/* eval_when([translate,load,loadfile,batch,demo],
 matchdeclare(a,nonzeroandfreeof(x),[b,c],freeof(x)))$ */

define_variable(takegcd,true,boolean,"used in gcdivide to decide the
gcd choice");

rempart&&  rempart(exp,n):=
    ?append(rest(exp,  /* combine two parts of exp
	   first part is beginning to part to be removed
	   specify that the first l-1 parts are retained */
	(if listp(n) then n[1] else n)-1-length(exp)),
	block([t],  /* last part is from removed part to end */
	if atom(t:rest(exp,  /* last m-1 parts are retained */
	    if listp(n) then n[2] else n))
	then ?list(t) else ?cdr(t)))$

wronskian&&  wronskian(functlist,var):=block([end],
    end:length(functlist)-1,
    functlist:[functlist],
    thru end do functlist:endcons(map(lambda([x],diff(x,var)),
	last(functlist)),functlist),
    apply('matrix,functlist))$

tracematrix&&  tracematrix(m):=block([sum,len],sum:0,len:length(m),
for i:1 thru len do sum:sum+part(m,i,i),sum)$

rational&&  rational(z):=block([n,d,cd,ratfac],
    ratfac:false,
    n:ratdisrep(ratnumer(z)*(cd:conjugate(d:ratdenom(z)))),
    d:rat(n/ratdisrep(d*cd)),
    if ratp(z) then d else ratdisrep(d))$

logical&&  logand(x,y):=?boole(1,x,y)$
	logor(x,y):=?boole(7,x,y)$
	logxor(x,y):=?boole(6,x,y)$

nonzeroandfreeof&&  nonzeroandfreeof(x,e):=is(e#0 and freeof(x,e))$

/* linear&& matchdeclare(a,nonzeroandfreeof(x),[b,c],freeof(x))$
    defmatch(linearize,a*x+b,x)$
    defmatch(quadraticize,a*x^2+b*x+c,x)$
    linear(exp,x):=block([a,b],if linearize(exp,x)=false then exp else
	a*x+b)$
    quadratic(exp,x):=block([a,b,c],if quadraticize(exp,x)=false then
	exp else a*x^2+b*x+c)$ */

lcm&& lcm([list]):=block([listconstvars:false],if listofvars(list)=[] then
lcm1(list) else factor(lcm1(list)))$

/* Replaced by the following routine
lcm1(list):=if list=[] then 1 else block([rlist:rest(list),flist:first(list),
frlist,partswitch:true,inflag:true,piece], if rlist=[] then flist else
lcm1(cons(flist*(frlist:first(rlist))/gcd(flist,frlist),rest(rlist))))$
*/

/* New implementation of the function lcm
 * Do not use a recursive, but an iterative algorithm.
 * Check more carefully the case division by zero.
 */

lcm1(list):=
   block 
   (
    [flist, rlist, result, keepfloat:true, ratprint:false],
    
    result : flatten(list),
    if result = [] then result : [1],
    unless rest(result)=[] do
    (
     flist : first(result),
     rlist : first(rest(result)),
     if rlist=0 then
        result : [0]
     else
        result : cons( (flist*rlist / gcd(flist, rlist)),
                       (rest(rest(result))))
    ),
    first(result)
)$

gcdivide&&  gcdivide(poly1,poly2):=block([gcdlist],
		gcdlist:if takegcd then ezgcd(poly1,poly2)
			else [1,poly1,poly2],
		gcdlist[2]/gcdlist[3])$

series&&  arithmetic(a,d,n):=a+(n-1)*d$
          geometric(a,r,n):=a*r^(n-1)$
          harmonic(a,b,c,n):=a/(b+(n-1)*c)$
          arithsum(a,d,n):=n*(a+(n-1)*d/2)$
          geosum(a,r,n):=block([],
            if r=1
                then if n='inf
                        then return(limit(i*a,i,'inf))
                        else return(n*a),
            if r=-1
                then error("The series is not convergent"),
            if n='inf 
                then if abs(r) < 1 
                        then a/(1-r)
                        else limit(a*(1-r^i)/(1-r),i,'inf)
                else a*(1-r^n)/(1-r)   )$

gauss&&  gaussprob(x):=1/sqrt(2*%pi)*%e^(-x^2/2)$

/* See, for example, http://en.wikipedia.org/wiki/Gudermannian_function */
gd&&  gd(x):=2*atan(%e^x)-%pi/2$
	agd(x):=log(tan(%pi/4+x/2))$

trig&&  vers(x):=1-cos(x)$
	covers(x):=1-sin(x)$
	exsec(x):=sec(x)-1$
	hav(x):=(1-cos(x))/2$

combination&&  combination(n,r):=binomial(n,r)$
	permutation(n,r):=binomial(n,r)*r!$
