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SumTools[IndefiniteSum][HomotopySum] - compute closed forms of indefinite sums of expressions containing unspecified functions
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Calling Sequence
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HomotopySum(E, k)
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Parameters
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E
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any algebraic expression
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k
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name, specifies the summation index
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Description
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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 ![u[k]](/support/helpjp/helpview.aspx?si=5460/file06640/math55.png) .
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HomotopySum uses discrete homotopy methods to find an anti-difference of the given expression - see the reeferences at the end.
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Notes
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This command is based on code written by Bernard Deconinck, Michael A. Nivala, and Matthew S. Patterson.
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Examples
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![with(SumTools[IndefiniteSum])](/support/helpjp/helpview.aspx?si=5460/file06640/math72.png)
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![E := u[k+1]-u[k]](/support/helpjp/helpview.aspx?si=5460/file06640/math76.png)
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![E := u[k+1]-u[k]](/support/helpjp/helpview.aspx?si=5460/file06640/math79.png)
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![u[k]](/support/helpjp/helpview.aspx?si=5460/file06640/math86.png)
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![E := 1/(u[k+2]+u[k+1])-1/(u[k+1]+u[k])](/support/helpjp/helpview.aspx?si=5460/file06640/math90.png)
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![E := 1/(u[k+2]+u[k+1])-1/(u[k+1]+u[k])](/support/helpjp/helpview.aspx?si=5460/file06640/math93.png)
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![1/(u[k+1]+u[k])](/support/helpjp/helpview.aspx?si=5460/file06640/math100.png)
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If no anti-difference is found, HomotopySum minimizes the number of terms remaining unsummed, as well as the order of their summation indices.
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![E := 2*u[k+3]^2*u[k+2]-u[k+1]^2*u[k]+u[k+2]](/support/helpjp/helpview.aspx?si=5460/file06640/math106.png)
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![E := 2*u[k+3]^2*u[k+2]-u[k+1]^2*u[k]+u[k+2]](/support/helpjp/helpview.aspx?si=5460/file06640/math109.png)
| (5) |
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![u[k]+u[k+1]+2*u[k+1]^2*u[k]+2*u[k+2]^2*u[k+1]+sum(u[k+1]^2*u[k]+u[k], k)](/support/helpjp/helpview.aspx?si=5460/file06640/math116.png)
| (6) |
The input expression may contain combinations of specified and unspecified functions of the summation index.
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![E := expand((k+1)^3*u[k+2]*v[k+1]^5+v[k+2]^3-k^3*u[k+1]*v[k]^5)](/support/helpjp/helpview.aspx?si=5460/file06640/math122.png)
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![E := u[k+2]*v[k+1]^5*k^3+3*u[k+2]*v[k+1]^5*k^2+3*u[k+2]*v[k+1]^5*k+u[k+2]*v[k+1]^5+v[k+2]^3-k^3*u[k+1]*v[k]^5](/support/helpjp/helpview.aspx?si=5460/file06640/math125.png)
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![v[k]^3+v[k+1]^3+k^3*u[k+1]*v[k]^5+sum(v[k]^3, k)](/support/helpjp/helpview.aspx?si=5460/file06640/math132.png)
| (8) |
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References
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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.
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