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/******************************************************************************
rtest_lambert_w.mac
Test for Lambert W function
******************************************************************************/
kill(all);
done;
(closeto(value,compare,tol):=
block(
[abse],
abse:abs(value-compare),if(abse<tol) then true else abse),
done);
done;
/* Exact values */
lambert_w(-%pi/2)$
%i*%pi/2;
lambert_w(-log(2)/2)$
-log(2);
lambert_w(-1/%e)$
-1;
lambert_w(0)$
0;
lambert_w(%e)$
1;
/* Numerical values. Results from Mathematica 6 */
/* This point -47/128 is slightly greater than the branch cut at x=-1/%e.
Known difficult region. */
closeto(lambert_w(-0.3671875),-0.93988639805243454196,5.0e-15);
true;
closeto(lambert_w(-0.25),-0.35740295618138890306,1.0e-16);
true;
closeto(lambert_w(0.0),0.0,1.0e-16);
true;
closeto(lambert_w(0.5),0.3517337112491958260,1.0e-16);
true;
closeto(lambert_w(2.0),0.85260550201372549134,2.0e-16);
true;
closeto(lambert_w(16.0),2.05319271746264858727,1.0e-16);
true;
closeto(lambert_w(128.0),3.57739529855165334288,1.0e-16);
true;
closeto(lambert_w(2048.0),5.85698416299949388816,1.0e-15);
true;
closeto(float(lambert_w(-%pi/2)),%i*%pi/2,1.0e-16);
true;
closeto(lambert_w(3.0+2*%i),
1.12605402258467335157 + 0.31512993765865848403*%i,1.0e-16);
true;
/* bfloat evaluation. Results from Mathematica 6 to 82 sf */
(oldfpprec:fpprec,fpprec:80,done);
done;
closeto(lambert_w(-0.3671875b0),
-0.9398863980524345419695464132107234809629689590715558376016258594821501408118726902b0,5.0b-79);
true;
closeto(lambert_w(-0.25b0),
-0.3574029561813889030688111040559047533165905550760120436276204485896714025961457963b0,5.0b-80);
true;
closeto(lambert_w(0.0b0),0.0b0,1.0e-80);
true;
closeto(lambert_w(0.5b0),
0.3517337112491958260249093009299510651714642155171118040466438461099606107203387109b0,1.0b-80);
true;
closeto(lambert_w(2.0b0),
0.8526055020137254913464724146953174668984533001514035087721073946525150656742630449b0,1.0b-80);
true;
closeto(lambert_w(16.0b0),
2.053192717462648587277573057065703715549879054683959196524461288893671477132320243b0,5.0b-80);
true;
closeto(lambert_w(128.0b0),
3.577395298551653342882928583858277466679549941663670990487212206804013161129017297b0,1.0b-80);
true;
closeto(lambert_w(2048.0b0),
5.856984162999493888162560618072936017287078565867036752738725003619707275074129900b0,5.0b-80);
true;
closeto(lambert_w(3.0b0+2.0b0*%i),
1.1260540225846733515720001139613054043549635214873248102633125218051215201783683422b0
+
0.3151299376586584840326916101687325819105911412081543422333577478256866198890713045b0
*%i ,1.0b-80);
true;
/* Check an argument > most-positive-double-float ~ 1.0e308 */
closeto(lambert_w(1.0b1000),
2294.846671683506869652792785993616789973426699478802684415757740164730983851156757b0,
1.0b-77);
true;
/* Arguments that are exact numbers are not evaluated numerically */
lambert_w(2);
lambert_w(2);
lambert_w(3+2*%i);
lambert_w(3+2*%i);
/* ... unless we set numer evflag*/
closeto(lambert_w(2),0.85260550201372549134,2.0e-16), numer;
true;
diff(lambert_w(x),x);
%e^-lambert_w(x)/(lambert_w(x)+1);
integrate(lambert_w(x),x);
x*(lambert_w(x)^2-lambert_w(x)+1)/lambert_w(x);
/* SF bug report 2468610 the integrator loops endlessly
errcatch() doesn't catch the error for clisp-2.46 on cygwin */
integrate(lambert_w(1/x),x);
'integrate(lambert_w(1/x),x);
taylor(lambert_w(x),x,0,6);
x-x^2+3*x^3/2-(8*x^4/3)+125*x^5/24-(54*x^6/5);
/******************************************************************************
Tests for Generalized Lambert W function
******************************************************************************/
closeto(generalized_lambert_w(0,-0.25),-0.35740295618138890306,1.0e-16);
true;
closeto(generalized_lambert_w(0,3.0+2*%i),
1.12605402258467335157 + 0.31512993765865848403*%i,1.0e-16);
true;
closeto(generalized_lambert_w(0,-0.25),-0.35740295618138890306,1.0e-16);
true;
/* This point -753/2048 is slightly greater than the branch cut at x=-1/%e.
