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% Tests of the root finding package. % Author: Stanley L. Kameny (stan%valley.uucp@rand.org) comment Answers are rounded to the value given by rootacc (default = 6) unless higher accuracy is needed to separate roots. Root order and format may differ from that given here, but values should agree. In the following, problems 20), 78) and 82) are time consuming and been commented out to speed up the test. The hard examples 111) through 115) almost double the test time but are necessary to test some logical paths.; roots x; % To load roots package. write "This is Roots Package test ", symbolic roots!-mod$ % Simple root finding. showtime; % 1) multiple real and imaginary roots plus two real roots. zz:= (x-3)**2*(100x**2+113)**2*(1000000x-10000111)*(x-1); roots zz; %{X=1.06301*I,X=1.06301*I,X=-1.06301*I,X=-1.06301*I, %X=3.0,X=3.0,X=1,X=10.0001} (rootacc caused rounding to 6 places) % Accuracy is increased whenever necessary to separate distinct roots. % 2) accuracy increase to 7 required for two roots. zz:=(x**2+1)*(x-2)*(1000000x-2000001); roots zz; %{X=2.0,X=I,X= - I,X=2.000001} % 3) accuracy increase to 8 required. zz:= (x-3)*(10000000x-30000001); roots zz; %{X=3.0,X=3.0000001} % 4) accuracy increase required here to separate repeated root from % simple root. zz := (x-3)*(1000000x-3000001)*(x-3)*(1000000x-3241234); roots zz; %{X=3.0,X=3.0,X=3.000001,X=3.24123} % other simple examples % 5) five real roots with widely different spacing. zz:= (x-1)*(10x-11)*(x-1000)*(x-1001)*(x-100000); roots zz; %{X=1,X=1.1,X=1000.0,X=1001.0,X=100000.0} % 6) a cluster of 5 roots in complex plane in vicinity of x=1. zz:= (x-1)*(10000x**2-20000x+10001)*(10000x**2-20000x+9999); roots zz; %{X=1,X=1.01,X=0.99,X=1 + 0.01*I,X=1 - 0.01*I} % 7) four closely spaced real roots. zz := (x-1)*(100x-101)*(100x-102)*(100x-103); roots zz; %{X=1.02,X=1.01,X=1,X=1.03} % 8) five closely spaced roots, 3 real + 1 complex pair. zz := (x-1)*(100x-101)*(100x-102)*(100x**2-200x+101); roots zz; %{X=1.01,X=1,X=1.02,X=1 + 0.1*I,X=1 - 0.1*I} % 9) symmetric cluster of 5 roots, 3 real + 1 complex pair. zz := (x-2)*(10000x**2-40000x+40001)*(10000x**2-40000x+39999); roots zz; %{X=2.0,X=2.01,X=1.99,X=2.0 + 0.01*I,X=2.0 - 0.01*I} % 10) closely spaced real and complex pair. ss:= (x-2)*(100000000x**2-400000000x+400000001); roots ss; %{X=2.0,X=2.0 + 0.0001*I,X=2.0 - 0.0001*I} % 11) Zero roots and multiple roots cause no problem. % Multiple roots are shown when the switch multiroot is on %(normally on.) zz:= x*(x-1)**2*(x-4)**3*(x**2+1); roots zz; %{X=0,X=4.0,X=4.0,X=4.0,X=1,X=1,X=I,X= - I} % 12) nearestroot will find a single root "near" a value, real or % complex. nearestroot(zz,2i); %{X=I} % More difficult examples. % Three examples in which root scaling is needed in the complex % iteration process. % 13) nine roots, 3 real and 3 complex pairs. zz:= x**9-45x-2; roots zz; %{X= - 0.0444444,X=1.61483, % X=0.00555 + 1.60944*I,X=0.00555 - 1.60944*I, % X= - 1.60371, % X=1.14348 + 1.13804*I,X=1.14348 - 1.13804*I, % X=-1.13237 + 1.13805*I,X=-1.13237 - 1.13805*I} comment In the next two examples, there are complex roots with extremely small real parts (new capability in Mod 1.91.); % 14) nine roots, 1 real and 4 complex pairs. zz:= x**9-9999x**2-0.01; roots zz; %{X=5.0E-29 + 0.00100005*I,X=5.0E-29 - 0.00100005*I, % X=3.72754, % X=-0.829456 + 3.63408*I, X=-0.829456 + 3.63408*I, % X=-3.3584 + 1.61732*I, X=-3.3584 - 1.61732*I, % X=2.32408 + 2.91431*I, X=2.32408 - 2.91431*I} comment Rootacc 7 produces 7 place accuracy. Answers will print in bigfloat format if floating point print >6 digits is not implemented.; % 15) nine roots, 1 real and 4 complex pairs. rootacc 7; zz:= x**9-500x**2-0.001; roots zz; %{X=1.6E-26 + 0.001414214*I,X=1.6E-26 - 0.001414214*I, % X=-0.540677 + 2.368861*I, X=-0.540677 - 2.368861*I, % X=-2.189157 + 1.054242*I, X=-2.189157 - 1.054242*I, % X=1.514944 + 1.899679*I, X=1.514944 - 1.899679*I} % the famous Wilkinson "ill-conditioned" polynomial and its family. % 16) W(6) four real roots plus one complex pair. zz:= 10000*(for j:=1:6 product(x+j))+27x**5; roots zz; %{X= - 2.950367,X=-4.452438 + 0.021235*I,X=-4.452438 - 0.021235*I, % X= - 0.9999775,X= - 2.003647,X= - 6.143833} % 17) W(8) 4 real roots plus 2 complex pairs. zz:= 1000*(for j:=1:8 product(x+j))+2x**7; roots zz; %{X=-4.295858 + 0.28151*I,X=-4.295858 - 0.28151*I, % X= - 2.982725, % X=-6.494828 + 1.015417*I,X=-6.494828 - 1.015417*I, % X= - 0.9999996,X= - 2.000356,X= - 8.437546} % 18) W(10) 6 real roots plus 2 complex pairs. zz:=1000*(for j:= 1:10 product (x+j))+x**9; roots zz; %{X= - 4.616444,X=-6.046279 + 1.134321*I,X=-6.046279 - 1.134321*I, % X= - 4.075943,X= - 2.998063, % X=-8.70405 + 1.691061*I,X=-8.70405 - 1.691061*I, % X= -1,X= - 2.000013,X= - 10.80988} % 19) W(12) 6 real roots plus 3 complex pairs. zz:= 10000*(for j:=1:12 product(x+j))+4x**11; roots zz; %{X=-5.985629 + 0.809425*I,X=-5.985629 - 0.809425*I, % X= - 4.880956,X= - 4.007117, % X=-7.953917 + 1.948001*I,X=-7.953917 - 1.948001*I, % X= -1,X=-11.02192 + 2.23956*I,X=-11.02192 - 2.23956*I, % X= - 2.0,X= - 2.999902,X= - 13.1895} % 20) W(20) 10 real roots plus 5 complex pairs. (The original problem) % This example is commented out, since it takes significant time: % zz:= x**19+10**7*for j:=1:20 product (x+j); roots zz; %{X=-10.12155 + 0.6013*I,X=-10.12155 - 0.6013*I, % X= - 8.928803, % X=-11.82101 + 1.59862*I,X=-11.82101 - 1.59862*I, % X= - 8.006075,X= - 6.999746, % X=-14.01105 + 2.44947*I,X=-14.01105 - 2.44947*I, % X=-1,X= - 6.000006, % X=-19.45964 + 1.87436*I,X=-19.45964 - 1.87436*I, % X= - 2.0,X= - 5.0, % X=-16.72504 + 2.73158*I,X=-16.72504 - 2.73158*I, % X= - 3.0,X= - 4.0,X= - 20.78881} rootacc 6; % 21) Finding one of a cluster of 8 roots. zz:= (10**16*(x-1)**8-1); nearestroot(zz,2); %{X=1.01} % 22) Six real roots spaced 0.01 apart. c := 100; zz:= (x-1)*for i:=1:5 product (c*x-(c+i)); roots zz; %{X=1.03,X=1.02,X=1.04,X=1.0,X=1.01,X=1.05} % 23) Six real roots spaced 0.001 apart. c := 1000; zz:= (x-1)*for i:=1:5 product (c*x-(c+i)); roots zz; %{X=1.003,X=1.002,X=1.004,X=1,X=1.001,X=1.005} % 24) Five real roots spaced 0.0001 apart. c := 10000; zz:= (x-1)*for i:=1:4 product (c*x-(c+i)); roots zz; %{X=1.0002,X=1.0003,X=1,X=1.0001,X=1.0004} % 25) A cluster of 9 roots, 5 real, 2 complex pairs; spacing 0.1. zz:= (x-1)*(10**8*(x-1)**8-1); roots zz; %{X=1,X=1.1,X=1 + 0.1*I,X=1 - 0.1*I,X=0.9, % X=0.929289 + 0.070711*I,X=0.929289 - 0.070711*I, % X=1.07071 + 0.07071*I,X=1.07071 - 0.07071*I} % 26) Same, but with spacing 0.01. zz:= (x-1)*(10**16*(x-1)**8-1); roots zz; %{X=1,X=1.01,X=1 + 0.01*I,X=1 - 0.01*I,X=0.99, % X=0.992929 + 0.007071*I,X=0.992929 - 0.007071*I, % X=1.00707 + 0.00707*I,X=1.00707 - 0.00707*I} % 27) Spacing reduced to 0.001. zz:= (x-1)*(10**24*(x-1)**8-1); roots zz; %{X=1,X=1.001,X=1 + 0.001*I,X=1 - 0.001*I,X=0.999, % X=0.999293 + 0.000707*I,X=0.999293 - 0.000707*I, % X=1.00071 + 0.000707*I,X=1.00071 - 0.000707*I} % 28) Eight roots divided into two clusters. zz:= (10**8*(x-1)**4-1)*(10**8*(x+1)**4-1); roots zz; %{X=1.01,X= - 1.01,X=1 + 0.01*I,X=-1 - 0.01*I, % X=-1 + 0.01*I,X=1 - 0.01*I,X=0.99,X= - 0.99} % 29) A cluster of 8 roots in a different configuration. zz:= (10**8*(x-1)**4-1)*(10**8*(100x-102)**4-1); roots zz; %{X=1.01, X=1.0199, X=1.02 + 0.0001*I, X=1.02 - 0.0001*I, % X=1 + 0.01*I, X=1 - 0.01*I, X=0.99, X=1.0201} % 30) A cluster of 8 complex roots. zz:= ((10x-1)**4+1)*((10x+1)**4+1); roots zz; %{X=0.0292893 + 0.0707107*I,X=-0.0292893 - 0.0707107*I, % X=-0.0292893 + 0.0707107*I,X=0.0292893 - 0.0707107*I, % X=0.170711 + 0.070711*I,X=-0.170711 - 0.070711*I, % X=-0.170711 + 0.070711*I,X=0.170711 - 0.070711*I} comment In these examples, accuracy increase is required to separate a repeated root from a simple root.; % 