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sin_cos_forward

Headings-> Differential Equation

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ Trigonometric and Hyperbolic Sine and Cosine Forward Theory
Differential Equation
The standard math function differential equation is @[@ B(u) * F^{(1)} (u) - A(u) * F (u) = D(u) @]@ In this sections we consider forward mode for the following choices:

		@(@ F(u) @)@		@(@ \sin(u) @)@		@(@ \cos(u) @)@		@(@ \sinh(u) @)@		@(@ \cosh(u) @)@
		@(@ A(u) @)@		@(@ 0 @)@		@(@ 0 @)@		@(@ 0 @)@		@(@ 0 @)@
		@(@ B(u) @)@		@(@ 1 @)@		@(@ 1 @)@		@(@ 1 @)@		@(@ 1 @)@
		@(@ D(u) @)@		@(@ \cos(u) @)@		@(@ - \sin(u) @)@		@(@ \cosh(u) @)@		@(@ \sinh(u) @)@

We use @(@ a @)@, @(@ b @)@, @(@ d @)@ and @(@ f @)@ for the Taylor coefficients of @(@ A [ X (t) ] @)@, @(@ B [ X (t) ] @)@, @(@ D [ X (t) ] @)@, and @(@ F [ X(t) ] @)@ respectively. It now follows from the general Taylor coefficients recursion formula that for @(@ j = 0 , 1, \ldots @)@, @[@ \begin{array}{rcl} f^{(0)} & = & D ( x^{(0)} ) \\ e^{(j)} & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * f^{(k)} \\ & = & d^{(j)} \\ f^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( \sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)} - \sum_{k=1}^j k f^{(k)} b^{(j+1-k)} \right) \\ & = & \frac{1}{j+1} \sum_{k=1}^{j+1} k x^{(k)} d^{(j+1-k)} \end{array} @]@ The formula above generates the order @(@ j+1 @)@ coefficient of @(@ F[ X(t) ] @)@ from the lower order coefficients for @(@ X(t) @)@ and @(@ D[ X(t) ] @)@.

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Input File: omh/appendix/theory/sin_cos_forward.omh