11:-module(plcomplex,[ 12 set_default_is/3, 13 op(700,xfx,iis), 14 iis/2, 15 op(700,xfx,c_equals), 16 c_equals/2, 17 complex_canonical/3, 18 is_canonical/3, 19 complex_exponential/3, 20 is_exponential/3, 21 complex_number/1 22 ]). 23:-op(700,xfx,iis). 24:-op(700,xfx,c_equals). 25 26:-dynamic a_default_is/3. 27a_default_is(is(A,B),A,B).
Example:
?- [user].
|: my_the_best_is(A,B):-
|: var(A),
|: nonvar(B),
|: A is B,
|: format('~w is ~w\n',[A,B]).
|: ^D% user://1 compiled 0.00 sec, 1 clauses
true.
?- set_default_is(my_the_best_is(A,B),A,B).
true.
?- X iis 3+i*5+15.
3 is 3
15 is 15
18 is 3+15
5 is 5+0
X = 18+5*i.
?- X iis 15**(3*i).
8.12415060330663 is log(15**3)
X = exp(i*8.12415060330663).
60set_default_is(F,A,B):- 61 retractall(a_default_is(_,_,_)), 62 asserta(a_default_is(F,A,B)). 63 64default_is(A,B):- 65 a_default_is(C,A,B), 66 call(C).
Examples:
?- X iis i*i. X = -1. ?- X iis 5*i*6*7. X = 210*i. ?- X iis 5*i+3. X = 3+5*i. ?- X iis phase(2+3*i). X = 0.982793723247329. ?- X iis abs(5+10*i). X = 11.180339887498949. ?- X iis conjugate(1+5*i). X = 1-5*i. ?- X iis imaginary(5+2*i). X = 2. ?- X iis imaginary(sqrt(2)*exp(45*pi/180*i)). X = 1.0. ?- X iis 6*i*5/(i*2*5). X = 3.
133% like normal (is)/2 134% --------------- 135 136(_ iis Y):- 137 \+ ground(Y), 138 throw(error(instantiation_error, context(complex:(iis)/2, _))). 139 140(X iis Y):- 141 catch(default_is(Z,Y),_,(!,fail)),!, 142 X = Z. 143 144(R+I*i iis X):- 145 complex_canonical(X,R,I),!. 146 147% Functions 148% ----------- 149(M iis abs(Z)):- 150 ZZ iis Z, 151 complex_exponential(ZZ,M,_),!. 152 153(F iis phase(Z)):- 154 ZZ iis Z, 155 complex_exponential(ZZ,_,F),!. 156 157(R iis real(Z)):- 158 ZZ iis Z, 159 complex_canonical(ZZ,R,_),!. 160 161(I iis imaginary(Z)):- 162 ZZ iis Z, 163 complex_canonical(ZZ,_,I),!. 164 165(R-I*i iis conjugate(Z)):- 166 ZZ iis Z, 167 complex_canonical(ZZ,R,I),!. 168 169(X iis reciprocal(Z)):- 170 ZZ iis Z, 171 complex_canonical(ZZ,R,I), 172 default_is(S,R*R+I*I), 173 X iis R/S + I/S*i,!. 174 175(X iis sqrt(Z)):-!, 176 X iis Z**0.5. 177 178(X iis exp(Z)):- 179 ZZ iis Z, 180 complex_canonical(ZZ,R,I), 181 X iis exp(R)*cos(I) + exp(R)*sin(I)*i,!. 182 183(X iis sin(Z)):- 184 ZZ iis Z, 185 complex_canonical(ZZ,R,I), 186 X iis sin(R)*cosh(I) + cos(R)*sinh(I)*i,!. 187 188(X iis cos(Z)):- 189 ZZ iis Z, 190 complex_canonical(ZZ,R,I), 191 X iis cos(R)*cosh(I) + sin(R)*sinh(I)*i,!. 192 193(X iis tan(Z)):- 194 ZZ iis Z, 195 complex_number(ZZ), 196 X iis sin(ZZ)/cos(ZZ),!. 197 198(X iis ctan(Z)):- 199 ZZ iis Z, 200 complex_number(ZZ), 201 X iis cos(ZZ)/sin(ZZ),!. 