31) Using allroots; zz:= (x-4)*(x-3)**2*(1000000x-3000001); roots zz; %{X=3.0,X=3.0,X=3.000001,X=4.0} % 32) Using realroots; realroots zz; %{X=3.0,X=3.0,X=3.000001,X=4.0} comment Tests of new capabilities in mod 1.87 for handling complex polynomials and polynomials with very small imaginary parts or very small real roots. A few real examples are shown, just to demonstrate that these still work.; % 33) A trivial complex case (but degrees 1 and 2 are special cases); zz:= x-i; roots zz; %{X=I} % 34) Real case. zz:= y-7; roots zz; %{Y=7.0} % 35) Roots with small imaginary parts (new capability); zz := 10**16*(x**2-2x+1)+1; roots zz; %{X=1 + 0.00000001*I,X=1 - 0.00000001*I} % 36) One real, one complex root. zz:=(x-9)*(x-5i-7); roots zz; %{X=9.0,X=7.0 + 5.0*I} % 37) Three real roots. zz:= (x-1)*(x-2)*(x-3); roots zz; %{X=2.0,X=1,X=3.0} % 38) 2 real + 1 imaginary root. zz:=(x**2-8)*(x-5i); roots zz; %{X=2.82843,X= - 2.82843,X=5.0*I} % 39) 2 complex roots. zz:= (x-1-2i)*(x+2+3i); roots zz; %{X=1 + 2.0*I,X=-2.0 - 3.0*I} % 40) 2 irrational complex roots. zz:= x**2+(3+2i)*x+7i; roots zz; %{X=0.14936 - 2.21259*I,X=-3.14936 + 0.21259*I} % 41) 2 complex roots of very different magnitudes with small imaginary % parts. zz:= x**2+(1000000000+12i)*x-1000000000; roots zz; %{X=1 - 0.000000012*I,X=-1.0E+9 - 12.0*I} % 42) Multiple real and complex roots cause no difficulty, provided % that input is given in integer or rational form, (or if in decimal % fraction format, with switch adjprec on and coefficients input % explicitly,) so that polynomial is stored exactly. zz :=(x**2-2i*x+5)**3*(x-2i)*(x-11/10)**2; roots zz; %{X=-1.44949*I, X=-1.44949*I, X=-1.44949*I, % X=3.44949*I, X=3.44949*I, X=3.44949*I, X=1.1, X=1.1, X=2.0*I} % 43) 2 real, 2 complex roots. zz:= (x**2-4)*(x**2+3i*x+5i); roots zz; %{X=2.0,X= - 2.0,X=-1.2714 + 0.466333*I,X=1.2714 - 3.46633*I} % 44) 4 complex roots. zz:= x**4+(0.000001i)*x-16; roots zz; %{X=2.0 - 0.0000000625*I,X=-2.0*I,X=-2.0 - 0.0000000625*I,X=2.0*I} % 45) 2 real, 2 complex roots. zz:= (x**2-4)*(x**2+2i*x+8); roots zz; %{X=2.0,X= - 2.0,X=2.0*I,X=-4.0*I} % 46) Using realroots to find only real roots. realroots zz; %{X= - 2.0,X=2.0} % 47) Same example, applying nearestroot to find a single root. zz:= (x**2-4)*(x**2+2i*x+8); nearestroot(zz,1); %{X=2.0} % 48) Same example, but focusing on imaginary point. nearestroot(zz,i); %{X=2.0*I} % 49) The seed parameter can be complex also. nearestroot(zz,1+i); %{X=2.0*I} % 50) One more nearestroot example. Nearest root to real point may be % complex. zz:= (x**2-4)*(x**2-i); roots zz; %{X=2.0,X= - 2.0,X=0.707107 + 0.707107*I,X=-0.707107 - 0.707107*I} nearestroot (zz,1); %{X=0.707107 + 0.707107*I} % 51) 1 real root plus 5 complex roots. zz:=(x**3-3i*x**2-5x+9)*(x**3-8); roots zz; %{X=2.0, X=-1 + 1.73205*I, X=-1 - 1.73205*I, % X=0.981383 - 0.646597*I, X=-2.41613 + 1.19385*I, % X=1.43475 + 2.45274*I} nearestroot(zz,1); %{X=0.981383 - 0.646597*I} % 52) roots can be computed to any accuracy desired, eg. (note that the % imaginary part of the second root is truncated because of its size, % and that the imaginary part of a complex root is never polished away, % even if it is smaller than the accuracy would require.) zz := x**3+10**(-20)*i*x**2+8; rootacc 12; roots zz; rootacc 0; %{X=1 + 1.73205080757*I,X=1 - 1.73205080757*I, % X=-2.0 - 3.33333E-21*I} % 53) Precision increase to 12 required to find small imaginary root, % but standard accuracy can be used. zz := x**2+123456789i*x+1; roots zz; %{X=0.0000000081*I,X=-123457000.0*I} % 54) Small real root is found with root 10*18 times larger(new). zz := (x+1)*(x**2+123456789*x+1); roots zz; %{X= - 0.0000000081,X= - 1.0,X= - 123457000.0} % 55) 2 complex, 3 real irrational roots. ss := (45*x**2+(-10i+12)*x-10i)*(x**3-5x**2+1); roots ss; %{X=0.469832,X= - 0.429174,X=4.95934,X=0.18139 + 0.417083*I, % X=-0.448056 - 0.19486*I} % 56) Complex