Lambert W Newton Approximation

Approximate either real Lambert W branch with safeguarded Newton/bisection and explicit equation residual; lower branch only on [−1/e,0), no complex continuation or false unique root.

Description

Lambert W solves w exp(w)=x.

Lambert W solves w exp(w)=x. The real principal and lower branches overlap on negative x but produce different values except at their common branch point.^{[1, 2]}

Approximate either real Lambert W branch with safeguarded Newton/bisection and explicit equation residual; lower branch only on [−1/e,0), no complex continuation or false unique root.

When to use Lambert W Newton Approximation

Use this tool to approximate either real Lambert W branch with safeguarded Newton/bisection and explicit equation residual; lower branch only on [−1/e,0), no complex continuation or false unique root.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Lambert W Newton Approximation and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter x, realBranch using the field table and calculation rules, then run the tool.
Read lambertW, absoluteEquationResidual. 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
x	X	number	Required
realBranch	Real branch	string	Required

Choose branch text principal or lower. Principal accepts −1/e≤x≤one million; lower accepts −1/e≤x<0. Use safeguarded Newton steps inside a bisection bracket for up to 200 iterations; near the branch point return −1. Return w and absolute equation residual.

w exp(w)=x; principal w≥−1; lower w≤−1

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
lambertW	Lambert w	number
absoluteEquationResidual	Absolute equation residual	number

json

{
  "input": {
    "x": 2.718281828459045,
    "realBranch": "principal"
  },
  "output": {
    "lambertW": 1,
    "absoluteEquationResidual": 0
  }
}

Limitations and assumptions

The residual is not a certified root error bound near the branch point. Complex branches are excluded.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Extremely large text inputs are rejected. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use 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. 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.

Further background: Lambert W function.

References

Lambert W Newton Approximation — tool.io
Lambert W function — NIST Digital Library of Mathematical Functions

Similar or alternative tools

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.
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.