reduce-historical/r34/xlog/roots.log

Sat Jun 29 13:48:36 PDT 1991
REDUCE 3.4, 15-Jul-91 ...

1: 1: 
2: 2: 
3: 3: % Tests of the root finding package.

% Author: Stanley L. Kameny (stan%valley.uucp@rand.org)

% 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.

roots x;


{X=0}
   % To load roots package.

write "This is Roots Package test ", symbolic roots!-mod$


This is Roots Package test Mod 1.92, 10 Oct 1990.


% Simple root finding.

showtime;


Time: 476 ms


% 1) multiple real and imaginary roots plus two real roots.
zz:= (x-3)**2*(100x**2+113)**2*(1000000x-10000111)*(x-1);


                   8                 7                 6
ZZ := 10000000000*X  - 170001110000*X  + 872607770000*X

                        5                  4                  3
       - 1974219158600*X  + 2833796550200*X  - 3810512046359*X

                        2
       + 3119397498913*X  - 2030292260385*X + 1149222756231
 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}

%{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);


               4            3            2
ZZ := 1000000*X  - 4000001*X  + 5000002*X  - 4000001*X + 4000002
 roots zz;


{X=2.0,X=I,X= - I,X=2.000001}

%{X=2.0,X=I,X= - I,X=2.000001}

% 3) accuracy increase to 8 required.
zz:= (x-3)*(10000000x-30000001);


                2
ZZ := 10000000*X  - 60000001*X + 90000003
 roots zz;


{X=3.0,X=3.0000001}

%{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);


                       4                  3                   2
ZZ := 2*(500000000000*X  - 6120617500000*X  + 28085557620617*X

          - 57256673223702*X + 43756673585553)
 roots zz;


{X=3.0,X=3.0,X=3.000001,X=3.24123}

%{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);


          5            4               3                  2
ZZ := 10*X  - 1020031*X  + 2013152032*X  - 1005224243011*X

       + 2104312111000*X - 1101100000000
 roots zz;


{X=1,X=1.1,X=1000.0,X=1001.0,X=100000.0}

%{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);


                 5              4               3               2
ZZ := 100000000*X  - 500000000*X  + 1000000000*X  - 1000000000*X

       + 499999999*X - 99999999
 roots zz;


{X=1,

 X=1.01,

 X=0.99,

 X=1 + 0.01*I,

 X=1 - 0.01*I}

%{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);


                 4            3            2
ZZ := 2*(500000*X  - 2030000*X  + 3090550*X  - 2091103*X + 530553)
 roots zz;


{X=1.02,X=1.01,X=1,X=1.03}

%{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);


                 5            4            3            2
ZZ := 2*(500000*X  - 2515000*X  + 5065100*X  - 5105450*X  + 2575601*X

          - 520251)
 roots zz;


{X=1.01,

 X=1,

 X=1.02,

 X=1 + 0.1*I,

 X=1 - 0.1*I}

%{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);


                 5               4               3               2
ZZ := 100000000*X  - 1000000000*X  + 4000000000*X  - 8000000000*X

       + 7999999999*X - 3199999998
 roots zz;


{X=2.0,

 X=2.01,

 X=1.99,

 X=2.0 + 0.01*I,

 X=2.0 - 0.01*I}

%{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);


                 3              2
SS := 100000000*X  - 600000000*X  + 1200000001*X - 800000002
 roots ss;


{X=2.0,X=2.0 + 0.0001*I,X=2.0 - 0.0001*I}

%{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);


          7       6       5        4        3        2
ZZ := X*(X  - 14*X  + 74*X  - 186*X  + 249*X  - 236*X  + 176*X - 64)
 roots zz;


{X=0,

 X=4.0,

 X=4.0,

 X=4.0,

 X=1,

 X=1,

 X=I,

 X= - I}

%{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}

%{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;


       9
ZZ := X  - 45*X - 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}

%{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}


% 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;


            9           2
       100*X  - 999900*X  - 1
ZZ := ------------------------
                100
 roots zz;