polynomial with floating coefficients. zz := x**2+1.2i*x+2.3i+6.7; roots zz; %{X=-0.42732 + 2.09121*I,X=0.42732 - 3.29121*I} % multiple roots will be found if coefficients read in exactly. ZZ := X**3 + (1.09 - 2.4*I)*X**2 + (-1.44 - 2.616*I)*X + -1.5696; roots zz; %{X=1.2*I,X=1.2*I,X= - 1.09} % 57) Realroots, isolater and rlrootno accept 1, 2 or 3 arguments: (new) zz:= for j:=-1:3 product (x-j); rlrootno zz; % 5 realroots zz; %{X=0,X= -1,X=1,X=2.0,X=3.0} rlrootno(zz,positive); %positive selects positive, excluding 0. % 3 rlrootno(zz,negative); %negative selects negative, excluding 0. % 1 realroots(zz,positive); %{X=1,X=2.0,X=3.0} rlrootno(zz,-1.5,2); %the format with 3 arguments selects a range. % 4 realroots(zz,-1.5,2); %the range is inclusive, except that: %{X=0,X=-1,X=1,X=2.0} % A specific limit b may be excluded by using exclude b. Also, the % limits infinity and -infinity can be specified. realroots(zz,exclude 0,infinity); % equivalent to realroots(zz,positive). %{X=1,X=2.0,X=3.0} rlrootno(zz,-infinity,exclude 0); % equivalent to rlrootno(zz,negative). % 1 rlrootno(zz,-infinity,0); % 2 rlrootno(zz,infinity,-infinity); %equivalent to rlrootno zz; (order of limits does not matter.) % 5 realroots(zz,1,infinity); % finds all real roots >= 1. %{X=1,X=2.0,X=3.0} realroots(zz,1,positive); % finds all real roots > 1. %{X=2.0,X=3.0} % New capabilities added in mod 1.88. % 58) 3 complex roots, with two separated by very small real difference. zz :=(x+i)*(x+10**8i)*(x+10**8i+1); roots zz; %{X= - I,X=-1.0E+8*I,X=-1 - 1.0E+8*I} % 59) Real polynomial with two complex roots separated by very small % imaginary part. zz:= (10**14x+123456789000000+i)*(10**14x+123456789000000-i); roots zz; %{X=-1.23457 + 1.0E-14*I,X=-1.23457 - 1.0E-14*I} % 60) Real polynomial with two roots extremely close together. zz:= (x+2)*(10**10x+12345678901)*(10**10x+12345678900); roots zz; %{X= - 1.2345678901,X= - 1.23456789,X= - 2.0} % 61) Real polynomial with multiple root extremely close to simple root. zz:= (x-12345678/10000000)*(x-12345679/10000000)**2; roots zz; %{X=1.2345679,X=1.2345679,X=1.2345678} % 62) Similar problem using realroots. zz:=(x-2**30/10**8)**2*(x-(2**30+1)/10**8); realroots zz; %{X=10.73741824,X=10.73741824,X=10.73741825} % 63) Three complex roots with small real separation between two. zz:= (x-i)*(x-1-10**8i)*(x-2-10**8i); roots zz; %{X=I,X=1 + 1.0E+8*I,X=2.0 + 1.0E+8*I} % 64) Use of nearestroot to isolate one of the close roots. nearestroot(zz,10**8i+99/100); %{X=1 + 1.0E+8*I} % 65) Slightly more complicated example with close complex roots. zz:= (x-i)*(10**8x-1234-10**12i)*(10**8x-1233-10**12i); roots zz; %{X=I,X=0.00001233 + 10000.0*I,X=0.00001234 + 10000.0*I} % 66) Four closely spaced real roots with varying spacings. zz:= (x-1+1/10**7)*(x-1+1/10**8)*(x-1)*(x-1-1/10**7); roots zz; %{X=1,X=0.9999999,X=0.99999999,X=1.0000001} % 67) Complex pair plus two close real roots. zz:= (x**2+1)*(x-12345678/10000000)*(x-12345679/10000000); roots zz; %{X=1.2345678,X=1.2345679,X=I,X= - I} % 68) Same problem using realroots to find only real roots. realroots zz; %{X=1.2345678,X=1.2345679} % The switch ratroot causes output to be given in rational form. % 69) Two complex roots with output in rational form. on ratroot,complex; zz:=x**2-(5i+1)*x+1; sss:= roots zz; % 17343 - 93179*I 96531 + 518636*I %SSS := {X=-----------------,X=------------------} % 500000 100000 % With roots in rational form, mkpoly can be used to reconstruct a % polynomial. zz1 := mkpoly sss; % 2 %ZZ1 := 50000000000*X - (49999800000 + 250000100000*I)*X + ( % % 50000120977 + 42099*I) % Finding the roots of the new polynomial zz1. rr:= roots zz1; % 17343 - 93179*I 96531 + 518636*I %RR := {X=-----------------,X=------------------} % 500000 100000 % The roots are stable to the extent that rr=ss, although zz1 and % zz may differ. zz1 - zz; % 2 %49999999999*X - (49999799999 + 250000099995*I)*X + (50000120976 + % % 42099*I) % 70) Same type of problem in which roots are found exactly. zz:=(x-10**8+i)*(x-10**8-i)*(x-10**8+3i/2)*(x-i); rr := roots zz; % 4 3 2 %ZZ := (2*X - (600000000 - I)*X + 60000000000000005*X - ( % % 2000000000000000800000000 + 29999999999999999*I)*X + ( % % 30000000000000003 + 2000000000000000200000000*I))/2 %RR := {X=100000000 + I,X=100000000 - I,X=I,X= % % 200000000 - 3*I % -----------------} % 2 % Reconstructing a polynomial from the roots. ss := mkpoly rr; % 4 3 2 %SS := 2*X - (600000000 - I)*X + 60000000000000005*X - ( % % 2000000000000000800000000 + 29999999999999999*I)*X + ( % % 30000000000000003 + 2000000000000000200000000*I) % In this case, the same polynomial is obtained. ss - num zz; % 0 % 71) Finding one of the complex roots using nearestroot. nearestroot(zz,10**8-2i); % 200000000 - 3*I %{X=-----------------} % 2 % Finding the other complex root using nearestroot. nearestroot(zz,10**8+2i); %{X=100000000 + I} % 72) A realroots problem which requires accuracy increase to avoid % confusion of two roots. zz:=(x+1)*(10000000x-19999999)*(1000000x-2000001)*(x-2); realroots zz; % 19999999 2000001 % {X=-1,X=----------,X=2,X=---------} % 10000000 1000000 % 73) Without the accuracy increase, this example would produce the % obviously incorrect answer 2. realroots(zz,3/2,exclude 2); % 19999999 % {X=----------} % 10000000 % Rlrootno also gives the correct answer in this case. rlrootno(zz,3/2,exclude 2); % 1 % 74) Roots works equally well in this problem. rr := roots zz; % 19999999 2000001 % RR := {X=----------,X=-1,X=2,X=---------} % 10000000 1000000 % 75) The function getroot is convenient for obtaining the value of a % root. rr1 := getroot(1,rr); % 19999999 % RR1 := ---------- % 10000000 % 76) For example, the value can be used as an argument to nearestroot. nearestroot(zz,rr1); % 19999999 % {X=----------} % 10000000 comment New capabilities added to Mod 1.90 for avoiding floating point exceptions and exceeding iteration limits.; % 77) This and the next example would previously have aborted because %of exceeding iteration limits: off ratroot; zz := x**16 - 900x**15 -2; roots zz; %{X= - 0.665423,X=0.069527 + 0.661817*I,X=0.069527 - 0.661817*I, % X=0.650944 + 0.138369*I,X=0.650944 - 0.138369*I, % X=-0.205664 + 0.632867*I,X=-0.205664 - 0.632867*I, % X=-0.607902 + 0.270641*I,X=-0.607902 - 0.270641*I, % X=0.332711 + 0.57633*I,X=0.332711 - 0.57633*I, % X=0.538375 + 0.391176*I,X=0.538375 - 0.391176*I, % X=-0.44528 + 0.494497*I,X=-0.44528 - 0.494497*I,X=900.0} % 78) a still harder example. % This example is commented out, since it takes significant time: % z := x**30 - 900x**29 - 2; roots zz; %{X= - 0.810021, % X=-0.04388 + 0.808856*I,X=-0.04388 - 0.808856*I, % X=0.805322 + 0.087587*I,X=0.805322 - 0.087587*I, % X=0.131027 + 0.799383*I,X=0.131027 - 0.799383*I, % X=-0.791085 + 0.174125*I,X=-0.791085 - 0.174125*I, % X=-0.216732 + 0.780507*I,X=-0.216732 - 0.780507*I, % X=0.767663 + 0.258664*I,X=0.767663 - 0.258664*I, % X=0.299811 + 0.752532*I,X=0.299811 - 0.752532*I, % X=-0.735162 + 0.340111*I,X=-0.735162 - 0.340111*I, % X=-0.379447 + 0.715665*I,X=-0.379447 - 0.715665*I, % X=0.694106 + 0.417645*I,X=0.694106 - 0.417645*I, % X=-0.524417 + 0.617362*I,X=-0.524417 - 0.617362*I, % X=0.454578 + 0.67049*I,X=0.454578 - 0.67049*I, % X=-0.644866 + 0.490195*I,X=-0.644866 - 0.490195*I, % X=0.588091 + 0.557094*I,X=0.588091 - 0.557094*I,X=900.0} % 79) this deceptively simple example previously caused floating point % overflows on some systems: aa := x**6 - 4*x**3 + 2; realroots aa; %{X=0.836719,X=1.50579} % 80) a harder problem, which would have failed on almost all systems: rr := x**16 - 90000x**15 - x**2 -2; realroots rr; %{X= - 0.493299,X=90000.0} % 81) this example would have failed because of floating point % exceptions on almost all computer systems. rr := X**30 - 9*10**10*X**29 - 2; realroots rr; %{X= - 