&DifferentialD;4&DifferentialD;x4fxx&equals;1&plus;5810961110&DifferentialD;4&DifferentialD;x4fxx&equals;0|&DifferentialD;4&DifferentialD;x4fxx&equals;0&comma;f10&equals;6036804565f01743288335&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;1&plus;6973153325&DifferentialD;&DifferentialD;xfxx&equals;0|&DifferentialD;&DifferentialD;xfxx&equals;0

(6)

See Also

LREtools, LREtools[AnalyticityConditions], LREtools[IsDesingularizable]

References

  

Abramov, S.A., and van Hoeij, M. "Set of Poles of Solutions of Linear Difference Equations with Polynomial Coefficients." Computation Mathematics and Mathematical Physics. Vol. 43 No. 1. (2003): 57-62.


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