SumTools[IndefiniteSum]
HomotopySum
compute closed forms of indefinite sums of expressions containing unspecified functions
Calling Sequence
Parameters
Description
Notes
Examples
References
HomotopySum(E, k)
E
-
any algebraic expression
k
name, specifies the summation index
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.
This command is based on code written by Bernard Deconinck, Michael A. Nivala, and Matthew S. Patterson.
withSumToolsIndefiniteSum:
E≔uk+1−uk
HomotopySumE,k
uk
E≔1uk+2+uk+1−1uk+1+uk
1uk+1+uk
If no anti-difference is found, HomotopySum minimizes the number of terms remaining unsummed, as well as the order of their summation indices.
E≔2uk+32uk+2−uk+12uk+uk+2
E≔−ukuk+12+2uk+32uk+2+uk+2
2ukuk+12+2uk+22uk+1+uk+uk+1+∑kukuk+12+uk
The input expression may contain combinations of specified and unspecified functions of the summation index.
E≔expandk+13uk+2vk+15+vk+23−k3uk+1vk5
E≔−k3uk+1vk5+k3uk+2vk+15+3k2uk+2vk+15+3kuk+2vk+15+uk+2vk+15+vk+23
k3uk+1vk5+vk3+vk+13+∑kvk3
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
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