202 203% Optimize 204% ---------- 205 206(i iis i):-!. 207 208(-1 iis i*i):-!. 209 210(0 iis 0*i):-!. 211 212(0 iis 0.0*i):-!. 213 214(i iis 1*i):-!. 215 216(i iis 1.0*i):-!. 217 218(-1 iis e**(i*pi)):-!. 219 220(-1 iis e**(pi*i)):-!. 221 222(AA*i iis A*i):- 223 number(A),!, 224 AA=A. 225 226(0 iis O1+O2*i):- 227 zero(O1), 228 zero(O2),!. 229(0 iis O1-O2*i):- 230 zero(O1), 231 zero(O2),!. 232 233(R iis R+O*i):- 234 number(R), 235 zero(O),!. 236(I*i iis O+I*i):- 237 number(I), 238 zero(O),!. 239(II*i iis O-I*i):- 240 number(I), 241 zero(O),!, 242 default_is(II,-I). 243 244(R-II*i iis R+I*i):- 245 number(R), 246 number(I), 247 I<0,!, 248 default_is(II,-I). 249 250(R+II*i iis R-I*i):- 251 number(R), 252 number(I), 253 I<0,!, 254 default_is(II,-I). 255 256/* 257(X iis X):- 258 complex_number(X),!. 259*/ 260 261% Plus 262% ------ 263(R iis R-I*i + I*i):-!. 264 265(RR+I*i iis A+Z):- 266 number(A), 267 complex_canonical(Z,R,I),!, 268 default_is(RR,A+R). 269 270(X*i iis A*i + B*i):- 271 number(A), 272 number(B),!, 273 default_is(X,A+B). 274 275(R+X*i iis R-A*i + B*i):- 276 number(R), 277 number(A), 278 number(B),!, 279 default_is(X,B-A). 280 281(RR+A*i iis B + R + A*i):- 282 number(R), 283 number(A), 284 number(B),!, 285 default_is(RR,R+B). 286 287 288(RR+A*i iis B + (R + A*i)):- 289 number(R), 290 number(A), 291 number(B),!, 292 default_is(RR,R+B). 293 294(R+X*i iis R+A*i + B*i):- 295 number(R), 296 number(A), 297 number(B),!, 298 default_is(X,A+B). 299 300(RR+II*i iis Z1+Z2):- 301 complex_canonical(Z1,R1,I1), 302 complex_canonical(Z2,R2,I2),!, 303 default_is(RR,R1+R2), 304 default_is(II,I1+I2). 305 306(X iis A+B):- 307 AA iis A, 308 BB iis B, 309 complex_number(AA), 310 complex_number(BB),!, 311 XX iis AA+BB, 312 X=XX. 313 314% Minus 315% ------- 316(0 iis I*i - I*i):- 317 number(I),!. 318 319(I*i iis R+I*i - R):-!. 320 321(R iis R+I*i - I*i):-!. 322 323(I*i iis I1*i - I2*i):- 324 number(I1), 325 number(I2),!, 326 default_is(I,I1-I2). 327 328(R-X*i iis R-A*i - B*i):- 329 number(R), 330 number(A), 331 number(B),!, 332 default_is(X,A+B). 333 334(R+X*i iis R+A*i - B*i):- 335 number(R), 336 number(A), 337 number(B),!, 338 default_is(X,A-B). 339 340 341(RR+I*i iis Z-A):- 342 number(A), 343 complex_canonical(Z,R,I),!, 344 default_is(RR,R-A). 345 346(RR+II*i iis Z1-Z2):- 347 complex_canonical(Z1,R1,I1), 348 complex_canonical(Z2,R2,I2),!, 349 default_is(RR,R1-R2), 350 default_is(II,I1-I2). 351 352(X iis A-B):- 353 AA iis A, 354 BB iis B, 355 complex_number(AA), 356 complex_number(BB),!, 357 XX iis AA-BB, 358 X=XX. 359 360% Time 361% ------ 362(A*i iis i*A):- 363 number(A),!. 