# Vocabulary/dcapco

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`x u D: n y`Secant Slope Conjunction

Rank *-- depends on the rank of u --*
WHY IS THIS IMPORTANT?

The slope of a straight line drawn through point `y` and point `y+x` on the curve of the mathematical function [implemented by verb] `u`

...That is for `n=1`. More generally ` D:n ` is an approximation to the `n`*-th derivative* of `u`

sin D.1 y=:i.6 NB. The 1st derivative, for comparison purposes 1 0.540302 _0.416147 _0.989992 _0.653644 0.283662 0.0001 sin D:1 y NB. Secant slope for interval(s) [y,y+0.0001] 1 0.54026 _0.416192 _0.99 _0.653606 0.28371 0.001 sin D:1 y NB. Secant slope for wider interval 1 0.539881 _0.416601 _0.990063 _0.653265 0.284142 0.01 sin D:1 y NB. Secant slope for still wider interval 0.999983 0.536086 _0.420686 _0.990682 _0.649849 0.288452

### Common Uses

1. To approximate a derivative, especially when better control over the step size `x` is needed

func =: verb define NB. function (x-2)^2 + (y-3)^2, with critical point at (2,3) 'x y' =. y (*: x-2) + (*: y-3) ) ]hessian =: func D. 2 (2 3) NB. Matrix of mixed second partial derivatives 2 _0.5 0.333333 2 NB. Whoops!! That's wrong. D. 2 chose a bad stepsize ]hessian =: 1e_6 func D: 2 (2 3) NB. Matrix of mixed second partial derivatives 2 0 0 2

### Related Primitives

Derivative (`u D. n`)

### More Information

1. `x` has the shape of an item of `y`, or is an atom that is replicated to that shape.
When the full `x` is given, the step size can be controlled for each direction.

2. Multiple slopes (corresponding to partial derivatives) are calculated when the cells of `u y` have rank greater than 0.
The result contains the slope of each result atom with respect to each input atom.

The shape of the overall result will be

(frame of y wrt verb u) , (shape of y) , (shape of result of u on a single cell) |<---- shape of result of single cell of u D: 1 ---->|

3. Higher derivatives are requested by a value of `n` greater than 1.
These results are sensitive to step size

0j13 ": 1e_3 sin D:3 (1) NB. Some bad results 111.0223024625157 0j13 ": 1e_2 sin D:3 (1) _0.5390394797189 0j13 ": 1e_2 sin D:3 (1) _0.5276133364651 0j13 ": sin"0 D.3 (1) NB. This is the correct result _0.5403023058681

4. Higher-order secants are allowed.

The second-order secant is the secant of a verb `u D: 1` whose result on each cell of `u` has shape

((shape of `y`) , (shape of result of `u` on a single cell)

The shape of `u D: 2 y` is

(frame of y wrt verb u) , (shape of y) , (shape of y) , (shape of result of u on a single cell) |<---- shape of result of single cell of u D: 1 ---->| |<----------- shape of result of single cell of u D: 2 ------------>|

The pattern continues for higher-order secants.
These approximate all the possible *mixed partial derivatives* of `u` at `y`.
The indexes in the "shape of y" sections of the above schematic select a particular mixed partial derivative.

### Details

1. If any atom of `n` is less than `0`, J signals an error.

`n` is converted to `1 >. '' $!.1 , n` before processing, i.e. the first atom of `n`, but with `0` changed to `1`.

The shape of the result is then

(frame of y wrt verb u) , (; n # <$y) , (shape of result of u on a single cell)