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.
The typical definition of a Taylor series at the point x is
$$f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!} (x-a)^2 + \ldots$$
All derivatives of the original function and this Taylor series match at the
point a.
The argument dx is instead interpreted as an incremental difference
from (x-a). So, passing 0 for dx would return the evaluation of the Taylor
series at the point a.
Returns a [[sicmutils.series/Series]] of the coefficients of the [Taylor
series](https://en.wikipedia.org/wiki/Taylor_series) of the function `f`
evaluated at `x`, with incremental quantity `dx`.
The typical definition of a Taylor series at the point `x` is
$$f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!} (x-a)^2 + \ldots$$
All derivatives of the original function and this Taylor series match at the
point `a`.
The argument `dx` is instead interpreted as an incremental difference
from `(x-a)`. So, passing 0 for `dx` would return the evaluation of the Taylor
series at the point `a`.
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