b --- \ / f(n) = f(a) + ... + f(b) --- n=aHere's what I have so far (both

*f*and

*next*are "functional objects", which in Tcl just means "strings that happen to be the name of a proc":

proc sum {f a next b} { expr {$a>$b? 0 : [$f $a] + [sum $f [$next $a] $next $b]} } # --------------------------- small building blocks: proc cube x {expr {$x*$x*$x}} proc inc x {incr x} ;# argument x is value, instead of name proc lambda {argl body} { set name [info level 0] proc $name $argl $body set name }This handful of code allows us to reproduce the Scheme results from SICP. For more info, see there.

- sum the cubes of 1..10:

sum cube 1 inc 10 ==> 3025, or: sum cube 1 [lambda x {incr x}] 10

- sum the integers from 1 to 10:

proc identity x {set x} sum identity 1 inc 10 ==> 55; or: sum [lambda x {set x}] 1 [lambda x {incr x}] 10

- approximate
*Pi*one slow way:

proc pi-term x {expr {1.0 / ($x * ($x+2))}} proc pi-next x {expr {$x + 4}} expr {[sum pi-term 1 pi-next 1000] *8 } ==> 3.1395926555897828whose run limit could be increased from 1000 until 2756 before raising the "too many nested calls..." error ;-( and still gave a less precise approximation than the good old

*atan(1)*4*...

**integrate a function**f between limits a and b:

proc integral0 {f a b dx} { set ::globaldx $dx expr {[sum $f [expr {$a + $::globaldx / 2}] add-dx $b] * $dx} } proc add-dx x {expr {$x+$::globaldx}} % integral0 cube 0 1 .0016 ==> 0.2499996800000055Here however I had to start to compromise: instead of Scheme's lexical scoping, where dx is visible everywhere inside integral's body, including the add-dx function, I had to elevate dx to global status, which is ugly; and Tcl's recursion depth limit caught me before I could try SICP's dx value of 0.001 - the result is still close (but no cigar) to the correct result of 0.25. Oh wait, at least in this case we can emulate lexical scoping more closely, by embodying $dx into a "live proc body" of add-dx:

proc integral {f a b dx} { proc add-dx x "expr {\$x+$dx}" expr {[sum $f [expr {$a+$dx/2}] add-dx $b] * $dx} } % integral cube 0 1 .00146 ==> 0.25009974849771255

A cleaner way to implement this "closure" would be an added default argument, like they do in Python - the body can remain braced, but the argument list of

*add-dx*now "inherits" from outside:

proc integral {f a b dx} { proc add-dx "x {dx $dx}" {expr {$x+$dx}} expr {[sum $f [expr {$a+$dx/2}] add-dx $b] * $dx} }

Slightly off-topic, but as all building blocks are there, here's a stint on the

*derivative*of a function, using the default args method, inheriting one argument from

*deriv*, and one from global namespace:

proc deriv g { lambda [list x [list g $g] [list dx $::dx]] \ {expr {([$g [expr {$x+$dx}]] - [$g $x])/$dx}} } % set dx 0.00001 ;# well, in SICP they have it global too... % [deriv [lambda x {expr $x*$x*$x}]] 5 ==> 75.0001499966Anyway, I'm again surprised how many steps towards functional programming are possible with Tcl .. and more to come.