Single variable function minimizer

The function must be continuous unimodal, taking a floating point argument and returning a floating point value.

The method is binary search based on the derivative at the splitting point. Convergence is judged by the change in function value.

First, words defined elsewhere.

~~~{{   :d:another-name (as-)     d:create &class:word reclass d:last d:xt swap d:xt fetch swap store ; ---reveal---   :d:aka (s-)_Also-Known-As,_make_alias_of_the_last_defined_word     [ d:last ]   dip d:another-name ; 'aka   d:aka   :d:alias (ss-)_make_alias_s2_of_s1     [ d:lookup ] dip d:another-name ; 'alias d:aka }}   'var-n  'var! (n-)  alias 'lt?    'n:<?       alias 'lteq?  'n:=<?      alias :v:put (a-) fetch n:put ; :e:fetch (a-__-f)_fetch_as_float  fetch e:to-f ; 'e:@ aka :e:store (a-__f-)_store_as_e f:to-e swap store ; 'e:! aka :e:call.vv (aaa-)_call_floating_point_function_variable-to-variable   rot e:@ (aa_n) call (a_n) e:! ;   :s:shout (s-) '!_%s s:format s:put ; {{   'Depth var   :message (-)     'abort_with__trail__invoked s:shout nl     'Do__reset__to_clear_stack. s:put   nl ;   :put-name (a-) fetch d:lookup-xt     dup n:-zero? [ d:name s:put nl ] [ drop ] choose ; ---reveal---   :trail repeat pop put-name again ;   :abort (-0) depth !Depth message trail ;   :s:abort (s-0) 's:abort_:_ s:prepend s:put nl abort ; }}   :assert         (q-) call [                  abort ] -if ; :assert.verbous (q-) call [ 'assert_:_fail s:abort ] -if ;   :dump-stacks (-)                    dump-stack   #0 f:depth  lt? [ nl 'f_  s:put f:dump-stack  ] if   #0 f:adepth lt? [ nl 'fa_ s:put f:dump-astack ] if ; '. aka ~~~

Program.

~~~{{   (input_variables     'F    var (Function_to_minimize_(-_n-n)_float_to_float     'L    var (e:low     'H    var (e:high       TRUE             'Tr    var!  (trace_flag     .0.000001 f:to-e 'TOL   const (=173=tolerance     #30              'ITMAX const (max_#_of_iterations     (working_variables     'Ly   var (e:low-value     'Hy   var     'X1y  var (y_at_upper_next_to_x     'Y    var (function_value_at_X     'Itr  var (iterations     (output_variable     'X    var (candidate     :trace  (-) @Tr [ ( Itr v:put sp )         ( @L e:put sp ) ( @X e:put sp ) ( @H e:put sp ) nl ] if ;   :y      (-) X Y  @F e:call.vv ;   :x      (-)_(L_H_->_X) @L @H + #2 / !X ;   :x1y    (-) @X n:inc e:to-f @F call X1y e:! ;     ---reveal---   :minimize (a-_nn-x)_(L_H_F_->_X)     (minimize_floating_point_function_F_over_[L,H]     (initialize ( !F f:to-e !H f:to-e !L )       ( L Ly @F e:call.vv ) ( H Hy @F e:call.vv ) x y ( #0 !Itr )     [ trace x1y         ( @Y @X1y n:<? ) [ @X !H  @Y !Hy ] [ @X !L  @Y !Ly ] choose       x y Itr v:inc [ @Itr ITMAX n:=<? ] assert.verbous       TOL @Y @X1y - n:abs n:<? ] while X e:@ ; }} ~~~

```:f (-_x-y)_(x-2)^2  .2. f:- f:dup f:* ; .-3 .10 &f minimize . nl ```

```:f (-_x-y) f:abs ; .-5. .5 &f minimize . nl ```

Note how x is defined without conversios as

:x (-)(LH->X) @L @H + 2 / !X ;

rather than

:x (-)(LH->X) ( L e:@ ) ( H e:@ ) f:+ .2. f:/ X e:! ;

Doing

```:f (-_x-y) ; .0 .5 &f minimize . nl ```

with the first definition gives

# low mid high 0 0.000000 1.249991 5.000009 1 0.000000 0.312492 1.249991 2 0.000000 0.078120 0.312492 3 0.000000 0.019530 0.078120 4 0.000000 0.004882 0.019530 5 0.000000 0.001220 0.004882 6 0.000000 0.000305 0.001220 7 0.000000 0.000076 0.000305 8 0.000000 0.000019 0.000076 9 0.000000 0.000005 0.000019 10 0.000000 0.000001 0.000005 final x = 0.000000

whereas with the second definition gives

# low mid high 0 0.000000 2.500004 5.000009 1 0.000000 1.249991 2.500004 2 0.000000 0.625001 1.249991 3 0.000000 0.312503 0.625001 4 0.000000 0.156254 0.312503 5 0.000000 0.078126 0.156254 6 0.000000 0.039062 0.078126 7 0.000000 0.019530 0.039062 8 0.000000 0.009765 0.019530 9 0.000000 0.004883 0.009765 10 0.000000 0.002441 0.004883 11 0.000000 0.001221 0.002441 12 0.000000 0.000611 0.001221 13 0.000000 0.000305 0.000611 14 0.000000 0.000153 0.000305 15 0.000000 0.000076 0.000153 16 0.000000 0.000038 0.000076 17 0.000000 0.000019 0.000038 18 0.000000 0.000009 0.000019 final x = 0.000005