0.429188,X=9.0E+10} % 82) a test of allroot on this example. % This example is commented out, since it takes significant time: % roots rr; %{X= - 0.429188, % X=-0.023236 + 0.428559*I,X=-0.023236 - 0.428559*I, % X=0.426672 + 0.046403*I,X=0.426672 - 0.046403*I, % X=0.069435 + 0.423534*I,X=0.069435 - 0.423534*I, % X=-0.419154 + 0.092263*I,X=-0.419154 - 0.092263*I, % X=-0.11482 + 0.413544*I,X=-0.11482 - 0.413544*I, % X=0.406722 + 0.13704*I,X=0.406722 - 0.13704*I, % X=0.158859 + 0.398706*I,X=0.158859 - 0.398706*I, % X=-0.389521 + 0.180211*I,X=-0.389521 - 0.180211*I, % X=-0.201035 + 0.379193*I,X=-0.201035 - 0.379193*I, % X=0.367753 + 0.22127*I,X=0.367753 - 0.22127*I, % X=-0.277851 + 0.327111*I,X=-0.277851 - 0.327111*I, % X=0.240855 + 0.355234*I,X=0.240855 - 0.355234*I, % X=-0.341674 + 0.259734*I,X=-0.341674 - 0.259734*I, % X=0.311589 + 0.295153*I,X=0.311589 - 0.295153*I,X=9.0E+10} % 83) test of starting point for iteration: no convergence if good % real starting point is not found. zz := x**30 -9*10**12x**29 -2; firstroot zz; %{X= - 0.36617} % 84) a case in which there are no real roots and good imaginary % starting point must be used or roots cannot be found. zz:= 9x**16 - x**5 +1; roots zz; %{X=0.182294 + 0.828368*I,X=0.182294 - 0.828368*I, % X=-0.697397 + 0.473355*I,X=-0.697397 - 0.473355*I, % X=0.845617 + 0.142879*I,X=0.845617 - 0.142879*I, % X=-0.161318 + 0.87905*I,X=-0.161318 - 0.87905*I, % X=-0.866594 + 0.193562*I,X=-0.866594 - 0.193562*I, % X=0.459373 + 0.737443*I,X=0.459373 - 0.737443*I, % X=0.748039 + 0.494348*I,X=0.748039 - 0.494348*I, % X=-0.510014 + 0.716449*I,X=-0.510014 - 0.716449*I} % 85) five complex roots. zz := x**5 - x**3 + i; roots zz; %{X=-0.83762*I,X=-0.664702 + 0.636663*I,X=0.664702 + 0.636663*I, % X=1.16695 - 0.21785*I,X=-1.16695 - 0.21785*I} % Additional capabilities in Mod 1.91. % 86) handling of polynomial with huge or infinitesimal coefficients. on rounded; zz:= 1.0e-500x**3+x**2+x; roots zz; off rounded,roundbf; %{X=0,X=-1,X= - 1.0E+500} comment Switch roundbf will have been turned on in the last example in most computer systems. This will inhibit the use of hardware floating point unless roundbf is turned off. Polynomials which make use of powergcd substitution and cascaded solutions. Uncomplicated cases.; switch powergcd; % introduced here to verify that same answers are % obtained with and without employing powergcd strategy. Roots are % found faster for applicable cases when !*powergcd=t (default state.) % 87) powergcd done at the top level. zz := x**12-5x**9+1; roots zz; %{X=0.848444,X=-0.424222 + 0.734774*I,X=-0.424222 - 0.734774*I, % X=0.152522 - 0.816316*I, % X=-0.783212 + 0.276071*I,X=0.63069 + 0.540246*I, % X=0.152522 + 0.816316*I, % X=-0.783212 - 0.276071*I,X=0.63069 - 0.540246*I, % X=1.70906,X=-0.85453 + 1.48009*I,X=-0.85453 - 1.48009*I} off powergcd; roots zz; on powergcd; %{X=0.848444,X=-0.424222 + 0.734774*I,X=-0.424222 - 0.734774*I, % X=1.70906,X=-0.783212 + 0.276071*I,X=-0.783212 - 0.276071*I, % X=0.152522 + 0.816316*I,X=0.152522 - 0.816316*I, % X=0.63069 + 0.540246*I,X=0.63069 - 0.540246*I, % X=-0.85453 + 1.48009*I,X=-0.85453 - 1.48009*I} % 88) powergcd done after square free factoring. zz := (x-1)**2*zz; roots zz; %{X=1,X=1, % X=0.848444,X=-0.424222 + 0.734774*I,X=-0.424222 - 0.734774*I, % X=0.152522 - 0.816316*I,X=-0.783212 + 0.276071*I, % X=0.63069 + 0.540246*I, % X=0.152522 + 0.816316*I,X=-0.783212 - 0.276071*I, % X=0.63069 - 0.540246*I, % X=1.70906,X=-0.85453 + 1.48009*I,X=-0.85453 - 1.48009*I} off powergcd; roots zz; on powergcd; %{X=1,X=1, % X=0.848444,X=-0.424222 + 0.734774*I,X=-0.424222 - 0.734774*I, % X=1.70906,X=-0.783212 + 0.276071*I,X=-0.783212 - 0.276071*I, % X=0.152522 + 0.816316*I,X=0.152522 - 0.816316*I, % X=0.63069 + 0.540246*I,X=0.63069 - 0.540246*I, % X=-0.85453 + 1.48009*I,X=-0.85453 - 1.48009*I} % 89) powergcd done after separation into real and complex polynomial. zz := x**5-i*x**4+x**3-i*x**2+x-i; roots zz; %{X=0.5 + 0.866025*I,X=-0.5 - 0.866025*I, % X=-0.5 + 0.866025*I,X=0.5 - 0.866025*I,X=I} off powergcd; roots zz; on powergcd; %{X=-0.5 + 0.866025*I,X=-0.5 - 0.866025*I, % X=0.5 + 0.866025*I,X=0.5 - 0.866025*I,X=I} % Cases where root separation requires accuracy and/or precision % increase. In some examples we get excess accuracy, but it is hard % avoid this and still get all roots separated. % 90) accuracy increase required to separate close roots; let x=y**2; zz:= (x-3)*(100000000x-300000001); roots zz; %{Y=1.732050808,Y= - 1.732050808,Y=1.73205081,Y= - 1.73205081} off powergcd; roots zz; on powergcd; %{Y=1.732050808,Y= - 1.732050808,Y= - 1.73205081,Y=1.73205081} % 91) roots to be separated are on different square free factors. zz:= (x-3)**2*(10000000x-30000001); roots zz; %{Y=1.73205081 ,Y=1.73205081 ,Y= - 1.73205081 ,Y= - 1.73205081 , % Y=1.73205084 ,Y= - 1.73205084} off powergcd; roots zz; on powergcd; %{Y=1.73205081 ,Y=1.73205081 ,Y= - 1.73205081,Y= - 1.73205081, % Y=1.73205084 ,Y= - 1.73205084} % 92) roots must be separated in the complex polynomial factor only. zz :=(y+1)*(x+10**8i)*(x+10**8i+1); roots zz; %{Y=-1, % Y=-7071.067812 + 7071.067812*I,Y=7071.067812 - 7071.067812*I, % Y=-7071.067777 + 7071.067847*I,Y=7071.067777 - 7071.067847*I} % 93) zz := (x-2)**2*(1000000x-2000001)*(y-1); roots zz; %{Y=1.4142136,Y=1.4142136,Y= - 1.4142136,Y= - 1.4142136, % Y=1,Y=1.4142139,Y= - 1.4142139} % 94) zz := (x-2)*(10000000x-20000001); roots zz; %{Y=1.41421356 ,Y= - 1.41421356 ,Y=1.4142136,Y= - 1.4142136} % 95) zz := (x-3)*(10000000x-30000001); roots zz; %{Y=1.73205081 ,Y= - 1.73205081 ,Y=1.73205084 ,Y= - 1.73205084} % 96) zz := (x-9)**2*(1000000x-9000001); roots zz; %{Y=3.0,Y=3.0,Y= - 3.0,Y= - 3.0,Y=3.00000017,Y= - 3.00000017} % 97) zz := (x-3)**2*(1000000x-3000001); roots zz; %{Y=1.7320508,Y=1.7320508,Y= - 1.7320508,Y= - 1.7320508, % Y=1.7320511,Y= - 1.7320511} % 98) the accuracy of the root -sqrt 3 depends upon another close root. on rounded; zz := (x-3)*(y+1.732058); roots zz; %{Y= - 1.732051,Y=1.73205,Y= - 1.732058} zz := (x-3)*(y+1.732051); roots zz; %{Y= - 1.73205081,Y=1.73205,Y= - 1.732051} zz := (x-3)*(y+1.73205081); roots zz; %{Y= - 1.732050808,Y=1.73205,Y= - 1.73205081} % 99) minimum accuracy specified using rootacc. rootacc 10; roots zz; %{Y= - 1.732050808,Y=1.732050808,Y= - 1.73205081} % 100) off rounded; rootacc 6; zz := (y-i)*(x-2)*(1000000x-2000001); roots zz; %{Y=1.4142136,Y= - 1.4142136,Y=1.4142139,Y= - 1.4142139,Y=I} % 101) this example requires accuracy 15. zz:= (y-2)*(100000000000000y-200000000000001); roots zz; %{Y=2.0,Y=2.00000000000001} % 102) still higher precision needed. zz:= (y-2)*(10000000000000000000y-20000000000000000001); roots zz; %{Y=2.000 00000 00000 00000 1,Y=2.0} % 103) increase in precision required for substituted polynomial. zz:= (x-2)*(10000000000x-20000000001); roots zz; %{Y=1.41421356241,Y= - 1.41421356241,Y=1.41421356237, % Y= - 1.41421356237} % 104) still higher precision required for substituted polynomial. zz:= (x-2)*(100000000000000x-200000000000001); roots zz; %{Y=1.414 21356 23730 99,Y= - 1.414 21356 23730 99, % Y=1.414 21356 23730 95,Y= - 1.414 21356 23730 95} % 105) accuracy must be increased to separate root of complex factor % from root of real factor. zz:=(9y-10)*(y-2)*(9y-10-9i/100000000); roots zz; %{Y=1.111111111,Y=2.0,Y=1.111111111 + 0.00000001*I} % 106) realroots does the same accuracy increase for real root based % upon the presence of a close complex root in the same polynomial. % The reason for this might not be obvious unless roots is called. realroots zz; %{Y=1.111111111,Y=2.0} % 107) realroots now uses powergcd logic whenever it is applicable. zz := (x-1)*(x-2)*(x-3); realroots zz; %{Y=-1,Y=1,Y= - 1.41421,Y=1.41421,Y= - 1.73205,Y=1.73205} realroots(zz,exclude 1,2); %{Y=1.41421,Y=1.73205} % 108) root of degree 1 polynomial