1% Demonstration of the REDUCE SOLVE package. 2 3on fullroots; 4 5 % To get complete solutions. 6 7% Simultaneous linear fractional equations. 8 9solve({(a*x+y)/(z-1)-3,y+b+z,x-y},{x,y,z}); 10 11 12 - 3*(b + 1) 13{{x=--------------, 14 a + 4 15 16 - 3*(b + 1) 17 y=--------------, 18 a + 4 19 20 - a*b - b + 3 21 z=----------------}} 22 a + 4 23 24 25 26% Use of square-free factorization together with recursive use of 27% quadratic and binomial solutions. 28 29solve((x**6-x**3-1)*(x**5-1)**2*x**2); 30 31 32Unknown: x 33 34 2*sqrt( - sqrt(5) - 5) + sqrt(10) - sqrt(2) 35{x=---------------------------------------------, 36 4*sqrt(2) 37 38 - 2*sqrt( - sqrt(5) - 5) + sqrt(10) - sqrt(2) 39 x=------------------------------------------------, 40 4*sqrt(2) 41 42 2*sqrt(sqrt(5) - 5) - sqrt(10) - sqrt(2) 43 x=------------------------------------------, 44 4*sqrt(2) 45 46 - 2*sqrt(sqrt(5) - 5) - sqrt(10) - sqrt(2) 47 x=---------------------------------------------, 48 4*sqrt(2) 49 50 x=1, 51 52 x=0, 53 54 1/3 55 ( - sqrt(5) + 1) *(sqrt(3)*i - 1) 56 x=-------------------------------------, 57 1/3 58 2*2 59 60 1/3 61 - ( - sqrt(5) + 1) *(sqrt(3)*i + 1) 62 x=----------------------------------------, 63 1/3 64 2*2 65 66 1/3 67 ( - sqrt(5) + 1) 68 x=---------------------, 69 1/3 70 2 71 72 1/3 73 (sqrt(5) + 1) *(sqrt(3)*i - 1) 74 x=----------------------------------, 75 1/3 76 2*2 77 78 1/3 79 - (sqrt(5) + 1) *(sqrt(3)*i + 1) 80 x=-------------------------------------, 81 1/3 82 2*2 83 84 1/3 85 (sqrt(5) + 1) 86 x=------------------} 87 1/3 88 2 89 90 91multiplicities!*; 92 93 94{2,2,2,2,2,2,1,1,1,1,1,1} 95 96 97 98% A singular equation without and with a consistent inhomogeneous term. 99 100solve(a,x); 101 102 103{} 104 105 106solve(0,x); 107 108 109{x=arbcomplex(1)} 110 111 112off solvesingular; 113 114 115 116solve(0,x); 117 118 119{} 120 121 122 123% Use of DECOMPOSE to solve high degree polynomials. 124 125solve(x**8-8*x**7+34*x**6-92*x**5+175*x**4-236*x**3+226*x**2-140*x+46); 126 127 128Unknown: x 129 130 sqrt( - sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 131{x=-------------------------------------------------, 132 2 133 134 - sqrt( - sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 135 x=----------------------------------------------------, 136 2 137 138 sqrt( - sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 139 x=----------------------------------------------, 140 2 141 142 - sqrt( - sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 143 x=-------------------------------------------------, 144 2 145 146 sqrt(sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 147 x=----------------------------------------------, 148 2 149 150 - sqrt(sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 151 x=-------------------------------------------------, 152 2 153 154 sqrt(sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 155 x=-------------------------------------------, 156 2 157 158 - sqrt(sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 159 x=----------------------------------------------} 160 2 161 162 163solve(x**8-88*x**7+2924*x**6-43912*x**5+263431*x**4-218900*x**3+ 164 65690*x**2-7700*x+234,x); 165 166 167{x=sqrt( - i + 116) + 11, 168 169 x= - sqrt( - i + 116) + 11, 170 171 x=sqrt(i + 116) + 11, 172 173 x= - sqrt(i + 116) + 11, 174 175 x=4*sqrt(7) + 11, 176 177 x= - 4*sqrt(7) + 11, 178 179 x=2*sqrt(30) + 11, 180 181 x= - 2*sqrt(30) + 11} 182 183 184 185% Recursive use of inverses, including multiple branches of rational 186% fractional powers. 187 188solve(log(acos(asin(x**(2/3)-b)-1))+2,x); 189 190 191 1 1 192{x=sqrt(sin(cos(----) + 1) + b)*(sin(cos(----) + 1) + b), 193 2 2 194 e e 195 196 1 1 197 x= - sqrt(sin(cos(----) + 1) + b)*(sin(cos(----) + 1) + b)} 198 2 2 199 e e 200 201 202 203% Square-free factors that are unsolvable, being of fifth degree, 204% transcendental, or without a defined inverse. 205 206operator f; 207 208 209 210solve((x-1)*(x+1)*(x-2)*(x+2)*(x-3)*(x*log(x)-1)*(f(x)-1),x); 211 212 213{x=root_of(f(x_) - 1,x_,tag_4), 214 215 x=root_of(log(x_)*x_ - 1,x_,tag_3), 216 217 x=3, 218 219 x=2, 220 221 x=1, 222 223 x=-1, 224 225 x=-2} 226 227 228multiplicities!*; 229 230 231{1,1,1,1,1,1,1} 232 233 234 235% Factors with more than one distinct top-level kernel, the first factor 236% a cubic. (Cubic solution suppressed since it is too messy to be of 237% much use). 238 239off fullroots; 240 241 242 243solve((x**(1/2)-(x-a)**(1/3))*(acos x-acos(2*x-b))* (2*log x 244 -log(x**2+x-c)-4),x); 245 246 247 2 4 4 2 248 e *(sqrt(4*c*e - 4*c + e ) - e ) 249{x=-----------------------------------, 250 4 251 2*(e - 1) 252 253 2 4 4 2 254 - e *(sqrt(4*c*e - 4*c + e ) + e ) 255 x=--------------------------------------, 256 4 257 2*(e - 1) 258 259 2/3 260 x=root_of(( - a + x_) - x_,x_,tag_9), 261 262 x=b} 263 264 265on fullroots; 266 267 268 269% Treatment of multiple-argument exponentials as polynomials. 270 271solve(a**(2*x)-3*a**x+2,x); 272 273 274 2*arbint(3)*i*pi + log(2) 275{x=---------------------------, 276 log(a) 277 278 2*arbint(2)*i*pi 279 x=------------------} 280 log(a) 281 282 283 284% A 12th degree reciprocal polynomial that is irreductible over the 285% integers, having a reduced polynomial that is also reciprocal. 286% (Reciprocal polynomials are those that have symmetric or antisymmetric 287% coefficient patterns.) We also demonstrate suppression of automatic 288% integer root extraction. 289 290solve(x**12-4*x**11+12*x**10-28*x**9+45*x**8-68*x**7+69*x**6-68*x**5+ 29145*x**4-28*x**3+12*x**2-4*x+1); 292 293 294Unknown: x 295 296 sqrt( - sqrt(5) - 3) 297{x=----------------------, 298 sqrt(2) 299 300 - sqrt( - sqrt(5) - 3) 301 x=-------------------------, 302 sqrt(2) 303 304 2*sqrt( - sqrt(3)*i - 9) - sqrt(6)*i + sqrt(2) 305 x=------------------------------------------------, 306 4*sqrt(2) 307 308 - 2*sqrt( - sqrt(3)*i - 9) - sqrt(6)*i + sqrt(2) 309 x=---------------------------------------------------, 310 4*sqrt(2) 311 312 2*sqrt( - 3*sqrt(5) - 1) - sqrt(10) + 3*sqrt(2) 313 x=-------------------------------------------------, 314 4*sqrt(2) 315 316 - 2*sqrt( - 3*sqrt(5) - 1) - sqrt(10) + 3*sqrt(2) 317 x=----------------------------------------------------, 318 4*sqrt(2) 319 320 2*sqrt(sqrt(3)*i - 9) + sqrt(6)*i + sqrt(2) 321 x=---------------------------------------------, 322 4*sqrt(2) 323 324 - 2*sqrt(sqrt(3)*i - 9) + sqrt(6)*i + sqrt(2) 325 x=------------------------------------------------, 326 4*sqrt(2) 327 328 2*sqrt(3*sqrt(5) - 1) + sqrt(10) + 3*sqrt(2) 329 x=----------------------------------------------, 330 4*sqrt(2) 331 332 - 2*sqrt(3*sqrt(5) - 1) + sqrt(10) + 3*sqrt(2) 333 x=-------------------------------------------------, 334 4*sqrt(2) 335 336 i*(sqrt(5) - 1) 337 x=-----------------, 338 2 339 340 i*( - sqrt(5) + 1) 341 x=--------------------} 342 2 343 344 345 346% The treatment of factors with non-unique inverses by introducing 347% unique new real or integer indeterminant kernels. 348 349solve((sin x-a)*(2**x-b)*(x**c-3),x); 350 351 352{x=2*arbint(6)*pi + asin(a), 353 354 x=2*arbint(6)*pi - asin(a) + pi, 355 356 2*arbint(5)*i*pi + log(b) 357 x=---------------------------, 358 log(2) 359 360 1/c 2*arbint(4)*pi 2*arbint(4)*pi 361 x=3 *(cos(----------------) + sin(----------------)*i)} 362 c c 363 364 365 366% Automatic restriction to principal branches. 367 368off allbranch; 369 370 371 372solve((sin x-a)*(2**x-b)*(x**c-3),x); 373 374 375{x=asin(a), 376 377 1/c 378 x=3 , 379 380 log(b) 381 x=--------} 382 log(2) 383 384 385 386% Regular system of linear equations. 387 388solve({2*x1+x2+3*x3-9,x1-2*x2+x3+2,3*x1+2*x2+2*x3-7}, {x1,x2,x3}); 389 390 391{{x1=-1,x2=2,x3=3}} 392 393 394 395% Underdetermined system of linear equations. 396 397on solvesingular; 398 399 400 401solve({x1-4*x2+2*x3+1,2*x1-3*x2-x3-5*x4+7,3*x1-7*x2+x3-5*x4+8}, 402 {x1,x2,x3,x4}); 403 404 405{{x1=4*arbcomplex(8) + 2*arbcomplex(7) - 5, 406 407 x2=arbcomplex(8) + arbcomplex(7) - 1, 408 409 x3=arbcomplex(7), 410 411 x4=arbcomplex(8)}} 412 413 414 415% Inconsistent system of linear equations. 416 417solve({2*x1+3*x2-x3-2,7*x1+4*x2+2*x3-8,3*x1-2*x2+4*x3-5}, 418 {x1,x2,x3}); 419 420 421{} 422 423 424 425% Overdetermined system of linear equations. 426 427solve({x1-x2+x3-12,2*x1+3*x2-x3-13,3*x2+4*x3-5,-3*x1+x2+4*x3+20}, 428 {x1,x2,x3}); 429 430 431{{x1=9,x2=-1,x3=2}} 432 433 434 435% Degenerate system of linear equations. 