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https://github.com/rreusser/complex-zeros-delves-lyness
[WIP] Compute the zeros of a complex analytic function using the method of Delves and Lyness
https://github.com/rreusser/complex-zeros-delves-lyness
Last synced: 2 months ago
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[WIP] Compute the zeros of a complex analytic function using the method of Delves and Lyness
- Host: GitHub
- URL: https://github.com/rreusser/complex-zeros-delves-lyness
- Owner: rreusser
- Created: 2016-05-23T06:40:31.000Z (over 8 years ago)
- Default Branch: master
- Last Pushed: 2016-11-25T02:42:09.000Z (about 8 years ago)
- Last Synced: 2024-10-20T12:44:08.559Z (2 months ago)
- Language: JavaScript
- Homepage: https://rreusser.github.io/complex-zeros-delves-lyness/zeros.html
- Size: 332 KB
- Stars: 5
- Watchers: 5
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# 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](http://www.ams.org/mcom/1967-21-100/S0025-5718-1967-0228165-4/S0025-5718-1967-0228165-4.pdf) [[1]](#References) to compute the zeros. That is, it computes
numerically using [adaptive Simpson's method](https://github.com/scijs/integrate-adaptive-simpson) integration. In the absence of poles, [Cauchy's argument principle](https://en.wikipedia.org/wiki/Argument_principle) states s0 is the number of zeros M. Using [Newton's Identities](https://en.wikipedia.org/wiki/Newton%27s_identities), the moments s1 through sM 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](http://github.com/scijs/durand-kerner) 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:
```javascript
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:
```javascript
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](http://www.ams.org/mcom/1967-21-100/S0025-5718-1967-0228165-4/S0025-5718-1967-0228165-4.pdf). Mathematics of Computation.
## License
© 2016 Ricky Reusser. MIT License.
[npm-image]: https://badge.fury.io/js/complex-zeros-delves-lyness.svg
[npm-url]: https://npmjs.org/package/complex-zeros-delves-lyness
[travis-image]: https://travis-ci.org/rreusser/complex-zeros-delves-lyness.svg?branch=master
[travis-url]: https://travis-ci.org//complex-zeros-delves-lyness
[daviddm-image]: https://david-dm.org/rreusser/complex-zeros-delves-lyness.svg?theme=shields.io
[daviddm-url]: https://david-dm.org//complex-zeros-delves-lyness