complex-deriv-fornberg

Compute the derivative of a complex analytic function using the method of Fornberg

Introduction

This module uses the method of Fornberg to compute the derivatives of a complex analytic function along with error bounds. The method uses a Fourier Transform to invert function evaluations around a circle into Taylor series coefficients, uses Richardson Extrapolation to improve and bound the estimate, then multiplies by a factorial to compute the derivatives. Unlike real-valued finite differences, the method searches for a desirable radius and so is reasonably insensitive to the initial radius—to within a number of orders of magnitude at least. For most cases, the default configuration is likely to succeed.

Restrictions

The method uses the coefficients themselves to control the truncation error, so the error will not be properly bounded for functions like low-order polynomials whose Taylor series coefficients are nearly zero. If the error cannot be bounded, degenerate flag will be set to true, and an answer will still be computed and returned but should be used with caution.

Example

To compute the first five derivatives of 1 / (1 - z) at z = 0:

var deriv = require('complex-deriv-fornberg');

function f(a, b) {
  var c = b * b + (1 - a) * (1 - a);
  return [(1 - a) / c, b / c];
}

deriv(f, 5, 0, 0)

// =>
// [ [ 1.0000000000000138,
//    1.0000000000000107,
//    2.0000000000000178,
//    6.000000000000274,
//    23.99999999999432,
//    120.00000000001907 ],
//  [ 3.260944019499375e-17,
//    -7.34322880255595e-17,
//    -1.0399525753674522e-15,
//    -1.9409005869118293e-14,
//    -1.3097221675086528e-13,
//    7.683410164000554e-13 ] ]

To output additional information about the computation, provide an empty object as the status argument:

var status = {};

deriv(f, 5, 0, 0, {}, status)

// status => 
// { truncationError: 
//   [ 3.1039630025864036e-13,
//     3.1039375685463187e-13,
//     6.207858181065558e-13,
//     1.8624780843579008e-12,
//     7.446002762584479e-12,
//     3.726089802919438e-11 ],
//  roundingError: 
//   [ 1.776399993410336e-15,
//     6.560675025011761e-15,
//     4.846032081004469e-14,
//     5.369270732187135e-13,
//     7.932006423296532e-12,
//     1.464740954531565e-10 ],
//  degenerate: false,
//  iterations: 7,
//  finalRadius: 0.27076482018055015,
// failed: false }

Installation

$ npm install complex-deriv-fornberg

API

require('complex-deriv-fornberg')([output, ]f, n, a, b[, options[, status]])

Compute the derivative of a complex analytic function f at a + b * i.

Parameters:

• output (optional). Optional array of arrays into which the output is written. If not provided, arrays will be allocated and returned.
• f: function of format function([out, ]a, b), that evaluates the function at a + b * i, either a into out[0] and b into out[1] or simply returning [a, b].
• n: Number of derivatives to compute where 0 represents the value of the function and n represents the nth derivative. Maximum number is 100.
• a: Real component of z at which to evaluate the derivatives
• b: Imaginary component of z at which to evaluate the derivatives
• options: Optional object of configuration parameters
  • r (default: 0.6580924658): Initial radius at which to evaluate. For well-behaved functions, the computation should be insensitive to the initial radius to within about four orders of magnitude.
  • maxIters (default: 30): Maximum number of iterations
  • taylor: (default: false): If false, output represents the derivatives of the function. If true, the output represents Taylor series coefficients, differing only in multiplication by a factorial.
  • minDegenerateIterations: (default: maxIters - 5): Minimum number of iterations before the solution may be deemed degenerate. A larger number allows the algorithm to correct a bad initial radius.
• status: Optional object into which output information is written. Fields are:
  • degenerate: True if the algorithm was unable to bound the error
  • iterations: Number of iterations executed
  • finalRadius: Ending radius of the algorithm
  • failed: True if the maximum number of iterations was reached
  • truncationError: An array containing approximate bounds of the truncation error achieved for each component of the solution
  • roundingError: An array containing approximate bounds of the rounding error achieved for each component of the solution

Returns: Returns the real and imaginary components of the derivatives in arrays, i.e. [[re1, re2, ...], [im1, im2, ...]], also writing the arrays to output, if provided.

Known Issues

• The logic is carefully verified against the referenced paper (see: derivation/*.py), but the error bounds seem not always strictly obeyed with about 1/100 evaluations losing as many as 1-2 digits of precision—which should be acceptable for most uses.
• Could be more robust in the neighborhood of pathologies like branch cuts. The degenerate argument should be checked to see if the algorithm was able to bound the error successfully, and for repeated application, the finalRadius output may be fed back into the next iteration to aid the radius search.

References

[1] Fornberg, B. (1981). Numerical Differentiation of Analytic Functions. ACM Transactions on Mathematical Software (TOMS), 7(4), 512–526. http://doi.org/10.1145/355972.355979

License

© 2016 Scijs Authors. MIT License.

Authors

Ricky Reusser