Examples
Examples
Try the starting example
- Proper numerator coefficients, highest degree first
- 1
- Distinct real roots of monic denominator
- -1,1
- Check x away from poles
- 2
- Residues matching denominator-root order
- -0.5, 0.5
- Direct rational value at check x
- 0.333333
- Partial-fraction reconstruction at check x
- 0.333333
- Absolute reconstruction difference
- 0
Try another scenario
- Proper numerator coefficients, highest degree first
- 2,1
- Distinct real roots of monic denominator
- -1,1
- Check x away from poles
- 0
- Residues matching denominator-root order
- 0.5, 1.5
- Direct rational value at check x
- -1
- Partial-fraction reconstruction at check x
- -1
- Absolute reconstruction difference
- 0
Calculation method
Calculation method
Residues matching denominator-root order
For a simple real pole r_i, residue=P(r_i)/product(r_i−r_j for j≠i). Denominator is product(x−r_i).
Direct rational value at check x
P(x) divided by the supplied monic factored denominator.
Partial-fraction reconstruction at check x
Sum residue_i/(x−r_i).
Absolute reconstruction difference
Absolute difference between direct and reconstructed values.
Symbols and units
- a
- Proper numerator coefficients, highest degree first
- b
- Distinct real roots of monic denominator
- c
- Check x away from poles
Use the units shown. A list keeps the order in which you enter its values. The symbol ^ means “raised to the power of”.
Assumptions & limits
Assumptions & limits
- Supports proper rational functions with distinct real linear denominator factors, up to 12 poles. Repeated or irreducible quadratic factors are outside this explicitly supported factor type. Double-precision results are approximate. Numeric tables use 12 significant digits and display components at roundoff scale (32 machine epsilons relative to their row or relevant calculation scale) as zero; scalar diagnostics retain full precision.
Are these inputs private?
These calculations run in your browser. A shared link contains the values you choose to share. Only questions submitted to the separate AI finder are sent to OpenAI.
Further reading
Background reading; this does not imply endorsement or independent review of SumFizz.