Known difficult region. */
closeto(generalized_lambert_w(-1,-753.0/2048),-1.033649565301979,5.0e-15);
true;
closeto(generalized_lambert_w(-1,-754.0/2048),
-0.9994844032397146 - 0.0393272270347577*%i,5.32e-15);
true;
closeto(generalized_lambert_w(1,-753.0/2048),
-3.089416043057986+7.461420464927938*%i,5.0e-15);
true;
closeto(generalized_lambert_w(1,-754.0/2048),
-3.088042730569348+7.461585404728464*%i,5.0e-15);
true;
/* Reduced precision when very near branch point z = -1/%e */
closeto(generalized_lambert_w(-1,z:float(-1/%e)-1.0e-12),
-0.999999999998187812 - 2.3316439815951875e-6*%i,1e-9);
true;
/* The branch test was wrong for this point due to roundoff */
closeto(generalized_lambert_w(-1,-1/float(%e)),-1.0,1e-15);
true;
/* bigfloat evaluation near the branch point failed to obtain a starting guess
There is significant (~ 14 decimal digits) loss of precision due to ill-conditioning
The expected answer below checked with 200 and 300 digit bfloats */
closeto(generalized_lambert_w(-1,bfloat(-1/%e)+1b-30),
- 1.000000000000002331643981597126015551421701533827747686376229738168073845444881159b0,
1.0b-64);
true;
closeto(generalized_lambert_w(1,-754.0b0/2048),
-3.088042730569348561456014556706168490039090862315553536854637271843252771228714625b0
+7.461585404728464111653134458076750679263104524925586270112477441975032123188737473b0*%i,
1.0b-77);
true;
/* Check an argument > most-positive-double-float ~ 1.0e308 */
closeto(generalized_lambert_w(0,1.0b1000),
2294.846671683506869652792785993616789973426699478802684415757740164730983851156757b0,
1.0b-77);
true;
closeto(generalized_lambert_w(0,1.0b1000*%i),
2.294846671449550599071991520727747213463470638397366444770394873636319358115746758b3
+1.570112136469240112764525899485121673623387222840770033671975542758566442609326195b0*%i
,1.0b-77);
true;
/* Arguments that are not evaluated numerically */
generalized_lambert_w(3,1/2);
generalized_lambert_w(3,1/2);
generalized_lambert_w(1/2,3.0);
generalized_lambert_w(1/2,3.0);
generalized_lambert_w(0.5,3.0);
generalized_lambert_w(0.5,3.0);
generalized_lambert_w(1.0,3.0);
generalized_lambert_w(1.0,3.0);
generalized_lambert_w(%pi,3.0);
generalized_lambert_w(%pi,3.0);
/* ... unless we set numer evflag */
closeto(generalized_lambert_w(0,2),0.85260550201372549134,2.0e-16), numer;
true;
diff(generalized_lambert_w(n,x),x);
%e^-generalized_lambert_w(n,x)/(generalized_lambert_w(n,x)+1);
integrate(generalized_lambert_w(n,x),x);
(generalized_lambert_w(n,x)^2-generalized_lambert_w(n,x)+1)*x
/generalized_lambert_w(n,x);
(fpprec:oldfpprec, kill(oldfpprec), done);
done;
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