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#### Comparison of Elliptic Integral Root Finding Algoritghms

A second example compares four root finding algorithms for locating the second radius of an ellipse with first radius 28 and arc length 300, for four floating-point types, `float`, `double`, `long double` and a Boost.Multiprecision type `cpp_bin_float_50`.

Which is to say we're solving:

    4xE(sqrt(1 - 282 / x2)) - 300 = 0

In each case the target accuracy was set using our "recomended" accuracy limits (or at least limits that make a good starting point - which is likely to give close to full accuracy without resorting to unnecessary iterations).

Function | Precision Requested
--- | ---
TOMS748 | numeric_limits<T>::digits - 2
Newton | floor(numeric_limits<T>::digits * 0.6)
Halley | floor(numeric_limits<T>::digits * 0.4)
Schröder | floor(numeric_limits<T>::digits * 0.4)

Tests used Microsoft Visual Studio 2013 (Update 1) and GCC 4.9.1 using source code root_elliptic_finding.cpp.

The timing uncertainty (especially using MSVC) is at least 5% of normalized time 'Norm'.

To pick out the 'best' and 'worst' algorithms are highlighted in blue and red. More than one result can be 'best' when normalized times are indistinguishable within the uncertainty.

##### Program root_elliptic_finding.cpp, Microsoft Visual C++ version 12.0, Dinkumware standard library version 610, Win32 Compiled in optimise mode., _X86_SSE2

Table 12.12. root with radius 28 and arc length 300) for float, double, long double and cpp_bin_float_50 types, using _X86_SSE2

| float |  |  |  |  | double |  |  |  |  | long d |  |  |  |  | cpp50 |  |  |  |
Algo | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |
TOMS748 | 5 | 515 | 1.43 | -1 |  | 9 | 968 | 1.82 | 1 |  | 9 | 968 | 1.82 | 1 |  | 11 | 871875 | 1.53 | -3 |
Newton | 3 | 453 | 1.26 | -1 |  | 4 | 640 | 1.21 | 1 |  | 4 | 640 | 1.21 | 1 |  | 5 | 685937 | 1.20 | 0 |
Halley | 2 | 359 | 1.00 | 0 |  | 3 | 531 | 1.00 | 3 |  | 3 | 531 | 1.00 | 3 |  | 4 | 570312 | 1.00 | 0 |
Schröder | 3 | 484 | 1.35 | -1 |  | 6 | 1000 | 1.88 | 1 |  | 6 | 984 | 1.85 | 1 |  | 5 | 742187 | 1.30 | -2 |

##### Program root_elliptic_finding.cpp, Microsoft Visual C++ version 12.0, Dinkumware standard library version 610, Win32 Compiled in optimise mode., _X64_AVX

Table 12.13. root with radius 28 and arc length 300) for float, double, long double and cpp_bin_float_50 types, using _X64_AVX

| float |  |  |  |  | double |  |  |  |  | long d |  |  |  |  | cpp50 |  |  |  |
Algo | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |
TOMS748 | 5 | 500 | 1.33 | -1 |  | 9 | 1046 | 1.72 | 1 |  | 9 | 1062 | 1.70 | 1 |  | 11 | 698437 | 1.54 | -3 |
Newton | 3 | 484 | 1.29 | -1 |  | 4 | 734 | 1.21 | 1 |  | 4 | 687 | 1.10 | 1 |  | 5 | 545312 | 1.20 | 0 |
Halley | 2 | 375 | 1.00 | 0 |  | 3 | 609 | 1.00 | 3 |  | 3 | 625 | 1.00 | 3 |  | 4 | 453125 | 1.00 | 0 |
Schröder | 3 | 546 | 1.46 | -1 |  | 6 | 1109 | 1.82 | 1 |  | 6 | 1187 | 1.90 | 1 |  | 5 | 564062 | 1.24 | -2 |

##### Program root_elliptic_finding.cpp, GNU C++ version 4.9.2, GNU libstdc++ version 20141030, Win32 Compiled in optimise mode., _X64_SSE2

Table 12.14. root with radius 28 and arc length 300) for float, double, long double and cpp_bin_float_50 types, using _X64_SSE2

| float |  |  |  |  | double |  |  |  |  | long d |  |  |  |  | cpp50 |  |  |  |
Algo | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |  | Its | Times | Norm | Dis |
TOMS748 | 5 | 328 | 1.31 | -1 |  | 8 | 875 | 1.51 | 0 |  | 8 | 1109 | 1.69 | 4 |  | 11 | 479687 | 1.49 | -3 |
Newton | 3 | 328 | 1.31 | -1 |  | 4 | 671 | 1.16 | 1 |  | 4 | 781 | 1.19 | 1 |  | 5 | 387500 | 1.20 | 0 |
Halley | 2 | 250 | 1.00 | 0 |  | 3 | 578 | 1.00 | 1 |  | 3 | 656 | 1.00 | 7 |  | 4 | 321875 | 1.00 | 0 |
Schröder | 3 | 375 | 1.50 | -1 |  | 4 | 734 | 1.27 | 0 |  | 4 | 828 | 1.26 | 3 |  | 5 | 414062 | 1.29 | -2 |

Remarks:

- The function being solved is now moderately expensive to call, and twice as expensive to call when obtaining the derivative than when not. Consequently there is very little improvement in moving from a derivative free method, to Newton iteration. However, once you've calculated the first derivative the second comes almost for free, consequently the third order methods (Halley) does much the best.
- Of the two second order methods, Halley does best as would be expected: the Schroder method offers better guarantees of quadratic convergence, while Halley relies on a smooth function with a single root to give cubic convergence. It's not entirely clear why Schroder iteration often does worse than Newton.