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*$m)} } ;# RS
Binomial coefficient: Perhaps the best (what criteria?) should move to the Binomial page and just a pointer to the page should be here? (This got too long; I'm keeping the best algorithm here, moving the previous discussion to Binomial Coefficients. This solution is called binom3 in that page.)
proc binom {m n} { set n [expr {(($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 } ;#RS
Linear 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 $r } ;#RS
Sign of a number:
proc sgn {a} {expr {$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 again
Actually,
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 $L]/2}] } ;# DKF
Logarithm to any base:
proc log {base x} {expr {log($x)/log($base)}} ;# RS
A faster logarithm to base two:
proc ld x "expr {log(\$x)/[expr log(2)]}"
This is an example of a "live" proc body - the divisor is computed only once, at definition time. With a single backslash escape needed, it's worth the fun ;-) (RS)
Epsilon: Comparing two floats x,y for equality is most safely done by testing abs($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 {$eps+1e-22}] } % eps 1 5.55112000002e-017 ;# on both my Win2K/P3 and Sun/Solaris % eps 0.1 6.93889999999e-018 % eps 0.01 8.674e-019 % eps 0.001 1.085e-019
CritLib (see the Critcl page) now includes an adapted version of Donal K. Fellows' extension which lets you write numerical functions for "expr" in Tcl. See the "mathf" readme [L1 ] - JCW
AM On the c.l.t. the other day [is this as of May 2003?], Martin Russell asked about how to define new math functions. If you want to do it without the help of DKF's extension [??] and CrtLib [ critcl's critlib?], then here is a receipe provided by Pat Thoyts:
"Something along these lines.
static Tcl_MathProc ArbLogProc; static int ArbLogProc(clientData, interp, args, resultPtr) ClientData clientData; Tcl_Interp *interp; /* current interpreter */ Tcl_Value *args; /* input arguments */ Tcl_Value *resultPtr; /* where to store the result */ { double b, n, d; b = args[0].doubleValue; n = args[1].doubleValue; /* do your maths and assign d to the result */ d = 1.0; resultPtr->type = TCL_DOUBLE; resultPtr->doubleValue = d; return TCL_OK; }
in your package initialisation...
Tcl_ValueType argblogArgs[2] = { TCL_DOUBLE, TCL_DOUBLE }; Tcl_CreateMathFunc(interp, "arblog", 2, arblogArgs, ArgLogProc, (ClientData)NULL);
"
Fibonacci numbers: tcllib::math has an iterative version, but here's the "closed form" if anyone cares:
proc fib n { expr {round(1/sqrt(5)*(pow((1+sqrt(5))/2,$n) - (pow((1-sqrt(5))/2,$n))))} } ;# RS
Lars H: Actually, you don't need to compute the second term, since it always contributes < 1/2 for non-negative n. You can simply do
proc fib2 n { expr {round(1/sqrt(5)*pow((1+sqrt(5))/2,$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