*** precision increased to 27 

{X=5.0E-29 + 0.00100005*I,

 X=5.0E-29 - 0.00100005*I,

 X=3.72754,

 X=-0.82946 + 3.63408*I,

 X=-0.82946 - 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}

%{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}

% 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;


7
 zz:= x**9-500x**2-0.001;


             9           2
       1000*X  - 500000*X  - 1
ZZ := -------------------------
                1000
 roots zz;


{X=1.6E-26 + 0.001414214*I,

 X=1.6E-26 - 0.001414214*I,

 X=2.429781,

 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}

%{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;


             6           5            4            3             2
ZZ := 10000*X  + 210027*X  + 1750000*X  + 7350000*X  + 16240000*X

       + 17640000*X + 7200000
 roots zz;


{X= - 2.950367,

 X=-4.452438 + 0.021235*I,

 X=-4.452438 - 0.021235*I,

 X= - 2.003647,

 X= - 0.9999775,

 X= - 6.143833}

%{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;


              8          7           6            5             4
ZZ := 2*(500*X  + 18001*X  + 273000*X  + 2268000*X  + 11224500*X

                      3             2
          + 33642000*X  + 59062000*X  + 54792000*X + 20160000)
 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}

%{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;


            10          9            8             7              6
ZZ := 1000*X   + 55001*X  + 1320000*X  + 18150000*X  + 157773000*X

                    5               4               3
       + 902055000*X  + 3416930000*X  + 8409500000*X

                      2
       + 12753576000*X  + 10628640000*X + 3628800000
 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}

%{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;


               12           11            10              9
ZZ := 4*(2500*X   + 195001*X   + 6792500*X   + 139425000*X

                        8                7                 6
          + 1873657500*X  + 17316585000*X  + 112475577500*X

                          5                  4                  3
          + 515175375000*X  + 1643017090000*X  + 3535037220000*X

                           2
          + 4828898880000*X  + 3716107200000*X + 1197504000000)
 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}

%{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;


6

% 21)  Finding one of a cluster of 8 roots.
zz:= (10**16*(x-1)**8-1);


                         8                      7
ZZ := 10000000000000000*X  - 80000000000000000*X

                             6                       5
       + 280000000000000000*X  - 560000000000000000*X

                             4                       3
       + 700000000000000000*X  - 560000000000000000*X

                             2
       + 280000000000000000*X  - 80000000000000000*X

       + 9999999999999999
 nearestroot(zz,2);


{X=1.01}

%{X=1.01}

% 22)  Six real roots spaced 0.01 apart.
c := 100;


C := 100
 zz:= (x-1)*for i:=1:5 product (c*x-(c+i));


                     6               5               4
ZZ := 40*(250000000*X  - 1537500000*X  + 3939625000*X

                         3               2
           - 5383556250*X  + 4137919435*X  - 1696170123*X + 289681938

          )
 roots zz;


{X=1.03,X=1.02,X=1.04,X=1.01,X=1,X=1.05}

%{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;


C := 1000
 zz:= (x-1)*for i:=1:5 product (c*x-(c+i));


                          6                    5                    4
ZZ := 40*(25000000000000*X  - 150375000000000*X  + 376877125000000*X

                              3                    2
           - 503758505625000*X  + 378762766881850*X

           - 151883516888703*X + 25377130631853)
 roots zz;


{X=1.003,X=1.002,X=1.004,X=1.001,X=1,X=1.005}

%{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;


C := 10000
 zz:= (x-1)*for i:=1:4 product (c*x-(c+i));


                           5                     4
ZZ := 8*(1250000000000000*X  - 6251250000000000*X

                               3                      2
          + 12505000437500000*X  - 12507501312562500*X

          + 6255001312625003*X - 1251250437562503)
 roots zz;


{X=1.0002,X=1.0003,X=1.0001,X=1,X=1.0004}

%{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);


                 9              8               7               6
ZZ := 100000000*X  - 900000000*X  + 3600000000*X  - 8400000000*X

                      5                4               3
       + 12600000000*X  - 12600000000*X  + 8400000000*X

                     2
       - 3600000000*X  + 899999999*X - 99999999
 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}