436 437operator xx,yy; 438 439 440 441yy(1) := -a**2*b**3-3*a**2*b**2-3*a**2*b+a**2*(xx(3)-2)-a*b-a*c+a*(xx(2) 442 -xx(5))-xx(4)-xx(5)+xx(1)-1; 443 444 445 2 2 3 446yy(1) := - xx(5)*a - xx(5) - xx(4) + xx(3)*a + xx(2)*a + xx(1) - a *b 447 448 2 2 2 2 449 - 3*a *b - 3*a *b - 2*a - a*b - a*c - 1 450 451 452yy(2) := -a*b**3-b**5+b**4*(-xx(4)-xx(5)+xx(1)-5)-b**3*c+b**3*(xx(2) 453 -xx(5)-3)+b**2*(xx(3)-1); 454 455 456 2 2 2 2 457yy(2) := b *( - xx(5)*b - xx(5)*b - xx(4)*b + xx(3) + xx(2)*b + xx(1)*b - a*b 458 459 3 2 460 - b - 5*b - b*c - 3*b - 1) 461 462 463yy(3) := -a*b**3*c-3*a*b**2*c-4*a*b*c+a*b*(-xx(4)-xx(5)+xx(1)-1) 464 +a*c*(xx(3)-1)-b**2*c-b*c**2+b*c*(xx(2)-xx(5)); 465 466 467yy(3) := - xx(5)*a*b - xx(5)*b*c - xx(4)*a*b + xx(3)*a*c + xx(2)*b*c 468 469 3 2 2 2 470 + xx(1)*a*b - a*b *c - 3*a*b *c - 4*a*b*c - a*b - a*c - b *c - b*c 471 472 473yy(4) := -a**2-a*c+a*(xx(2)-xx(4)-2*xx(5)+xx(1)-1)-b**4-b**3*c-3*b**3 474 -3*b**2*c-2*b**2-2*b*c+b*(xx(3)-xx(2)-xx(4)+xx(1)-2) 475 +c*(xx(3)-1); 476 477 478yy(4) := - 2*xx(5)*a - xx(4)*a - xx(4)*b + xx(3)*b + xx(3)*c + xx(2)*a 479 480 2 4 3 3 481 - xx(2)*b + xx(1)*a + xx(1)*b - a - a*c - a - b - b *c - 3*b 482 483 2 2 484 - 3*b *c - 2*b - 2*b*c - 2*b - c 485 486 487yy(5) := -2*a-3*b**3-9*b**2-11*b-2*c+3*xx(3)+2*xx(2)-xx(4)-3*xx(5)+xx(1) 488 -4; 489 490 491 3 2 492yy(5) := - 3*xx(5) - xx(4) + 3*xx(3) + 2*xx(2) + xx(1) - 2*a - 3*b - 9*b 493 494 - 11*b - 2*c - 4 495 496 497soln := solve({yy(1),yy(2),yy(3),yy(4),yy(5)}, 498 {xx(1),xx(2),xx(3),xx(4),xx(5)}); 499 500 501soln := {{xx(1)=arbcomplex(10) + arbcomplex(9) + 1, 502 503 xx(2)=arbcomplex(10) + a + b + c, 504 505 3 2 506 xx(3)=b + 3*b + 3*b + 1, 507 508 xx(4)=arbcomplex(9), 509 510 xx(5)=arbcomplex(10)}} 511 512 513for i := 1:5 do xx(i) := part(soln,1,i,2); 514 515 516 517for i := 1:5 do write yy(i); 518 519 5200 521 5220 523 5240 525 5260 527 5280 529 530 531 532% Single equations liftable to polynomial systems. 533 534solve ({a*sin x + b*cos x},{x}); 535 536 537 2 2 538 sqrt(a + b ) - a 539{x= - 2*atan(-------------------), 540 b 541 542 2 2 543 sqrt(a + b ) + a 544 x=2*atan(-------------------)} 545 b 546 547 548solve ({a*sin(x+1) + b*cos(x+1)},{x}); 549 550 551 2 2 552 sqrt(a + b ) - a 553{x= - 2*atan(-------------------) - 1, 554 b 555 556 2 2 557 sqrt(a + b ) + a 558 x=2*atan(-------------------) - 1} 559 b 560 561 562% Intersection of 2 curves: system with a free parameter. 563 564solve ({sqrt(x^2 + y^2)=r,0=sqrt(x)+ y**3-1},{x,y,r}); 565 566 567 6 3 568{{x=y - 2*y + 1, 569 570 y=arbcomplex(12), 571 572 12 9 6 3 2 573 r=sqrt(y - 4*y + 6*y - 4*y + y + 1)}, 574 575 6 3 576 {x=y - 2*y + 1, 577 578 y=arbcomplex(11), 579 580 12 9 6 3 2 581 r= - sqrt(y - 4*y + 6*y - 4*y + y + 1)}} 582 583 584solve ({e^x - e^(1/2 * x) - 7},{x}); 585 586 587 - sqrt(29) + 1 588{x=2*log(-----------------), 589 2 590 591 sqrt(29) + 1 592 x=2*log(--------------)} 593 2 594 595 596% Generally not liftable. 597 598 % variable inside and outside of sin. 599 600 solve({sin x + x - 1/2},{x}); 601 602 603{x=root_of(2*sin(x_) + 2*x_ - 1,x_,tag_15)} 604 605 606 % Variable inside and outside of exponential. 607 608 solve({e^x - x**2},{x}); 609 610 611 - 1 612{x= - 2*lambert_w(-------------------------)} 613 2*plus_or_minus(tag_16) 614 615 616 % Variable inside trigonometrical functions with different forms. 617 618 solve ({a*sin(x+1) + b*cos(x+2)},{x}); 619 620 621 2 2 622{x=2*atan((cos(1)*a - sqrt(2*cos(2)*sin(1)*a*b - 2*cos(1)*sin(2)*a*b + a + b ) 623 624 - sin(2)*b)/(cos(2)*b + sin(1)*a)), 625 626 2 2 627 x=2*atan((cos(1)*a + sqrt(2*cos(2)*sin(1)*a*b - 2*cos(1)*sin(2)*a*b + a + b ) 628 629 - sin(2)*b)/(cos(2)*b + sin(1)*a))} 630 631 632 % Undetermined exponents. 633 634 solve({x^a - 2},{x}); 635 636 637 1/a 638{x=2 } 639 640 641 642% Example taken from M.L. Griss, ACM Trans. Math. Softw. 2 (1976) 1. 643 644e1 := x1 - l/(3*k)$ 645 646 647 648e2 := x2 - 1$ 649 650 651 652e3 := x3 - 35*b6/(6*l)*x4 + 33*b11/(2*l)*x6 - 715*b15/(14*l)*x8$ 653 654 655 656e4 := 14*k/(3*l)*x1 - 7*b4/(2*l)*x3 + x4$ 657 658 659 660e5 := x5 - 891*b11/(40*l)*x6 +3861*b15/(56*l)*x8$ 661 662 663 664e6 := -88*k/(15*l)*x1 + 22*b4/(5*l)*x3 - 99*b9/(8*l)*x5 +x6$ 665 666 667 668e7 := -768*k/(5005*b13)*x1 + 576*b4/(5005*b13)*x3 - 669 324*b9/(1001*b13)*x5 + x7 - 16*l/(715*b13)*x8$ 670 671 672 673e8 := 7*l/(143*b15)*x1 + 49*b6/(429*b15)*x4 - 21*b11/(65*b15)*x6 + 674 x8 - 7*b2/(143*b15)$ 675 676 677 678solve({e1,e2,e3,e4,e5,e6,e7,e8},{x1,x2,x3,x4,x5,x6,x7,x8}); 679 680 681 l 682{{x1=-----, 683 3*k 684 685 x2=1, 686 687 2 688 5*(3*b2*k - l ) 689 x3=-----------------, 690 6*k*l 691 692 2 2 693 7*(45*b2*b4*k - 15*b4*l - 8*k*l ) 694 x4=------------------------------------, 695 2 696 36*k*l 697 698 2 2 2 4 699 2205*b2*b4*b6*k - 108*b2*k*l - 735*b4*b6*l - 392*b6*k*l + 36*l 700 x5=--------------------------------------------------------------------, 701 3 702 32*k*l 703 704 2 2 705 x6=(11*(893025*b2*b4*b6*b9*k - 11520*b2*b4*k*l - 43740*b2*b9*k*l 706 707 2 4 2 4 708 - 297675*b4*b6*b9*l + 3840*b4*l - 158760*b6*b9*k*l + 14580*b9*l 709 710 4 4 711 + 2048*k*l ))/(11520*k*l ), 712 713 2 714 x7=(47652707025*b11*b2*b4*b6*b9*k - 614718720*b11*b2*b4*k*l 715 716 2 2 717 - 2334010140*b11*b2*b9*k*l - 15884235675*b11*b4*b6*b9*l 718 719 4 2 4 720 + 204906240*b11*b4*l - 8471592360*b11*b6*b9*k*l + 778003380*b11*b9*l 721 722 4 723 + 109283328*b11*k*l + 172398476250*b15*b2*b4*b6*b9*k 724 725 2 2 726 - 2223936000*b15*b2*b4*k*l - 8444007000*b15*b2*b9*k*l 727 728 2 4 729 - 57466158750*b15*b4*b6*b9*l + 741312000*b15*b4*l 730 731 2 4 4 732 - 30648618000*b15*b6*b9*k*l + 2814669000*b15*b9*l + 395366400*b15*k*l 733 734 2 4 4 735 - 172872000*b2*b4*b6*k*l + 8467200*b2*k*l + 57624000*b4*b6*l 736 737 4 6 3 738 + 30732800*b6*k*l - 2822400*l )/(7729722000*b13*b15*k*l ), 739 740 2 741 x8=(7*(972504225*b11*b2*b4*b6*b9*k - 12545280*b11*b2*b4*k*l 742 743 2 2 744 - 47632860*b11*b2*b9*k*l - 324168075*b11*b4*b6*b9*l 745 746 4 2 4 747 + 4181760*b11*b4*l - 172889640*b11*b6*b9*k*l + 15877620*b11*b9*l 748 749 4 2 4 750 + 2230272*b11*k*l - 3528000*b2*b4*b6*k*l + 172800*b2*k*l 751 752 4 4 6 4 753 + 1176000*b4*b6*l + 627200*b6*k*l - 57600*l ))/(24710400*b15*k*l )}} 754 755 756 757f1 := x1 - x*x2 - y*x3 + 1/2*x**2*x4 + x*y*x5 + 1/2*y**2*x6 + 758 1/6*x**3*x7 + 1/2*x*y*(x - y)*x8 - 1/6*y**3*x9$ 759 760 761 762f2 := x1 - y*x3 + 1/2*y**2*x6 - 1/6*y**3*x9$ 763 764 765 766f3 := x1 + y*x2 - y*x3 + 1/2*y**2*x4 - y**2*x5 + 1/2*y**2*x6 + 767 1/6*y**3*x7 + 1/2*y**3*x8 - 1/6*y**3*x9$ 768 769 770 771f4 := x1 + (1 - x)*x2 - x*x3 + 1/2*(1 - x)**2*x4 - y*(1 - x)*x5 + 772 1/2*y**2*x6 + 1/6*(1 - x)**3*x7 + 1/2*y*(1 - x - y)*(1 - x)*x8 773 - 1/6*y**3*x9$ 774 775 776 777f5 := x1 + (1 - x - y)*x2 + 1/2*(1 - x - y)**2*x4 + 778 1/6*(1 - x - y)**3*x7$ 779 780 781 782f6 := x1 + (1 - x - y)*x3 + 1/2*(1 - x - y)*x6 + 783 1/6*(1 - x - y)**3*x9$ 784 785 786 787f7 := x1 - x*x2 + (1 - y)*x3 + 1/2*x*x4 - x*(1 - y)*x5 + 788 1/2*(1 - y)**2*x6 - 1/6*x**3*x7 + 1/2*x*(1 - y)*(1 - y + x)*x8 789 + 1/6*(1-y)**3*x9$ 790 791 792 793f8 := x1 - x*x2 + x*x3 + 1/2*x**2*x4 - x**2*x5 + 1/2*x**2*x6 + 794 1/6*x**3*x7 - 1/2*x**3*x8 + 1/6*x**3*x9$ 795 796 797 798f9 := x1 - x*x2 + 1/2*x**2*x4 + 1/6*x**3*x7$ 799 800 801 802solve({f1,f2,f3,f4,f5,f6,f7,f8,f9},{x1,x2,x3,x4,x5,x6,x7,x8,x9}); 803 804 805{{x1=0,x2=0,x3=0,x4=0,x5=0,x6=0,x7=0,x8=0,x9=0}} 806 807 808solve({f1 - 1,f2,f3,f4,f5,f6,f7,f8,f9},{x1,x2,x3,x4,x5,x6,x7,x8,x9}); 809 810 811 8 8 7 3 7 2 7 7 6 4 812{{x1=(y*( - 8*x *y + 10*x + 9*x *y - 49*x *y + 85*x *y - 43*x + 23*x *y 813 814 6 3 6 2 6 6 5 5 5 4 815 - 128*x *y + 266*x *y - 246*x *y + 77*x + 20*x *y - 145*x *y 816 817 5 3 5 2 5 5 4 6 4 5 818 + 383*x *y - 512*x *y + 329*x *y - 75*x + 9*x *y - 84*x *y 819 820 4 4 4 3 4 2 4 4 3 7 821 + 276*x *y - 469*x *y + 464*x *y - 233*x *y + 43*x + 3*x *y 822 823 3 6 3 5 3 4 3 3 3 2 3 824 - 23*x *y + 97*x *y - 196*x *y + 245*x *y - 201*x *y + 87*x *y 825 826 3 2 8 2 7 2 6 2 5 2 4 827 - 14*x - 2*x *y + 13*x *y - 25*x *y + 23*x *y - 10*x *y 828 829 2 3 2 2 2 2 9 8 7 830 - 17*x *y + 31*x *y - 15*x *y + 2*x - 2*x*y + 10*x*y - 24*x*y 831 832 6 5 4 3 2 6 5 833 + 41*x*y - 57*x*y + 53*x*y - 24*x*y + 2*x*y + x*y - 2*y + 7*y 834 835 4 3 2 10 10 9 2 9 9 836 - 9*y + 5*y - y ))/(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x 837 838 8 3 8 2 8 8 7 4 7 3 7 2 839 + x *y - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y 840 841 7 7 6 5 6 4 6 3 6 2 842 + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y 843 844 6 6 5 6 5 5 5 4 5 3 845 - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y 846 847 5 2 5 5 4 7 4 6 4 5 4 4 848 - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y 849 850 4 3 4 2 4 4 3 8 3 7 851 - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y 852 853 3 6 3 5 3 4 3 3 3 2 3 854 + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y 855 856 2 9 2 8 2 7 2 6 2 5 2 4 857 + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y 858 859 2 3 2 2 10 9 8 7 860 + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y 861 862 6 5 4 3 9 8 7 6 863 + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y 864 865 5 866 + 2*y ), 867 868 10 10 9 2 9 9 8 3 8 2 8 869 x2=(2*x *y - 2*x + 5*x *y - 12*x *y + 7*x - 8*x *y + 9*x *y + 2*x *y 870 871 8 7 4 7 3 7 2 7 7 6 5 872 - x - 15*x *y + 65*x *y - 83*x *y + 52*x *y - 17*x + 5*x *y 873 874 6 4 6 3 6 2 6 6 5 6 5 5 875 - 5*x *y - 20*x *y + 46*x *y - 54*x *y + 20*x + 23*x *y - 151*x *y 876 877 5 4 5 3 5 2 5 5 4 7 878 + 321*x *y - 338*x *y + 166*x *y - 13*x *y - 8*x + 29*x *y 879 880 4 6 4 5 4 4 4 3 4 2 4 881 - 207*x *y + 523*x *y - 676*x *y + 522*x *y - 222*x *y + 36*x *y 882 883 4 3 8 3 7 3 6 3 5 3 4 884 + x + 16*x *y - 103*x *y + 300*x *y - 463*x *y + 433*x *y 885 886 3 3 3 2 3 2 9 2 7 2 6 2 5 887 - 268*x *y + 98*x *y - 15*x *y - x *y + 22*x *y - 54*x *y + 60*x *y 888 889 2 4 2 3 2 2 2 10 9 8 890 - 56*x *y + 44*x *y - 17*x *y + 2*x *y - 2*x*y + 10*x*y - 22*x*y 891 892 7 6 5 4 3 2 7 6 893 + 34*x*y - 48*x*y + 48*x*y - 23*x*y + 2*x*y + x*y - 2*y + 7*y 894 895 5 4 3 10 10 9 2 9 9 896 - 9*y + 5*y - y )/(x*(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x 897 898 8 3 8 2 8 8 7 4 7 3 899 + x *y - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y 900 901 7 2 7 7 6 5 6 4 6 3 902 - 105*x *y + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y 903 904 6 2 6 6 5 6 5 5 5 4 905 + 308*x *y - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y 906 907 5 3 5 2 5 5 4 7 4 6 908 + 401*x *y - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y 909 910 4 5 4 4 4 3 4 2 4 4 911 + 14*x *y + 90*x *y - 149*x *y + 97*x *y - 24*x *y + x 912 913 3 8 3 7 3 6 3 5 3 4 914 + 20*x *y - 118*x *y + 244*x *y - 237*x *y + 117*x *y 915 916 3 3 3 2 3 2 9 2 8 2 7 917 - 21*x *y - 7*x *y + 2*x *y + 13*x *y - 86*x *y + 228*x *y 918 919 2 6 2 5 2 4 2 3 2 2 10 920 - 294*x *y + 204*x *y - 86*x *y + 23*x *y - 2*x *y + 4*x*y 921 922 9 8 7 6 5 4 923 - 31*x*y + 84*x*y - 121*x*y + 100*x*y - 48*x*y + 15*x*y 924 925 3 9 8 7 6 5 926 - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y )), 927 928 9 9 8 2 8 8 7 3 7 2 7 929 x3=(2*x *y - 4*x + 8*x *y - 32*x *y + 26*x + 9*x *y - 70*x *y + 131*x *y 930 931 7 6 4 6 3 6 2 6 6 5 5 932 - 66*x + 7*x *y - 73*x *y + 226*x *y - 253*x *y + 89*x + 11*x *y 933 934 5 4 5 3 5 2 5 5 4 6 935 - 81*x *y + 244*x *y - 383*x *y + 280*x *y - 73*x + 13*x *y 936 937 4 5 4 4 4 3 4 2 4 4 938 - 89*x *y + 235*x *y - 367*x *y + 360*x *y - 189*x *y + 39*x 939 940 3 7 3 6 3 5 3 4 3 3 3 2 941 + 9*x *y - 59*x *y + 156*x *y - 227*x *y + 231*x *y - 171*x *y 942 943 3 3 2 8 2 7 2 6 2 5 2 4 944 + 74*x *y - 13*x + 3*x *y - 21*x *y + 62*x *y - 78*x *y + 51*x *y 945 946 2 3 2 2 2 2 8 7 6 947 - 35*x *y + 30*x *y - 14*x *y + 2*x - 5*x*y + 18*x*y - 22*x*y 948 949 5 4 3 2 8 7 6 5 4 950 - x*y + 21*x*y - 13*x*y + x*y + x*y + 2*y - 6*y + 6*y + y - 6*y 951 952 3 2 10 10 9 2 9 9 8 3 953 + 4*y - y )/(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + x *y 954 955 8 2 8 8 7 4 7 3 7 2 956 - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y 957 958 7 7 6 5 6 4 6 3 6 2 959 + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y 960 961 6 6 5 6 5 5 5 4 5 3 962 - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y 963 964 5 2 5 5 4 7 4 6 4 5 4 4 965 - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y 966 967 4 3 4 2 4 4 3 8 3 7 968 - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y 969 970 3 6 3 5 3 4 3 3 3 2 3 971 + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y 972 973 2 9 2 8 2 7 2 6 2 5 2 4 974 + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y 975 976 2 3 2 2 10 9 8 7 977 + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y 978 979 6 5 4 3 9 8 7 6 980 + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y 981 982 5 983 + 2*y ), 984 985 9 9 8 2 8 8 7 3 7 2 7 986 x4=(2*(2*x *y - 2*x + 4*x *y - 10*x *y + 6*x - 9*x *y + 21*x *y - 13*x *y 987 988 7 6 4 6 3 6 2 6 6 5 5 989 + x - 18*x *y + 88*x *y - 130*x *y + 74*x *y - 14*x - 10*x *y 990 991 5 4 5 3 5 2 5 5 4 6 992 + 74*x *y - 180*x *y + 191*x *y - 90*x *y + 15*x + 4*x *y 993 994 4 5 4 4 4 3 4 2 4 4 995 - 18*x *y - 20*x *y + 105*x *y - 111*x *y + 47*x *y - 7*x 996 997 3 7 3 6 3 5 3 4 3 3 3 2 998 + 16*x *y - 96*x *y + 188*x *y - 155*x *y + 44*x *y + 8*x *y 999 1000 3 3 2 8 2 7 2 6 2 5 1001 - 6*x *y + x + 10*x *y - 62*x *y + 164*x *y - 219*x *y 1002 1003 2 4 2 3 2 2 2 9 8 7 1004 + 154*x *y - 56*x *y + 10*x *y - x *y + x*y - 13*x*y + 45*x*y 1005 1006 6 5 4 3 2 8 7 6 1007 - 72*x*y + 64*x*y - 35*x*y + 12*x*y - 2*x*y + 2*y - 7*y + 9*y 1008 1009 5 4 10 10 9 2 9 9 8 3 1010 - 5*y + y ))/(x*(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + x *y 1011 1012 8 2 8 8 7 4 7 3 7 2 1013 - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y 1014 1015 7 7 6 5 6 4 6 3 6 2 1016 + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y 1017 1018 6 6 5 6 5 5 5 4 5 3 1019 - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y 1020 1021 5 2 5 5 4 7 4 6 4 5 1022 - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y 1023 1024 4 4 4 3 4 2 4 4 3 8 1025 + 90*x *y - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y 1026 1027 3 7 3 6 3 5 3 4 3 3 3 2 1028 - 118*x *y + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y 1029 1030 3 2 9 2 8 2 7 2 6 2 5 1031 + 2*x *y + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y 1032 1033 2 4 2 3 2 2 10 9 8 1034 - 86*x *y + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y 1035 1036 7 6 5 4 3 9 8 1037 - 121*x*y + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y 1038 1039 7 6 5 1040 + 15*y - 9*y + 2*y )), 1041 1042 10 10 9 2 9 9 8 3 8 2 8 1043 x5=(2*x *y - 2*x + 7*x *y - 16*x *y + 7*x - 3*x *y - 11*x *y + 21*x *y 1044 1045 8 7 4 7 3 7 2 7 7 6 5 1046 - x - 18*x *y + 60*x *y - 46*x *y + 23*x *y - 17*x - 4*x *y 1047 1048 6 4 6 3 6 2 6 6 5 6 5 5 1049 + 38*x *y - 70*x *y + 40*x *y - 36*x *y + 20*x + 14*x *y - 86*x *y 1050 1051 5 4 5 3 5 2 5 5 4 7 1052 + 164*x *y - 182*x *y + 114*x *y - 14*x *y - 8*x + 24*x *y 1053 1054 4 6 4 5 4 4 4 3 4 2 4 1055 - 167*x *y + 387*x *y - 455*x *y + 348*x *y - 164*x *y + 32*x *y 1056 1057 4 3 8 3 7 3 6 3 5 3 4 1058 + x + 21*x *y - 130*x *y + 339*x *y - 458*x *y + 370*x *y 1059 1060 3 3 3 2 3 2 9 2 8 2 7 1061 - 211*x *y + 81*x *y - 14*x *y + 5*x *y - 43*x *y + 140*x *y 1062 1063 2 6 2 5 2 4 2 3 2 2 2 1064 - 209*x *y + 165*x *y - 86*x *y + 42*x *y - 16*x *y + 2*x *y 1065 1066 9 8 7 6 5 4 3 2 1067 - 5*x*y + 20*x*y - 32*x*y + 16*x*y + 8*x*y - 9*x*y + x*y + x*y 1068 1069 9 8 7 6 5 4 3 10 10 1070 + 2*y - 6*y + 6*y + y - 6*y + 4*y - y )/(x*y*(2*x *y - 4*x 1071 1072 9 2 9 9 8 3 8 2 8 8 1073 + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y - 31*x 1074 1075 7 4 7 3 7 2 7 7 6 5 1076 - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x - 28*x *y 1077 1078 6 4 6 3 6 2 6 6 5 6 1079 + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x - 14*x *y 1080 1081 5 5 5 4 5 3 5 2 5 5 1082 + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y - 5*x 1083 1084 4 7 4 6 4 5 4 4 4 3 4 2 1085 + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y + 97*x *y 1086 1087 4 4 3 8 3 7 3 6 3 5 1088 - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y - 237*x *y 1089 1090 3 4 3 3 3 2 3 2 9 2 8 1091 + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y - 86*x *y 1092 1093 2 7 2 6 2 5 2 4 2 3 2 2 1094 + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y - 2*x *y 1095 1096 10 9 8 7 6 5 1097 + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y - 48*x*y 1098 1099 4 3 9 8 7 6 5 1100 + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y )), 1101 1102 9 9 8 2 8 8 7 3 7 2 1103 x6=(2*(2*x *y - 4*x + 8*x *y - 24*x *y + 16*x - 2*x *y - 19*x *y 1104 1105 7 7 6 4 6 3 6 2 6 6 1106 + 50*x *y - 23*x - 20*x *y + 71*x *y - 46*x *y - 15*x *y + 12*x 1107 1108 5 5 5 4 5 3 5 2 5 5 1109 - 8*x *y + 82*x *y - 195*x *y + 155*x *y - 46*x *y + 2*x 1110 1111 4 6 4 5 4 4 4 3 4 2 4 1112 + 8*x *y - 11*x *y - 81*x *y + 184*x *y - 142*x *y + 46*x *y 1113 1114 4 3 6 3 5 3 4 3 3 3 2 3 1115 - 4*x - 21*x *y + 50*x *y + x *y - 60*x *y + 49*x *y - 14*x *y 1116 1117 3 2 8 2 7 2 6 2 5 2 4 2 3 1118 + x + 6*x *y - 34*x *y + 82*x *y - 99*x *y + 54*x *y - 8*x *y 1119 1120 2 2 2 8 7 6 5 4 1121 - 4*x *y + x *y - 6*x*y + 38*x*y - 79*x*y + 78*x*y - 41*x*y 1122 1123 3 2 7 6 5 4 3 10 1124 + 11*x*y - x*y - 4*y + 10*y - 10*y + 5*y - y ))/(y*(2*x *y 1125 1126 10 9 2 9 9 8 3 8 2 8 1127 - 4*x + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y 1128 1129 8 7 4 7 3 7 2 7 7 1130 - 31*x - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x 1131 1132 6 5 6 4 6 3 6 2 6 6 1133 - 28*x *y + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x 1134 1135 5 6 5 5 5 4 5 3 5 2 5 1136 - 14*x *y + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y 1137 1138 5 4 7 4 6 4 5 4 4 4 3 1139 - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y 1140 1141 4 2 4 4 3 8 3 7 3 6 1142 + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y 1143 1144 3 5 3 4 3 3 3 2 3 2 9 1145 - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y 1146 1147 2 8 2 7 2 6 2 5 2 4 2 3 1148 - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y 1149 1150 2 2 10 9 8 7 6 1151 - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y 1152 1153 5 4 3 9 8 7 6 5 1154 - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y )), 1155 1156 7 2 7 7 6 3 6 2 6 6 5 4 1157 x7=(6*(x *y - 2*x *y + x + x *y - 4*x *y + 5*x *y - 2*x - 6*x *y 1158 1159 5 3 5 2 5 5 4 5 4 4 1160 + 26*x *y - 38*x *y + 21*x *y - 3*x - 8*x *y + 49*x *y 1161 1162 4 3 4 2 4 4 3 6 3 5 3 4 1163 - 106*x *y + 101*x *y - 41*x *y + 5*x - x *y + 12*x *y - 42*x *y 1164 1165 3 3 3 2 3 3 2 7 2 6 2 5 1166 + 69*x *y - 52*x *y + 15*x *y - x + 4*x *y - 27*x *y + 59*x *y 1167 1168 2 4 2 3 2 2 2 8 7 6 1169 - 52*x *y + 14*x *y + 3*x *y - x *y + 3*x*y - 18*x*y + 39*x*y 1170 1171 5 4 3 2 7 6 5 4 3 1172 - 48*x*y + 34*x*y - 11*x*y + x*y + 2*y - 5*y + 6*y - 4*y + y ) 1173 1174 10 10 9 2 9 9 8 3 8 2 1175 )/(x*(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y 1176 1177 8 8 7 4 7 3 7 2 7 1178 + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y + 18*x *y 1179 1180 7 6 5 6 4 6 3 6 2 6 1181 + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y - 104*x *y 1182 1183 6 5 6 5 5 5 4 5 3 5 2 1184 + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y - 278*x *y 1185 1186 5 5 4 7 4 6 4 5 4 4 1187 + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y 1188 1189 4 3 4 2 4 4 3 8 3 7 1190 - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y 1191 1192 3 6 3 5 3 4 3 3 3 2 3 1193 + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y 1194 1195 2 9 2 8 2 7 2 6 2 5 1196 + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y 1197 1198 2 4 2 3 2 2 10 9 8 1199 - 86*x *y + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y 1200 1201 7 6 5 4 3 9 8 1202 - 121*x*y + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y 1203 1204 7 6 5 1205 + 15*y - 9*y + 2*y )), 1206 1207 9 8 2 8 8 7 3 7 2 7 1208 x8=(2*( - 2*x + x *y - 10*x *y + 13*x + 5*x *y - 24*x *y + 49*x *y 1209 1210 7 6 4 6 3 6 2 6 6 5 5 1211 - 30*x + 8*x *y - 41*x *y + 75*x *y - 78*x *y + 32*x + 7*x *y 1212 1213 5 4 5 3 5 2 5 5 4 6 4 5 1214 - 35*x *y + 61*x *y - 56*x *y + 41*x *y - 16*x - x *y + 9*x *y 1215 1216 4 4 4 3 4 2 4 4 3 7 3 6 1217 - 10*x *y + 15*x *y - 22*x *y + 6*x *y + 3*x - 10*x *y + 57*x *y 1218 1219 3 5 3 4 3 3 3 2 3 2 8 1220 - 107*x *y + 91*x *y - 55*x *y + 34*x *y - 10*x *y - 8*x *y 1221 1222 2 7 2 6 2 5 2 4 2 3 2 2 1223 + 46*x *y - 105*x *y + 116*x *y - 63*x *y + 23*x *y - 11*x *y 1224 1225 2 9 8 7 6 5 4 1226 + 2*x *y - 2*x*y + 16*x*y - 42*x*y + 54*x*y - 34*x*y + 6*x*y 1227 1228 3 2 8 7 6 5 4 3 10 1229 + x*y + x*y - 2*y + 6*y - 7*y + 3*y + y - y ))/(x*y*(2*x *y 1230 1231 10 9 2 9 9 8 3 8 2 8 1232 - 4*x + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y 1233 1234 8 7 4 7 3 7 2 7 7 1235 - 31*x - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x 1236 1237 6 5 6 4 6 3 6 2 6 6 1238 - 28*x *y + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x 1239 1240 5 6 5 5 5 4 5 3 5 2 5 1241 - 14*x *y + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y 1242 1243 5 4 7 4 6 4 5 4 4 4 3 1244 - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y 1245 1246 4 2 4 4 3 8 3 7 3 6 1247 + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y 1248 1249 3 5 3 4 3 3 3 2 3 2 9 1250 - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y 1251 1252 2 8 2 7 2 6 2 5 2 4 2 3 1253 - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y 1254 1255 2 2 10 9 8 7 6 1256 - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y 1257 1258 5 4 3 9 8 7 6 5 1259 - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y )), 1260 1261 7 2 7 7 6 3 6 2 6 6 5 4 1262 x9=(6*( - 2*x *y + 2*x *y + 4*x - 4*x *y + 16*x *y - 6*x *y - 8*x + x *y 1263 1264 5 3 5 2 5 5 4 5 4 4 4 3 1265 + 18*x *y - 56*x *y + 26*x *y + 3*x + 4*x *y - 6*x *y - 40*x *y 1266 1267 4 2 4 4 3 6 3 5 3 4 3 3 1268 + 82*x *y - 38*x *y + 2*x - 6*x *y + 15*x *y - 9*x *y + 32*x *y 1269 1270 3 2 3 3 2 7 2 5 2 4 2 3 1271 - 46*x *y + 19*x *y - x + x *y - 5*x *y + 2*x *y - 7*x *y 1272 1273 2 2 2 8 7 6 5 4 1274 + 10*x *y - 3*x *y - 2*x*y + 9*x*y - 4*x*y - 16*x*y + 22*x*y 1275 1276 3 7 6 5 4 3 10 10 1277 - 9*x*y - 2*y + 2*y + 2*y - 4*y + 2*y ))/(y*(2*x *y - 4*x 1278 1279 9 2 9 9 8 3 8 2 8 8 1280 + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y - 31*x 1281 1282 7 4 7 3 7 2 7 7 6 5 1283 - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x - 28*x *y 1284 1285 6 4 6 3 6 2 6 6 5 6 1286 + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x - 14*x *y 1287 1288 5 5 5 4 5 3 5 2 5 5 1289 + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y - 5*x 1290 1291 4 7 4 6 4 5 4 4 4 3 4 2 1292 + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y + 97*x *y 1293 1294 4 4 3 8 3 7 3 6 3 5 1295 - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y - 237*x *y 1296 1297 3 4 3 3 3 2 3 2 9 2 8 1298 + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y - 86*x *y 1299 1300 2 7 2 6 2 5 2 4 2 3 2 2 1301 + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y - 2*x *y 1302 1303 10 9 8 7 6 5 1304 + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y - 48*x*y 1305 1306 4 3 9 8 7 6 5 1307 + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y ))}} 1308 1309 1310 1311% The following examples were discussed in Char, B.W., Fee, G.J., 1312% Geddes, K.O., Gonnet, G.H., Monagan, M.B., Watt, S.M., "On the 1313% Design and Performance of the Maple System", Proc. 1984 Macsyma 1314% Users' Conference, G.E., Schenectady, NY, 1984, 199-219. 1315 1316% Problem 1. 1317 1318solve({ -22319*x0+25032*x1-83247*x2+67973*x3+54189*x4 1319 -67793*x5+81135*x6+22293*x7+27327*x8+96599*x9-15144, 1320 79815*x0+37299*x1-28495*x2-52463*x3+25708*x4 -55333*x5- 1321 2742*x6+83127*x7-29417*x8-43202*x9+93314, -29065*x0-77803*x1- 1322 49717*x2-64748*x3-68324*x4 -50162*x5-64222*x6- 1323 4716*x7+30737*x8+22971*x9+90348, 62470*x0+59658*x1- 1324 46120*x2+58376*x3-28208*x4 -74506*x5+28491*x6+21099*x7+29149*x8- 1325 20387*x9+36254, -98233*x0-26263*x1-63227*x2+34307*x3+92294*x4 1326 +10148*x5+3192*x6+24044*x7-83764*x8-1121*x9+13871, 1327 -20427*x0+62666*x1+27330*x2-78670*x3+9036*x4 +56024*x5-4525*x6- 1328 50589*x7-62127*x8-32846*x9+38466, 1329 -85609*x0+5424*x1+86992*x2+59651*x3-60859*x4 -55984*x5- 1330 6061*x6+44417*x7+92421*x8+6701*x9-9459, 1331 -68255*x0+19652*x1+92650*x2-93032*x3-30191*x4 -31075*x5- 1332 89060*x6+12150*x7-78089*x8-12462*x9+1027, 55526*x0- 1333 91202*x1+91329*x2-25919*x3-98215*x4 +30554*x5+913*x6- 1334 35751*x7+17948*x8-58850*x9+66583, 40612*x0+84364*x1- 1335 83317*x2+10658*x3+37213*x4 +50489*x5+72040*x6- 1336 21227*x7+60772*x8+95114*x9-68533}); 1337 1338 1339Unknowns: {x0,x1,x2,x3,x4,x5,x6,x7,x8,x9} 1340 1341 4352444991703786550093529782474564455970663240687 1342{{x0=---------------------------------------------------, 1343 8420785423059099972039395927798127489505890997055 1344 1345 459141297061698284317621371232198410031030658042 1346 x1=---------------------------------------------------, 1347 1684157084611819994407879185559625497901178199411 1348 1349 1068462443128238131632235196977352568525519548284 1350 x2=---------------------------------------------------, 1351 1684157084611819994407879185559625497901178199411 1352 1353 1645748379263608982132912334741766606871657041427 1354 x3=---------------------------------------------------, 1355 1684157084611819994407879185559625497901178199411 1356 1357 25308331428404990886292916036626876985377936966579 1358 x4=----------------------------------------------------, 1359 42103927115295499860196979638990637447529454985275 1360 1361 17958909252564152456194678743404876001526265937527 1362 x5=----------------------------------------------------, 1363 42103927115295499860196979638990637447529454985275 1364 1365 - 50670056205024448621117426699348037457452368820774 1366 