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| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.bessel.bessel_first"></a><a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">Bessel Functions of |
| the First and Second Kinds</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.bessel.bessel_first.h0"></a> |
| <span class="phrase"><a name="math_toolkit.bessel.bessel_first.synopsis"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.synopsis">Synopsis</a> |
| </h5> |
| <p> |
| <code class="computeroutput"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code> |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_j</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_j</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_neumann</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_neumann</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.bessel.bessel_first.h1"></a> |
| <span class="phrase"><a name="math_toolkit.bessel.bessel_first.description"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.description">Description</a> |
| </h5> |
| <p> |
| The functions <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> |
| and <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> return |
| the result of the Bessel functions of the first and second kinds respectively: |
| </p> |
| <p> |
| cyl_bessel_j(v, x) = J<sub>v</sub>(x) |
| </p> |
| <p> |
| cyl_neumann(v, x) = Y<sub>v</sub>(x) = N<sub>v</sub>(x) |
| </p> |
| <p> |
| where: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span> |
| </p> |
| <p> |
| The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result |
| type calculation rules</em></span></a> when T1 and T2 are different types. |
| The functions are also optimised for the relatively common case that T1 is |
| an integer. |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can |
| be used to control the behaviour of the function: how it handles errors, |
| what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <p> |
| The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| whenever the result is undefined or complex. For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> |
| this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span> |
| <span class="number">0</span></code> and v is not an integer, or when |
| <code class="computeroutput"><span class="identifier">x</span> <span class="special">==</span> |
| <span class="number">0</span></code> and <code class="computeroutput"><span class="identifier">v</span> |
| <span class="special">!=</span> <span class="number">0</span></code>. |
| For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> this |
| occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><=</span> |
| <span class="number">0</span></code>. |
| </p> |
| <p> |
| The following graph illustrates the cyclic nature of J<sub>v</sub>: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_j.svg" align="middle"></span> |
| </p> |
| <p> |
| The following graph shows the behaviour of Y<sub>v</sub>: this is also cyclic for large |
| <span class="emphasis"><em>x</em></span>, but tends to -∞   for small <span class="emphasis"><em>x</em></span>: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/cyl_neumann.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.bessel.bessel_first.h2"></a> |
| <span class="phrase"><a name="math_toolkit.bessel.bessel_first.testing"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.testing">Testing</a> |
| </h5> |
| <p> |
| There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>, |
| and a much larger set of tests computed using a simplified version of this |
| implementation (with all the special case handling removed). |
| </p> |
| <h5> |
| <a name="math_toolkit.bessel.bessel_first.h3"></a> |
| <span class="phrase"><a name="math_toolkit.bessel.bessel_first.accuracy"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following tables show how the accuracy of these functions varies on various |
| platforms, along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> |
| and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. |
| Note that the cyclic nature of these functions means that they have an infinite |
| number of irrational roots: in general these functions have arbitrarily large |
| <span class="emphasis"><em>relative</em></span> errors when the arguments are sufficiently |
| close to a root. Of course the absolute error in such cases is always small. |
| Note that only results for the widest floating-point type on the system are |
| given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively |
| zero error</a>. All values are relative errors in units of epsilon. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.bessel.bessel_first.errors_rates_in_cyl_bessel_j"></a><p class="title"><b>Table 6.21. Errors Rates in cyl_bessel_j</b></p> |
| <div class="table-contents"><table class="table" summary="Errors Rates in cyl_bessel_j"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| J<sub>0</sub>   and J<sub>1</sub> |
| </p> |
| </th> |
| <th> |
| <p> |
| J<sub>v</sub> |
| </p> |
| </th> |
| <th> |
| <p> |
| J<sub>v</sub>   (large values of x > 1000) |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32 / Visual C++ 8.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.5 Mean=1.1 |
| </p> |
| <p> |
| GSL Peak=6.6 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=2.5 |
| Mean=1.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=11 Mean=2.2 |
| </p> |
| <p> |
| GSL Peak=11 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=17 |
| Mean=2.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=59 Mean=10 |
| </p> |
| <p> |
