complex-zeros-delves-lyness

Compute the zeros of a complex analytic function using the method of Delves and Lyness

Introduction

Given a complex analytic function and its derivative, this module uses the method of Delves and Lyness [1] to compute the zeros. That is, it computes

numerically using adaptive Simpson's method integration. In the absence of poles, Cauchy's argument principle states s₀ is the number of zeros M. Using Newton's Identities, the moments s₁ through s_M are transformed into a polynomial, the roots of which correspond to the encircled zeros of the function f and which may be computed with a complex polynomial root-finder like the Weierstrass method.

To Do

Currently locates multiple zeros with recursive subdivision, but needs checks for near-zero along the contour and check for poor S₀ integration. Optional derivative-based post-processing refinement (i.e. Newton-Raphson) would also be a nice plus.

Installation

Can be installed from github, but not currently published.

Example

The single zero of the function cos(z) + sin(z) inside the unit circle is z₀ = -π / 4:

var zeros = require('complex-zeros-delves-lyness');

function f (out, a, b) {
  out[0] = Math.cosh(b) * (Math.cos(a) + Math.sin(a));
  out[1] = Math.sinh(b) * (Math.cos(a) - Math.sin(a));
};

function fp (out, a, b) {
  out[0] = Math.cosh(b) * (Math.cos(a) - Math.sin(a));
  out[1] = -Math.sinh(b) * (Math.cos(a) + Math.sin(a));
};

zeros(f, fp, [0, 0], 1);
// => [ [ -0.785398350762156 ], [ 3.289335470668675e-11 ] ]

Since f and f' are used 1-1 and often repeat many identical operations, they may be computed together and returned as the third and fourth entries of the output array:

function f (out, a, b) {
  var chb = Math.cosh(b);
  var shb = Math.sinh(b);
  var ca = Math.cos(a);
  var sa = Math.sin(a);
  out[0] = chb * (ca + sa);
  out[1] = shb * (ca - sa);
  out[2] = chb * (ca - sa);
  out[3] = -shb * (ca + sa);
};

zeros(f, null, [0, 0], 1);
// => [ [ -0.785398350762156 ], [ 3.289335470668675e-11 ] ]

Usage

require('complex-zeros-delves-lyness')(f, fp, z0, r, tol, maxDepth)

Compute the zeros of a complex analytic function.

Arguments:

• f: function(out: Array, a: Number, b: Number): a function that places the real and imaginary components of f(a + ib) into the first and second elements of the output array.
• fp: function(out: Array, a: Number, b: Number): a function that places the real and imaginary components of f'(a + ib) into the first and second elements of the output array. If fp is null, argument f may instead place the real and imaginary components of f' into the third and fourth elements of the output of f.
• z0: Array: an Array specifying the real and imaginary components of the center of the contour around which to integrate. Default is [0, 0].
• r: Number: the radius of the contour. Default is 1.
• tol: tolerance used in the integration and polynomial root-finding steps. Default is 1e-8.
• maxDepth: maximum recursion depth of the adaptive Simpson integration. Default is 20.

Returns: Returns false on failure, otherwise an array containing an Array containing Arrays of the real and imaginary components of the zeros, respectively. That is, 1 + 2i and 3 + 4i would be returned as [[1, 3], [2, 4]].

References

[1] Delves, L. M., & Lyness, J. N. (1967). A numerical method for locating the zeros of an analytic function. Mathematics of Computation.

License

© 2016 Ricky Reusser. MIT License.