Artifact 7ae7217dff0221f71c17e3176eddacde74a07259a79711bd75c3a5ff4fea68c2:
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r36/xlog/NUMERIC.LOG
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2011-10-17 17:58:33
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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... on errcont; bounds (x,x=(1 .. 2)); 1 .. 2 bounds (2*x,x=(1 .. 2)); 2 .. 4 bounds (x**3,x=(1 .. 2)); 1 .. 8 bounds (x*y,x=(1 .. 2),y=(-1 .. 0)); - 2 .. 0 bounds (x**3+y,x=(1 .. 2),y=(-1 .. 0)); 0 .. 8 bounds (x**3/y,{x=(1 .. 2),y=(-1 .. -0.5)}); - 16 .. -1 bounds (x**3/y,x=(1 .. 2),y=(-1 .. -0.5)); - 16 .. -1 % unbounded expression (pole at y=0) bounds (x**3/y,x=(1 .. 2),y=(-1 .. 0.5)); ***** unbounded in range on rounded; bounds(e**x,x=(1 .. 2)); 2.71828182846 .. 7.38905609893 bounds((1/2)**x,x=(1 .. 2)); 0.25 .. 0.5 off rounded; bounds(abs x,x=(1 .. 2)); 1 .. 2 bounds(abs x,x=(-3 .. 2)); 0 .. 3 bounds(abs x,x=(-3 .. -2)); 2 .. 3 bounds(sin x,x=(1 .. 2)); - 1 .. 1 on rounded; bounds(sin x,x=(1 .. 2)); 0.841470984808 .. 1 bounds(sin x,x=(1 .. 10)); - 1 .. 1 bounds(sin x,x=(1001 .. 1002)); 0.167266541974 .. 0.919990597586 bounds(log x,x=(1 .. 10)); 0 .. 2.30258509299 bounds(tan x,x=(1 .. 1.1)); 1.55740772465 .. 1.96475965725 bounds(cot x,x=(1 .. 1.1)); 0.508968105239 .. 0.642092615934 bounds(asin x,x=(-0.6 .. 0.6)); - 0.643501108793 .. 0.643501108793 bounds(acos x,x=(-0.6 .. 0.6)); 0.927295218002 .. 2.21429743559 bounds(sqrt(x),x=(1 .. 1.1)); 1 .. 1.04880884817 bounds(x**(7/3),x=(1 .. 1.1)); 1 .. 1.2490589397 bounds(x**y,x=(1 .. 1.1),y=(2 .. 4)); 1 .. 1.4641 off rounded; % MINIMA (steepest descent) % Rosenbrock function (minimum extremely hard to find). fktn := 100*(x1^2-x2)^2 + (1-x1)^2; 4 2 2 2 fktn := 100*x1 - 200*x1 *x2 + x1 - 2*x1 + 100*x2 + 1 num_min(fktn, x1=-1.2, x2=1, accuracy=6); {4.30709181812,{x1= - 1.01283672884,x2=1.0763931719}} % infinitely many local minima num_min(sin(x)+x/5, x=1); { - 1.31699384718,{x= - 1.96120922347}} % bivariate polynomial num_min(x^4 + 3 x^2 * y + 5 y^2 + x + y, x=0.1, y=0.2); { - 0.0682649733788,{x= - 0.027563873416,y= - 0.143877701468}} % ROOTS (non polynomial: damped Newton) num_solve (cos x -x, x=0,accuracy=6); {x=0.739085133215} % automatically randomized starting point num_solve (cos x -x,x, accuracy=6); {x=0.739085133215} % syntactical errors: forms do not evaluate to purely % numerical values num_solve (cos x -x, x=a); ***** a invalid as number num_solve (cos x -a, x=0); ***** a invalid as number ***** error during function evaluation (e.g. singularity) num_solve (sin x = 0, x=3); {x=3.14159265359} % blows up: no real solution exists num_solve(sin x = 2, x=1); ***** Newton method does not converge % solution in complex plane(only fond with complex starting point): on complex; *** Domain mode rounded changed to complex-rounded num_solve(sin x = 2, x=1+i); {x=1.57079632542 + 1.31695789681*i} off complex; *** Domain mode complex-rounded changed to rounded % blows up for derivative 0 in starting point num_solve(x^2-1, x=0); ***** division by zero % succeeds because of perturbed starting point num_solve(x^2-1, x=0.1); {x=1.00000000033} % bivariate equation system num_solve({sin x=cos y, x + y = 1},{x=1,y=2}); {x= - 