x6=-------------------------------------------------------, 1367 42103927115295499860196979638990637447529454985275 1368 1369 - 11882862555847887107599498171234654114612212813799 1370 x7=-------------------------------------------------------, 1371 42103927115295499860196979638990637447529454985275 1372 1373 - 273286267131634194631661772113331181980867938658 1374 x8=-----------------------------------------------------, 1375 8420785423059099972039395927798127489505890997055 1376 1377 46816360472823082478331070276129336252954604132203 1378 x9=----------------------------------------------------}} 1379 42103927115295499860196979638990637447529454985275 1380 1381 1382solve({ -22319*x0+25032*x1-83247*x2+67973*x3+54189*x4 1383 -67793*x5+81135*x6+22293*x7+27327*x8+96599*x9-15144, 1384 79815*x0+37299*x1-28495*x2-52463*x3+25708*x4 -55333*x5- 1385 2742*x6+83127*x7-29417*x8-43202*x9+93314, -29065*x0-77803*x1- 1386 49717*x2-64748*x3-68324*x4 -50162*x5-64222*x6- 1387 4716*x7+30737*x8+22971*x9+90348, 62470*x0+59658*x1- 1388 46120*x2+58376*x3-28208*x4-74506*x5+28491*x6+21099*x7+29149*x8- 1389 20387*x9+36254,-98233*x0-26263*x1-63227*x2+34307*x3+92294*x4 1390 +10148*x5+3192*x6+24044*x7-83764*x8-1121*x9+13871, 1391 -20427*x0+62666*x1+27330*x2-78670*x3+9036*x4 +56024*x5-4525*x6- 1392 50589*x7-62127*x8-32846*x9+38466, 1393 -85609*x0+5424*x1+86992*x2+59651*x3-60859*x4 -55984*x5- 1394 6061*x6+44417*x7+92421*x8+6701*x9-9459, 1395 -68255*x0+19652*x1+92650*x2-93032*x3-30191*x4 -31075*x5- 1396 89060*x6+12150*x7-78089*x8-12462*x9+1027, 55526*x0- 1397 91202*x1+91329*x2-25919*x3-98215*x4 +30554*x5+913*x6- 1398 35751*x7+17948*x8-58850*x9+66583, 40612*x0+84364*x1- 1399 83317*x2+10658*x3+37213*x4 +50489*x5+72040*x6- 1400 21227*x7+60772*x8+95114*x9-68533}); 1401 1402 1403Unknowns: {x0,x1,x2,x3,x4,x5,x6,x7,x8,x9} 1404 1405 4352444991703786550093529782474564455970663240687 1406{{x0=---------------------------------------------------, 1407 8420785423059099972039395927798127489505890997055 1408 1409 459141297061698284317621371232198410031030658042 1410 x1=---------------------------------------------------, 1411 1684157084611819994407879185559625497901178199411 1412 1413 1068462443128238131632235196977352568525519548284 1414 x2=---------------------------------------------------, 1415 1684157084611819994407879185559625497901178199411 1416 1417 1645748379263608982132912334741766606871657041427 1418 x3=---------------------------------------------------, 1419 1684157084611819994407879185559625497901178199411 1420 1421 25308331428404990886292916036626876985377936966579 1422 x4=----------------------------------------------------, 1423 42103927115295499860196979638990637447529454985275 1424 1425 17958909252564152456194678743404876001526265937527 1426 x5=----------------------------------------------------, 1427 42103927115295499860196979638990637447529454985275 1428 1429 - 50670056205024448621117426699348037457452368820774 1430 x6=-------------------------------------------------------, 1431 42103927115295499860196979638990637447529454985275 1432 1433 - 11882862555847887107599498171234654114612212813799 1434 x7=-------------------------------------------------------, 1435 42103927115295499860196979638990637447529454985275 1436 1437 - 273286267131634194631661772113331181980867938658 1438 x8=-----------------------------------------------------, 1439 8420785423059099972039395927798127489505890997055 1440 1441 46816360472823082478331070276129336252954604132203 1442 x9=----------------------------------------------------}} 1443 42103927115295499860196979638990637447529454985275 1444 1445 1446 1447% The next two problems give the current routines some trouble and 1448% have therefore been commented out. 1449 1450% Problem 2. 1451 1452comment 1453solve({ 81*x30-96*x21-45, -36*x4+59*x29+26, 1454 -59*x26+5*x3-33, -81*x19-92*x23-21*x17-9, -46*x29- 1455 13*x22+22*x24+83, 47*x4-47*x14-15*x26-40, 83*x30+70*x17+56*x10- 1456 31, 10*x27-90*x9+52*x21+52, -33*x20-97*x26+20*x6-76, 1457 97*x16+41*x8-13*x12+66, 16*x16-52*x10-73*x28+49, -28*x1-53*x24- 1458 x27-67, -22*x26-29*x24+73*x10+8, 88*x18+61*x19-98*x9-55, 99*x28- 1459 91*x26+26*x21-95, -6*x18+25*x7-77*x2+99, 28*x13-50*x17-52*x14-64, 1460 -50*x20+26*x11+93*x2+77, -70*x8+74*x19-94*x26+86, -18*x18-2*x16- 1461 79*x23+91, 36*x26-13*x11-53*x25-5, 10*x7+57*x16-85*x10-14, 1462 -3*x27+44*x4+52*x22-1, 21*x11+20*x25-30*x4-83, 70*x2-97*x19- 1463 41*x26-50, -51*x8+95*x12-85*x26+45, 83*x30+41*x12+50*x2+53, 1464 -4*x26+69*x8-58*x5-95, 59*x27-78*x30-66*x23+16, -10*x20-36*x11- 1465 60*x1-59}); 1466 1467 1468 1469% Problem 3. 1470comment 1471solve({ 115*x40+566*x41-378*x42+11401086415/6899901, 1472 560*x0-45*x1-506*x2-11143386403/8309444, -621*x1- 1473 328*x2+384*x3+1041841/64675, -856*x2+54*x3+869*x4-41430291/24700, 1474 596*x3-608*x4-560*x5-10773384/11075, 1475 -61*x4+444*x5+924*x6+4185100079/11278780, 67*x5-95*x6- 1476 682*x7+903866812/6618863, 196*x6+926*x7-930*x8- 1477 2051864151/2031976, -302*x7-311*x8-890*x9-14210414139/27719792, 1478 121*x8-781*x9-125*x10-4747129093/39901584, 10*x9+555*x10- 1479 912*x11+32476047/3471829, -151*x38+732*x39- 1480 397*x40+327281689/173242, 913*x10-259*x11-982*x12- 1481 18080663/5014020, 305*x11+9*x12-357*x13+1500752933/1780680, 1482 179*x12-588*x13+665*x14+8128189/51832, 406*x13+843*x14- 1483 833*x15+201925713/97774, 107*x14+372*x15+505*x16- 1484 5161192791/3486415, 720*x15-212*x16+607*x17-31529295571/7197760, 1485 951*x16-685*x17+148*x18+1034546543/711104, -654*x17- 1486 899*x18+543*x19+1942961717/1646560, 1487 -448*x18+673*x19+702*x20+856422818/1286375, 396*x19- 1488 196*x20+218*x21-4386267866/21303625, -233*x20-796*x21-373*x22- 1489 85246365829/57545250, 921*x21-368*x22+730*x23- 1490 93446707622/51330363, -424*x22+378*x23+727*x24- 1491 6673617931/3477462, -633*x23+565*x24-208*x25+8607636805/4092942, 1492 971*x24+170*x25-865*x26-25224505/18354, 937*x25+333*x26-463*x27- 1493 339307103/1025430, 494*x26-8*x27-50*x28+57395804/34695, 1494 530*x27+631*x28-193*x29-8424597157/680022, 1495 -435*x28+252*x29+916*x30+196828511/19593, 327*x29+403*x30- 1496 845*x31+8458823325/5927971, 246*x30+881*x31- 1497 394*x32+13624765321/156546826, 946*x31+169*x32-43*x33- 1498 53594199271/126093183, -146*x32+503*x33- 1499 363*x34+66802797635/15234909, -132*x33- 1500 686*x34+376*x35+8167530636/902635, -38*x34-188*x35- 1501 583*x36+1814153743/1124240, 389*x35+562*x36-688*x37- 1502 12251043951/5513560, -769*x37-474*x38-89*x39-2725415872/1235019, 1503 -625*x36-122*x37+468*x38+7725682775/4506736, 1504 839*x39+936*x40+703*x41+1912091857/1000749, 1505 -314*x41+102*x42+790*x43+7290073150/8132873, -905*x42- 1506 454*x43+524*x44-10110944527/4538233, 379*x43+518*x44-328*x45- 1507 2071620692/519645, 284*x44-979*x45+690*x46-915987532/16665, 1508 198*x45-650*x46-763*x47+548801657/11220, 974*x46+12*x47+410*x48- 1509 3831097561/51051, -498*x47-135*x48-230*x49-18920705/9282, 1510 665*x48+156*x49+34*x0-27714736/156585, -519*x49-366*x0-730*x1- 1511 2958446681/798985}); 1512 1513 1514 1515% Problem 4. 1516 1517% This one needs the Cramer code --- it takes forever otherwise. 1518 1519on cramer; 1520 1521 1522 1523solve({ -b*k8/a+c*k8/a, -b*k11/a+c*k11/a, 1524 -b*k10/a+c*k10/a+k2, 1525 -k3-b*k9/a+c*k9/a, -b*k14/a+c*k14/a, -b*k15/a+c*k15/a, 1526 -b*k18/a+c*k18/a-k2, -b*k17/a+c*k17/a, -b*k16/a+c*k16/a+k4, 1527 -b*k13/a+c*k13/a-b*k21/a+c*k21/a+b*k5/a-c*k5/a, 1528 b*k44/a-c*k44/a, -b*k45/a+c*k45/a, -b*k20/a+c*k20/a, 1529 -b*k44/a+c*k44/a, b*k46/a-c*k46/a, 1530 b**2*k47/a**2-2*b*c*k47/a**2+c**2*k47/a**2, 1531 k3, -k4, -b*k12/a+c*k12/a-a*k6/b+c*k6/b, 1532 -b*k19/a+c*k19/a+a*k7/c-b*k7/c, b*k45/a-c*k45/a, 1533 -b*k46/a+c*k46/a, -k48+c*k48/a+c*k48/b-c**2*k48/(a*b), 1534 -k49+b*k49/a+b*k49/c-b**2*k49/(a*c), a*k1/b-c*k1/b, 1535 a*k4/b-c*k4/b, a*k3/b-c*k3/b+k9, -k10+a*k2/b-c*k2/b, 1536 a*k7/b-c*k7/b, -k9, k11, b*k12/a-c*k12/a+a*k6/b-c*k6/b, 1537 a*k15/b-c*k15/b, k10+a*k18/b-c*k18/b, 1538 -k11+a*k17/b-c*k17/b, a*k16/b-c*k16/b, 1539 -a*k13/b+c*k13/b+a*k21/b-c*k21/b+a*k5/b-c*k5/b, 1540 -a*k44/b+c*k44/b, a*k45/b-c*k45/b, 1541 a*k14/c-b*k14/c+a*k20/b-c*k20/b, a*k44/b-c*k44/b, 1542 -a*k46/b+c*k46/b, -k47+c*k47/a+c*k47/b-c**2*k47/(a*b), 1543 a*k19/b-c*k19/b, -a*k45/b+c*k45/b, a*k46/b-c*k46/b, 1544 a**2*k48/b**2-2*a*c*k48/b**2+c**2*k48/b**2, 1545 -k49+a*k49/b+a*k49/c-a**2*k49/(b*c), k16, -k17, 1546 -a*k1/c+b*k1/c, -k16-a*k4/c+b*k4/c, -a*k3/c+b*k3/c, 1547 k18-a*k2/c+b*k2/c, b*k19/a-c*k19/a-a*k7/c+b*k7/c, 1548 -a*k6/c+b*k6/c, -a*k8/c+b*k8/c, -a*k11/c+b*k11/c+k17, 1549 -a*k10/c+b*k10/c-k18, -a*k9/c+b*k9/c, 1550 -a*k14/c+b*k14/c-a*k20/b+c*k20/b, 