%{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);


                         9                      8
ZZ := 10000000000000000*X  - 90000000000000000*X

                             7                       6
       + 360000000000000000*X  - 840000000000000000*X

                              5                        4
       + 1260000000000000000*X  - 1260000000000000000*X

                             3                       2
       + 840000000000000000*X  - 360000000000000000*X

       + 89999999999999999*X - 9999999999999999
 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}

%{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);


                                 9                              8
ZZ := 1000000000000000000000000*X  - 9000000000000000000000000*X

                                     7
       + 36000000000000000000000000*X

                                     6
       - 84000000000000000000000000*X

                                      5
       + 126000000000000000000000000*X

                                      4
       - 126000000000000000000000000*X

                                     3
       + 84000000000000000000000000*X

                                     2
       - 36000000000000000000000000*X  + 8999999999999999999999999*X

       - 999999999999999999999999
 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}

%{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);


                         8                      6
ZZ := 10000000000000000*X  - 40000000000000000*X

                            4                      2
       + 59999999800000000*X  - 40000001200000000*X

       + 9999999800000001
 roots zz;


{X=1.01,

 X= - 1.01,

 X=0.99,

 X= - 0.99,

 X=1 + 0.01*I,

 X=-1 - 0.01*I,

 X=-1 + 0.01*I,

 X=1 - 0.01*I}

%{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);


                                 8                              7
ZZ := 1000000000000000000000000*X  - 8080000000000000000000000*X

                                     6
       + 28562400000000000000000000*X

                                     5
       - 57694432000000000000000000*X

                                     4
       + 72836160149999999900000000*X

                                     3
       - 58848320599199999600000000*X

                                     2
       + 29716320897575999400000000*X  - 8574560597551679600000000*X

       + 1082432149175678300000001
 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}

%{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);


                   8            6          4        2
ZZ := 4*(25000000*X  - 1000000*X  + 20000*X  + 200*X  + 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}

%{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}

% 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);


ZZ := 

         4             3             2
1000000*X  - 13000001*X  + 63000010*X  - 135000033*X + 108000036
 roots zz;


{X=3.0,X=3.0,X=3.000001,X=4.0}

%{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}

%{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;


ZZ :=  - I + X
 roots zz;


{X=I}

%{X=I}

% 34) Real case.
zz:= y-7;


ZZ := Y - 7
 roots zz;


{Y=7.0}

%{Y=7.0}

% 35) Roots with small imaginary parts (new capability);
zz := 10**16*(x**2-2x+1)+1;


                         2
ZZ := 10000000000000000*X  - 20000000000000000*X + 10000000000000001
 roots zz;


{X=1 + 0.00000001*I,X=1 - 0.00000001*I}

%{X=1 + 0.00000001*I,X=1 - 0.00000001*I}

% 36) One real, one complex root.
zz:=(x-9)*(x-5i-7);


                         2
ZZ :=  - 5*I*X + 45*I + X  - 16*X + 63
 roots zz;


{X=9.0,X=7.0 + 5.0*I}

%{X=9.0,X=7.0 + 5.0*I}

% 37) Three real roots.
zz:= (x-1)*(x-2)*(x-3);


       3      2
ZZ := X  - 6*X  + 11*X - 6
 roots zz;


{X=2.0,X=1,X=3.0}

%{X=2.0,X=1,X=3.0}

% 38) 2 real + 1 imaginary root.
zz:=(x**2-8)*(x-5i);


              2           3
ZZ :=  - 5*I*X  + 40*I + X  - 8*X
 roots zz;


{X=2.82843,X= - 2.82843,X=5.0*I}

%{X=2.82843,X= - 2.82843,X=5.0*I}

% 39) 2 complex roots.
zz:= (x-1-2i)*(x+2+3i);


                   2
ZZ := I*X - 7*I + X  + X + 4
 roots zz;


{X=1 + 2.0*I,X=-2.0 - 3.0*I}

%{X=1 + 2.0*I,X=-2.0 - 3.0*I}

% 40) 2 irrational complex roots.
zz:= x**2+(3+2i)*x+7i;