364 365(-A iis A*i*i):- 366 number(A),!. 367 368(X iis A*i*B*i):- 369 number(A), 370 number(B),!, 371 default_is(X,-A*B). 372 373(I*i iis A*i*B):- 374 number(A), 375 number(B),!, 376 default_is(I,A*B). 377 378(C*exp(i*H) iis A*exp(i*F)*(B*exp(i*G))):- 379 number(A), 380 number(B), 381 number(F), 382 number(G),!, 383 default_is(C,A*B), 384 default_is(H,F+G). 385 386% different brackets 387(C*exp(i*H) iis A*exp(i*F)*B*exp(i*G)):- 388 number(A), 389 number(B), 390 number(F), 391 number(G),!, 392 default_is(C,A*B), 393 default_is(H,F+G). 394 395(C*exp(i*G) iis A*(B*exp(i*G))):- 396 number(A), 397 number(B), 398 number(G),!, 399 default_is(C,A*B). 400 401% different brackets 402(C*exp(i*G) iis A*B*exp(i*G)):- 403 number(A), 404 number(B), 405 number(G),!, 406 default_is(C,A*B). 407 408 409(RR+II*i iis A*Z):- 410 number(A), 411 complex_canonical(Z,R,I),!, 412 default_is(RR,A*R), 413 default_is(II,A*I). 414 415(RR+II*i iis Z1*Z2):- 416 complex_canonical(Z1,R1,I1), 417 complex_canonical(Z2,R2,I2),!, 418 default_is(RR,R1*R2 - I1*I2), 419 default_is(II,R1*I2 + I1*R2). 420 421(X iis A*B):- 422 AA iis A, 423 BB iis B, 424 complex_number(AA), 425 complex_number(BB),!, 426 XX iis AA*BB, 427 X=XX. 428 429% Divide 430% ------- 431(1 iis i/i):-!. 432(C*i iis i/A):- 433 number(A),!, 434 default_is(C,1/A). 435(C iis i/(A*i)):- 436 number(A),!, 437 default_is(C,1/A). 438(A iis A*i/i):- 439 number(A),!. 440(C iis A*i/(B*i)):- 441 number(A), 442 number(B),!, 443 default_is(C,A/B). 444(C*i iis A*i/B):- 445 number(A), 446 number(B),!, 447 default_is(C,A/B). 448 449 450(RR+II*i iis A/B):- 451 number(B), 452 AA iis A, 453 complex_canonical(AA,R,I),!, 454 default_is(RR,R/B), 455 default_is(II,I/B). 456 457(X iis A/B):- 458 complex_number(A), 459 complex_number(B), 460 X iis A*reciprocal(B),!. 461 462(X iis A/B):- 463 AA iis A, 464 BB iis B, 465 complex_number(AA), 466 complex_number(BB),!, 467 XX iis AA/BB, 468 X=XX. 469 470% Exponentiation 471% ---------------- 472(W iis i**N):- 473 integer(N), 474 default_is(K,N mod 4), 475 ( K=0,!, 476 W=1 477 ; K=1,!, 478 W=i 479 ; K=2,!, 480 W= -1 481 ; K=3,!, 482 W= -i 483 ). 484 485(exp(i*P) iis i**A):- 486 number(A),!, 487 default_is(P,pi*A/2). 488 489(X iis -1**i):-!, 490 default_is(X,exp(-pi)). 491(X iis -1.0**i):-!, 492 default_is(X,exp(-pi)). 493 494(X iis A**i):- 495 number(A),!, 496 ( A>0,!, 497 default_is(L,log(A)), 498 X = exp(i*L) 499 ; A<0,!, 500 default_is(L,log(-A)), 501 default_is(M,exp(-pi)), 502 X = M*exp(i*L) 503 ; X=0 504 ). 