tests/rtest4.mac

   1 /*************** -*- Mode: MACSYMA; Package: MAXIMA -*-  ******************/
   2 /***************************************************************************
   3 ***                                                                    *****
   4 ***     Copyright (c) 1984 by William Schelter,University of Texas     *****
   5 ***     All rights reserved                                            *****
   6 ***************************************************************************/
   7
   8
   9 kill(functions,arrays,values);
  10 done$
  11 ev(sin(x),exponentialize);
  12 -%i*(%e^(%i*x)-%e^-(%i*x))/2$
  13 taylor(sin(x)/x,x,0,4);
  14 ''(taylor(1-x^2/6+x^4/120,x,0,4))$
  15 ev(cos(x)^2-sin(x)^2,sin(x)^2 = 1-cos(x)^2);
  16 2*cos(x)^2-1$
  17 (sqrt(-4)+sqrt(2.25))^2;
  18 (2*%i+1.5)^2$
  19 expand(%);
  20 6.0*%i-1.75$
  21 expand(sqrt(2*%i));
  22 (-1)^(1/4)*sqrt(2)$
  23 kill(all);
  24 done$
  25 sin(x)+cos(y)+(w+1)^2+'diff(sin(w),w);
  26 cos(y)+sin(x)+'diff(sin(w),w,1)+(w+1)^2$
  27 ev(%,sin,expand,diff,x = 2,y = 1);
  28 cos(w)+w^2+2*w+cos(1)+sin(2)+1;
  29 ev(x+y,x:a+y,y:2);
  30 y+a+2$
  31 'diff(y^2+x*y+x^2,x,2,y,1);
  32 'diff(y^2+x*y+x^2,x,2,y,1)$
  33 ev(%,diff);
  34 0$
  35 exp:2*x-3*y = 3;
  36 2*x-3*y = 3$
  37 -3*x+2*y = -4;
  38 2*y-3*x = -4$
  39 solve([exp,%]);
  40 [[y = -1/5,x = 6/5]]$
  41 ev(exp,%);
  42 3 = 3$
  43 x+1/x > gamma(1/2);
  44 x+1/x > sqrt(%pi)$
  45 ev(%,numer,x = 1/2);
  46 2.5 > 1.772453850905516;
  47 /* tops 20 :2.5 > 1.77245384$ */
  48 ev(%,pred);
  49 true$
  50 zeroequiv(sin(2*x)-2*sin(x)*cos(x),x);
  51 true$
  52 zeroequiv(%e^x+x,x);
  53 false$
  54 zeroequiv(log(a*b)-log(a)-log(b),a);
  55 dontknow$
  56 (1/(x+y)^4-3/(y+z)^3)^2;
  57 (1/(y+x)^4-3/(z+y)^3)^2$
  58 expand(%,2,0);
  59 -6/((y+x)^4*(z+y)^3)+9/(z+y)^6+1/(y+x)^8$
  60 expand(a . (b+c . (d+e)+f));
  61 a . f+a . c . e+a . c . d+a . b$
  62 expand((x+1)^3);
  63 x^3+3*x^2+3*x+1$
  64 exp:(x+1)^7;
  65 (x+1)^7$
  66 expand(%);
  67 x^7+7*x^6+21*x^5+35*x^4+35*x^3+21*x^2+7*x+1$
  68 expand(exp,7,7);
  69 x^7+7*x^6+21*x^5+35*x^4+35*x^3+21*x^2+7*x+1$
  70 ev(a*(b+c)+a*(b+c)^2,expop:1);
  71 a*(c+b)^2+a*c+a*b$