factor must be evaluated at % precision 18 and accuracy 10 in order to separate it from a root of % another real factor. clear x; zz:=(9x-10)**2*(9x-10-9/100000000)*(x-2); roots zz; %{X=1.111111111,X=1.111111111,X=1.111111121,X=2.0} nearestroot(zz,1); %{X=1.111111111} nearestroot(zz,1.5); %{X=1.111111121} nearestroot(zz,1.65); %{X=2.0} % 109) in this example, precision >=40 is used and two roots need to be % found to accuracy 16 and two to accuracy 14. zz := (9x-10)*(7x-8)*(9x-10-9/10**12)*(7x-8-7/10**14); roots zz; %{X=1.1111111111121,X=1.142 85714 28571 43, % X=1.1111111111111,X=1.142 85714 28571 53} % 110) very small real or imaginary parts of roots require high % precision or exact computations, or they will be lost or incorrectly % found. zz := 1000000*R**18 + 250000000000*R**4 - 1000000*R**2 + 1; roots zz; %{R=0.00141421 + 1.6E-26*I,R=-0.00141421 - 1.6E-26*I, % R=0.00141421 - 1.6E-26*I,R=-0.00141421 + 1.6E-26*I, % R=2.36886 + 0.54068*I,R=-2.36886 - 0.54068*I, % R=-2.36886 + 0.54068*I,R=2.36886 - 0.54068*I, % R=1.05424 + 2.18916*I,R=-1.05424 - 2.18916*I, % R=-1.05424 + 2.18916*I,R=1.05424 - 2.18916*I, % R=2.42978*I,R=-2.42978*I, % R=1.89968 + 1.51494*I,R=-1.89968 - 1.51494*I, % R=-1.89968 + 1.51494*I,R=1.89968 - 1.51494*I} comment These five examples are very difficult root finding problems for automatic root finding (not employing problem-specific procedures.) They require extremely high precision and high accuracy to separate almost multiple roots (multiplicity broken by a small high order perturbation.) The examples are roughly in ascending order of difficulty.; % 111) Two simple complex roots with extremely small real separation. c := 10^-6; zz:=(x-3c^2)^2+i*c*x^7; roots zz; %{X=2.999 99999 99999 99999 99999 99999 9997 E -12 % + 3.306 81115 27572 904 E -44*I, % X=3.000 00000 00000 00000 00000 00000 0003 E -12 % - 3.306 81115 27572 904 E -44*I, % X=15.0732 + 4.8976*I, X=-15.0732 + 4.8976*I, % X=-9.3158 - 12.8221*I, X=9.3158 - 12.8221*I, % X=-1.2E-12 + 15.8489*I} % 112) Four simple complex roots in two close sets. c := 10^-4; zz:=(x^2-3c^2)^2+i*c^2*x^9; roots zz; %{X=-0.00 01732 05080 75688 5 + 2.4177823E-18*I, % X=0.000 17320 50807 56885 + 2.4177823E-18*I, % X=-0.00017320508075689 - 2.4177823E-18*I, % X=0.00017320508075689 - 2.4177823E-18*I, % X=37.8622 + 12.3022*I, X=-37.8622 + 12.3022*I, % X=-23.4002 - 32.2075*I, X=23.4002 - 32.2075*I, X=39.8107*I} % 113) Same example, but with higher minimum root accuracy specified. rootacc 20; roots zz; %{X=-0.00 01732 05080 75688 53115 7 + 2.417782347E-18*I, % X=0.000 17320 50807 56885 31157 + 2.417782347E-18*I, % X=-0.00 01732 05080 75689 01471 4 - 2.417782347E-18*I, % X=0.000 17320 50807 56890 14714 - 2.417782347E-18*I, % X=37.86 22418 73586 29052 6 + 12.30 21881 28448 77534 5*I, % X=-37.8 62241 87358 62905 26 + 12.30 21881 28448 77534 5*I, % X=23.40 01523 68145 82711 8 - 32.20 75466 56274 35106 9*I, % X=-23.4 00152 36814 58271 18 - 32.20 75466 56274 35106 9*I, % X=39.81 07170 55651 15144 9*I} rootacc 6; % 114) Two extremely close real roots plus a complex pair with extremely % small imaginary part. c := 10^-6; zz:=(x^2-3c^2)^2+c^2*x^9; roots zz; %{X= - 0.000 00173 20508 07568 87729 3524, % X=0.00000173205 + 3.42E-27*I, X=0.00000173205 - 3.42E-27*I, % X= - 0.000 00173 20508 07568 87729 3531, % X=203.216 + 147.645*I, X=203.216 - 147.645*I, X= - 251.189, % X=-77.622 + 238.895*I, X=-77.622 - 238.895*I} % 115) Four simple complex roots in two extremely close sets. c := 10^-6; zz:=(x^2-3c^2)^2+i*c^2*x^9; roots zz; %{X=-0.00 00017 32050 80756 88772 93525 + 2.4177823466E-27*I, % X=0.000 00173 20508 07568 87729 3525 + 2.4177823466E-27*I, % X=0.000 00173 20508 07568 87729 353 - 2.4177823466E-27*I, % X=-0.00 00017 32050 80756 88772 9353 - 2.4177823466E-27*I, % X=238.895 + 77.622*I, X=-238.895 + 77.622*I, % X=-147.645 - 203.216*I, X=147.645 - 203.216*I, X=251.189*I} showtime; end;