505 506(X iis Z**i):- 507 complex_exponential(Z,M,P),!, 508 default_is(A,e**(-P)), 509 X iis A*M**i. 510 511(exp(i*L) iis A**(B*i)):- 512 number(A), 513 number(B),!, 514 catch( 515 catch( 516 default_is(L,log(A**B)), 517 error(evaluation_error(float_overflow),_), 518 L is B*log(A) 519 ), 520 error(evaluation_error(undefined), context(log/1, _)), 521 ( A = 0, 522 L = 0 523 ; L is log(A) 524 ) 525 ). 526 527(A*E iis Z**i):- 528 complex_exponential(Z,M,P),!, 529 default_is(A,e**(-P)), 530 E iis M**i. 531 532 533(AA*exp(i*PP) iis Z**N):- 534 number(N), 535 complex_exponential(Z,A,P),!, 536 default_is(AA,A**N), 537 default_is(PP,P*N). 538 539(XX iis N**Z):- 540 number(N), 541 complex_canonical(Z,A,B),!, 542 default_is(M,N**A), 543 E iis N**(B*i), 544 ( E = exp(_), 545 XX = M*E 546 ; XX iis M*E 547 ). 548 549(X iis Z**Z2):- 550 complex_exponential(Z,M,P), 551 complex_canonical(Z2,A,B),!, 552 default_is(MM,M**A), 553 exp(P1) iis M**(B*i), 554 P2 iis i*P*Z2, 555 PP iis P1+P2, 556 X iis MM*exp(PP). 557 558 559(X iis N**Z):- 560 number(N), 561 complex_canonical(Z,R,I),!, 562 default_is(A,N**R), 563 B iis N**(I*i), 564 X iis A*B. 565 566(X iis A**B):- 567 AA iis A, 568 BB iis B, 569 complex_number(AA), 570 complex_number(BB),!, 571 XX iis AA**BB, 572 XX = X. 573 574(X iis A^B):- 575 X iis A ** B.
582c_equals(A,B):-
583 Z1 iis A,
584 complex_canonical(Z1,R1,I1),
585 Z2 iis B,
586 complex_canonical(Z2,R2,I2),
587 R1==R2,
588 I1==I2.complex_number(Complex) in the same time.595% canonical 596complex_canonical(i,0,1):-!. 597complex_canonical(-i,0,-1):-!. 598complex_canonical(R,R,0):- 599 number(R),!. 600complex_canonical(I*i,0,I):- 601 number(I),!. 602complex_canonical(R+i,R,1):- 603 number(R),!. 604complex_canonical(R+I*i,R,I):- 605 number(R), 606 number(I),!. 607complex_canonical(R-i,R,-1):- 608 number(R),!. 609complex_canonical(R-II*i,R,I):- 610 number(R), 611 number(II),!, 612 default_is(I,-II). 613 614% exponential 615complex_canonical(exp(i),R,I):-!, 616 default_is(R,cos(1)), 617 default_is(I,sin(1)). 618complex_canonical(exp(i*F),R,I):- 619 number(F),!, 620 default_is(R,cos(F)), 621 default_is(I,sin(F)). 622complex_canonical(M*exp(i),R,I):- 623 number(M),!, 624 default_is(R,M*cos(1)), 625 default_is(I,M*sin(1)). 626complex_canonical(M*exp(i*F),R,I):- 627 number(M), 628 number(F),!, 629 pol_rec(R,I,M,F). 630 631% trigonometric 632complex_canonical(cos(F)+i*sin(F),R,I):- 633 number(F),!, 634 pol_rec(R,I,1,F). 635complex_canonical(M*(cos(F)+i*sin(F)),R,I):- 636 number(M), 637 number(F),!, 638 pol_rec(R,I,M,F).