  72 ratexpand((2*x-3*y)^3);
  73 -27*y^3+54*x*y^2-36*x^2*y+8*x^3$
  74 exp:(x-1)/(x+1)^2+1/(x-1);
  75 (x-1)/(x+1)^2+1/(x-1)$
  76 expand(%);
  77 x/(x^2+2*x+1)-1/(x^2+2*x+1)+1/(x-1)$
  78 ratexpand(exp);
  79 2*x^2/(x^3+x^2-x-1)+2/(x^3+x^2-x-1)$
  80 sin(x/(x^2+x)) = %e^((log(x)+1)^2-log(x)^2);
  81 sin(x/(x^2+x)) = %e^((log(x)+1)^2-log(x)^2)$
  82 ratsimp(%);
  83 sin(1/(x+1)) = %e*x^2$
  84 b*(a/b-x)+b*x+a;
  85 b*x+b*(a/b-x)+a$
  86 ratsimp(%);
  87 2*a$
  88 ((x-1)^(3/2)-(x+1)*sqrt(x-1))/sqrt(x-1)/sqrt(x+1);
  89 ((x-1)^(3/2)-sqrt(x-1)*(x+1))/(sqrt(x-1)*sqrt(x+1))$
  90 ratsimp(%);
  91 -2/sqrt(x+1)$
  92 ev(x^(a+1/a),ratsimpexpons);
  93 x^((a^2+1)/a)$
  94 (log(x^2+x)-log(x))^a/log(x+1)^(a/2);
  95 (log(x^2+x)-log(x))^a/log(x+1)^(a/2)$
  96 radcan(%);
  97 log(x+1)^(a/2)$
  98 log(a^(2*x)+2*a^x+1)/log(a^x+1);
  99 log(a^(2*x)+2*a^x+1)/log(a^x+1)$
 100 radcan(%);
 101 2$
 102 (%e^x-1)/(%e^(x/2)+1);
 103 (%e^x-1)/(%e^(x/2)+1)$
 104 radcan(%);
 105 %e^(x/2)-1$
 106 kill(all);
 107 done$
 108 combine(a/x+b/x+a/y+b/y);
 109 (b+a)/y+(b+a)/x$
 110 x/(x-y)^2-1/(x-y)-f(x)/(x-y)^3;
 111 -1/(x-y)+x/(x-y)^2-f(x)/(x-y)^3$
 112 multthru((x-y)^3,%);
 113 -(x-y)^2+x*(x-y)-f(x)$
 114 ratexpand(%);
 115 -y^2+x*y-f(x)$
 116 ((a+b)^10*s^2+2*a*b*s+(a*b)^2)/(a*b*s^2);
 117 ((b+a)^10*s^2+2*a*b*s+a^2*b^2)/(a*b*s^2)$
 118 multthru(%);
 119 2/s+a*b/s^2+(b+a)^10/(a*b)$
 120 multthru(a . (b+c . (d+e)+f));
 121 a . f+a . c . (e+d)+a . b$
 122 ((x+2)^20-2*y)/(x+y)^20+(x+y)^-19-x/(x+y)^20;
 123 1/(y+x)^19+((x+2)^20-2*y)/(y+x)^20-x/(y+x)^20$
 124 xthru(%);
 125 ((x+2)^20-y)/(y+x)^20$
 126 2/(x+2)-2/(x+1)+1/(x+1)^2;
 127 2/(x+2)-2/(x+1)+1/(x+1)^2$
 128 ratsimp(%);
 129 -x/(x^3+4*x^2+5*x+2)$
 130 partfrac(%,x);
 131 2/(x+2)-2/(x+1)+1/(x+1)^2$
 132 sort(args(factor(2^63-1)));
 133 ''(sort([7^2,73,127,337,92737,649657]))$
 134 factor(z^2*(x+2*y)-4*x-8*y);
 135 (2*y+x)*(z-2)*(z+2)$
 136 x^2*y^2+2*x*y^2+y^2-x^2-2*x-1;
 137 x^2*y^2+2*x*y^2+y^2-x^2-2*x-1$
 138 block([dontfactor:[x]],factor(%/36/(y^2+2*y+1)));
 139 (x^2+2*x+1)*(y-1)/(36*(y+1))$
 140 factor(%e^(3*x)+1);
 141 (%e^x+1)*(%e^(2*x)-%e^x+1)$
 142 factor(x^4+1,a^2-2);
 143 (x^2-a*x+1)*(x^2+a*x+1)$
 144 factor(x^3+x^2*y^2-x*z^2-y^2*z^2);