52.1216769476,y=53.1216769476} on rounded,evallhseqp; sub(ws,{sin x=cos y, x + y = 1}); { - 0.959549629985= - 0.959549629985,1=1} off rounded,evallhseqp; % temporal member of the Barry Simon test sequence sys :={sin (x) + y^2 + log(z) = 7, 3*x + 2^y - z^3 = -90, x^2 + y^2 + z^(1/2) = 16}; 2 sys := {sin(x) + y + log(z)=7, y 3 3*x + 2 - z =-90, 2 2 1/2 x + y + z =16} sol:=num_solve(sys,{x=1,y=1,z=1}); sol := {x=2.93087675819,y= - 2.29328251176,z=4.62601269017} on rounded; for each s in sys collect sub(sol,lhs s-rhs s); {0,0,0} off rounded; clear sys,sol; % 2 examples taken from Nowak/Weimann (Tech.Rep TR91-10, ZIB Berlin) % #1: exp/sin combination on rounded; sys := {e**(x1**2 + x2**2)-3, x1 + x2 - sin(3(x1 + x2))}; 2 2 x1 + x2 sys := {e - 3, - sin(3*x1 + 3*x2) + x1 + x2} num_solve(sys,x1=0.81, x2=0.82); *** precision increased to 14 *** precision increased to 18 *** precision increased to 21 *** precision increased to 24 {x1= - 0.256625076922,x2=1.01624596361} sub(ws,sys); {0,0} % 2nd example (semiconductor simulation), here computed with % intermediate steps printed alpha := 38.683; alpha := 38.683 ni := 1.22e10; ni := 1.22e+10 v := 100; v := 100 d := 1e17; d := 1.0e+17 sys := { e**(alpha*(x3-x1)) - e**(alpha*(x1-x2)) - d/ni, x2, x3, e**(alpha*(x6-x4)) - e**(alpha*(x4-x5)) + d/ni, x5 - v, x6 - v}; 77.366*x1 38.683*x1 + 38.683*x2 sys := {( - e - 8.19672131148e+6*e 38.683*x2 + 38.683*x3 38.683*x1 + 38.683*x2 + e )/e , x2, x3, 77.366*x4 38.683*x4 + 38.683*x5 ( - e + 8.19672131148e+6*e 38.683*x5 + 38.683*x6 38.683*x4 + 38.683*x5 + e )/e , x5 - 100, x6 - 100} on trnumeric; num_solve(sys,x1=1,x2=2,x3=3,x4=4,x5=5,x6=6,iterations=100); *** computing symbolic Jacobian *** starting Newton iteration 1. residue=(1.46329673989e+33 , 0 , 0 , 1.46329673988e+33 , 0.0 , 0.0) , step length=163.473870223 at ( - 1.97414885092 , 0 , 0 , 98.0258511491 , 100.0 , 100.0) 2. residue=(5.38316786938e+32 , 0.0 , 0.0 , 5.38316786935e+32 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.94829770183 , 0.0 , 0.0 , 98.0517022982 , 100.0 , 100.0) 3. residue=(1.98035678752e+32 , 0.0 , 0.0 , 1.98035678751e+32 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.92244655275 , 0.0 , 0.0 , 98.0775534473 , 100.0 , 100.0) 4. residue=(7.28532548313e+31 , 0.0 , 0.0 , 7.28532548309e+31 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.89659540367 , 0.0 , 0.0 , 98.1034045963 , 100.0 , 100.0) 5. residue=(2.68012146749e+31 , 0.0 , 0.0 , 2.68012146747e+31 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.87074425458 , 0.0 , 0.0 , 98.1292557454 , 100.0 , 100.0) 6. residue=(9.8596158773e+30 , 0.0 , 0.0 , 9.85961587725e+30 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.8448931055 , 0.0 , 0.0 , 98.1551068945 , 100.0 , 100.0) 7. residue=(3.62714997911e+30 , 0.0 , 0.0 , 3.62714997909e+30 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.81904195641 , 0.0 , 0.0 , 98.1809580436 , 100.0 , 100.0) 8. residue=(1.33435390736e+30 , 0.0 , 0.0 , 1.33435390735e+30 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.79319080733 , 0.0 , 0.0 , 98.2068091927 , 100.0 , 100.0) 9. residue=(4.90881369764e+29 , 0.0 , 0.0 , 4.90881369762e+29 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.76733965825 , 0.0 , 0.0 , 