| GSL Peak=6x10<sup>11</sup> |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=2x10<sup>5</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA64 / G++ 3.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=7 Mean=3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=117 Mean=10 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2x10<sup>4</sup>   Mean=6x10<sup>3</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| SUSE Linux AMD64 / G++ 4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=7 Mean=3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=400 Mean=40 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2x10<sup>4</sup>   Mean=1x10<sup>4</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HP-UX / HP aCC 6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=14 Mean=6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=29 Mean=3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2700 Mean=450 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.bessel.bessel_first.errors_rates_in_cyl_neumann"></a><p class="title"><b>Table 6.22. Errors Rates in cyl_neumann</b></p> |
| <div class="table-contents"><table class="table" summary="Errors Rates in cyl_neumann"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| Y<sub>0</sub>   and Y<sub>1</sub> |
| </p> |
| </th> |
| <th> |
| <p> |
| Y<sub>n</sub> (integer orders) |
| </p> |
| </th> |
| <th> |
| <p> |
| Y<sub>v</sub> (fractional orders) |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32 / Visual C++ 8.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.7 Mean=1.7 |
| </p> |
| <p> |
| GSL Peak=34 Mean=9 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=330 |
| Mean=54 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=117 Mean=10 |
| </p> |
| <p> |
| GSL Peak=500 Mean=54 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=923 |
| Mean=83 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=800 Mean=40 |
| </p> |
| <p> |
| GSL Peak=1.4x10<sup>6</sup>   Mean=7x10<sup>4</sup>   |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=+INF |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA64 / G++ 3.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=470 Mean=56 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=843 Mean=51 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=741 Mean=51 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| SUSE Linux AMD64 / G++ 4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1300 Mean=424 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2x10<sup>4</sup>   Mean=8x10<sup>3</sup> |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1x10<sup>5</sup>   Mean=6x10<sup>3</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HP-UX / HP aCC 6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=180 Mean=63 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=340 Mean=150 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2x10<sup>4</sup>   Mean=1200 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><p> |
| Note that for large <span class="emphasis"><em>x</em></span> these functions are largely dependent |
| on the accuracy of the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">sin</span></code> and |
| <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cos</span></code> functions. |
| </p> |
| <p> |
| Comparison to GSL and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> |
| is interesting: both <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> |
| and this library optimise the integer order case - leading to identical results |
| - simply using the general case is for the most part slightly more accurate |
| though, as noted by the better accuracy of GSL in the integer argument cases. |
| This implementation tends to perform much better when the arguments become |
| large, <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> in particular |
| produces some remarkably inaccurate results with some of the test data (no |
| significant figures correct), and even GSL performs badly with some inputs |
| to J<sub>v</sub>. Note that by way of double-checking these results, the worst performing |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> and GSL cases were |
| recomputed using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>, |
| and the result checked against our test data: no errors in the test data |
| were found. |
| </p> |
| <h5> |
| <a name="math_toolkit.bessel.bessel_first.h4"></a> |
| <span class="phrase"><a name="math_toolkit.bessel.bessel_first.implementation"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.implementation">Implementation</a> |
| </h5> |
| <p> |
| The implementation is mostly about filtering off various special cases: |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is negative, then the order <span class="emphasis"><em>v</em></span> |
| must be an integer or the result is a domain error. If the order is an integer |
| then the function is odd for odd orders and even for even orders, so we reflect |
| to <span class="emphasis"><em>x > 0</em></span>. |
| </p> |
| <p> |
| When the order <span class="emphasis"><em>v</em></span> is negative then the reflection formulae |
| can be used to move to <span class="emphasis"><em>v > 0</em></span>: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span> |
| </p> |
| <p> |
| Note that if the order is an integer, then these formulae reduce to: |
| </p> |
| <p> |
| J<sub>-n</sub> = (-1)<sup>n</sup>J<sub>n</sub> |
| </p> |
| <p> |
| Y<sub>-n</sub> = (-1)<sup>n</sup>Y<sub>n</sub> |
| </p> |
| <p> |
| However, in general, a negative order implies that we will need to compute |
| both J and Y. |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is large compared to the order <span class="emphasis"><em>v</em></span> |
| then the asymptotic expansions for large <span class="emphasis"><em>x</em></span> in M. Abramowitz |
| and I.A. Stegun, <span class="emphasis"><em>Handbook of Mathematical Functions</em></span> |
| 9.2.19 are used (these were found to be more reliable than those in A&S |
| 9.2.5). |
| </p> |
| <p> |
| When the order <span class="emphasis"><em>v</em></span> is an integer the method first relates |
| the result to J<sub>0</sub>, J<sub>1</sub>, Y<sub>0</sub>   and Y<sub>1</sub>   using either forwards or backwards recurrence |
| (Miller's algorithm) depending upon which is stable. The values for J<sub>0</sub>, J<sub>1</sub>, |
| Y<sub>0</sub>   and Y<sub>1</sub>   are calculated using the rational minimax approximations on root-bracketing |