 145 -((y^2+x)*(z-x)*(z+x))$
 146 (x+2)/(x+3)/(x+b)/(x+c)^2;
 147 (x+2)/((x+3)*(x+b)*(x+c)^2)$
 148 ratsimp(%);
 149 (x+2)/(x^4+(2*c+b+3)*x^3+(c^2+(2*b+6)*c+3*b)*x^2+((b+3)*c^2+6*b*c)*x+3*b*c^2)$
 150 partfrac(%,x);
 151 -((c^2-4*c-b+6)/((c^4+(-2*b-6)*c^3+(b^2+12*b+9)*c^2+(-6*b^2-18*b)*c+9*b^2)
 152                *(x+c)))
 153  -((c-2)/((c^2+(-b-3)*c+3*b)*(x+c)^2))
 154  +(b-2)/(((b-3)*c^2+(6*b-2*b^2)*c+b^3-3*b^2)*(x+b))
 155  -(1/(((b-3)*c^2+(18-6*b)*c+9*b-27)*(x+3)))$
 156 map('factor,%);
 157 -((c^2-4*c-b+6)/((c-3)^2*(c-b)^2*(x+c)))-((c-2)/((c-3)*(c-b)*(x+c)^2))
 158                                       +(b-2)/((b-3)*(c-b)^2*(x+b))
 159                                       -(1/((b-3)*(c-3)^2*(x+3)))$
 160 exp:ratsimp((x^5-1)/(x-1));
 161 x^4+x^3+x^2+x+1$
 162 subst(a,x,%);
 163 a^4+a^3+a^2+a+1$
 164 factor(exp,%);
 165 (x-a)*(x-a^2)*(x-a^3)*(x+a^3+a^2+a+1)$
 166 factor(x^12+1);
 167 (x^4+1)*(x^8-x^4+1)$
 168 factor(x^99+1);
 169 (x+1)*(x^2-x+1)*(x^6-x^3+1)*(x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1)
 170      *(x^20+x^19-x^17-x^16+x^14+x^13-x^11-x^10-x^9+x^7+x^6-x^4-x^3+x+1)
 171      *(x^60+x^57-x^51-x^48+x^42+x^39-x^33-x^30-x^27+x^21+x^18-x^12-x^9+x^3+1)$
 172 ev((x+1)*((u+v)^2+a*(w+z)^2),expand);
 173 a*x*z^2+a*z^2+2*a*w*x*z+2*a*w*z+a*w^2*x+v^2*x+2*u*v*x+u^2*x+a*w^2+v^2+2*u*v
 174        +u^2$
 175 factorsum(%);
 176 (x+1)*(a*(z+w)^2+(v+u)^2)$
 177 sqfr(4*x^4+4*x^3-3*x^2-4*x-1);
 178 (2*x+1)^2*(x^2-1)$
 179 gfactor(x^4-1);
 180 (x-1)*(x+1)*(x-%i)*(x+%i)$
 181 factor(y^20-7*y^19+11*y^18-8*y^17+59*y^16-461*y^15+1694*y^14+198*y^13-3541*
 182  y^12+2818*y^11-33327*y^10-11272*y^9-56656*y^8-12672*y^7+433664*y^6+472064*y^5
 183  +241664*y^4+131072*y^3+720896*y^2+1835008*y+1048576);
 184 (y^8-y^7-11*y^6-19*y^5+101*y^4+76*y^3-176*y^2+64*y+256)
 185       *(y^12-6*y^11+16*y^10-39*y^9-19*y^8-75*y^7-315*y^6+300*y^5-304*y^4
 186             +2496*y^3+4096*y^2+6144*y+4096)$
 187
 188 /* sf bug 1650226 */
 189 factor(156*x^7+4808*x^6-182041*x^5-1266489*x^4+43104271*x^3+29839285*x^2-2542327662*x+7826952672);
 190 (x-16)*(x+14)*(x+49)*(2*x-11)*(2*x+21)*(3*x-49)*(13*x-63)$
 191
 192 /* sf bug 681191 */
 193 gfactor(x^4+y^4);