                     2
ZZ := 2*I*X + 7*I + X  + 3*X
 roots zz;


{X=0.14936 - 2.21259*I,X=-3.14936 + 0.21259*I}

%{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;


                2
ZZ := 12*I*X + X  + 1000000000*X - 1000000000
 roots zz;


{X=1 - 0.000000012*I,X=-1.0E+9 - 12.0*I}

%{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 and mode is complex.
zz :=(x**2-2i*x+5)**3*(x-2i)*(x-11/10)**2;


                 8           7           6            5            4
ZZ := ( - 800*I*X  + 1760*I*X  - 6768*I*X  + 12760*I*X  - 25018*I*X

                   3            2                              9
        + 39600*I*X  - 46780*I*X  + 55000*I*X - 30250*I + 100*X

               8        7         6         5          4          3
        - 220*X  - 779*X  + 1980*X  - 9989*X  + 19580*X  - 28269*X

                 2
        + 38500*X  - 21175*X)/100
 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}

%{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);


           3        2                    4      2
ZZ := 3*I*X  + 5*I*X  - 12*I*X - 20*I + X  - 4*X
 roots zz;


{X=2.0,

 X= - 2.0,

 X=-1.2714 + 0.46633*I,

 X=1.2714 - 3.46633*I}

%{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;


                      4
       I*X + 1000000*X  - 16000000
ZZ := -----------------------------
                 1000000
 roots zz;


{X=2.0 - 0.0000000625*I,

 X=-2.0 - 0.0000000625*I,

 X=-2.0*I,

 X=2.0*I}

%{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);


           3            4      2
ZZ := 2*I*X  - 8*I*X + X  + 4*X  - 32
 roots zz;


{X=2.0,X= - 2.0,X=2.0*I,X=-4.0*I}

%{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}

%{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);


           3            4      2
ZZ := 2*I*X  - 8*I*X + X  + 4*X  - 32
 nearestroot(zz,1);


{X=2.0}

%{X=2.0}

% 48) Same example, but focusing on imaginary point.
nearestroot(zz,i);


{X=2.0*I}

%{X=2.0*I}

% 49) The seed parameter can be complex also.
nearestroot(zz,1+i);


{X=2.0*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);


            2          4      2
ZZ :=  - I*X  + 4*I + X  - 4*X
 roots zz;


{X=2.0,

 X= - 2.0,

 X=0.707107 + 0.707107*I,

 X=-0.707107 - 0.707107*I}

%{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}

%{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);


              5         2    6      4    3
ZZ :=  - 3*I*X  + 24*I*X  + X  - 5*X  + X  + 40*X - 72
 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}

%{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}

%{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;


                 1               2    3
ZZ := -----------------------*I*X  + X  + 8
       100000000000000000000
 rootacc 12;


12
 roots zz;


{X=1 + 1.73205080757*I,X=1 - 1.73205080757*I,X=-2.0 - 3.33333E-21*I}
 rootacc 0;


6

%{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;


                       2
ZZ := 123456789*I*X + X  + 1
 roots zz;


{X=0.0000000081*I,X=-123457000.0*I}

%{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);


       3              2
ZZ := X  + 123456790*X  + 123456790*X + 1
 roots zz;


{X= - 0.0000000081,X=-1,X= - 123457000.0}

%{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);


               4         3         2                       5        4
SS :=  - 10*I*X  + 40*I*X  + 50*I*X  - 10*I*X - 10*I + 45*X  - 213*X

             3       2
       - 60*X  + 45*X  + 12*X
 roots ss;


{X=0.469832,

 X= - 0.429174,

 X=4.95934,

 X=0.18139 + 0.417083*I,

 X=-0.448056 - 0.19486*I}

%{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;


                           2
       12*I*X + 23*I + 10*X  + 67
ZZ := ----------------------------
                   10
 roots zz;


{X=-0.42732 + 2.09121*I,X=0.42732 - 3.29121*I}

%{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;