1551 -a*k13/c+b*k13/c+a*k21/c-b*k21/c-a*k5/c+b*k5/c, 1552 a*k44/c-b*k44/c, -a*k45/c+b*k45/c, -a*k44/c+b*k44/c, 1553 a*k46/c-b*k46/c, -k47+b*k47/a+b*k47/c-b**2*k47/(a*c), 1554 -a*k12/c+b*k12/c, a*k45/c-b*k45/c, -a*k46/c+b*k46/c, 1555 -k48+a*k48/b+a*k48/c-a**2*k48/(b*c), 1556 a**2*k49/c**2-2*a*b*k49/c**2+b**2*k49/c**2, k8, k11, -k15, 1557 k10-k18, -k17, k9, -k16, -k29, k14-k32, -k21+k23-k31, 1558 -k24-k30, -k35, k44, -k45, k36, k13-k23+k39, -k20+k38, 1559 k25+k37, b*k26/a-c*k26/a-k34+k42, -2*k44, k45, k46, 1560 b*k47/a-c*k47/a, k41, k44, -k46, -b*k47/a+c*k47/a, 1561 k12+k24, -k19-k25, -a*k27/b+c*k27/b-k33, k45, -k46, 1562 -a*k48/b+c*k48/b, a*k28/c-b*k28/c+k40, -k45, k46, 1563 a*k48/b-c*k48/b, a*k49/c-b*k49/c, -a*k49/c+b*k49/c, 1564 -k1, -k4, -k3, k15, k18-k2, k17, k16, k22, k25-k7, 1565 k24+k30, k21+k23-k31, k28, -k44, k45, -k30-k6, k20+k32, 1566 k27+b*k33/a-c*k33/a, k44, -k46, -b*k47/a+c*k47/a, -k36, 1567 k31-k39-k5, -k32-k38, k19-k37, k26-a*k34/b+c*k34/b-k42, 1568 k44, -2*k45, k46, a*k48/b-c*k48/b, a*k35/c-b*k35/c-k41, 1569 -k44, k46, b*k47/a-c*k47/a, -a*k49/c+b*k49/c, -k40, k45, 1570 -k46, -a*k48/b+c*k48/b, a*k49/c-b*k49/c, k1, k4, k3, -k8, 1571 -k11, -k10+k2, -k9, k37+k7, -k14-k38, -k22, -k25-k37, -k24+k6, 1572 -k13-k23+k39, -k28+b*k40/a-c*k40/a, k44, -k45, -k27, -k44, 1573 k46, b*k47/a-c*k47/a, k29, k32+k38, k31-k39+k5, -k12+k30, 1574 k35-a*k41/b+c*k41/b, -k44, k45, -k26+k34+a*k42/c-b*k42/c, 1575 k44, k45, -2*k46, -b*k47/a+c*k47/a, -a*k48/b+c*k48/b, 1576 a*k49/c-b*k49/c, k33, -k45, k46, a*k48/b-c*k48/b, 1577 -a*k49/c+b*k49/c }, 1578 {k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14, 1579 k15, k16, k17, k18, k19, k20, k21, k22, k23, k24, k25, k26, 1580 k27, k28, k29, k30, k31, k32, k33, k34, k35, k36, k37, k38, 1581 k39, k40, k41, k42, k43, k44, k45, k46, k47, k48, k49}); 1582 1583 1584{{k1=0, 1585 1586 k2=0, 1587 1588 k3=0, 1589 1590 k4=0, 1591 1592 k5=0, 1593 1594 k6=0, 1595 1596 k7=0, 1597 1598 k8=0, 1599 1600 k9=0, 1601 1602 k10=0, 1603 1604 k11=0, 1605 1606 k12=0, 1607 1608 k13=0, 1609 1610 k14=0, 1611 1612 k15=0, 1613 1614 k16=0, 1615 1616 k17=0, 1617 1618 k18=0, 1619 1620 k19=0, 1621 1622 k20=0, 1623 1624 k21=0, 1625 1626 k22=0, 1627 1628 k23=arbcomplex(14), 1629 1630 k24=0, 1631 1632 k25=0, 1633 1634 arbcomplex(15)*a 1635 k26=------------------, 1636 c 1637 1638 k27=0, 1639 1640 k28=0, 1641 1642 k29=0, 1643 1644 k30=0, 1645 1646 k31=arbcomplex(14), 1647 1648 k32=0, 1649 1650 k33=0, 1651 1652 arbcomplex(15)*b 1653 k34=------------------, 1654 c 1655 1656 k35=0, 1657 1658 k36=0, 1659 1660 k37=0, 1661 1662 k38=0, 1663 1664 k39=arbcomplex(14), 1665 1666 k40=0, 1667 1668 k41=0, 1669 1670 k42=arbcomplex(15), 1671 1672 k43=arbcomplex(16), 1673 1674 k44=0, 1675 1676 k45=0, 1677 1678 k46=0, 1679 1680 k47=0, 1681 1682 k48=0, 1683 1684 k49=0}} 1685 1686 1687off cramer; 1688 1689 1690 1691% Problem 5. 1692 1693solve ({2*a3*b3+a5*b3+a3*b5, a5*b3+2*a5*b5+a3*b5, 1694 a5*b5, a2*b2, a4*b4, a5*b1+b5+a4*b3+a3*b4, 1695 a5*b3+a5*b5+a3*b5+a3*b3, a0*b2+b2+a4*b2+a2*b4+c2+a2*b0+a2*b1, 1696 a0*b0+a0*b1+a0*b4+a3*b2+b0+b1+b4+a4*b0+a4*b1+a2*b5+a4*b4+c1+c4 1697 +a5*b2+a2*b3+c0, 1698 -1+a3*b0+a0*b3+a0*b5+a5*b0+b3+b5+a5*b4+a4*b3+a4*b5+a3*b4+a5*b1 1699 +a3*b1+c3+c5, 1700 b4+a4*b1, a5*b3+a3*b5, a2*b1+b2, a4*b5+a5*b4, a2*b4+a4*b2, 1701 a0*b5+a5*b0+a3*b4+2*a5*b4+a5*b1+b5+a4*b3+2*a4*b5+c5, 1702 a4*b0+2*a4*b4+a2*b5+b4+a4*b1+a5*b2+a0*b4+c4, 1703 c3+a0*b3+2*b3+b5+a4*b3+a3*b0+2*a3*b1+a5*b1+a3*b4, 1704 c1+a0*b1+2*b1+a4*b1+a2*b3+b0+a3*b2+b4}); 1705 1706 1707Unknowns: {a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5} 1708 1709{{a0=arbcomplex(25), 1710 1711 a2=0, 1712 1713 - 1 1714 a3=------, 1715 b1 1716 1717 a4=0, 1718 1719 a5=0, 1720 1721 b0=arbcomplex(24), 1722 1723 b1=arbcomplex(23), 1724 1725 b2=0, 1726 1727 b3=0, 1728 1729 b4=0, 1730 1731 b5=0, 1732 1733 c0= - a0*b0 + b1, 1734 1735 c1= - a0*b1 - b0 - 2*b1, 1736 1737 c2=0, 1738 1739 b0 + 2*b1 1740 c3=-----------, 1741 b1 1742 1743 c4=0, 1744 1745 c5=0}, 1746 1747 {a0=arbcomplex(20), 1748 1749 a2=arbcomplex(21), 1750 1751 a3=0, 1752 1753 a4=0, 1754 1755 a5=0, 1756 1757 b0=arbcomplex(22), 1758 1759 b1=0, 1760 1761 b2=0, 1762 1763 b3=-1, 1764 1765 b4=0, 1766 1767 b5=0, 1768 1769 c0= - a0*b0, 1770 1771 c1=a2 - b0, 1772 1773 c2= - a2*b0, 1774 1775 c3=a0 + 2, 1776 1777 c4=0, 1778 1779 c5=0}, 1780 1781 {a0=arbcomplex(17), 1782 1783 a2=0, 1784 1785 a3=0, 1786 1787 a4=0, 1788 1789 a5=0, 1790 1791 b0=arbcomplex(18), 1792 1793 b1=arbcomplex(19), 1794 1795 b2=0, 1796 1797 b3=-1, 1798 1799 b4=0, 1800 1801 b5=0, 1802 1803 c0= - a0*b0 + b1, 1804 1805 c1= - a0*b1 - b0 - 2*b1, 1806 1807 c2=0, 1808 1809 c3=a0 + 2, 1810 1811 c4=0, 1812 1813 c5=0}} 1814 1815 1816 1817% Problem 6. 1818 1819solve({2*a3*b3+a5*b3+a3*b5, a5*b3+2*a5*b5+a3*b5, 1820 a4*b4, a5*b3+a5*b5+a3*b5+a3*b3, b1, a3*b3, a2*b2, a5*b5, 1821 a5*b1+b5+a4*b3+a3*b4, a0*b2+b2+a4*b2+a2*b4+c2+a2*b0+a2*b1, 1822 b4+a4*b1, b3+a3*b1, a5*b3+a3*b5, a2*b1+b2, a4*b5+a5*b4, 1823 a2*b4+a4*b2, a0*b0+a0*b1+a0*b4+a3*b2+b0+b1+b4+a4*b0+a4*b1 1824 +a2*b5+a4*b4+c1+c4+a5*b2+a2*b3+c0,-1+a3*b0+a0*b3+a0*b5+a5*b0 1825 +b3+b5+a5*b4+a4*b3+a4*b5+a3*b4+a5*b1+a3*b1+c3+c5, 1826 a0*b5+a5*b0+a3*b4+2*a5*b4+a5*b1+b5+a4*b3+2*a4*b5+c5, 1827 a4*b0+2*a4*b4+a2*b5+b4+a4*b1+a5*b2+a0*b4+c4, 1828 c3+a0*b3+2*b3+b5+a4*b3+a3*b0+2*a3*b1+a5*b1+a3*b4, 1829 c1+a0*b1+2*b1+a4*b1+a2*b3+b0+a3*b2+b4}); 1830 1831 1832Unknowns: {a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5} 1833 1834{} 1835 1836 1837% Example cited by Bruno Buchberger 1838% in R.Janssen: Trends in Computer Algebra, 1839% Springer, 1987 1840% Geometry of a simple robot, 1841% l1,l2 length of arms 1842% ci,si cos and sin of rotation angles 1843 1844 1845solve( { c1*c2 -cf*ct*cp + sf*sp, 1846 s1*c2 - sf*ct*cp - cf*sp, 1847 s2 + st*cp, 1848 -c1*s2 - cf*ct*sp + sf*cp, 1849 -s1*s2 + sf*ct*sp - cf*cp, 1850 c2 - st*sp, 1851 s1 - cf*st, 1852 -c1 - sf*st, 1853 ct, 1854 l2*c1*c2 - px, 1855 l2*s1*c2 - py, 1856 l2*s2 + l1 - pz, 1857 c1**2 + s1**2 -1, 1858 c2**2 + s2**2 -1, 1859 cf**2 + sf**2 -1, 1860 ct**2 + st**2 -1, 1861 cp**2 + sp**2 -1}, 1862 {c1,c2,s1,s2,py,cf,ct,cp,sf,st,sp}); 1863 1864 1865 2 2 2 1866 sqrt( - l1 + 2*l1*pz + l2 - pz )*px 1867{{c1=---------------------------------------, 1868 2 2 2 1869 l1 - 2*l1*pz - l2 + pz 1870 1871 2 2 2 1872 - sqrt( - l1 + 2*l1*pz + l2 - pz ) 1873 c2=---------------------------------------, 1874 l2 1875 1876 2 2 2 2 1877 sqrt(l1 - 2*l1*pz - l2 + px + pz ) 1878 s1=---------------------------------------, 1879 2 2 2 1880 sqrt(l1 - 2*l1*pz - l2 + pz ) 1881 1882 - l1 + pz 1883 s2=------------, 1884 l2 1885 1886 py 1887 1888 2 2 2 2 2 2 2 1889 - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) 1890 =----------------------------------------------------------------------------- 1891 2 2 2 1892 sqrt(l1 - 2*l1*pz - l2 + pz ) 1893 1894 , 1895 1896 2 2 2 2 1897 sqrt(l1 - 2*l1*pz - l2 + px + pz ) 1898 cf=---------------------------------------, 1899 2 2 2 1900 sqrt(l1 - 2*l1*pz - l2 + pz ) 1901 1902 ct=0, 1903 1904 l1 - pz 1905 cp=---------, 1906 l2 1907 1908 2 2 2 1909 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px 1910 sf=------------------------------------------, 1911 2 2 2 1912 l1 - 2*l1*pz - l2 + pz 1913 1914 st=1, 1915 1916 2 2 2 1917 - sqrt( - l1 + 2*l1*pz + l2 - pz ) 1918 sp=---------------------------------------}, 1919 l2 1920 1921 2 2 2 1922 sqrt( - l1 + 2*l1*pz + l2 - pz )*px 1923 {c1=---------------------------------------, 1924 2 2 2 1925 l1 - 2*l1*pz - l2 + pz 1926 1927 2 2 2 1928 - sqrt( - l1 + 2*l1*pz + l2 - pz ) 1929 c2=---------------------------------------, 1930 l2 1931 1932 2 2 2 2 1933 sqrt(l1 - 2*l1*pz - l2 + px + pz ) 1934 s1=---------------------------------------, 1935 2 2 2 1936 sqrt(l1 - 2*l1*pz - l2 + pz ) 1937 1938 - l1 + pz 1939 s2=------------, 1940 l2 1941 1942 py 1943 1944 2 2 2 2 2 2 2 1945 - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) 1946 =----------------------------------------------------------------------------- 1947 2 2 2 1948 sqrt(l1 - 2*l1*pz - l2 + pz ) 1949 1950 , 1951 1952 2 2 2 2 1953 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) 1954 cf=------------------------------------------, 1955 2 2 2 1956 sqrt(l1 - 2*l1*pz - l2 + pz ) 1957 1958 ct=0, 1959 1960 - l1 + pz 1961 cp=------------, 1962 l2 1963 1964 2 2 2 1965 sqrt( - l1 + 2*l1*pz + l2 - pz )*px 1966 sf=---------------------------------------, 1967 2 2 2 1968 l1 - 2*l1*pz - l2 + pz 1969 1970 st=-1, 1971 1972 2 2 2 1973 sqrt( - l1 + 2*l1*pz + l2 - pz ) 1974 sp=------------------------------------}, 1975 l2 1976 1977 