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The Theory of Forward Mode

Taylor Notation
In Taylor notation, each variable corresponds to a function of a single argument which we denote by t (see Section 10.2 of Evaluating Derivatives ). Here and below @(@ X(t) @)@, @(@ Y(t) @)@, and Z(t) are scalar valued functions and the corresponding p-th order Taylor coefficients row vectors are @(@ x @)@, @(@ y @)@ and @(@ z @)@; i.e., @[@ \begin{array}{lcr} X(t) & = & x^{(0)} + x^{(1)} * t + \cdots + x^{(p)} * t^p + o( t^p ) \\ Y(t) & = & y^{(0)} + y^{(1)} * t + \cdots + y^{(p)} * t^p + o( t^p ) \\ Z(t) & = & z^{(0)} + z^{(1)} * t + \cdots + z^{(p)} * t^p + o( t^p ) \end{array} @]@ For the purposes of this section, we are given @(@ x @)@ and @(@ y @)@ and need to determine @(@ z @)@.

Binary Operators

Addition
@[@ \begin{array}{rcl} Z(t) & = & X(t) + Y(t) \\ \sum_{j=0}^p z^{(j)} * t^j & = & \sum_{j=0}^p x^{(j)} * t^j + \sum_{j=0}^p y^{(j)} * t^j + o( t^p ) \\ z^{(j)} & = & x^{(j)} + y^{(j)} \end{array} @]@
Subtraction
@[@ \begin{array}{rcl} Z(t) & = & X(t) - Y(t) \\ \sum_{j=0}^p z^{(j)} * t^j & = & \sum_{j=0}^p x^{(j)} * t^j - \sum_{j=0}^p y^{(j)} * t^j + o( t^p ) \\ z^{(j)} & = & x^{(j)} - y^{(j)} \end{array} @]@
Multiplication
@[@ \begin{array}{rcl} Z(t) & = & X(t) * Y(t) \\ \sum_{j=0}^p z^{(j)} * t^j & = & \left( \sum_{j=0}^p x^{(j)} * t^j \right) * \left( \sum_{j=0}^p y^{(j)} * t^j \right) + o( t^p ) \\ z^{(j)} & = & \sum_{k=0}^j x^{(j-k)} * y^{(k)} \end{array} @]@
Division
@[@ \begin{array}{rcl} Z(t) & = & X(t) / Y(t) \\ x & = & z * y \\ \sum_{j=0}^p x^{(j)} * t^j & = & \left( \sum_{j=0}^p z^{(j)} * t^j \right) * \left( \sum_{j=0}^p y^{(j)} * t^j \right) + o( t^p ) \\ x^{(j)} & = & \sum_{k=0}^j z^{(j-k)} y^{(k)} \\ z^{(j)} & = & \frac{1}{y^{(0)}} \left( x^{(j)} - \sum_{k=1}^j z^{(j-k)} y^{(k)} \right) \end{array} @]@
Standard Math Functions
Suppose that @(@ F @)@ is a standard math function and @[@ Z(t) = F[ X(t) ] @]@

Differential Equation
All of the standard math functions satisfy a differential equation of the form @[@ B(u) * F^{(1)} (u) - A(u) * F (u) = D(u) @]@ We use @(@ a @)@, @(@ b @)@ and @(@ d @)@ to denote the p-th order Taylor coefficient row vectors for @(@ A [ X (t) ] @)@, @(@ B [ X (t) ] @)@ and @(@ D [ X (t) ] @)@ respectively. We assume that these coefficients are known functions of @(@ x @)@, the p-th order Taylor coefficients for @(@ X(t) @)@.

Taylor Coefficients Recursion Formula
Our problem here is to express @(@ z @)@, the p-th order Taylor coefficient row vector for @(@ Z(t) @)@, in terms of these other known coefficients. It follows from the formulas above that @[@ \begin{array}{rcl} Z^{(1)} (t) & = & F^{(1)} [ X(t) ] * X^{(1)} (t) \\ B[ X(t) ] * Z^{(1)} (t) & = & \{ D[ X(t) ] + A[ X(t) ] * Z(t) \} * X^{(1)} (t) \\ B[ X(t) ] * Z^{(1)} (t) & = & E(t) * X^{(1)} (t) \end{array} @]@ where we define @[@ E(t) = D[X(t)] + A[X(t)] * Z(t) @]@ We can compute the value of @(@ z^{(0)} @)@ using the formula @[@ z^{(0)} = F ( x^{(0)} ) @]@ Suppose by induction (on @(@ j @)@) that we are given the Taylor coefficients of @(@ E(t) @)@ up to order @(@ j-1 @)@; i.e., @(@ e^{(k)} @)@ for @(@ k = 0 , \ldots , j-1 @)@ and the coefficients @(@ z^{(k)} @)@ for @(@ k = 0 , \ldots , j @)@. We can compute @(@ e^{(j)} @)@ using the formula @[@ e^{(j)} = d^{(j)} + \sum_{k=0}^j a^{(j-k)} * z^{(k)} @]@ We need to complete the induction by finding formulas for @(@ z^{(j+1)} @)@. It follows for the formula for the multiplication operator that @[@ \begin{array}{rcl} \left( \sum_{k=0}^j b^{(k)} t^k \right) * \left( \sum_{k=1}^{j+1} k z^{(k)} * t^{k-1} \right) & = & \left( \sum_{k=0}^j e^{(k)} * t^k \right) * \left( \sum_{k=1}^{j+1} k x^{(k)} * t^{k-1} \right) + o( t^p ) \\ z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( \sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)} - \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)} \right) \\ z^{(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 z^{(k)} b^{(j+1-k)} \right) \end{array} @]@ This completes the induction that computes @(@ e^{(j)} @)@ and @(@ z^{(j+1)} @)@.

Cases that Apply Recursion Above
exp_forward Exponential Function Forward Mode Theory
log_forward Logarithm Function Forward Mode Theory
sqrt_forward Square Root Function Forward Mode Theory
sin_cos_forward Trigonometric and Hyperbolic Sine and Cosine Forward Theory
atan_forward Inverse Tangent and Hyperbolic Tangent Forward Mode Theory
asin_forward Inverse Sine and Hyperbolic Sine Forward Mode Theory
acos_forward Inverse Cosine and Hyperbolic Cosine Forward Mode Theory

Special Cases
tan_forward Tangent and Hyperbolic Tangent Forward Taylor Polynomial Theory

Input File: omh/theory/forward_theory.omh