98.2326603418 , 100.0 , 100.0) 10. residue=(1.8058516399e+29 , 0.0 , 0.0 , 1.80585163991e+29 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.74148850916 , 0.0 , 0.0 , 98.2585114908 , 100.0 , 100.0) 11. residue=(6.64335692126e+28 , 0.0 , 0.0 , 6.64335692127e+28 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.71563736008 , 0.0 , 0.0 , 98.2843626399 , 100.0 , 100.0) 12. residue=(2.4439544317e+28 , 0.0 , 0.0 , 2.4439544317e+28 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.689786211 , 0.0 , 0.0 , 98.310213789 , 100.0 , 100.0) 13. residue=(8.99080590581e+27 , 0.0 , 0.0 , 8.99080590582e+27 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.66393506191 , 0.0 , 0.0 , 98.3360649381 , 100.0 , 100.0) 14. residue=(3.30753265231e+27 , 0.0 , 0.0 , 3.30753265232e+27 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.63808391283 , 0.0 , 0.0 , 98.3619160872 , 100.0 , 100.0) 15. residue=(1.21677326379e+27 , 0.0 , 0.0 , 1.21677326379e+27 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.61223276375 , 0.0 , 0.0 , 98.3877672363 , 100.0 , 100.0) 16. residue=(4.47625868315e+26 , 0.0 , 0.0 , 4.47625868316e+26 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.58638161466 , 0.0 , 0.0 , 98.4136183853 , 100.0 , 100.0) 17. residue=(1.64672354289e+26 , 0.0 , 0.0 , 1.6467235429e+26 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.56053046558 , 0.0 , 0.0 , 98.4394695344 , 100.0 , 100.0) 18. residue=(6.05795736724e+25 , 0.0 , 0.0 , 6.05795736725e+25 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.5346793165 , 0.0 , 0.0 , 98.4653206835 , 100.0 , 100.0) 19. residue=(2.2285979709e+25 , 0.0 , 0.0 , 2.2285979709e+25 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.50882816741 , 0.0 , 0.0 , 98.4911718326 , 100.0 , 100.0) 20. residue=(8.19855376131e+24 , 0.0 , 0.0 , 8.19855376132e+24 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.48297701833 , 0.0 , 0.0 , 98.5170229817 , 100.0 , 100.0) 21. residue=(3.01607937612e+24 , 0.0 , 0.0 , 3.01607937613e+24 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.45712586924 , 0.0 , 0.0 , 98.5428741308 , 100.0 , 100.0) 22. residue=(1.10955359542e+24 , 0.0 , 0.0 , 1.10955359542e+24 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.43127472016 , 0.0 , 0.0 , 98.5687252798 , 100.0 , 100.0) 23. residue=(4.08181956632e+23 , 0.0 , 0.0 , 4.08181956633e+23 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.40542357108 , 0.0 , 0.0 , 98.5945764289 , 100.0 , 100.0) 24. residue=(1.50161750102e+23 , 0.0 , 0.0 , 1.50161750102e+23 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.37957242199 , 0.0 , 0.0 , 98.620427578 , 100.0 , 100.0) 25. residue=(5.52414207128e+22 , 0.0 , 0.0 , 5.52414207133e+22 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.35372127291 , 0.0 , 0.0 , 98.6462787271 , 100.0 , 100.0) 26. residue=(2.03221829814e+22 , 0.0 , 0.0 , 2.03221829815e+22 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.32787012383 , 0.0 , 0.0 , 98.6721298762 , 100.0 , 100.0) 27. residue=(7.47611331856e+21 , 0.0 , 0.0 , 7.47611331863e+21 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.30201897474 , 0.0 , 0.0 , 98.6979810253 , 100.0 , 100.0) 28. residue=(2.75030838977e+21 , 0.0 , 0.0 , 2.75030838979e+21 