 194 (y^2-%i*x^2)*(y^2+%i*x^2)$
 195
 196 /* sf bug 696804 */
 197 factor(expand((35*x*y^2+19*x^2+25) * (35*x*y^2+25*x^2+19)));
 198 (35*x*y^2+19*x^2+25) * (35*x*y^2+25*x^2+19)$
 199
 200 /* bug reported on mailing list circa 2007-11-27; bignum in exponent */
 201
 202 is (errcatch (rat (x^(2^128))) = []);
 203 false;
 204
 205 /* [ 1943756 ] ezgcd() never returns. */
 206 ezgcd(n^2+(-2*d+c+b+2)*n+d^2+(-c-b-2)*d+c+b+1, -n^2+(2*d-c-b-2)*n-d^2+(c+b+2)*d+(-b-1)*c-b-1);
 207 [1,n^2+(-2*d+c+b+2)*n+d^2+(-c-b-2)*d+c+b+1,
 208        -n^2+(2*d-c-b-2)*n-d^2+(c+b+2)*d+(-b-1)*c-b-1];
 209
 210 /* mailing list 2015-12-10: "problem with poly_pseudo_divide" */
 211
 212 (kill (p, q, a, b, c),
 213  p : 50688*y[4]^2*y[5]^10+198144*y[4]^4*y[5]^8+292680*y[4]^6*y[5]^6
 214          +193464*y[4]^8*y[5]^4+48240*y[4]^10*y[5]^2,
 215  q : (-768*y[5]^7)-2016*y[4]^2*y[5]^5-1788*y[4]^4*y[5]^3
 216      -540*y[4]^6*y[5],
 217  [a, b, c] : ezgcd(p, q));
 218 [12*y[5]^3+12*y[4]^2*y[5],
 219  4224*y[4]^2*y[5]^7+12288*y[4]^4*y[5]^5+12102*y[4]^6*y[5]^3
 220  +4020*y[4]^8*y[5],
 221  (-64*y[5]^4)-104*y[4]^2*y[5]^2-45*y[4]^4];
 222
 223 is (equal (p, a*b));
 224 true;
 225
 226 is (equal (q, a*c));
 227 true;
 228
 229 /* bug reported in mailing list thread "when might resimplify be needed?"
 230  * on 2019-03-16
 231  */
 232 3*(2/3^(3/2))^(2/3);
 233 2^(2/3);
 234
 235 /* [ 1980715 ] wrong radcan-simplification */
 236 radcan(3^(1/6)*9^(1/3));
 237 3^(5/6);
 238
 239 radcan(81^(1/5)*9^(1/3));
 240 3*3^(7/15);
 241
 242 /* SF #2905 Assigning variable twice yields different results */
 243
 244 block ([var1, var5, var6, diff_expr3],
 245   var1 : D^5*Some,
 246   var5 : sqrt(var1)*z[D]*r^D,
 247   var6 : -sqrt(var1)*r^D/D,
 248   diff_expr3 : diff(var5,r),
 249   var6*diff_expr3,
 250   [var1, var5, var6, diff_expr3, %%]);
 251 [D^5*Some,
 252  r^D*sqrt(Some*D^5)*z[D],
 253  -sqrt(Some*D^5)*r^D/D,
 254  D*r^(D - 1)*sqrt(Some*D^5)*z[D],
 255  -r^(2*D - 1)*Some*D^5*z[D]];
 256
 257 /* SF #4308: "gfactor throws Lisp error" */
 258
 259 gfactor (((%i*t-1)*z*(z^2-3*t^2-3))/(sqrt(2)*sqrt(3)*(t^4+4*%i*t^3-6*t^2-4*%i*t+1)));
 260 (%i*z*(z^2-3*t^2-3))/(sqrt(2)*sqrt(3)*(t+%i)^3)$