sicmutils.calculus.derivative
This namespace implements a number of differential operators like D, and
the machinery to apply D to various structures.
This namespace implements a number of differential operators like [[D]], and the machinery to apply [[D]] to various structures.
Dclj/s
Derivative operator. Takes some function f and returns a function
whose value at some point can multiply an increment in the arguments, to
produce the best linear estimate of the increment in the function value.
For univariate functions, D computes a derivative. For vector-valued
functions, D computes
the Jacobian
of f.
The related [[Grad]] returns a function that produces a structure of the
opposite orientation as D. Both of these functions use forward-mode
automatic differentiation.
Derivative operator. Takes some function `f` and returns a function whose value at some point can multiply an increment in the arguments, to produce the best linear estimate of the increment in the function value. For univariate functions, [[D]] computes a derivative. For vector-valued functions, [[D]] computes the [Jacobian](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant) of `f`. The related [[Grad]] returns a function that produces a structure of the opposite orientation as [[D]]. Both of these functions use forward-mode automatic differentiation.
derivativeclj/s
(derivative f)Returns a single-argument function of that, when called with an argument x,
returns the derivative of f at x using forward-mode automatic
differentiation.
For numerical differentiation,
see sicmutils.numerical.derivative/D-numeric.
f must be built out of generic operations that know how to
handle [[d/Differential]] inputs in addition to any types that a normal (f x) call would present. This restriction does not apply to operations like
putting x into a container or destructuring; just primitive function calls.
Returns a single-argument function of that, when called with an argument `x`, returns the derivative of `f` at `x` using forward-mode automatic differentiation. For numerical differentiation, see [[sicmutils.numerical.derivative/D-numeric]]. `f` must be built out of generic operations that know how to handle [[d/Differential]] inputs in addition to any types that a normal `(f x)` call would present. This restriction does _not_ apply to operations like putting `x` into a container or destructuring; just primitive function calls.
partialclj/s
(partial & selectors)Returns an operator that, when applied to a function f, produces a function
that computes the partial derivative of f at the (zero-based) slot index
provided via selectors.
Returns an operator that, when applied to a function `f`, produces a function that computes the partial derivative of `f` at the (zero-based) slot index provided via `selectors`.
taylor-seriesclj/s
(taylor-series f x dx)Returns a sicmutils.series/Series of the coefficients of the taylor series
of the function f evaluated at x, with incremental quantity dx.
NOTE: The (constantly dx) term is what allows this to work with arbitrary
structures of x and dx. Without this wrapper, ((g/* dx D) f) with dx
== (up 'dx 'dy) would expand to this:
(fn [x] (* (s/up ('dx x) ('dy x))
((D f) x)))
constantly delays the interpretation of dx one step:
(fn [x] (* (s/up 'dx 'dy)
((D f) x)))
Returns a [[sicmutils.series/Series]] of the coefficients of the taylor series
of the function `f` evaluated at `x`, with incremental quantity `dx`.
NOTE: The `(constantly dx)` term is what allows this to work with arbitrary
structures of `x` and `dx`. Without this wrapper, `((g/* dx D) f)` with `dx`
== `(up 'dx 'dy)` would expand to this:
```clojure
(fn [x] (* (s/up ('dx x) ('dy x))
((D f) x)))
```
`constantly` delays the interpretation of `dx` one step:
```clojure
(fn [x] (* (s/up 'dx 'dy)
((D f) x)))
```
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