2 2 2 1978 sqrt( - l1 + 2*l1*pz + l2 - pz )*px 1979 {c1=---------------------------------------, 1980 2 2 2 1981 l1 - 2*l1*pz - l2 + pz 1982 1983 2 2 2 1984 - sqrt( - l1 + 2*l1*pz + l2 - pz ) 1985 c2=---------------------------------------, 1986 l2 1987 1988 2 2 2 2 1989 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) 1990 s1=------------------------------------------, 1991 2 2 2 1992 sqrt(l1 - 2*l1*pz - l2 + pz ) 1993 1994 - l1 + pz 1995 s2=------------, 1996 l2 1997 1998 2 2 2 2 2 2 2 1999 sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2000 py=--------------------------------------------------------------------------, 2001 2 2 2 2002 sqrt(l1 - 2*l1*pz - l2 + pz ) 2003 2004 2 2 2 2 2005 sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2006 cf=---------------------------------------, 2007 2 2 2 2008 sqrt(l1 - 2*l1*pz - l2 + pz ) 2009 2010 ct=0, 2011 2012 - l1 + pz 2013 cp=------------, 2014 l2 2015 2016 2 2 2 2017 sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2018 sf=---------------------------------------, 2019 2 2 2 2020 l1 - 2*l1*pz - l2 + pz 2021 2022 st=-1, 2023 2024 2 2 2 2025 sqrt( - l1 + 2*l1*pz + l2 - pz ) 2026 sp=------------------------------------}, 2027 l2 2028 2029 2 2 2 2030 sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2031 {c1=---------------------------------------, 2032 2 2 2 2033 l1 - 2*l1*pz - l2 + pz 2034 2035 2 2 2 2036 - sqrt( - l1 + 2*l1*pz + l2 - pz ) 2037 c2=---------------------------------------, 2038 l2 2039 2040 2 2 2 2 2041 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2042 s1=------------------------------------------, 2043 2 2 2 2044 sqrt(l1 - 2*l1*pz - l2 + pz ) 2045 2046 - l1 + pz 2047 s2=------------, 2048 l2 2049 2050 2 2 2 2 2 2 2 2051 sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2052 py=--------------------------------------------------------------------------, 2053 2 2 2 2054 sqrt(l1 - 2*l1*pz - l2 + pz ) 2055 2056 2 2 2 2 2057 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2058 cf=------------------------------------------, 2059 2 2 2 2060 sqrt(l1 - 2*l1*pz - l2 + pz ) 2061 2062 ct=0, 2063 2064 l1 - pz 2065 cp=---------, 2066 l2 2067 2068 2 2 2 2069 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2070 sf=------------------------------------------, 2071 2 2 2 2072 l1 - 2*l1*pz - l2 + pz 2073 2074 st=1, 2075 2076 2 2 2 2077 - sqrt( - l1 + 2*l1*pz + l2 - pz ) 2078 sp=---------------------------------------}, 2079 l2 2080 2081 2 2 2 2082 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2083 {c1=------------------------------------------, 2084 2 2 2 2085 l1 - 2*l1*pz - l2 + pz 2086 2087 2 2 2 2088 sqrt( - l1 + 2*l1*pz + l2 - pz ) 2089 c2=------------------------------------, 2090 l2 2091 2092 2 2 2 2 2093 sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2094 s1=---------------------------------------, 2095 2 2 2 2096 sqrt(l1 - 2*l1*pz - l2 + pz ) 2097 2098 - l1 + pz 2099 s2=------------, 2100 l2 2101 2102 2 2 2 2 2 2 2 2103 sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2104 py=--------------------------------------------------------------------------, 2105 2 2 2 2106 sqrt(l1 - 2*l1*pz - l2 + pz ) 2107 2108 2 2 2 2 2109 sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2110 cf=---------------------------------------, 2111 2 2 2 2112 sqrt(l1 - 2*l1*pz - l2 + pz ) 2113 2114 ct=0, 2115 2116 l1 - pz 2117 cp=---------, 2118 l2 2119 2120 2 2 2 2121 sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2122 sf=---------------------------------------, 2123 2 2 2 2124 l1 - 2*l1*pz - l2 + pz 2125 2126 st=1, 2127 2128 2 2 2 2129 sqrt( - l1 + 2*l1*pz + l2 - pz ) 2130 sp=------------------------------------}, 2131 l2 2132 2133 2 2 2 2134 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2135 {c1=------------------------------------------, 2136 2 2 2 2137 l1 - 2*l1*pz - l2 + pz 2138 2139 2 2 2 2140 sqrt( - l1 + 2*l1*pz + l2 - pz ) 2141 c2=------------------------------------, 2142 l2 2143 2144 2 2 2 2 2145 sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2146 s1=---------------------------------------, 2147 2 2 2 2148 sqrt(l1 - 2*l1*pz - l2 + pz ) 2149 2150 - l1 + pz 2151 s2=------------, 2152 l2 2153 2154 2 2 2 2 2 2 2 2155 sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2156 py=--------------------------------------------------------------------------, 2157 2 2 2 2158 sqrt(l1 - 2*l1*pz - l2 + pz ) 2159 2160 2 2 2 2 2161 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2162 cf=------------------------------------------, 2163 2 2 2 2164 sqrt(l1 - 2*l1*pz - l2 + pz ) 2165 2166 ct=0, 2167 2168 - l1 + pz 2169 cp=------------, 2170 l2 2171 2172 2 2 2 2173 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2174 sf=------------------------------------------, 2175 2 2 2 2176 l1 - 2*l1*pz - l2 + pz 2177 2178 st=-1, 2179 2180 2 2 2 2181 - sqrt( - l1 + 2*l1*pz + l2 - pz ) 2182 sp=---------------------------------------}, 2183 l2 2184 2185 2 2 2 2186 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2187 {c1=------------------------------------------, 2188 2 2 2 2189 l1 - 2*l1*pz - l2 + pz 2190 2191 2 2 2 2192 sqrt( - l1 + 2*l1*pz + l2 - pz ) 2193 c2=------------------------------------, 2194 l2 2195 2196 2 2 2 2 2197 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2198 s1=------------------------------------------, 2199 2 2 2 2200 sqrt(l1 - 2*l1*pz - l2 + pz ) 2201 2202 - l1 + pz 2203 s2=------------, 2204 l2 2205 2206 py 2207 2208 2 2 2 2 2 2 2 2209 - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2210 =----------------------------------------------------------------------------- 2211 2 2 2 2212 sqrt(l1 - 2*l1*pz - l2 + pz ) 2213 2214 , 2215 2216 2 2 2 2 2217 sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2218 cf=---------------------------------------, 2219 2 2 2 2220 sqrt(l1 - 2*l1*pz - l2 + pz ) 2221 2222 ct=0, 2223 2224 - l1 + pz 2225 cp=------------, 2226 l2 2227 2228 2 2 2 2229 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2230 sf=------------------------------------------, 2231 2 2 2 2232 l1 - 2*l1*pz - l2 + pz 2233 2234 st=-1, 2235 2236 2 2 2 2237 - sqrt( - l1 + 2*l1*pz + l2 - pz ) 2238 sp=---------------------------------------}, 2239 l2 2240 2241 2 2 2 2242 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2243 {c1=------------------------------------------, 2244 2 2 2 2245 l1 - 2*l1*pz - l2 + pz 2246 2247 2 2 2 2248 sqrt( - l1 + 2*l1*pz + l2 - pz ) 2249 c2=------------------------------------, 2250 l2 2251 2252 2 2 2 2 2253 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2254 s1=------------------------------------------, 2255 2 2 2 2256 sqrt(l1 - 2*l1*pz - l2 + pz ) 2257 2258 - l1 + pz 2259 s2=------------, 2260 l2 2261 2262 py 2263 2264 2 2 2 2 2 2 2 2265 - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2266 =----------------------------------------------------------------------------- 2267 2 2 2 2268 sqrt(l1 - 2*l1*pz - l2 + pz ) 2269 2270 , 2271 2272 2 2 2 2 2273 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) 2274 cf=------------------------------------------, 2275 2 2 2 2276 sqrt(l1 - 2*l1*pz - l2 + pz ) 2277 2278 ct=0, 2279 2280 l1 - pz 2281 cp=---------, 2282 l2 2283 2284 2 2 2 2285 sqrt( - l1 + 2*l1*pz + l2 - pz )*px 2286 sf=---------------------------------------, 2287 2 2 2 2288 l1 - 2*l1*pz - l2 + pz 2289 2290 st=1, 2291 2292 2 2 2 2293 sqrt( - l1 + 2*l1*pz + l2 - pz ) 2294 sp=------------------------------------}} 2295 l2 2296 2297 2298% Steady state computation of a prototypical chemical 2299% reaction network (the "Edelstein" network) 2300 2301solve( 2302 { alpha * c1 - beta * c1**2 - gamma*c1*c2 + epsilon*c3, 2303 -gamma*c1*c2 + (epsilon+theta)*c3 -eta *c2, 2304 gamma*c1*c2 + eta*c2 - (epsilon+theta) * c3}, 2305 {c3,c2,c1}); 2306 2307 2308 2 2309 c1*( - c1 *beta*gamma + c1*alpha*gamma - c1*beta*eta + alpha*eta) 2310{{c3=-------------------------------------------------------------------, 2311 c1*gamma*theta - epsilon*eta 2312 2313 c1*( - c1*beta*epsilon - c1*beta*theta + alpha*epsilon + alpha*theta) 2314 c2=-----------------------------------------------------------------------, 2315 c1*gamma*theta - epsilon*eta 2316 2317 c1=arbcomplex(26)}} 2318 2319 2320solve( 2321{( - 81*y1**2*y2**2 + 594*y1**2*y2 - 225*y1**2 + 594*y1*y2**2 - 3492* 2322y1*y2 - 750*y1 - 225*y2**2 - 750*y2 + 14575)/81, 2323( - 81*y2**2*y3**2 + 594*y2**2*y3 - 225*y2**2 + 594*y2*y3**2 - 3492* 2324y2*y3 - 750*y2 - 225*y3**2 - 750*y3 + 14575)/81, 2325( - 81*y1**2*y3**2 + 594*y1**2*y3 - 225*y1**2 + 594*y1*y3**2 - 