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.27616782566 , 0.0 , 0.0 , 98.7238321743 , 100.0 , 100.0) 29. residue=(1.01178191348e+21 , 0.0 , 0.0 , 1.01178191349e+21 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.25031667658 , 0.0 , 0.0 , 98.7496833234 , 100.0 , 100.0) 30. residue=(3.72213764917e+20 , 0.0 , 0.0 , 3.72213764921e+20 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.22446552749 , 0.0 , 0.0 , 98.7755344725 , 100.0 , 100.0) 31. residue=(1.36929791834e+20 , 0.0 , 0.0 , 1.36929791835e+20 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.19861437841 , 0.0 , 0.0 , 98.8013856216 , 100.0 , 100.0) 32. residue=(5.03736552996e+19 , 0.0 , 0.0 , 5.03736553001e+19 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.17276322933 , 0.0 , 0.0 , 98.8272367707 , 100.0 , 100.0) 33. residue=(1.85314321614e+19 , 0.0 , 0.0 , 1.85314321616e+19 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.14691208024 , 0.0 , 0.0 , 98.8530879198 , 100.0 , 100.0) 34. residue=(6.81733290764e+18 , 0.0 , 0.0 , 6.81733290771e+18 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.12106093116 , 0.0 , 0.0 , 98.8789390688 , 100.0 , 100.0) 35. residue=(2.50795662034e+18 , 0.0 , 0.0 , 2.50795662037e+18 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.09520978207 , 0.0 , 0.0 , 98.9047902179 , 100.0 , 100.0) 36. residue=(9.2262567997e+17 , 0.0 , 0.0 , 9.22625679991e+17 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.06935863299 , 0.0 , 0.0 , 98.930641367 , 100.0 , 100.0) 37. residue=(3.39415019556e+17 , 0.0 , 0.0 , 3.39415019568e+17 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.04350748391 , 0.0 , 0.0 , 98.9564925161 , 100.0 , 100.0) 38. residue=(1.24863807717e+17 , 0.0 , 0.0 , 1.24863807725e+17 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 1.01765633483 , 0.0 , 0.0 , 98.9823436652 , 100.0 , 100.0) 39. residue=(4.59348278034e+16 , 0.0 , 0.0 , 4.59348278107e+16 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.991805185743 , 0.0 , 0.0 , 99.0081948143 , 100.0 , 100.0) 40. residue=(1.68984787804e+16 , 0.0 , 0.0 , 1.68984787874e+16 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.965954036664 , 0.0 , 0.0 , 99.0340459633 , 100.0 , 100.0) 41. residue=(6.21660292823e+15 , 0.0 , 0.0 , 6.21660293516e+15 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.940102887593 , 0.0 , 0.0 , 99.0598971124 , 100.0 , 100.0) 42. residue=(2.28696040906e+15 , 0.0 , 0.0 , 2.28696041594e+15 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.914251738544 , 0.0 , 0.0 , 99.0857482616 , 100.0 , 100.0) 43. residue=(8.41325715099e+14 , 0.0 , 0.0 , 8.41325721962e+14 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.888400589553 , 0.0 , 0.0 , 99.1115994107 , 100.0 , 100.0) 44. residue=(3.09506431748e+14 , 0.0 , 0.0 , 3.09506438604e+14 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.862549440721 , 0.0 , 0.0 , 99.1374505601 , 100.0 , 100.0) 45. residue=(1.13861050984e+14 , 0.0 , 0.0 , 1.13861057839e+14 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.836698292322 , 0.0 , 0.0 , 99.1633017098 , 100.0 , 100.0) 46. residue=(4.18871376415e+13 , 0.0 , 0.0 , 4.18871444947e+13 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.8108471451 , 0.0 , 0.0 , 99.1891528608 , 100.0 , 100.0) 