| intervals for small <span class="emphasis"><em>|x|</em></span> and Hankel asymptotic expansion |
| for large <span class="emphasis"><em>|x|</em></span>. The coefficients are from: |
| </p> |
| <p> |
| W.J. Cody, <span class="emphasis"><em>ALGORITHM 715: SPECFUN - A Portable FORTRAN Package |
| of Special Function Routines and Test Drivers</em></span>, ACM Transactions |
| on Mathematical Software, vol 19, 22 (1993). |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| J.F. Hart et al, <span class="emphasis"><em>Computer Approximations</em></span>, John Wiley |
| & Sons, New York, 1968. |
| </p> |
| <p> |
| These approximations are accurate to around 19 decimal digits: therefore |
| these methods are not used when type T has more than 64 binary digits. |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is smaller than machine epsilon then the following |
| approximations for Y<sub>0</sub>(x), Y<sub>1</sub>(x), Y<sub>2</sub>(x) and Y<sub>n</sub>(x) can be used (see: <a href="http://functions.wolfram.com/03.03.06.0037.01" target="_top">http://functions.wolfram.com/03.03.06.0037.01</a>, |
| <a href="http://functions.wolfram.com/03.03.06.0038.01" target="_top">http://functions.wolfram.com/03.03.06.0038.01</a>, |
| <a href="http://functions.wolfram.com/03.03.06.0039.01" target="_top">http://functions.wolfram.com/03.03.06.0039.01</a> |
| and <a href="http://functions.wolfram.com/03.03.06.0040.01" target="_top">http://functions.wolfram.com/03.03.06.0040.01</a>): |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel_y0_small_z.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel_y1_small_z.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel_y2_small_z.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel_yn_small_z.svg"></span> |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span> and |
| <span class="emphasis"><em>v</em></span> is not an integer, then the following series approximation |
| can be used for Y<sub>v</sub>(x), this is also an area where other approximations are |
| often too slow to converge to be used (see <a href="http://functions.wolfram.com/03.03.06.0034.01" target="_top">http://functions.wolfram.com/03.03.06.0034.01</a>): |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel_yv_small_z.svg"></span> |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span>, |
| J<sub>v</sub>x   is best computed directly from the series: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span> |
| </p> |
| <p> |
| In the general case we compute J<sub>v</sub>   and Y<sub>v</sub>   simultaneously. |
| </p> |
| <p> |
| To get the initial values, let μ   = ν - floor(ν + 1/2), then μ   is the fractional part |
| of ν   such that |μ| <= 1/2 (we need this for convergence later). The idea |
| is to calculate J<sub>μ</sub>(x), J<sub>μ+1</sub>(x), Y<sub>μ</sub>(x), Y<sub>μ+1</sub>(x) and use them to obtain J<sub>ν</sub>(x), Y<sub>ν</sub>(x). |
| </p> |
| <p> |
| The algorithm is called Steed's method, which needs two continued fractions |
| as well as the Wronskian: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel11.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel12.svg"></span> |
| </p> |
| <p> |
| See: F.S. Acton, <span class="emphasis"><em>Numerical Methods that Work</em></span>, The Mathematical |
| Association of America, Washington, 1997. |
| </p> |
| <p> |
| The continued fractions are computed using the modified Lentz's method (W.J. |
| Lentz, <span class="emphasis"><em>Generating Bessel functions in Mie scattering calculations |
| using continued fractions</em></span>, Applied Optics, vol 15, 668 (1976)). |
| Their convergence rates depend on <span class="emphasis"><em>x</em></span>, therefore we need |
| different strategies for large <span class="emphasis"><em>x</em></span> and small <span class="emphasis"><em>x</em></span>. |
| </p> |
| <p> |
| <span class="emphasis"><em>x > v</em></span>, CF1 needs O(<span class="emphasis"><em>x</em></span>) iterations |
| to converge, CF2 converges rapidly |
| </p> |
| <p> |
| <span class="emphasis"><em>x <= v</em></span>, CF1 converges rapidly, CF2 fails to converge |
| when <span class="emphasis"><em>x</em></span> <code class="literal">-></code> 0 |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is large (<span class="emphasis"><em>x</em></span> > 2), both |
| continued fractions converge (CF1 may be slow for really large <span class="emphasis"><em>x</em></span>). |
| J<sub>μ</sub>, J<sub>μ+1</sub>, Y<sub>μ</sub>, Y<sub>μ+1</sub> can be calculated by |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel13.svg"></span> |
| </p> |
| <p> |
| where |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel14.svg"></span> |
| </p> |
| <p> |
| J<sub>ν</sub> and Y<sub>μ</sub> are then calculated using backward (Miller's algorithm) and forward |
| recurrence respectively. |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is small (<span class="emphasis"><em>x</em></span> <= 2), CF2 |
| convergence may fail (but CF1 works very well). The solution here is Temme's |
| series: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel15.svg"></span> |
| </p> |
| <p> |
| where |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/bessel16.svg"></span> |
| </p> |
| <p> |
| g<sub>k</sub>   and h<sub>k</sub>   |
| are also computed by recursions (involving gamma functions), but |
| the formulas are a little complicated, readers are refered to N.M. Temme, |
| <span class="emphasis"><em>On the numerical evaluation of the ordinary Bessel function of |
| the second kind</em></span>, Journal of Computational Physics, vol 21, 343 |
| (1976). Note Temme's series converge only for |μ| <= 1/2. |
| </p> |
| <p> |
| As the previous case, Y<sub>ν</sub>   is calculated from the forward recurrence, so is Y<sub>ν+1</sub>. |
| With these two values and f<sub>ν</sub>, the Wronskian yields J<sub>ν</sub>(x) directly without backward |
| recurrence. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
| </div></td> |
| </tr></table> |
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