3492* 2326y1*y3 - 750*y1 - 225*y3**2 - 750*y3 + 14575)/81, 2327(2*(81*y1**2*y2**2*y3 + 81*y1**2*y2*y3**2 - 594*y1**2*y2*y3 - 225*y1 2328**2*y2 - 225*y1**2*y3 + 1650*y1**2 + 81*y1*y2**2*y3**2 - 594*y1* 2329y2**2*y3 - 225*y1*y2**2 - 594*y1*y2*y3**2 + 2592*y1*y2*y3 + 2550 2330*y1*y2 - 225*y1*y3**2 + 2550*y1*y3 - 3575*y1 - 225*y2**2*y3 + 23311650*y2**2 - 225*y2*y3**2 + 2550*y2*y3 - 3575*y2 + 1650*y3**2 - 23323575*y3 - 30250))/81}, {y1,y2,y3}); 2333 2334 2335 2 2336{{y1=(99*y3 - 582*y3 2337 2338 4 3 2 2339 + 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 2340 2341 2 2342 )/(3*(9*y3 - 66*y3 + 25)), 2343 2344 2 2345 y2=(99*y3 - 582*y3 2346 2347 4 3 2 2348 - 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 2349 2350 2 2351 )/(3*(9*y3 - 66*y3 + 25)), 2352 2353 y3=arbcomplex(27)}, 2354 2355 2 2356 {y1=(99*y3 - 582*y3 2357 2358 4 3 2 2359 - 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 2360 2361 2 2362 )/(3*(9*y3 - 66*y3 + 25)), 2363 2364 2 2365 y2=(99*y3 - 582*y3 2366 2367 4 3 2 2368 + 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 2369 2370 2 2371 )/(3*(9*y3 - 66*y3 + 25)), 2372 2373 y3=arbcomplex(28)}, 2374 2375 11 11 11 2376 {y1=----,y2=----,y3=----}, 2377 3 3 3 2378 2379 - 5 - 5 - 5 2380 {y1=------,y2=------,y3=------}} 2381 3 3 3 2382 2383 2384% Another nice nonlinear system. 2385 2386solve({y=x+t^2,x=y+u^2},{x,y,u,t}); 2387 2388 2389 2 2390{{x=y - t , 2391 2392 y=arbcomplex(32), 2393 2394 u=t*i, 2395 2396 t=arbcomplex(31)}, 2397 2398 2 2399 {x=y - t , 2400 2401 y=arbcomplex(30), 2402 2403 u= - t*i, 2404 2405 t=arbcomplex(29)}} 2406 2407 2408% Example from Stan Kameny (relation between Gamma function values) 2409% containing surds in the coefficients. 2410 2411solve({x54=x14/4,x54*x34=sqrt pi/sqrt 2*x32,x32=x12/2, 2412 x12=sqrt pi, x14*x34=pi*sqrt 2}); 2413 2414 2415Unknowns: {x12,x14,x32,x34,x54} 2416 2417{{x12=sqrt(pi), 2418 2419 x14=4*arbcomplex(33), 2420 2421 sqrt(pi) 2422 x32=----------, 2423 2 2424 2425 sqrt(2)*pi 2426 x34=------------, 2427 4*x54 2428 2429 x54=arbcomplex(33)}} 2430 2431 2432% A system given by J. Hietarinta with complex coefficients. 2433 2434on complex; 2435 2436 2437 2438apu := {2*a - a6,2*b*c3 - 1,i - 2*x + 1,2*x**2 - 2*x + 1,n1 + 1}$ 2439 2440 2441 2442solve apu; 2443 2444 2445Unknowns: {a,a6,b,c3,n1,x} 2446 2447 arbcomplex(35) 2448{{a=----------------, 2449 2 2450 2451 a6=arbcomplex(35), 2452 2453 1 2454 b=------, 2455 2*c3 2456 2457 c3=arbcomplex(34), 2458 2459 n1=-1, 2460 2461 1 2462 x=-------}} 2463 1 - i 2464 2465 2466clear apu; 2467 2468 2469 2470off complex; 2471 2472 2473 2474% a trivial system which led to a wrong result: 2475 2476{a**2*b - a*b**2 + 1, a**2*b + a*b**2 - 1}$ 2477 2478 2479 2480solve ws; 2481 2482 2483Unknowns: {a,b} 2484 2485{} 2486 2487 2488% also communicated by Jarmo Hietarinta 2489 2490% More examples that can now be solved. 2491 2492solve({e^(x+y)-1,x-y},{x,y}); 2493 2494 2495{{x=log(-1),y=log(-1)},{x=0,y=0}} 2496 2497 2498solve({e^(x+y)+sin x,x-y},{x,y}); 2499 2500 2501 2*y_ 2502{{x=y,y=root_of(e + sin(y_),y_,tag_18)}} 2503 % no algebraic solution exists. 2504 2505solve({e^(x+y)-1,x-y**2},{x,y}); 2506 2507 2508 2 2 2509{{x=y ,y=0},{x=y ,y=-1}} 2510 2511 2512solve(e^(y^2) * e^y -1,y); 2513 2514 2515{y=0} 2516 2517 2518solve(e^(y^2 +y)-1,y); 2519 2520 2521{y=0} 2522 2523 2524solve(e^(y^2)-1,y); 2525 2526 2527{y=0} 2528 2529 2530solve(e^(y^2+1)-1,y); 2531 2532 2533{y=i,y= - i} 2534 2535 2536solve({e^(x+y+z)-1,x-y**2=1,x**2-z=2},{x,y,z}); 2537 2538 2539 2 2540{{x=y + 1, 2541 2542 1 1 2543 asinh(---) asinh(---) 2544 2 2 2545 y=sqrt(3)*cosh(------------)*i + sinh(------------), 2546 3 3 2547 2548 4 2 2549 z=y + 2*y - 1}, 2550 2551 2 2552 {x=y + 1, 2553 2554 1 1 2555 asinh(---) asinh(---) 2556 2 2 2557 y= - sqrt(3)*cosh(------------)*i + sinh(------------), 2558 3 3 2559 2560 4 2 2561 z=y + 2*y - 1}, 2562 2563 2 2564 {x=y + 1, 2565 2566 1 2567 asinh(---) 2568 2 2569 y= - 2*sinh(------------), 2570 3 2571 2572 4 2 2573 z=y + 2*y - 1}, 2574 2575 2 4 2 2576 {x=y + 1,y=0,z=y + 2*y - 1}} 2577 2578 2579solve(e^(y^4+3y^2+y)-1,y); 2580 2581 2582 2/3 1/3 1/3 2583{y=(sqrt( - 3*(sqrt(5) + 3) - 12*(sqrt(5) + 3) *2 + 2*sqrt( 2584 2585 2/3 2/3 1/3 1/3 1/6 2586 9*(sqrt(5) + 3) *2 + (sqrt(5) + 3) *sqrt(15)*3 *3 2587 2588 1/3 1/3 1/6 1/3 2589 + 3*(sqrt(5) + 3) *sqrt(3)*3 *3 + 12*(sqrt(5) + 3) 2590 2591 1/3 1/6 1/3 1/6 1/3 1/3 1/6 2592 + 2*6 *sqrt(15)*3 + 6*6 *sqrt(3)*3 + 6*2 )*3 *3 2593 2594 2/3 1/3 1/3 2595 - 3*2 ) + (sqrt(5) + 3) *sqrt(3) - 2 *sqrt(3))/(2 2596 2597 1/6 1/6 2598 *(sqrt(5) + 3) *2 *sqrt(3))} 2599 2600 2601% Transcendental equations proposed by Roger Germundsson 2602% <roger@isy.liu.se> 2603 2604eq1 := 2*asin(x) + asin(2*x) - PI/2; 2605 2606 2607 2*asin(2*x) + 4*asin(x) - pi 2608eq1 := ------------------------------ 2609 2 2610 2611eq2 := 2*asin(x) - acos(3*x); 2612 2613 2614eq2 := - acos(3*x) + 2*asin(x) 2615 2616eq3 := acos(x) - atan(x); 2617 2618 2619eq3 := acos(x) - atan(x) 2620 2621eq4 := acos(2*x**2 - 4*x -x) - 2*asin(x); 2622 2623 2624 2 2625eq4 := acos(2*x - 5*x) - 2*asin(x) 2626 2627eq5 := 2*atan(x) - atan( 2*x/(1-x**2) ); 2628 2629 2630 2*x 2631eq5 := atan(--------) + 2*atan(x) 2632 2 2633 x - 1 2634 2635 2636sol1 := solve(eq1,x); 2637 2638 2639 sqrt(3) - 1 2640sol1 := {x=-------------} 2641 2 2642 2643sol2 := solve(eq2,x); 2644 2645 2646 sqrt(17) - 3 2647sol2 := {x=--------------} 2648 4 2649 2650sol3 := solve(eq3,x); 2651 2652 2653 sqrt(sqrt(5) - 1) 2654sol3 := {x=-------------------} 2655 sqrt(2) 2656 2657sol4 := solve(eq4,x); 2658 2659 2660sol4 := {} 2661 2662sol5 := solve(eq5,x); 2663 2664 2665sol5 := {x=arbcomplex(37)} 2666 % This solution should be the open interval 2667 % (-1,1). 2668 2669% Example 52 of M. Wester: the function has no real zero although 2670% REDUCE 3.5 and Maple tend to return 3/4. 2671 2672if solve(sqrt(x^2 +1) - x +2,x) neq {} then rederr "Illegal result"; 2673 2674 2675 2676% Using a root_of expression as an algebraic number. 2677 2678solve(x^5 - x - 1,x); 2679 2680 2681 5 2682{x=root_of(x_ - x_ - 1,x_,tag_24)} 2683 2684 2685w:=rhs first ws; 2686 2687 2688 5 2689w := root_of(x_ - x_ - 1,x_,tag_24) 2690 2691 2692w^5; 2693 2694 2695 5 2696root_of(x_ - x_ - 1,x_,tag_24) + 1 2697 2698 2699w^5-w; 2700 2701 27021 2703 2704 2705clear w; 2706 2707 2708 2709% The following examples come from Daniel Lichtblau of WRI and were 2710% communicated by Laurent.Bernardin from ETH Zuerich. 2711 2712solve(x-Pi/2 = cos(x+Pi),x); 2713 2714 2715{x=root_of(2*cos(x_) - pi + 2*x_,x_,tag_26)} 2716 2717 2718solve(exp(x^2+x+2)-1,x); 2719 2720 2721 sqrt(7)*i - 1 2722{x=---------------} 2723 2 2724 2725 2726solve(log(sqrt(1+z)/sqrt(z-1))=x,z); 2727 2728 2729 2*x 2730 e + 1 2731{z=----------} 2732 2*x 2733 e - 1 2734 2735 2736solve({exp(x+3*y-2)=7,3^(2*x-y+4)=2},{x,y}); 2737 2738 2739 x + 3*y 2 2740{{e - 7*e =0}, 2741 2742 2*x y 2743 {81*3 - 2*3 =0}} 2744 2745 2746solve(a*3^(c*t)+b*3^((c+a)*t),t); 2747 2748 2749 - a 2750 log(------) 2751 b 2752{t=-------------} 2753 log(3)*a 2754 2755 2756solve(log(x+sqrt(x^2+a))=b,{x}); 2757 2758 2759 2*b 2760 e - a 2761{x=----------} 2762 b 2763 2*e 2764 2765 2766solve(z=log(w)/log(2)+w^2,w); 2767 2768 2769 2 2770{w=root_of(log(w_) + log(2)*w_ - log(2)*z,w_,tag_29)} 2771 2772 2773solve(w*2^(w^2)=5,w); 2774 2775 2776 2 2777 w_ 2778{w=root_of(2 *w_ - 5,w_,tag_31)} 2779 2780 2781solve(log(x/y)=1/y^2*(x+(1/x)),y); 2782 2783 2784 x 2 2 2785{y=root_of(log(----)*x*y_ - x - 1,y_,tag_33)} 2786 y_ 2787 2788 2789solve(exp(z)=w*z^(-n),z); 2790 2791 2792 n z_ 2793{z=root_of(z_ *e - w,z_,tag_35)} 2794 2795 2796solve(-log(3)+log(2+y/3)/2-log(y/3)/2=(-I)/2*Pi,y); 2797 2798 2799 - 3 2800{y=------} 2801 5 2802 2803 2804solve(-log(x)-log(y/x)/2+log(2+y/x)/2=(-3*I)/2*Pi,y); 2805 2806 2807 - 2*x 2808{y=--------} 2809 2 2810 x + 1 2811 2812 2813solve((I+1)*log(x)+(3*I+3)*log(x+3)=7,x); 2814 2815 2816 i 3*i 4 i 3*i 3 i 3*i 2 2817{x=root_of(x_ *(x_ + 3) *x_ + 9*x_ *(x_ + 3) *x_ + 27*x_ *(x_ + 3) *x_ 2818 2819 i 3*i 7 2820 + 27*x_ *(x_ + 3) *x_ - e ,x_,tag_37)} 2821 2822 2823solve(x+sqrt(x)=1,x); 2824 2825 2826 - sqrt(5) + 3 2827{x=----------------} 2828 2 2829 2830 2831solve({cos(1/5+alpha+x)=5,cos(2/5+alpha-x)=6},{alpha,x}); 2832 2833 2834 5*alpha - 5*x + 2 2835{{cos(-------------------) - 6=0}, 2836 5 2837 2838 5*alpha + 5*x + 1 2839 {cos(-------------------) - 5=0}} 2840 5 2841 2842 2843end; 2844 2845Tested on x86_64-pc-windows CSL 2846Time (counter 1): 688 ms plus GC time: 46 ms 2847 2848End of Lisp run after 0.68+0.09 seconds 2849real 0.99 2850user 0.00 2851sys 0.04 2852