continuedFractionEval

continuedFractionEval(fraction, options?): ConvergenceResult<number>

Defined in: packages/math-analysis/src/continued-fraction-eval.ts:79

Evaluates a continued fraction — an expression of the form b₀ + a₁/(b₁ + a₂/(b₂ + ⋯)) that in principle never ends.

Many functions have a continued fraction that converges far faster than their power series, especially away from the origin, which is why the incomplete gamma and beta in this package are built on one. This evaluates such a fraction without having to decide up front how deep to go: the running value is multiplied by a factor tending to 1, and the iteration stops when that factor is within tolerance of 1.

It reports rather than throws. A fraction that exhausts maxIterations still returns its best estimate, with converged: false — check it before trusting the value.

O(maxIterations) calls to each of ContinuedFraction.numerator and ContinuedFraction.denominator, and the same count of divisions. O(1) memory: the coefficients are functions of the index, so an unbounded fraction needs no array.

Parameters

fraction

ContinuedFraction

The partial numerators and denominators.

options?

ConvergenceOptions

Termination criteria. tolerance is compared against the distance from 1 of the multiplicative update, so it is relative; it defaults to Number.EPSILON and maxIterations to 300.

Returns

ConvergenceResult<number>

The fraction’s value, the iteration count, and whether the tolerance was met.

Example

  // 1 + 1/(1 + 1/(1 + ⋯)) is the golden ratio.
  const phi = continuedFractionEval({
    numerator: () => 1,
    denominator: () => 1,
  });
  phi.value; // 1.618033988749895
  phi.iterations; // 38
  phi.converged; // true
  ```;

## See

[Continued fractions](https://anthochamp.pages.grenadine.io/node-essentials/topics/math/calculus/series/)