Padé Approximant Calculator
Solve bounded [m/n] Padé approximant from exactly m+n+1 caller Taylor coefficients, denominator constant one; singular/unresolved systems refused rather than invent a unique approximation.
Description
A Padé approximation matches a finite Taylor expansion with a ratio of polynomials.
A Padé approximation matches a finite Taylor expansion with a ratio of polynomials. Fixing the denominator's constant coefficient to one sets the normalization.1
Solve bounded [m/n] Padé approximant from exactly m+n+1 caller Taylor coefficients, denominator constant one; singular/unresolved systems refused rather than invent a unique approximation.
When to use Padé Approximant Calculator
Use this tool to solve bounded [m/n] Padé approximant from exactly m+n+1 caller Taylor coefficients, denominator constant one; singular/unresolved systems refused rather than invent a unique approximation.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Padé Approximant Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter taylorCoefficients, numeratorDegree, denominatorDegree using the field table and calculation rules, then run the tool.
- Read numeratorCoefficients, denominatorCoefficients. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| taylorCoefficients | Taylor coefficients | list of number | Required |
| numeratorDegree | Numerator degree | number | Required |
| denominatorDegree | Denominator degree | number | Required |
Enter integer numerator and denominator degrees 0–12 and exactly m+n+1 constant-first Taylor coefficients. Solve denominator constraints by pivoted elimination, then convolve with the series to obtain numerator coefficients. Singular or numerically unresolved pivots are rejected.
Q(x)series(x)−P(x) has zero coefficients through order m+n; Q0=1 Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| numeratorCoefficients | Numerator coefficients | list of number |
| denominatorCoefficients | Denominator coefficients | list of number |
{
"input": {
"taylorCoefficients": [
1,
1,
0.5
],
"numeratorDegree": 1,
"denominatorDegree": 1
},
"output": {
"numeratorCoefficients": [
1,
0.5
],
"denominatorCoefficients": [
1,
-0.5
]
}
}Limitations and assumptions
A finite matching system does not establish approximation accuracy away from the expansion point. Singular data do not define a unique returned approximant.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Partial Fraction Decomposition Tool: Decompose a strictly proper rational function whose monic denominator is caller supplied product of distinct simple real (x−pole) factors; residues computed by p(pole)/q′(pole), not a general factoring claim. For a complementary task, use Best Rational Approximation Calculator: Find closest rational to supplied binary floating-point value over denominators 1..caller bound, with earliest denominator ties; unlike continued fractions, checks global finite denominator inventory, not irrationality proof.
References
-
Padé Approximant Calculator — tool.io
Similar or alternative tools
- Partial Fraction Decomposition Tool
Decompose a strictly proper rational function whose monic denominator is caller supplied product of distinct simple real (x−pole) factors; residues computed by p(pole)/q′(pole), not a general factoring claim.
- Best Rational Approximation Calculator
Find closest rational to supplied binary floating-point value over denominators 1..caller bound, with earliest denominator ties; unlike continued fractions, checks global finite denominator inventory, not irrationality proof.