share/pdiff/rtest_pdiff.mac

   1 (load(pdiff),kill(all), 0);
   2 0$
   3
   4 pderivop(f);
   5 f$
   6
   7 pderivop(f,0);
   8 f$
   9
  10 pderivop(42,0)(x);
  11 42$
  12
  13 pderivop(42,1)(x);
  14 0$
  15
  16 op(pderivop(%pi));
  17 lambda$
  18
  19 pderivop(%pi)(a,b,2);
  20 %pi$
  21
  22 pderivop(12.1,1)(x);
  23 0$
  24
  25 pderivop('log,1)(x);
  26 1/x$
  27
  28 pderivop('abs,1)(x);
  29 x/abs(x)$
  30
  31 pderivop('acos,2)(x);
  32 ''(diff(acos(x),x,2))$
  33
  34 pderivop(bessel_i,0,1)(n,x);
  35 ''(diff(bessel_i(n,x),x))$
  36
  37 pderivop(bessel_j,0,1)(n,x);
  38 ''(diff(bessel_j(n,x),x))$
  39
  40 pderivop(bessel_k,0,1)(n,x);
  41 ''(diff(bessel_k(n,x),x))$
  42
  43 pderivop(bessel_y,0,1)(n,x);
  44 ''(diff(bessel_y(n,x),x))$
  45
  46 makelist(pderivop('sqrt,k)(x),k,0,5);
  47 ''(makelist(diff(sqrt(x),x,k),k,0,5))$
  48
  49 pderivop('erf,1)(x);
  50 ''(diff(erf(x),x))$
  51
  52 [pderivop("^",1,0)(a,b),pderivop("^",0,1)(a,b), pderivop("^",1,1)(a,b)];
  53 ''([diff(a^b,a), diff(a^b,b),diff(a^b,a,1,b,1)])$
  54
  55 [pderivop("^^",1,0)(a,b),pderivop("^^",0,1)(a,b), pderivop("^^",1,1)(a,b)];
  56 ''([diff(a^^b,a), diff(a^^b,b),diff(a^^b,a,1,b,1)])$
  57
  58 [pderivop(".",1,0)(a,b),pderivop(".",0,1)(a,b), pderivop(".",1,1)(a,b)];
  59 ''([diff(a.b,a), diff(a.b,b),diff(a.b,a,1,b,1)])$
  60
  61 is(op(pderivop(lambda([[x]], x),1)) = 'pderivop);
  62 true$
  63
  64 (gradef(ff(a,b),a+b,a-b), [pderivop(ff,1,0)(x,y), pderivop(ff,0,1)(x,y), pderivop(ff,1,1)(x,y)]);
  65 [y+x,x-y,1]$
  66
  67 errcatch(pderivop(ff,1));
  68 []$
  69
  70 (remove(ff,gradef),0);
  71 0$
  72
  73 (gradef(ff(a), ff(a-1)), [pderivop(ff)(x), pderivop(ff,1)(x), pderivop(ff,2)(x)]);
  74 [ff(x),ff(x-1),ff(x-2)]$
  75
  76 (remove(ff,gradef),0);
  77 0$
  78
  79 convert_to_diff(pderivop(f[5],1,1)(x,y));
  80 ''(diff(f[5](x,y),x,1,y,1))$
  81
  82 errcatch(pderivop(f,1,2)(x));
  83 []$
  84
  85 errcatch(pderivop(f,1,2)());
  86 []$
  87
  88 errcatch(pderivop(f,1,2)(x,x,x));
  89 []$
  90
  91 (pderivop(f,x), subst(x = 5,%%));
  92 pderivop(f,5)$
  93
  94 (pderivop(f,x,y), ratsubst(5,x,%%), ratsubst(7,y,%%));
  95 pderivop(f,5,7)$
  96
  97 (clear_rules(), tellsimpafter(pderivop(f,1)(a),1), tellsimpafter(pderivop(f,2)(a),1),0);
  98 0$
  99
 100 subst(x=a, diff(f(x),x,2) + diff(f(x),x));
 101 2$
 102
 103 (clear_rules(),0);
 104 0$
 105
 106 (commute_partial_derivatives : true,0);
 107 0$
 108
 109 pderivop(pderivop(f,1,0),0,1) - pderivop(f,1,1);
 110 0$
