I see a factorial function on 3-4 different pages -some not even about math. And yet none in the tcllib math library. Perhaps one should be submitted. How to determine best?
KBK: There is indeed a factorial in ::tcllib::math. It's in some sense 'better' than any of the ones I've seen here on the Wiki:
RS I like this one, compact but recursive:
proc fac n {expr {$n<2? 1: $n*[fac [expr {$n-1}]]}}However, this one runs 1/3 faster:
proc fac2 n {expr $n<2? 1: [join [iota 1 $n] *]+0}given an index generator iota, e.g. iota 1 5 => {1 2 3 4 5}
proc iota {base n} {
set res {}
for {set i $base} {$i<$n+$base} {incr i} {lappend res $i}
set res
}However, factorials computed in terms of expr are correct only until 12!; above that you get "false positives", negatives, or zeroes.. Of course one could use doubles, which seem to be exact up to 18! (at the maximum tcl_precision 17). But the fastest fac is still tabulated:
proc fac3 n {
lindex {
1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600
479001600.0 87178291200.0 1307674368000.0 20922789888000.0
355687428096000.0 6402373705728000.0
} $n
} ;#-)Perhaps this function should move to the Stats page mentioned above? Square mean and standard deviation:
proc mean2 list {
set sum 0
foreach i $list {set sum [expr {$sum+$i*$i}]}
expr {double($sum)/[llength $list]}
}
proc stddev list {
set m [mean $list] ;# see below for [mean]
expr {sqrt([mean2 $list]-$m*$m)}
} ;# RSBinomial coefficient: Perhaps the best (what criteria?) should move to the Binomial page and just a pointer to the page should be here? (This got too long; I'm keeping the best algorithm here, moving the previous discussion to Binomial Coefficients. This solution is called binom3 in that page.)
proc binom {m n} {
set n [expr {(($m-$n) > $n) ? $m-$n : $n}]
if {$n > $m} {return 0}
if {$n == $m} {return 1}
set res 1
set d 0
while {$n < $m} {
set res [expr {($res*[incr n])/[incr d]}]
}
set res
}Prime factors of an integer:
proc primefactors n {
# a number x is prime if [llength [primefactors $x]]==1
set res {}
set f 2
while {$f<=$n} {
while {$n%$f==0} {
set n [expr {$n/$f}]
lappend res $f
}
set f [expr {$f+2-($f==2)}]
}
set res
} ;#RSLinear regression and correlation coefficient:
proc reg,cor points {
# linear regression y=ax+b for {{x0 y0} {x1 y1}...}
# returns {a b r}, where r: correlation coefficient
foreach i {N Sx Sy Sxy Sx2 Sy2} {set $i 0.0}
foreach point $points {
foreach {x y} $point break
set Sx [expr {$Sx + $x}]
set Sy [expr {$Sy + $y}]
set Sx2 [expr {$Sx2 + $x*$x}]
set Sy2 [expr {$Sy2 + $y*$y}]
set Sxy [expr {$Sxy + $x*$y}]
incr N
}
set t1 [expr {$N*$Sxy - $Sx*$Sy}]
set t2 [expr {$N*$Sx2 - $Sx*$Sx}]
set a [expr {double($t1)/$t2}]
set b [expr {double($Sy-$a*$Sx)/$N}]
set r [expr {$t1/(sqrt($t2)*sqrt($N*$Sy2-$Sy*$Sy))}]
list $a $b $r
} ;#RSSign of a number:
proc sgn {a} {expr {$a>0 ? 1 : $a<0 ? -1 : 0}} ;# rmax
proc sgn x {expr {$x<0? -1: $x>0}} ;# RS
proc sgn x {expr {($x>0)+($x>>31)}} ;# jcw (32-bit arch)
proc sgn x {expr {($x>0)-($x<0)}} ;# rmax againActually,
string compare $a 0
seems to give the correct result for all integer values and floating point values not equal to 0.
0.0 (and 0.00 etc) [string compare 0.0 0] returns 1, however.