47. residue=(1.54094146219e+13 , 0.0 , 0.0 , 1.54094214749e+13 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.784996001075 , 0.0 , 0.0 , 99.2150040149 , 100.0 , 100.0) 48. residue=(5.66880467397e+12 , 0.0 , 0.0 , 5.6688115269e+12 , 0.0 , 0.0) , step length=0.0365590456369 at ( - 0.759144865742 , 0.0 , 0.0 , 99.2408551778 , 100.0 , 100.0) 49. residue=(2.08543452966e+12 , 0.0 , 0.0 , 2.08544138253e+12 , 0.0 , 0.0) , step length=0.036559045637 at ( - 0.733293754037 , 0.0 , 0.0 , 99.2667063642 , 100.0 , 100.0) 50. residue=(767186323467.0 , 0.0 , 0.0 , 767193176319.0 , 0.0 , 0.0) , step length=0.0365590456375 at ( - 0.70744270656 , 0.0 , 0.0 , 99.2925576149 , 100.0 , 100.0) 51. residue=(282229910057.0 , 0.0 , 0.0 , 282236762901.0 , 0.0 , 0.0) , step length=0.0365590456414 at ( - 0.681591833671 , 0.0 , 0.0 , 99.3184090402 , 100.0 , 100.0) 52. residue=(103824415727.0 , 0.0 , 0.0 , 103831268569.0 , 0.0 , 0.0) , step length=0.0365590456703 at ( - 0.655741435353 , 0.0 , 0.0 , 99.3442609401 , 100.0 , 100.0) 53. residue=(3.81927022457e+10 , 0.0 , 0.0 , 3.8199555087e+10 , 0.0 , 0.0) , step length=0.0365590458835 at ( - 0.629892327003 , 0.0 , 0.0 , 99.3701141301 , 100.0 , 100.0) 54. residue=(1.40481443717e+10 , 0.0 , 0.0 , 1.40549972128e+10 , 0.0 , 0.0) , step length=0.0365590474585 at ( - 0.604046724769 , 0.0 , 0.0 , 99.3959708274 , 100.0 , 100.0) 55. residue=(5.16585846951e+9 , 0.0 , 0.0 , 5.17271131074e+9 , 0.0 , 0.0) , step length=0.0365590590969 at ( - 0.578210650353 , 0.0 , 0.0 , 99.4218370614 , 100.0 , 100.0) 56. residue=(1.89824960589e+9 , 0.0 , 0.0 , 1.90510244878e+9 , 0.0 , 0.0) , step length=0.0365591450932 at ( - 0.552400454575 , 0.0 , 0.0 , 99.4477292394 , 100.0 , 100.0) 57. residue=(6.96167585052e+8 , 0.0 , 0.0 , 7.03020440594e+8 , 0.0 , 0.0) , step length=0.0365597805245 at ( - 0.526660451901 , 0.0 , 0.0 , 99.4736920939 , 100.0 , 100.0) 58. residue=(2.53957444815e+8 , 0.0 , 0.0 , 2.60810394004e+8 , 0.0 , 0.0) , step length=0.0365644757288 at ( - 0.501110133883 , 0.0 , 0.0 , 99.4998482048 , 100.0 , 100.0) 59. residue=(9.13074482936e+7 , 0.0 , 0.0 , 9.81610893493e+7 , 0.0 , 0.0) , step length=0.036599167075 at ( - 0.476067267452 , 0.0 , 0.0 , 99.526538163 , 100.0 , 100.0) 60. residue=(3.15519019482e+7 , 0.0 , 0.0 , 3.8410646839e+7 , 0.0 , 0.0) , step length=0.0368554086395 at ( - 0.452345623749 , 0.0 , 0.0 , 99.5547446298 , 100.0 , 100.0) 61. residue=(9.77481469301e+6 , 0.0 , 0.0 , 1.66708124709e+7 , 0.0 , 0.0) , step length=0.0387445972031 at ( - 0.431825342687 , 0.0 , 0.0 , 99.5876089247 , 100.0 , 100.0) 62. residue=(2.23533931681e+6 , 0.0 , 0.0 , 9.38172386018e+6 , 0.0 , 0.0) , step length=0.0527640781991 at ( - 0.417764764235 , 0.0 , 0.0 , 99.6384650755 , 100.0 , 100.0) 63. residue=(2.23262484734e+5 , 0.0 , 0.0 , 8.19715321429e+6 , 0.0 , 0.0) , step length=0.204739774363 at ( - 0.41222548566 , 0.0 , 0.0 , 99.843129903 , 100.0 , 100.0) 64. residue=(2.23088062724e+5 , 0.0 , 0.0 , 8.19035290355e+6 , 0.0 , 0.0) , step length=0.383303021247 at ( - 0.412224950141 , 0.0 , 0.0 , 100.226432924 , 100.0 , 100.0) 65. residue=(2.22216670074e+5 , 0.0 , 0.0 , 7.2288078e+6 , 0.0 , 0.0) , step length=0.129870827173 at ( - 0.412222274586 , 0.0 , 0.0 , 100.356303751 , 100.0 , 100.0) 66. residue=(1.66845393097e+5 , 0.0 , 0.0 , 1.93472870727e+6 , 0.0 , 0.0) , step length=0.048267265871 at ( - 0.412051690235 , 0.0 , 0.0 , 100.404570716 , 100.0 , 100.0) 67. residue=(1653.19388834 , 0.0 , 0.0 , - 3.3219398641e+5 , 0.0 , 0.0) , step length=0.00800369958771 at ( - 0.411535983805 , 0.0 , 0.0 , 100.412557784 , 100.0 , 100.0) 68. residue=(0.166671264823 , 0.0 , 0.0 , - 6386.15622619 , 0.0 , 0.0) , step length=0.00100689373229 at ( - 0.411530770947 , 0.0 , 0.0 , 100.411550903 , 100.0 , 100.0) 69. residue=( - 0.000000122 , 0.0 , 0.0 , - 2.48519530414 , 0.0 , 0.0) , step length=0.0000201252363669 at ( - 0.411530770421 , 0.0 , 0.0 , 100.411530778 , 100.0 , 100.0) {x1= - 0.411530770421,x2=0.0,x3=0.0,x4=100.41153077,x5=100.0,x6=100.0} off trnumeric; clear alpha,ni,v,d,sys; off rounded; % INTEGRALS num_int( x**2,x=(1 .. 2),accuracy=3); 7 --- 3 % 1st case: using formal integral needle := 1/(10**-4 + x**2); 10000 needle := -------------- 2 10000*x + 1 num_int(needle,x=(-1 .. 1),accuracy=3); 312.159332022 % 312.16 % no formal integral, but easy Chebyshev fit num_int(sin x/x,x=(1 .. 10)); 0.712264523852 % using a Chebyshev fit of order 60 num_int(exp(-x**2),x=(-10 .. 10),accuracy=3); 1.77245387654 % 1.772 % cases with singularities num_int(1/sqrt x ,x=(0 .. 1),accuracy=2); 2 % 1.999 num_int(1/sqrt abs x ,x=(-1 .. 1),iterations=50); 3.99999231465 % 3.999 % simple multidimensional integrals num_int(x+y,x=(0 .. 1),y=(2 .. 3)); 3.0 num_int(sin(x+y),x=(0 .. 1),y=(0 .. 1)); 0.773147731572 % some integrals with infinite bounds on rounded; % for the error function num_int(e^(-x) ,x=(0 .. infinity)); 1.00000034605 % 1.000 2/sqrt(pi)* num_int(e^(-x^2) ,x=(0 .. infinity)); 1.00000003784 % 1.00 2/sqrt(pi)* num_int(e^(-x^2), x=(-infinity .. infinity)); 2.00000007569 % 2.00 num_int(sin(x) * e^(-x), x=(0 .. infinity)); 0.500000522701 % 0.500 off rounded; % APPROXIMATION %approximate sin x by a cubic polynomial num_fit(sin x,{1,x,x**2,x**3},x=for i:=0:20 collect 0.1*i); 3 2 { - 0.0847539694989*x - 0.134641944765*x + 1.06263064633*x - 0.00519313406463, { - 0.00519313406463,1.06263064633, - 0.134641944765, - 0.0847539694989}} % approximate x**2 by a harmonic series in the interval [0,1] num_fit(x**2,1 . for i:=1:5 join {sin(i*x)}, x=for i:=0:10 collect i/10); { - 1.3095780871*sin(5*x) + 7.1637556683*sin(4*x) - 18.549018248*sin(3*x) + 26.5601709095*sin(2*x) - 19.4492185507*sin(x) - 0.00197199704297, { - 0.00197199704297, - 19.4492185507,26.5601709095, - 18.549018248,7.163755668 3, - 1.3095780871}} % approximate a set of points by a polynomial pts:=for i:=1 step 0.1 until 3 collect i$ vals:=for each p in pts collect (p+2)**3$ num_fit(vals,{1,x,x**2,x**3},x=pts); 3 2 {1.0*x + 5.99999999998*x + 12.0*x + 7.99999999998,{7.99999999998,12.0,5.999999 99998,1.0}} % compute the approximation error on rounded; first ws - (x+2)**3; 3 2 2.50954812486e-12*x - 0.0000000000155884194442*x + 0.0000000000306474845502*x - 0.0000000000188205007134 off rounded; % ODE SOLUTION (Runge-Kutta) depend(y,x); % approximate y=y(x) with df(y,x)=2y in interval [0 : 5] num_odesolve(df(y,x)=y,y=2,x=(0 .. 