source: trunk/omh/theory/acos_forward.omh @ 3679

Last change on this file since 3679 was 3679, checked in by bradbell, 5 years ago

merge to branch: trunk
from repository: https://github.com/coin-or/CppAD
start hash code: 5e2f77aac80262cd007effafcc48ecf2d313e467
end hash code: 071875a4beba3363e5fa9752426aec4762cd1caa

commit 071875a4beba3363e5fa9752426aec4762cd1caa
Author: Brad Bell <bradbell@…>
Date: Tue May 5 12:03:54 2015 -0700

Corrections to previous edit of test_one.sh.in.

commit 7c2bedb5c54ecca1727514bfc4ef457dcff83e9d
Author: Brad Bell <bradbell@…>
Date: Tue May 5 11:54:49 2015 -0700

Fix the Auto-tools build as follows:

  1. Include asinh files in auto-tools distribution and build.
  2. Update auto-tools for including cppad_compiler_has_asinh flag (as false).
  3. Convert from automake 1.13.4 to automake 1.14.1.


acos_forward.omh: add theory for acosh.

  • Property svn:keywords set to Id
File size: 2.9 KB
Line 
1$Id: acos_forward.omh 3679 2015-05-05 19:10:59Z bradbell $
2// BEGIN SHORT COPYRIGHT
3/* --------------------------------------------------------------------------
4CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-15 Bradley M. Bell
5
6CppAD is distributed under multiple licenses. This distribution is under
7the terms of the
8                    Eclipse Public License Version 1.0.
9
10A copy of this license is included in the COPYING file of this distribution.
11Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
12-------------------------------------------------------------------------- */
13// END SHORT COPYRIGHT
14
15$begin acos_forward$$
16$spell
17        acos
18        acosh
19        Arccosine
20        Taylor
21$$
22
23$section Arc Cosine and Hyperbolic Cosine Forward Taylor Polynomial Theory$$
24$mindex acos, acosh$$
25
26$head Derivatives$$
27$latex \[
28\begin{array}{rcl}
29\R{arccos}^{(1)} (x)  & = & - 1 / \sqrt{ 1 - x * x }
30\\
31\R{arccosh}^{(1)} (x) & = & + 1 / \sqrt{ x * x - 1}
32\end{array}
33\] $$
34If $latex F(x)$$ is $latex \R{arccos} (x)$$ or $latex \R{arccosh} (x)$$
35the corresponding derivative satisfies the equation
36$latex \[
37        \sqrt{ \mp ( x * x - 1 ) } * F^{(1)} (x) - 0 * F (u)  = \mp 1
38\] $$
39and in the
40$cref/standard math function differential equation
41        /ForwardTheory
42        /Standard Math Functions
43        /Differential Equation
44/$$,
45$latex A(x) = 0$$,
46$latex B(x) = \sqrt{ \mp( x * x - 1 ) }$$,
47and $latex D(x) = \mp 1$$.
48We use $latex a$$, $latex b$$, $latex d$$ and $latex z$$ to denote the
49Taylor coefficients for
50$latex A [ X (t) ] $$,
51$latex B [ X (t) ]$$,
52$latex D [ X (t) ] $$,
53and $latex F [ X(t) ] $$ respectively.
54$pre
55
56$$
57We define $latex Q(x) = \mp ( x * x - 1 )$$
58and let $latex q$$ be the corresponding Taylor coefficients for
59$latex Q[ X(t) ]$$.
60It follows that
61$latex \[
62q^{(j)} = \left\{ \begin{array}{ll}
63        \mp ( x^{(0)} * x^{(0)} - 1 )      & {\rm if} \; j = 0 \\
64        \mp \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise}
65\end{array} \right.
66\] $$
67It follows that
68$latex B[ X(t) ] = \sqrt{ Q[ X(t) ] }$$ and
69from the equations for the
70$cref/square root/SqrtForward/$$
71that for $latex j = 0 , 1, \ldots$$,
72$latex \[
73\begin{array}{rcl}
74b^{(0)}   & = & \sqrt{ q^{(0)} }
75\\
76b^{(j+1)} & = &
77        \frac{1}{j+1} \frac{1}{ b^{(0)} }
78        \left(
79                \frac{j+1}{2} q^{(j+1) }
80                - \sum_{k=1}^j k b^{(k)} b^{(j+1-k)}
81        \right)
82\end{array}
83\] $$
84It now follows from the general
85$xref/
86        ForwardTheory/
87        Standard Math Functions/
88        Taylor Coefficients Recursion Formula/
89        Taylor coefficients recursion formula/
90        1
91/$$
92that for $latex j = 0 , 1, \ldots$$,
93$latex \[
94\begin{array}{rcl}
95z^{(0)} & = & F ( x^{(0)} )
96\\
97e^{(j)}
98& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)}
99\\
100& = & \left\{ \begin{array}{ll}
101        \mp 1 & {\rm if} \; j = 0 \\
102        0 & {\rm otherwise}
103\end{array} \right.
104\\
105z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
106\left(
107        \sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)}
108        - \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)}
109\right)
110\\
111z^{(j+1)} & = & - \frac{1}{j+1} \frac{1}{ b^{(0)} }
112\left(
113        (j+1) x^{(j+1)}
114        + \sum_{k=1}^j k z^{(k)}  b^{(j+1-k)}
115\right)
116\end{array}
117\] $$
118
119
120$end
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