                  2                    3         2
        - 6000*I*X  - 6540*I*X + 2500*X  + 2725*X  - 3600*X - 3924
ZZ := -------------------------------------------------------------
                                  2500

roots zz;


{X=1.2*I,X=1.2*I,X= - 1.09}

%{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);


          4      3      2
ZZ := X*(X  - 5*X  + 5*X  + 5*X - 6)
 rlrootno zz;


5

% 5

realroots zz;


{X=0,

 X= - 1,

 X=1,

 X=2.0,

 X=3.0}

%{X=0,X= -1,X=1,X=2.0,X=3.0}

rlrootno(zz,positive);


3
 %positive selects positive, excluding 0.
% 3

rlrootno(zz,negative);


1
 %negative selects negative, excluding 0.
% 1

realroots(zz,positive);


{X=1,X=2.0,X=3.0}

%{X=1,X=2.0,X=3.0}

rlrootno(zz,-1.5,2);


4
 %the format with 3 arguments selects a range.
% 4

realroots(zz,-1.5,2);


{X=0,X= - 1,X=1,X=2.0}
 %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);


{X=1,X=2.0,X=3.0}

% equivalent to realroots(zz,positive).
%{X=1,X=2.0,X=3.0}

rlrootno(zz,-infinity,exclude 0);


1
 % equivalent to rlrootno(zz,negative).
% 1

rlrootno(zz,-infinity,0);


2

% 2

rlrootno(zz,infinity,-infinity);


5

%equivalent to rlrootno zz; (order of limits does not matter.)
% 5

realroots(zz,1,infinity);


{X=1,X=2.0,X=3.0}
 % finds all real roots >= 1.
%{X=1,X=2.0,X=3.0}

realroots(zz,1,positive);


{X=2.0,X=3.0}
 % 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);


                   2                                          3    2
ZZ := 200000001*I*X  + 100000001*I*X - 10000000000000000*I + X  + X

       - 10000000200000000*X - 100000000
 roots zz;


{X= - I,X=-1.0E+8*I,X=-1 - 1.0E+8*I}

%{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);


                                     2
ZZ := 10000000000000000000000000000*X

       + 24691357800000000000000000000*X

       + 15241578750190521000000000001
 roots zz;


{X=-1.23457 + 1.0E-14*I,X=-1.23457 - 1.0E-14*I}

%{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);


                                3                        2
ZZ := 100*(1000000000000000000*X  + 4469135780100000000*X

            + 6462429435342508889*X + 3048315750285017778)
 roots zz;


{X= - 1.2345678901,X= - 1.23456789,X= - 2.0}

%{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;


                              3                           2
ZZ := (500000000000000000000*X  - 1851851800000000000000*X

        + 2286236726108825000000*X - 940838132549050755399)/

      500000000000000000000
 roots zz;


{X=1.2345679,X=1.2345679,X=1.2345678}

%{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);


                           3                         2
ZZ := (610351562500000000*X  - 19660800006103515625*X

        + 211106232664064000000*X - 755578637962830675968)/

      610351562500000000
 realroots zz;


{X=10.73741824,X=10.73741824,X=10.73741825}

%{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);


                      2                                         3
ZZ :=  - 200000001*I*X  + 300000003*I*X + 9999999999999998*I + X

            2
       - 3*X  - 10000000199999998*X + 300000000
 roots zz;


{X=I,X=1 + 1.0E+8*I,X=2.0 + 1.0E+8*I}

%{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}

% {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);


                                     2
ZZ := 2*( - 100005000000000000000*I*X  + 1233623350000000*I*X

                                                           3
          + 499999999999999999239239*I + 5000000000000000*X

                          2
          - 123350000000*X  - 500099999999999999239239*X

          + 1233500000000000)
 roots zz;


{X=I,X=0.00001233 + 10000.0*I,X=0.00001234 + 10000.0*I}

%{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);


                                4                            3
ZZ := (10000000000000000000000*X  - 39999999900000000000000*X

                                   2
        + 59999999699999900000000*X  - 39999999699999800000001*X

        + 9999999899999900000001)/10000000000000000000000
 roots zz;


