# 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
start hash code: 5e2f77aac80262cd007effafcc48ecf2d313e467
end hash code: 071875a4beba3363e5fa9752426aec4762cd1caa

commit 071875a4beba3363e5fa9752426aec4762cd1caa
Date: Tue May 5 12:03:54 2015 -0700

Corrections to previous edit of test_one.sh.in.

commit 7c2bedb5c54ecca1727514bfc4ef457dcff83e9d
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.

• Property svn:keywords set to Id
File size: 2.9 KB
Line
1$Id: acos_forward.omh 3679 2015-05-05 19:10:59Z bradbell$
3/* --------------------------------------------------------------------------
5
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.
12-------------------------------------------------------------------------- */
14
15$begin acos_forward$$16spell 17 acos 18 acosh 19 Arccosine 20 Taylor 21$$ 22 23$section Arc Cosine and Hyperbolic Cosine Forward Taylor Polynomial Theory$$24mindex acos, acosh$$
25
26$head Derivatives$$27latex $28\begin{array}{rcl} 29\R{arccos}^{(1)} (x) & = & - 1 / \sqrt{ 1 - x * x } 30\\ 31\R{arccosh}^{(1)} (x) & = & + 1 / \sqrt{ x * x - 1} 32\end{array} 33$$$ 34If$latex F(x)$$is latex \R{arccos} (x)$$ or $latex \R{arccosh} (x)$$35the corresponding derivative satisfies the equation 36latex $37 \sqrt{ \mp ( x * x - 1 ) } * F^{(1)} (x) - 0 * F (u) = \mp 1 38$$$ 39and in the 40$cref/standard math function differential equation
41        /ForwardTheory
42        /Standard Math Functions
43        /Differential Equation
44/$$, 45latex A(x) = 0$$,
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) ] $$, 51latex B [ X (t) ]$$,
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$$be the corresponding Taylor coefficients for 59latex Q[ X(t) ]$$. 60It follows that 61$latex $62q^{(j)} = \left\{ \begin{array}{ll} 63 \mp ( x^{(0)} * x^{(0)} - 1 ) & {\rm if} \; j = 0 \\ 64 \mp \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise} 65\end{array} \right. 66$ $$67It follows that 68latex B[ X(t) ] = \sqrt{ Q[ X(t) ] }$$ and
69from the equations for the
70$cref/square root/SqrtForward/$$71that for latex j = 0 , 1, \ldots$$, 72$latex $73\begin{array}{rcl} 74b^{(0)} & = & \sqrt{ q^{(0)} } 75\\ 76b^{(j+1)} & = & 77 \frac{1}{j+1} \frac{1}{ b^{(0)} } 78 \left( 79 \frac{j+1}{2} q^{(j+1) } 80 - \sum_{k=1}^j k b^{(k)} b^{(j+1-k)} 81 \right) 82\end{array} 83$ $$84It now follows from the general 85xref/ 86 ForwardTheory/ 87 Standard Math Functions/ 88 Taylor Coefficients Recursion Formula/ 89 Taylor coefficients recursion formula/ 90 1 91/$$
92that for $latex j = 0 , 1, \ldots$$, 93latex $94\begin{array}{rcl} 95z^{(0)} & = & F ( x^{(0)} ) 96\\ 97e^{(j)} 98& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)} 99\\ 100& = & \left\{ \begin{array}{ll} 101 \mp 1 & {\rm if} \; j = 0 \\ 102 0 & {\rm otherwise} 103\end{array} \right. 104\\ 105z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } 106\left( 107 \sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)} 108 - \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)} 109\right) 110\\ 111z^{(j+1)} & = & - \frac{1}{j+1} \frac{1}{ b^{(0)} } 112\left( 113 (j+1) x^{(j+1)} 114 + \sum_{k=1}^j k z^{(k)} b^{(j+1-k)} 115\right) 116\end{array} 117$$$ 118 119 120$end
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