1 | $Id: asin_forward.omh 3680 2015-05-07 19:17:37Z bradbell $ |
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2 | // BEGIN SHORT COPYRIGHT |
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3 | /* -------------------------------------------------------------------------- |
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4 | CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-15 Bradley M. Bell |
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5 | |
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6 | CppAD is distributed under multiple licenses. This distribution is under |
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7 | the terms of the |
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8 | Eclipse Public License Version 1.0. |
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9 | |
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10 | A copy of this license is included in the COPYING file of this distribution. |
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11 | Please visit http://www.coin-or.org/CppAD/ for information on other licenses. |
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12 | -------------------------------------------------------------------------- */ |
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13 | // END SHORT COPYRIGHT |
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14 | |
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15 | $begin asin_forward$$ |
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16 | $spell |
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17 | asinh |
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18 | asin |
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19 | Taylor |
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20 | $$ |
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21 | |
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22 | $section Inverse Sine and Hyperbolic Sine Forward Mode Theory$$ |
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23 | $mindex asin, asinh$$ |
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24 | |
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25 | $head Derivatives$$ |
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26 | $latex \[ |
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27 | \begin{array}{rcl} |
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28 | \R{asin}^{(1)} (x) & = & 1 / \sqrt{ 1 - x * x } |
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29 | \\ |
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30 | \R{asinh}^{(1)} (x) & = & 1 / \sqrt{ 1 + x * x } |
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31 | \end{array} |
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32 | \] $$ |
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33 | If $latex F(x)$$ is $latex \R{asin} (x) $$ or $latex \R{asinh} (x)$$ |
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34 | the corresponding derivative satisfies the equation |
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35 | $latex \[ |
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36 | \sqrt{ 1 \mp x * x } * F^{(1)} (x) - 0 * F (u) = 1 |
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37 | \] $$ |
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38 | and in the |
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39 | $cref/standard math function differential equation |
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40 | /ForwardTheory |
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41 | /Standard Math Functions |
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42 | /Differential Equation |
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43 | /$$, |
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44 | $latex A(x) = 0$$, |
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45 | $latex B(x) = \sqrt{1 \mp x * x }$$, |
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46 | and $latex D(x) = 1$$. |
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47 | We use $latex a$$, $latex b$$, $latex d$$ and $latex z$$ to denote the |
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48 | Taylor coefficients for |
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49 | $latex A [ X (t) ] $$, |
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50 | $latex B [ X (t) ]$$, |
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51 | $latex D [ X (t) ] $$, |
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52 | and $latex F [ X(t) ] $$ respectively. |
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53 | $pre |
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54 | |
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55 | $$ |
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56 | We define $latex Q(x) = 1 \mp x * x$$ |
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57 | and let $latex q$$ be the corresponding Taylor coefficients for |
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58 | $latex Q[ X(t) ]$$. |
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59 | It follows that |
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60 | $latex \[ |
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61 | q^{(j)} = \left\{ \begin{array}{ll} |
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62 | 1 \mp x^{(0)} * x^{(0)} & {\rm if} \; j = 0 \\ |
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63 | \mp \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise} |
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64 | \end{array} \right. |
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65 | \] $$ |
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66 | It follows that |
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67 | $latex B[ X(t) ] = \sqrt{ Q[ X(t) ] }$$ and |
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68 | from the equations for the |
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69 | $cref/square root/SqrtForward/$$ |
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70 | that for $latex j = 0 , 1, \ldots$$, |
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71 | $latex \[ |
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72 | \begin{array}{rcl} |
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73 | b^{(0)} & = & \sqrt{ q^{(0)} } |
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74 | \\ |
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75 | b^{(j+1)} & = & |
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76 | \frac{1}{j+1} \frac{1}{ b^{(0)} } |
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77 | \left( |
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78 | \frac{j+1}{2} q^{(j+1) } |
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79 | - \sum_{k=1}^j k b^{(k)} b^{(j+1-k)} |
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80 | \right) |
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81 | \end{array} |
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82 | \] $$ |
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83 | It now follows from the general |
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84 | $xref/ |
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85 | ForwardTheory/ |
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86 | Standard Math Functions/ |
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87 | Taylor Coefficients Recursion Formula/ |
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88 | Taylor coefficients recursion formula/ |
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89 | 1 |
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90 | /$$ |
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91 | that for $latex j = 0 , 1, \ldots$$, |
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92 | $latex \[ |
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93 | \begin{array}{rcl} |
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94 | z^{(0)} & = & F ( x^{(0)} ) |
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95 | \\ |
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96 | e^{(j)} |
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97 | & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)} |
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98 | \\ |
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99 | & = & \left\{ \begin{array}{ll} |
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100 | 1 & {\rm if} \; j = 0 \\ |
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101 | 0 & {\rm otherwise} |
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102 | \end{array} \right. |
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103 | \\ |
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104 | z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } |
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105 | \left( |
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106 | \sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)} |
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107 | - \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)} |
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108 | \right) |
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109 | \\ |
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110 | z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } |
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111 | \left( |
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112 | (j+1) x^{(j+1)} |
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113 | - \sum_{k=1}^j k z^{(k)} b^{(j+1-k)} |
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114 | \right) |
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115 | \end{array} |
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116 | \] $$ |
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117 | |
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118 | |
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119 | $end |
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