 111
 112 (commute_partial_derivatives : false,0);
 113 0$
 114
 115 is(pderivop(pderivop(f,1,0),0,1) = pderivop(f,1,1));
 116 false$
 117
 118 (reset(commute_partial_derivatives),0);
 119 0$
 120
 121 pderivop(lambda([s], s^2),1)(x);
 122 2 *x$
 123
 124 pderivop(lambda([s], 42),1)(x);
 125 0$
 126
 127 pderivop(lambda([a,b], f(a,b)))(x,y);
 128 f(x,y)$
 129
 130 pderivop(lambda([a,b], f(a,b)),1,0)(x,y) - diff(f(x,y),x);
 131 0$
 132
 133 pderivop(lambda([a,b], f(a,b)),1,1)(x,y) - diff(f(x,y),x,1,y,1);
 134 0$
 135
 136 (diff(f(x,y),x,1,y,1), subst([x=1,y=2],%%), %% - at(diff(f(x,y),x,1,y,1),[x=1,y=2]));
 137 0$
 138
 139 (f(x) := x^3, g : pderivop(f,1), [op(g), g(6)]);
 140 [lambda,108]$
 141
 142 (remfunction(f), remvalue(g),0);
 143 0$
 144
 145 (commute_partial_derivatives : true, pderivop(pderivop(f,0,1),1,0));
 146 pderivop(f,1,1)$
 147
 148 /* examples from pdiff-doc.pdf */
 149
 150 ratdisrep(taylor(f(x+x^2),x,0,2));
 151 ''(f(0) + at(diff(f(x + x^2),x),x = 0) * x + at(diff(f(x + x^2),x,2),x = 0) * x^2/2)$
 152
 153 (f(x - c*t) + f(x + c * t), expand(diff(%%,t,2) - c^2 * diff(%%,x,2)));
 154 0$
 155
 156 (diff(f(x,y),x,1,y,1), subst(p(s),y, %%),
 157   convert_to_diff(%%), ev(%%, 'diff, 'at));
 158 ''(subst(p(s),y, diff(f(x,y),x,1,y,1)))$
 159
 160 (tellsimpafter(pderivop(f,1)(1),1),
 161   tellsimpafter(pderivop(f,2)(1),2),
 162   diff(f(x),x,2) + diff(f(x),x),
 163   subst(1,x,%%));
 164 3$
 165
 166 (clear_rules(),0);
 167 0$
 168
 169 (tellsimpafter(pderivop(f,1,0)(0,0),a),
 170   tellsimpafter(pderivop(f,0,1)(0,0),b),
 171   sublis([x = 0, y = 0], diff(f(x,y),x) + diff(f(x,y),y)));
 172 a+b$
 173
 174 (clear_rules(),0);
 175 0$
 176
 177 (commute_partial_derivatives : true, pderivop(pderivop(f,3,4),1,2));
 178 pderivop(f,4,6)$
 179
 180 (pderivop(f,1), apply(%%,[z]));
 181 ''(diff(f(z),z))$
 182
 183 (f(x) := x^2, pderivop(f,2), apply(%%,[10]));
 184 2$
 185
 186 (remfunction(f),0);
 187 0$
 188
 189 (pderivop(lambda([x],x^2),1), op(%%));
 190 lambda$
 191
 192 (de : 4*x^2 * 'diff(y,x,2) + 4*x * 'diff(y,x,1) + (x-1)*y = 0,
 193  de : subst(g(x^n),y,de),
 194  de : ev(de, diff),
 195  de : radcan(subst(x^(1/n),x, de)),
 196  de : block ([ctxt:newcontext(), foo], assume(x >= 0), foo:subst(1/2,n, de), killcontext(ctxt), foo),
 197  convert_to_diff(de)),logexpand;
 198 x^2 * 'diff(g(x),x,2)+x*'diff(g(x),x,1)+(x^2-1)*g(x)=0$
 199
 200 (remvalue(de),0);
 201 0$
 202
 203 /* end of examples from pdiff-doc.pdf */
 204