Traditional degrees: clock format can be put to un-timely uses. As degrees especially in geography are also subdivided in minutes and seconds, how's this one-liner for formatting decimal degrees:
proc dec2deg x {concat [expr int($x)] [clock format [expr round($x*3600)] -format "%M' %S\""]}An additional -gmt 1 switch is needed if you happen to live in a non-integer timezone. (RS)
Cross-sum of non-negative integers:
proc crosssum {x} {expr [join [split $x ""] +]}Note that this expression may not be braced. (RS)
Should this function move to the Stats page? Means of a number list: (arithmetic, geometric, quadratic, harmonic)
proc mean L {expr ([join $L +])/[llength $L].}
proc gmean L {expr pow([join $L *],1./[llength $L])}
proc qmean L {expr sqrt((pow([join $L ,2)+pow(],2))/[llength $L])}
proc hmean L {expr [llength $L]/(1./[join $L +1./])}where qmean is the best braintwister... For a list of {1 2} the string
sqrt((pow( 1 ,2)+pow( 2 ,2))/ 2)
(blanks added for clarity) is built up and fed to expr, where it makes a perfectly well-formed expression if not braced. (RS)
proc median L {lindex $L [expr {[llength $L]/2}] } ;# DKFThat's not a median, that's the middle of a list. THIS is a median:
proc median {l} {
if {[set len [llength $l]] % 2} then {
return [lindex [lsort -real $l] [expr ($len - 1) / 2]]
} else {
return [expr ([lindex [set sl [lsort -real $l]] [expr ($len / 2) - 1]] + \
[lindex $sl [expr $len / 2]]) / 2.0]
}
}Logarithm to any base:
proc log {base x} {expr {log($x)/log($base)}} ;# RSA faster logarithm to base two:
proc ld x "expr {log(\$x)/[expr log(2)]}"This is an example of a "live" proc body - the divisor is computed only once, at definition time. With a single backslash escape needed, it's worth the fun ;-) (RS)
Epsilon: Comparing two floats x,y for equality is most safely done by testing abs($x-$y)<$eps, where eps is a sufficiently small number. You can find out which eps is good for your machine with the following code:
proc eps {{base 1}} {
set eps 1e-20
while {$base-$eps==$base} {
set eps [expr {$eps+1e-22}]
}
set eps [expr {$eps+1e-22}]
}
% eps 1
5.55112000002e-017 ;# on both my Win2K/P3 and Sun/Solaris
% eps 0.1
6.93889999999e-018
% eps 0.01
8.674e-019
% eps 0.001
1.085e-019CritLib (see the Critcl page) now includes an adapted version of Donal K. Fellows' extension which lets you write numerical functions for "expr" in Tcl. See the "mathf" readme [L1 ] - JCW
AM On the c.l.t. the other day [is this as of May 2003?], Martin Russell asked about how to define new math functions. If you want to do it without the help of DKF's extension [??] and CrtLib [ critcl's critlib?], then here is a receipe provided by Pat Thoyts:
"Something along these lines.
static Tcl_MathProc ArbLogProc;
static int
ArbLogProc(clientData, interp, args, resultPtr)
ClientData clientData;
Tcl_Interp *interp; /* current interpreter */
Tcl_Value *args; /* input arguments */
Tcl_Value *resultPtr; /* where to store the result */
{
double b, n, d;
b = args[0].doubleValue;
n = args[1].doubleValue;
/* do your maths and assign d to the result */
d = 1.0;
resultPtr->type = TCL_DOUBLE;
resultPtr->doubleValue = d;
return TCL_OK;
}in your package initialisation...
Tcl_ValueType argblogArgs[2] = { TCL_DOUBLE, TCL_DOUBLE };
Tcl_CreateMathFunc(interp, "arblog", 2, arblogArgs, ArgLogProc,
(ClientData)NULL);"
Fibonacci numbers: tcllib::math has an iterative version, but here's the "closed form" if anyone cares:
proc fib n {
expr {round(1/sqrt(5)*(pow((1+sqrt(5))/2,$n) - (pow((1-sqrt(5))/2,$n))))}
} ;# RSLars H: Actually, you don't need to compute the second term, since it always contributes < 1/2 for non-negative n. You can simply do
proc fib2 n {
expr {round(1/sqrt(5)*pow((1+sqrt(5))/2,$n))}
}For negative n it is instead the first term that can be ignored, but one rarely needs those Fibonacci numbers. BTW, I also changed an "int" to a "round" in RS's proc (if you're unlucky with the numerics, "int" can give you one less than the correct answer).
Mathematically oriented extensions - Arts and crafts of Tcl-Tk programming - Category Mathematics