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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

Parameters

Description

Examples

References

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;

L1xE23x3E&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;I110400ID3f3&plus;I4226896811156640625If3&plus;I&plus;1071845000ID2f2&plus;I&plus;6041699111156640625If2&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;6578491098500000ID2f3&plus;I4226896811156640625IDf3&plus;I2578072904729007265625000If3&plus;I141600ID4f3&plus;I&plus;1669431549250000ID2f2&plus;I&plus;6881034150358014531250000If2&plus;I&plus;357845000ID3f2&plus;I&plus;6041699111156640625IDf2&plus;I&plus;35200ID4f2&plus;I&plus;549845000ID3f3&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;2113448411156640625ID2f3&plus;I2578072904729007265625000IDf3&plus;I&plus;326000ID5f2&plus;I&plus;4696284071715318854722656250000If3&plus;I&plus;5493380000ID4f3&plus;I&plus;6041699122313281250ID2f2&plus;I4883949953396118854722656250000If2&plus;I&plus;556477549250000ID3f2&plus;I&plus;6881034150358014531250000IDf2&plus;I&plus;3573380000ID4f2&plus;I&plus;2192831098500000ID3f3&plus;I1208000ID5f3&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;1&plus;3486576665&DifferentialD;2&DifferentialD;x2fxx&equals;0|&DifferentialD;2&DifferentialD;x2fxx&equals;017432883310&DifferentialD;2&DifferentialD;x2fxx&equals;1|&DifferentialD;2&DifferentialD;x2fxx&equals;1&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;1&plus;1162192225&DifferentialD;3&DifferentialD;x3fxx&equals;0|&DifferentialD;3&DifferentialD;x3fxx&equals;05810961110&DifferentialD;3&DifferentialD;x3fxx&equals;1|&DifferentialD;3&DifferentialD;x3fxx&equals;1&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;1&plus;5810961110&DifferentialD;4&DifferentialD;x4fxx&equals;0|&DifferentialD;4&DifferentialD;x4fxx&equals;05810961140&DifferentialD;4&DifferentialD;x4fxx&equals;1|&DifferentialD;4&DifferentialD;x4fxx&equals;1&comma;f10&equals;6036804565f0&plus;6973153325&DifferentialD;&DifferentialD;xfxx&equals;0|&DifferentialD;&DifferentialD;xfxx&equals;01743288335&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;1

(6)

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.

See Also

LREtools

LREtools[AnalyticityConditions]

LREtools[IsDesingularizable]

 


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