{X=1,X=0.99999999,X=0.9999999,X=1.0000001}

%{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);


                       4                    3                    2
ZZ := (50000000000000*X  - 123456785000000*X  + 126207888812681*X

        - 123456785000000*X + 76207888812681)/50000000000000
 roots zz;


{X=1.2345678,X=1.2345679,X=I,X= - I}

%{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}

%{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;


       2
ZZ := X  - (1 + 5*I)*X + 1
 sss:= roots zz;


           17343 - 93179*I     96531 + 518636*I
SSS := {X=-----------------,X=------------------}
               500000               100000


%           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)


%                    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


%          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)


%             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);


          4                    3                      2
ZZ := (2*X  - (600000000 - I)*X  + 60000000000000005*X

        - (2000000000000000800000000 + 29999999999999999*I)*X

        + (30000000000000003 + 2000000000000000200000000*I))/2
 rr := roots zz;


                                              200000000 - 3*I
RR := {X=100000000 + I,X=100000000 - I,X=I,X=-----------------}
                                                     2


%          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)


%         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

% 0

% 71) Finding one of the complex roots using nearestroot.
nearestroot(zz,10**8-2i);


    200000000 - 3*I
{X=-----------------}
           2


%    200000000 - 3*I
%{X=-----------------}
%           2

% Finding the other complex root using nearestroot.
nearestroot(zz,10**8+2i);


{X=100000000 + I}

%{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);


                      4                   3                   2
ZZ := 10000000000000*X  - 50000009000000*X  + 60000026999999*X

       + 40000000000001*X - 80000035999998
 realroots zz;


         19999999         2000001
{X=-1,X=----------,X=2,X=---------}
         10000000         1000000


%          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


%     19999999
% {X=----------}
%     10000000

% Rlrootno also gives the correct answer in this case.
rlrootno(zz,3/2,exclude 2);


1

% 1

% 74) Roots works equally well in this problem.
rr := roots zz;


          19999999              2000001
RR := {X=----------,X=2,X=-1,X=---------}
          10000000              1000000


%           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


%         19999999
% RR1 := ----------
%         10000000

% 76) For example, the value can be used as an argument to nearestroot.
nearestroot(zz,rr1);


    19999999
{X=----------}
    10000000


%     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;


       16        15
ZZ := X   - 900*X   - 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}

%{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;


       6      3
AA := X  - 4*X  + 2
 realroots aa;


{X=0.836719,X=1.50579}

%{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;


       16          15    2
RR := X   - 90000*X   - X  - 2
 realroots rr;


{X= - 0.493299,X=90000.0}

%{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;


       30                29
RR := X   - 90000000000*X   - 2
  realroots rr;


{X= - 0.429188,X=9.0E+10}

%{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;


       30                  29
ZZ := X   - 9000000000000*X   - 2
 firstroot zz;


{X= - 0.36617}

%{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;


         16    5
ZZ := 9*X   - X  + 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}

%{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;


       5    3
ZZ := X  - X  + 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}

%{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;


*** Domain mode COMPLEX changed to COMPLEX-ROUNDED 

zz:= 1.0e-500x**3+x**2+x;


                   2
ZZ := X*(1.0E-500*X  + X + 1)
 roots zz;


{X=0,X=-1,X= - 1.0E+500}
 off rounded;


*** Domain mode COMPLEX-ROUNDED changed to COMPLEX 

%{X=0,X=-1,X= - 1.0E+500}

% 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;


       12      9
ZZ := X   - 5*X  + 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}

%{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;


{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}
 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;


       14      13    12      11       10      9    2
ZZ := X   - 2*X   + X   - 5*X   + 10*X   - 5*X  + X  - 2*X + 1
 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}

%{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;


{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}
 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;


       5      4    3      2
ZZ := X  - I*X  + X  - I*X  + 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}

%{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;


{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}
 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);


                 4              2
ZZ := 100000000*Y  - 600000001*Y  + 900000003
 roots zz;