5),iterations=20); {{x,y}, {0.0,2.0}, {0.25,2.56805083337}, {0.5,3.2974425414}, {0.75,4.23400003322}, {1.0,5.43656365691}, {1.25,6.98068591491}, {1.5,8.96337814065}, {1.75,11.509205352}, {2.0,14.7781121978}, {2.25,18.9754716726}, {2.5,24.3649879213}, {2.75,31.2852637682}, {3.0,40.1710738461}, {3.25,51.5806798341}, {3.5,66.2309039169}, {3.75,85.0421639995}, {4.0,109.196300065}, {4.25,140.210824692}, {4.5,180.034262599}, {4.75,231.168569052}, {5.0,296.826318202}} % same with negative direction num_odesolve(df(y,x)=y,y=2,x=(0 .. -5),iterations=20); {{x,y}, {0.0,2.0}, {-0.25,1.55760156614}, {-0.5,1.21306131943}, {-0.75,0.944733105483}, {-1.0,0.735758882344}, {-1.25,0.573009593722}, {-1.5,0.446260320298}, {-1.75,0.347547886902}, {-2.0,0.270670566474}, {-2.25,0.210798449125}, {-2.5,0.164169997249}, {-2.75,0.127855722414}, {-3.0,0.0995741367363}, {-3.25,0.0775484156639}, {-3.5,0.060394766845}, {-3.75,0.0470354917124}, {-4.0,0.0366312777778}, {-4.25,0.0285284678182}, {-4.5,0.0222179930767}, {-4.75,0.0173033904064}, {-5.0,0.0134758939983}} % giving a nice picture when plotted num_odesolve(df(y,x)=1- x*y**2 ,y=0,x=(0 .. 4),iterations=20); {{x,y}, {0.0,0.0}, {0.2,0.199600912188}, {0.4,0.393714914166}, {0.6,0.569482634406}, {0.8,0.710657687564}, {1.0,0.805480022354}, {1.2,0.852604291055}, {1.4,0.860563377356}, {1.6,0.842333334456}, {1.8,0.80999200878}, {2.0,0.772211952811}, {2.2,0.734163640068}, {2.4,0.698433235122}, {2.6,0.666019196492}, {2.8,0.637070046905}, {3.0,0.611341375657}, {3.2,0.588447372601}, {3.4,0.567985133759}, {3.6,0.549587947292}, {3.8,0.532942255624}, {4.0,0.517787833735}} % system of ordinary differential equations depend(y,x); depend(z,x); num_odesolve( {df(z,x) = y, df(y,x)= y+x}, {z=2, y=4}, x=(0 .. 5),iterations=20); {{x,z,y}, {0.0,2.0,4.0}, {0.25,3.13887708344,5.17012708344}, {0.5,4.61860635349,6.74360635349}, {0.75,6.55375008305,8.83500008305}, {1.0,9.09140914227,11.5914091423}, {1.25,12.4204647873,15.2017147873}, {1.5,16.7834453516,19.9084453516}, {1.75,22.4917633799,26.0230133799}, {2.0,29.9452804945,33.9452804945}, {2.25,39.6574291816,44.1886791816}, {2.5,52.2874698032,57.4124698032}, {2.75,68.6819094205,74.4631594205}, {3.0,89.9276846154,96.4276846154}, {3.25,117.420449585,124.701699585}, {3.5,152.952259792,161.077259792}, {3.75,198.824159999,207.855409999}, {4.0,257.990750164,267.990750164}, {4.25,334.245811731,345.277061731}, {4.5,432.460656499,444.585656499}, {4.75,558.890172631,572.171422631}, {5.0,721.565795506,736.065795506}} %----------------- Chebyshev fit ------------------------- on rounded; func := x**2 * (x**2 - 2) * sin x; 2 2 func := sin(x)*x *(x - 2) ord := 15; ord := 15 cx:=chebyshev_fit(func,x=(0 .. 2),ord)$ cp:=first cx; 13 12 11 cp := 0.000000620105096185*x + 0.0000168737305191*x - 0.000269014332288*x 10 9 8 + 0.000155646029006*x + 0.00848163760265*x + 0.000272748604876*x 7 6 5 - 0.183540091904*x + 0.00010680840443*x + 1.33329694616*x 4 3 2 + 0.00000770692780683*x - 2.00000091554*x + 0.0000000501515695639*x - 0.000000000784989184766*x - 4.86055640181e-13 cc:=second cx; cc := {2.69320512829, 2.76751928466, 2.25642507569, 0.955452569949, 0.0509075944268, - 0.0868248678183, - 0.0170919216091, 0.00104527137626, 0.000349190502034, - 0.00000253521592323, - 0.00000280798840641, - 0.0000000157676044858, 0.0000000121753402195, 0.000000000118269801846, - 0.0000000000331230439026} for u:=0 step 0.2 until 2 do write "x:",u," true value:",sub(x=u,func), " Chebyshev eval:", chebyshev_eval(cc,x=(0 .. 