158 lines
4.2 KiB
Forth
158 lines
4.2 KiB
Forth
# Single variable function minimizer
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The function must be continuous unimodal,
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taking a floating point argument and returning a floating point value.
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The method is binary search based on the derivative at the splitting point.
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Convergence is judged by the change in function value.
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First, words defined elsewhere.
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~~~
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{{
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:d:another-name (as-)
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d:create &class:word reclass d:last d:xt swap d:xt fetch swap store ;
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---reveal---
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:d:aka (s-)_Also-Known-As,_make_alias_of_the_last_defined_word
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[ d:last ] dip d:another-name ; 'aka d:aka
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:d:alias (ss-)_make_alias_s2_of_s1
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[ d:lookup ] dip d:another-name ; 'alias d:aka
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}}
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'var-n 'var! (n-) alias
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'lt? 'n:<? alias
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'lteq? 'n:=<? alias
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:v:put (a-) fetch n:put ;
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:e:fetch (a-__-f)_fetch_as_float fetch e:to-f ; 'e:@ aka
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:e:store (a-__f-)_store_as_e f:to-e swap store ; 'e:! aka
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:e:call.vv (aaa-)_call_floating_point_function_variable-to-variable
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rot e:@ (aa_n) call (a_n) e:! ;
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:s:shout (s-) '!_%s s:format s:put ;
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{{
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'Depth var
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:message (-)
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'abort_with__trail__invoked s:shout nl
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'Do__reset__to_clear_stack. s:put nl ;
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:put-name (a-) fetch d:lookup-xt
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dup n:-zero? [ d:name s:put nl ] [ drop ] choose ;
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---reveal---
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:trail repeat pop put-name again ;
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:abort (-0) depth !Depth message trail ;
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:s:abort (s-0) 's:abort_:_ s:prepend s:put nl abort ;
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}}
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:assert (q-) call [ abort ] -if ;
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:assert.verbous (q-) call [ 'assert_:_fail s:abort ] -if ;
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:dump-stacks (-) dump-stack
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#0 f:depth lt? [ nl 'f_ s:put f:dump-stack ] if
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#0 f:adepth lt? [ nl 'fa_ s:put f:dump-astack ] if ; '. aka
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~~~
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Program.
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~~~
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{{
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(input_variables
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'F var (Function_to_minimize_(-_n-n)_float_to_float
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'L var (e:low
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'H var (e:high
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TRUE 'Tr var! (trace_flag
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.0.000001 f:to-e 'TOL const (=173=tolerance
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#30 'ITMAX const (max_#_of_iterations
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(working_variables
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'Ly var (e:low-value
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'Hy var
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'X1y var (y_at_upper_next_to_x
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'Y var (function_value_at_X
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'Itr var (iterations
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(output_variable
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'X var (candidate
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:trace (-) @Tr [ ( Itr v:put sp )
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( @L e:put sp ) ( @X e:put sp ) ( @H e:put sp ) nl ] if ;
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:y (-) X Y @F e:call.vv ;
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:x (-)_(L_H_->_X) @L @H + #2 / !X ;
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:x1y (-) @X n:inc e:to-f @F call X1y e:! ;
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---reveal---
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:minimize (a-_nn-x)_(L_H_F_->_X)
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(minimize_floating_point_function_F_over_[L,H]
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(initialize ( !F f:to-e !H f:to-e !L )
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( L Ly @F e:call.vv ) ( H Hy @F e:call.vv ) x y ( #0 !Itr )
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[ trace x1y
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( @Y @X1y n:<? ) [ @X !H @Y !Hy ] [ @X !L @Y !Ly ] choose
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x y Itr v:inc [ @Itr ITMAX n:=<? ] assert.verbous
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TOL @Y @X1y - n:abs n:<? ] while X e:@ ;
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}}
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~~~
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```
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:f (-_x-y)_(x-2)^2 .2. f:- f:dup f:* ;
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.-3 .10 &f minimize . nl
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```
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```
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:f (-_x-y) f:abs ;
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.-5. .5 &f minimize . nl
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```
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Note how `x` is defined without conversios as
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:x (-)_(L_H_->_X) @L @H + 2 / !X ;
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rather than
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:x (-)_(L_H_->_X) ( L e:@ ) ( H e:@ ) f:+ .2. f:/ X e:! ;
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Doing
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```
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:f (-_x-y) ;
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.0 .5 &f minimize . nl
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```
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with the first definition gives
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# low mid high
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0 0.000000 1.249991 5.000009
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1 0.000000 0.312492 1.249991
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2 0.000000 0.078120 0.312492
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3 0.000000 0.019530 0.078120
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4 0.000000 0.004882 0.019530
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5 0.000000 0.001220 0.004882
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6 0.000000 0.000305 0.001220
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7 0.000000 0.000076 0.000305
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8 0.000000 0.000019 0.000076
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9 0.000000 0.000005 0.000019
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10 0.000000 0.000001 0.000005
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final x = 0.000000
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whereas with the second definition gives
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# low mid high
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0 0.000000 2.500004 5.000009
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1 0.000000 1.249991 2.500004
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2 0.000000 0.625001 1.249991
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3 0.000000 0.312503 0.625001
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4 0.000000 0.156254 0.312503
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5 0.000000 0.078126 0.156254
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6 0.000000 0.039062 0.078126
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7 0.000000 0.019530 0.039062
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8 0.000000 0.009765 0.019530
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9 0.000000 0.004883 0.009765
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10 0.000000 0.002441 0.004883
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11 0.000000 0.001221 0.002441
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12 0.000000 0.000611 0.001221
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13 0.000000 0.000305 0.000611
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14 0.000000 0.000153 0.000305
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15 0.000000 0.000076 0.000153
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16 0.000000 0.000038 0.000076
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17 0.000000 0.000019 0.000038
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18 0.000000 0.000009 0.000019
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final x = 0.000005
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