This is a dual number expression RPN (reverse Polish notation) calculator for automatic differentiation, modeled on an old-style scientific calculator. Dual numbers are numbers of the form  , where  ,  are real and  . (  is displayed with  and  in adjacent boxes. Use the calculator to calculate an arithmetic expression in  and reals  ,  , and  ). Clicking a value puts the value on top of a pile. Clicking an "op" applies the "op" to the top member(s) of the pile as argument(s), and replaces them on the pile with its value. This is the RPN method of evaluation. The list of buttons clicked will be the reverse Polish notation for the expression. If the result is  for a function  , its value will be  . Thus,  is calculated "automatically". Moreover, this will be the result if  is any function computable by the calculator. In fact,  . It is as if  were expanded in a Maclaurin series in  , since  . Click "x + e", "x + e", "*" to get  . Click "x + e", "x + e", *, "a", +, "x + e", "*" to get  . Click "x + e", "x^2", "x + e", "1/x", "sin", "*" to get  . Click "x + h e", "cos", "x^2" to get  . Calculating an expression  in dual numbers  and  results in  . For example, to compute the partial derivatives of  , click "x + h e", "y + k e", "x^2", "*", "x + h e", "x^2", "y + k e", "x^2", "+", "/". The partial derivatives are the coefficients of  and  in the second box. Built-in generic functions  and  of one variable and  and  of two variables can be used to derive general differentiation formulas. For example, compute the formula for the derivative of  by clicking "x + e", "f", "x + e", "g", "/" and for  by clicking "a", "x + e", "*", "f". The expression pile contains symbolic expressions and the value pile contains these expressions evaluated at  ,  ,  ,  , and  . (To get decimal values, set an involved slider to a decimal value.) Click "clear" to start a new calculation. The calculator can also be used to calculate real expressions and their values in  ,  ,  ,  , and  .
The calculator works by extending built-in functions  of one or two real variables to duals by  and  . The chain rule implies these relations hold for every computable function. For example, if  and  , then  . In Dual Number, a dual number is a number of the form  , where  and  are real, and  is a matrix such that  , such as  Our calculator represents the dual number  as the expression  , with operations defined by  ,  ,  , which defines negation (  ) and hence subtraction,  for real  , which defines reciprocal (  ) and hence division,  , for functions  of one variable,  for functions  of two variables,  . These rules are implemented in Mathematica using "up values". With operations thus overloaded, the calculator is implemented as if it were defined only for reals. [2] L. B. Rall, "The Arithmetic of Differentiation," Mathematics Magazine, 59(5), 1986, pp. 275–282. [3] R. D. Neidinger, "Automatic Differentiation and APL," The College Mathematics Journal, 20(3), 1989 pp. 238–251. [4] R. D. Neidinger, "Introduction to Automatic Differentiation and MATLAB Obect Oriented Programming," SIAM Review, 52(3), 2010 pp. 545–563.
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