646is_canonical(i,0,1):-!. 647is_canonical(-i,0,-1):-!. 648is_canonical(R,R,0):- 649 number(R),!. 650is_canonical(I*i,0,I):- 651 number(I),!. 652is_canonical(R+i,R,1):- 653 number(R),!. 654is_canonical(R+I*i,R,I):- 655 number(R), 656 number(I),!. 657is_canonical(R-i,R,-1):- 658 number(R),!. 659is_canonical(R-II*i,R,I):- 660 number(R), 661 number(II),!, 662 default_is(I,-II).
complex_number(Complex) in the same time670% canonical 671complex_exponential(i,1,F):-!, 672 default_is(F,pi/2). 673complex_exponential(-i,-1,F):-!, 674 default_is(F,pi/2). 675complex_exponential(M,M,0):- 676 number(M),!. 677complex_exponential(M*i,M,F):- 678 number(M),!, 679 default_is(F,pi/2). 680complex_exponential(R+i,M,F):- 681 number(R),!, 682 pol_rec(R,1,M,F). 683complex_exponential(R+I*i,M,F):- 684 number(R), 685 number(I),!, 686 pol_rec(R,I,M,F). 687complex_exponential(R-i,M,F):- 688 number(R),!, 689 pol_rec(R,-1,M,F). 690complex_exponential(R-I*i,M,F):- 691 number(R), 692 number(I),!, 693 default_is(II,-I), 694 pol_rec(R,II,M,F). 695 696% exponential 697complex_exponential(exp(i),1,1):-!. 698complex_exponential(exp(i*F),1,F):- 699 number(F),!. 700complex_exponential(M*exp(i),M,1):- 701 number(M),!. 702complex_exponential(M*exp(i*F),M,F):- 703 number(M), 704 number(F),!. 705 706% trigonometric 707complex_exponential(cos(F)+i*sin(F),1,F):- 708 number(F),!. 709complex_exponential(M*(cos(F)+i*sin(F)),M,F):- 710 number(M), 711 number(F),!.
718% exponential 719is_exponential(exp(i),1,1):-!. 720is_exponential(exp(i*F),1,F):- 721 number(F),!. 722is_exponential(M*exp(i),M,1):- 723 number(M),!. 724is_exponential(M*exp(i*F),M,F):- 725 number(M), 726 number(F),!. 727 728% trigonometric 729is_exponential(cos(F)+i*sin(F),1,F):- 730 number(F),!. 731is_exponential(M*(cos(F)+i*sin(F)),M,F):- 732 number(M), 733 number(F),!. 734 735 736pol_rec(R,I,M,F):- 737 number(R), 738 number(I),!, 739 default_is(M,sqrt(R*R+I*I)), 740 default_is(F,atan2(I,R)). 741pol_rec(R,I,M,F):- 742 number(M), 743 number(F),!, 744 default_is(R,M*cos(F)), 745 default_is(I,M*sin(F)).
4, i, -1, 2*i, 3+2*i, 8-4*i, 3+ -2*i
12*exp(i*5), exp(i*12)
10*(cos(3)+i*sin(3)), cos(4)+i*sin(4)
765complex_number(A):- 766 var(A),!,fail. 767 768complex_number(i):-!. 769complex_number(-i):-!. 770complex_number(R):- 771 number(R),!. 772complex_number(R + I*i):- 773 number(R), 774 number(I),!. 775complex_number(R - I*i):- 776 number(R), 777 number(I),!. 778complex_number(R - i):- 779 number(R),!. 780complex_number(exp(i*F)):- 781 number(F),!. 782complex_number(M*exp(i*F)):- 783 number(M), 784 number(F),!. 785complex_number(cos(F)+i*sin(F)):- 786 number(F),!. 787complex_number(M*(cos(F)+i*sin(F))):- 788 number(M), 789 number(F),!. 790complex_number(I):- 791 imaginary_number(I,_),!. 792 793imaginary_number(i,1):-!. 794imaginary_number(I*i,I):- 795 number(I),!. 796 797zero(0):-!. 798zero(0.0):-!. 799zero(-0):-!. 800zero(-0.0):-!. 801 802one(1):-!. 803one(1.0):-!
Complex number
This is module to calculate imaginary number.
The imaginary unit is represented by
i./