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SumTools[IndefiniteSum]

  

HomotopySum

  

compute closed forms of indefinite sums of expressions containing unspecified functions

 

Calling Sequence

Parameters

Description

Notes

Examples

References

Calling Sequence

HomotopySum(E, k)

Parameters

E

-

any algebraic expression

k

-

name, specifies the summation index

Description

• 

The HomotopySum command allows for the symbolic summation of expressions containing unspecified functions of a discrete variable. A typical example is HomotopySum(u[k+1]-u[k], k), which returns uk.

• 

HomotopySum uses discrete homotopy methods to find an anti-difference of the given expression - see the references at the end.

Notes

• 

This command is based on code written by Bernard Deconinck, Michael A. Nivala, and Matthew S. Patterson.

Examples

withSumTools[IndefiniteSum]:

Euk+1uk

E:=uk+1uk

(1)

HomotopySumE,k

uk

(2)

E1uk+2+uk+11uk+1+uk

E:=1uk+2+uk+11uk+1+uk

(3)

HomotopySumE,k

1uk+1+uk

(4)

If no anti-difference is found, HomotopySum minimizes the number of terms remaining unsummed, as well as the order of their summation indices.

E2uk+32uk+2uk+12uk+uk+2

E:=ukuk+12+2uk+2uk+32+uk+2

(5)

HomotopySumE,k

2ukuk+12+2uk+22uk+1+uk+uk+1+kukuk+12+uk

(6)

The input expression may contain combinations of specified and unspecified functions of the summation index.

Eexpandk+13uk+2vk+15+vk+23k3uk+1vk5

E:=k3uk+1vk5+k3uk+2vk+15+3k2uk+2vk+15+3kuk+2vk+15+uk+2vk+15+vk+23

(7)

HomotopySumE,k

k3uk+1vk5+vk3+vk+13+kvk3

(8)

References

  

Hereman, W.; Colagrosso, M.; Sayers, R.; Ringler, A.; Deconinck, B.; Nivala, M.; and Hickman, M. "Continuous and Discrete Homotopy Operators with Applications in Integrability Testing." In Differential Equations with Symbolic computation, pp. 255-290. Edited by D. Wang and Z. Zheng. Birkhauser, 2005.

See Also

SumTools

SumTools[IndefiniteSum]

 


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