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forward_theory
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.,
For the purposes of this section, we are given \(x\) and \(y\) and need to determine \(z\).
Binary Operators
Addition
Subtraction
Multiplication
Division
Standard Math Functions
Suppose that \(F\) is a standard math function and
Differential Equation
All of the standard math functions satisfy a differential equation of the form
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
where we define
We can compute the value of \(z^{(0)}\) using the formula
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
We need to complete the induction by finding formulas for \(z^{(j+1)}\). It follows from the definition of \(E(t)\) that
Setting the left and right side coefficients of \(t^j\) equal, and using the formula for Multiplication , we obtain
This completes the induction that computes \(e^{(j)}\) and \(z^{(j+1)}\).
Cases that Apply Recursion Above
exp_forward |
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log_forward |
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sqrt_forward |
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sin_cos_forward |
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atan_forward |
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asin_forward |
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acos_forward |
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pow_forward |
Special Cases
tan_forward |
Tangent and Hyperbolic Tangent Forward Taylor Polynomial Theory |
erf_forward |