2),x=u), " Chebyshev polynomial:",sub(x=u,cp); x:0 true value:0 Chebyshev eval: - 4.85167461761e-13 Chebyshev polynomial: - 4.86055640181e-13 x:0.2 true value: - 0.0155756755343 Chebyshev eval: - 0.0155756755339 Chebyshev polynomial: - 0.015575675548 x:0.4 true value: - 0.114644759976 Chebyshev eval: - 0.114644759976 Chebyshev polynomial: - 0.114644759974 x:0.6 true value: - 0.333364916292 Chebyshev eval: - 0.333364916292 Chebyshev polynomial: - 0.333364916295 x:0.8 true value: - 0.624386741519 Chebyshev eval: - 0.624386741519 Chebyshev polynomial: - 0.624386741504 x:1 true value: - 0.841470984808 Chebyshev eval: - 0.841470984808 Chebyshev polynomial: - 0.841470984841 x:1.2 true value: - 0.751596318924 Chebyshev eval: - 0.751596318924 Chebyshev polynomial: - 0.751596318876 x:1.4 true value: - 0.0772592588311 Chebyshev eval: - 0.0772592588311 Chebyshev polynomial: - 0.0772592588864 x:1.6 true value:1.43298871732 Chebyshev eval:1.43298871732 Chebyshev polynomial:1.43298871738 x:1.8 true value:3.91253024182 Chebyshev eval:3.91253024182 Chebyshev polynomial:3.91253024177 x:2.0 true value:7.27437941461 Chebyshev eval:7.27437941461 Chebyshev polynomial:7.27437941467 % integral % integrate coefficients ci := chebyshev_int(cc,x=(0 .. 2)); ci := {0.0310113015322, 0.2183900263, 0.453016678678, 0.367586246877, 0.130284679721, 0.00679995160359, - 0.00732251159954, - 0.00124579372222, 0.0000654879120115, 0.0000195554716911, - 0.000000125972415937, - 0.000000128189261211, - 0.000000000661911428653, 0.000000000469556279362, 4.22392149449e-12} % compare with true values (normalized absolute term) ci0:=chebyshev_eval(ci,x=(0 .. 2),x=0)$ ifunc := int(func,x)$ if0 := sub(x=0,ifunc); if0 := - 28.0 for u:=0 step 0.2 until 2 do write {u,sub(x=u,ifunc) - if0, chebyshev_eval(ci,x=(0 .. 2),x=u) - ci0}; {0,0,0} {0.2, - 0.000785836355117, - 0.00078583635293} {0.4, - 0.0119047051867, - 0.0119047051858} {0.6, - 0.0548116700418, - 0.0548116700408} {0.8, - 0.150297976106, - 0.150297976105} {1, - 0.299838223412, - 0.29983822341} {1.2, - 0.466528961073, - 0.466528961072} {1.4, - 0.561460555384, - 0.561460555383} {1.6, - 0.441445769516, - 0.441445769514} {1.8,0.0768452822437,0.0768452822437} {2.0,1.18309971762,1.18309971762} % derivative % differentiate coefficients cd := chebyshev_df(cc,x=(0 .. 2))$ % compute coefficients of derivative cds := second chebyshev_fit(df(func,x),x=(0 .. 2),ord)$ % compare coefficients for i:=1:ord do write {part(cd,i),part(cds,i)}; {10.4140931324,10.4140931324} {9.23338917839,9.2333891784} {4.87905456308,4.87905456307} {0.207688875651,0.207688875654} { - 0.853660856614, - 0.853660856625} { - 0.199571879764, - 0.19957187976} {0.0145878215687,0.0145878215579} {0.00553117954514,0.00553117954883} { - 0.000045977698902, - 0.0000459777097756} { - 0.0000558684874082, - 0.0000558684837245} { - 0.00000034381228384, - 0.00000034382315144} {0.000000291280720039,0.00000029128440905} {0.00000000307501484799,0.00000000306414359587} { - 0.000000000927445229273, - 0.000000000923750392908} {0, - 0.0000000000109242472928} clear func,ord,cc,cx,cd,cds,ci,ci0; off rounded; end; (TIME: numeric 34570 36600)