*** Roots precision increase to  22 

{Y=1.732050808,Y= - 1.732050808,Y=1.73205081,Y= - 1.73205081}

%{Y=1.732050808,Y= - 1.732050808,Y=1.73205081,Y= - 1.73205081}

off powergcd;

 roots zz;


{Y=1.732050808,Y= - 1.732050808,Y= - 1.73205081,Y=1.73205081}
 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);


                6             4              2
ZZ := 10000000*Y  - 90000001*Y  + 270000006*Y  - 270000009
 roots zz;


{Y=1.73205081,

 Y=1.73205081,

 Y= - 1.73205081,

 Y= - 1.73205081,

 Y=1.73205084,

 Y= - 1.73205084}

%{Y=1.73205081 ,Y=1.73205081 ,Y= - 1.73205081 ,Y= - 1.73205081 ,
% Y=1.73205084 ,Y= - 1.73205084}

off powergcd;

 roots zz;


{Y=1.73205081,

 Y=1.73205081,

 Y= - 1.73205081,

 Y= - 1.73205081,

 Y=1.73205084,

 Y= - 1.73205084}
 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);


       5    4                      3                      2
ZZ := Y  + Y  + (1 + 200000000*I)*Y  + (1 + 200000000*I)*Y

       - (10000000000000000 - 100000000*I)*Y

       - (10000000000000000 - 100000000*I)
 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}

%{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);


               7            6            5            4             3
ZZ := 1000000*Y  - 1000000*Y  - 6000001*Y  + 6000001*Y  + 12000004*Y

                   2
       - 12000004*Y  - 8000004*Y + 8000004
 roots zz;


{Y=1.4142136,

 Y=1.4142136,

 Y= - 1.4142136,

 Y= - 1.4142136,

 Y=1,

 Y=1.4142139,

 Y= - 1.4142139}

%{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);


                4             2
ZZ := 10000000*Y  - 40000001*Y  + 40000002
 roots zz;


{Y=1.41421356,Y= - 1.41421356,Y=1.4142136,Y= - 1.4142136}

%{Y=1.41421356 ,Y= - 1.41421356 ,Y=1.4142136,Y= - 1.4142136}

% 95)
zz := (x-3)*(10000000x-30000001);


                4             2
ZZ := 10000000*Y  - 60000001*Y  + 90000003
 roots zz;


{Y=1.73205081,Y= - 1.73205081,Y=1.73205084,Y= - 1.73205084}

%{Y=1.73205081 ,Y= - 1.73205081 ,Y=1.73205084 ,Y= - 1.73205084}

% 96)
zz := (x-9)**2*(1000000x-9000001);


               6             4              2
ZZ := 1000000*Y  - 27000001*Y  + 243000018*Y  - 729000081
 roots zz;


{Y=3.0,

 Y=3.0,

 Y= - 3.0,

 Y= - 3.0,

 Y=3.00000017,

 Y= - 3.00000017}

%{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);


               6            4             2
ZZ := 1000000*Y  - 9000001*Y  + 27000006*Y  - 27000009
 roots zz;


{Y=1.7320508,

 Y=1.7320508,

 Y= - 1.7320508,

 Y= - 1.7320508,

 Y=1.7320511,

 Y= - 1.7320511}

%{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;


*** Domain mode COMPLEX changed to COMPLEX-ROUNDED 

zz := (x-3)*(y+1.732058);


       3             2
ZZ := Y  + 1.732058*Y  - 3*Y - 5.196174
 roots zz;


{Y= - 1.732051,Y=1.73205,Y= - 1.732058}

%{Y= - 1.732051,Y=1.73205,Y= - 1.732058}

zz := (x-3)*(y+1.732051);


       3             2
ZZ := Y  + 1.732051*Y  - 3*Y - 5.196153
 roots zz;


{Y= - 1.73205081,Y=1.73205,Y= - 1.732051}

%{Y= - 1.73205081,Y=1.73205,Y= - 1.732051}

zz := (x-3)*(y+1.73205081);


       3               2
ZZ := Y  + 1.73205081*Y  - 3*Y - 5.19615243
 roots zz;