 205 convert_to_diff([-1,42, 123.0, -7.8b0, x, cos(q), a+b, [], a # b, f(x)]);
 206 [-1,42, 123.0, -7.8b0, x, cos(q), a+b, [], a # b, f(x)]$
 207
 208 convert_to_diff(f[5](x));
 209 f[5](x)$
 210
 211 convert_to_diff(f[g[5]](x));
 212 f[g[5]](x)$
 213
 214 convert_to_diff(pderviop(f,4,5));
 215 pderviop(f,4,5)$
 216
 217 convert_to_diff(diff(f(x),x));
 218 'diff(f(x),x,1)$
 219
 220 convert_to_diff(diff(f(x),x,5));
 221 'diff(f(x),x,5)$
 222
 223 convert_to_diff(42 + diff(f(x),x,5) + diff(g(x),x));
 224 42 + 'diff(f(x),x,5) + 'diff(g(x),x)$
 225
 226 (diff(f(x),x), subst(x=10,%%), convert_to_diff(%%), ev(%%, f(x) := x^7, 'diff, 'at));
 227 7000000$
 228
 229 (diff(f(x),x), subst(x=a,%%), convert_to_diff(%%), ev(%%, f(x) := x^7, 'diff, 'at));
 230 7 * a^6$
 231
 232 (diff(f(x,y),x,1,y,1), subst([x=a,y=b], %%), convert_to_diff(%%), ev(%%, f(x,y) := x^2 * y^2, 'diff, 'at));
 233 4*a*b$
 234
 235 (clear_rules(),0);
 236 0$
 237
 238 (tellsimpafter(f(0),6),tellsimpafter(''(subst(x=0, diff(f(x),x))), 28), taylor(f(x),x,0,1));
 239 6+28*x$
 240
 241 (clear_rules(),0);
 242 0$
 243
 244 /*chain rule tests */
 245 is(equal(diff(f(g(x)),x), diff(g(x),x) * pderivop(f,1)(g(x))));
 246 true$
 247
 248 is(equal(diff(f(g(x), h(x)),x), diff(g(x),x) * pderivop(f,1,0)(g(x), h(x)) +
 249     diff(h(x),x) * pderivop(f,0,1)(g(x), h(x))));
 250 true$
 251
 252 (diff(f(x^3),x), convert_to_diff(%%), ev(%%, f(x) := x^5, diff, at));
 253 ''(diff((x^5)^3,x))$
 254
 255 (diff(f(x^2,x*y),x), subst(x = 1, %%), ev(%%, f(x,y) := x^2 * y^2, diff, at));
 256 ''(subst(x=1, diff((x^2)^2 * (x*y)^2,x)))$
 257
 258 /* subscripted functions */
 259
 260 pderivop(f[2]);
 261 f[2]$
 262
 263 pderivop(f[2],0);
 264 f[2]$
 265
 266 (pderivop(f[2],1), %%(x), convert_to_diff(%%));
 267 'diff(f[2](x),x,1)$
 268
 269 (pderivop(f[x],1), %%(x), convert_to_diff(%%));
 270 'diff(f[x](x),x,1)$
 271
 272 (pderivop(f[2],1,1), %%(x,y), convert_to_diff(%%));
 273 'diff(f[2](x,y),x,1,y,1)$
 274
 275 (pderivop(f,1,1), %%(x,y), subst(y=10,%%), convert_to_diff(%%), ev(%%, f(x,y) := x^2 * y^2, diff, at));
 276 ''(subst(y=10, diff(x^2*y^2,x,1,y,1)))$
 277
 278 [pderivop(%pi)(), pderivop(%pi)(a), pderivop(%pi)(a,b,c)];
 279 [%pi, %pi, %pi]$
 280
 281 (diff(f(x^2),x), ev(%%, f = sin));
 282 2*x*cos(x^2)$
 283
 284 /* check that the default for use_pdiff is false */
 285 (reset(use_pdiff),0);
 286 0$
 287
 288 use_pdiff;
 289 false$
 290
 291
 292
 293
 294
 295
 296
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 298
 299
 300