A Dual-Number Type

Our first step will be creating a new dual-number data type to perform "forward-mode” automatic differentiation. For dual-numbers in Scheme, an implementation using pairs makes sense. The first element of the pair will be x and the second will be the infinitesimal

x'

: (x . x').

(define (make-dual n d) (cons n d))

(define (val x) (car x))

(define (dval x) (cdr x))

(define (make-variable x) (make-dual x 1))

(This implementation is reminiscent of the rational number data type outlined in 2.1.1: Arithmetic Operations for Rational Numbers.)

To find the derivative of an addition expression, for example,

f(x) = x + x

;

f'(2)

, we apply the procedure add-dual:

(define (add-dual u v)

(make-dual (+ (val u) (val v))

(+ (dval u) (dval v))))

(define x (make-variable 2))

(add-dual x x) --> (4 . 2)

Warmup

Write the make-constant constructor (remember the derivative of any constant is zero).

Use the add-dual to write the scheme expression for finding $g(x) = x + x + x + 4$ .

Find $g(10$ ) and $g'(10)$ .

Dual Number Arithmetic

Your dual numbers should support the following mathematical functions:

For example,

f(x) = 1/\cos(\sin(x)^2)

(quo-dual (make-constant 1)

(cos-dual (pow-dual (sin-dual x) (make-constant 2))))

-->(1.48 . -1.21)

A Derivative Interpreter

Building upon the program in 2.3.2: Symbolic Differentiation from SICP and our dual numbers, create a function deval to allow more declarative evaluation of derivative expressions.

deval: (expression x symbol x value) → dual-number

(deval '(* x 2) 'x 2) --> (4 . 2)

(deval '(/ 1 (cos (^ (sin x) 2))) 'x 2) --> (1.48 . -1.21)

(Optional) Next Steps

Use an assoc list, instead of a single variable and value, in the deval procedure to allow multivariate equations.

(deval '(* x y) '((x 2) (y 2))) --> (4 2 2)

Use shunting yard or pratt parsing to convert infix expressions to prefix, then compute the derivative using deval.

(deval (infix->prefix "1/cos((sin x)^2)") 'x 2) --> (1.48 . -1.21)

Allow procedures and implement the chain rule.

​​A Dual-Number Type

​​Warmup

​​Dual Number Arithmetic

​​A Derivative Interpreter

​​(Optional) Next Steps

A Dual-Number Type

Warmup

Dual Number Arithmetic

A Derivative Interpreter

(Optional) Next Steps