{Y= - 1.732050808,Y=1.73205,Y= - 1.73205081}

%{Y= - 1.732050808,Y=1.73205,Y= - 1.73205081}

% 99) minimum accuracy specified using rootacc.
rootacc 10;


10
 roots zz;


{Y= - 1.732050808,Y=1.732050808,Y= - 1.73205081}

%{Y= - 1.732050808,Y=1.732050808,Y= - 1.73205081}

% 100)
off rounded;


*** Domain mode COMPLEX-ROUNDED changed to COMPLEX 
 rootacc 6;


6

zz := (y-i)*(x-2)*(1000000x-2000001);


               5                4            3                2
ZZ := 1000000*Y  - (1000000*I)*Y  - 4000001*Y  + (4000001*I)*Y

       + 4000002*Y - 4000002*I
 roots zz;


{Y=1.4142136,Y= - 1.4142136,Y=1.4142139,Y= - 1.4142139,Y=I}

%{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);


                       2
ZZ := 100000000000000*Y  - 400000000000001*Y + 400000000000002
 roots zz;


{Y=2.00000000000001,Y=2.0}

%{Y=2.0,Y=2.00000000000001}

% 102) still higher precision needed.
zz:= (y-2)*(10000000000000000000y-20000000000000000001);


                            2
ZZ := 10000000000000000000*Y  - 40000000000000000001*Y

       + 40000000000000000002
 roots zz;


{Y=2.000 00000 00000 00000 1,Y=2.0}

%{Y=2.000 00000 00000 00000 1,Y=2.0}

% 103) increase in precision required for substituted polynomial.
zz:= (x-2)*(10000000000x-20000000001);


                   4                2
ZZ := 10000000000*Y  - 40000000001*Y  + 40000000002
 roots zz;


{Y=1.41421356241,Y= - 1.41421356241,Y=1.41421356237,Y

 = - 1.41421356237}

%{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);


                       4                    2
ZZ := 100000000000000*Y  - 400000000000001*Y  + 400000000000002
 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}

%{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);


                   3                         2
ZZ := (8100000000*Y  - (34200000000 + 81*I)*Y

        + (46000000000 + 252*I)*Y - (20000000000 + 180*I))/100000000
 roots zz;


{Y=1.111111111,Y=2.0,Y=1.111111111 + 0.00000001*I}

%{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}

%{Y=1.111111111,Y=2.0}

% 107) realroots now uses powergcd logic whenever it is applicable.
zz := (x-1)*(x-2)*(x-3);


       6      4       2
ZZ := Y  - 6*Y  + 11*Y  - 6
  realroots zz;


{Y= - 1,

 Y=1,

 Y= - 1.41421,

 Y=1.41421,

 Y= - 1.73205,

 Y=1.73205}

%{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}

%{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);


                    4                 3                 2
ZZ := (72900000000*X  - 388800000729*X  + 756000003078*X

        - 640000004140*X + 200000001800)/100000000
 roots zz;


{X=1.111111111,X=1.111111111,X=1.111111121,X=2.0}

%{X=1.111111111,X=1.111111111,X=1.111111121,X=2.0}

nearestroot(zz,1);


{X=1.111111111}

%{X=1.111111111}

nearestroot(zz,1.5);


{X=1.111111121}

%{X=1.111111121}

nearestroot(zz,1.65);


{X=2.0}

%{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);


                                       4
ZZ := (396900000000000000000000000000*X

                                           3
        - 1789200000000400869000000000000*X

                                           2
        + 3024400000001361556000000003969*X

        - 2272000000001541380000000008946*X

        + 640000000000581600000000005040)/100000000000000000000000000

roots zz;


{X=1.1111111111121,X=1.142 85714 28571 43,X=1.1111111111111,X

 =1.142 85714 28571 53}

%{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;


               18                 4            2
ZZ := 1000000*R   + 250000000000*R  - 1000000*R  + 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}

%{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}

showtime;


Time: 573342 ms


end;

4: 4: 
Quitting
Sat Jun 29 13:58:15 PDT 1991