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)