reduce-historical/r36/xlog/COMPLEX.LOG

REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...


% Test of Complex Number mode in REDUCE.

on complex;



(31+i)/74;


 31 + i
--------
   74


ws/(b+1);


   31 + i
------------
 74*(b + 1)
  % this now comes out right!

w:=(x+3*i)**2;


      2
w := x  + 6*i*x - 9


on gcd;



(x**3-7*x**2+x-7)/(x**2+(3+i)*x+3*i);


  2
 x  - (7 + i)*x + 7*i
----------------------
        x + 3


off gcd;



sqrt(x**4+14*i*x**3-51*x**2-14*i*x+1);


     2
abs(x  + 7*i*x - 1)


% All rounded tests are done twice:  first, they are done at the default
% precision, in which all rounded operations use standard floating point
% logic.  Then precision is increased, causing all rounded operations to
% use extended precision bigfloat arithmetic.  This is necessary to
% exercise and test the bigfloat-based arithmetic functions.

prec0 := precision 0;


prec0 := 12
  % to determine the nominal default precision.

% Tests using default precision:

on rounded;


*** Domain mode complex changed to complex-rounded 


(3.25 + 8.5i) + (6.75 - 8.5i);


10.0


(3.25 + 8.5i) - (6.0 - 9.5i);


 - 2.75 + 18.0*i


(1.0 + 10.0*i)*(-6.5 + 2.5*i);


 - 31.5 - 62.5*i


(1.2 - 3.4*i)*(-5.6 + 7.8*i);


19.8 + 28.4*i


(19.8 + 28.4*i)/(-5.6 + 7.8*i);


1.2 - 3.4*i


e;


2.71828182846


pi;


3.14159265359


17*i**2;


-17


(-7.0 + 24.0*i)**(1/2);


3.0 + 4.0*i


sqrt(-7.0 + 24.0*i);


3.0 + 4.0*i


sqrt(-10.12 - 8.16*i);


1.2 - 3.4*i


sin(0.0 + 0.0*i);


0


sin(1.0 + 0.0*i);


0.841470984808


sin(1.0 + 1.0*i);


1.29845758142 + 0.634963914785*i


cos(0.0 + 0.0*i);


1


cos(1.0 - 0.0*i);


0.540302305868


cos(1.0 + 1.0*i);


0.833730025131 - 0.988897705763*i


tan(0.0 + 0.0*i);


0


tan(1.0 + 0.0*i);


1.55740772465


tan(1.0 + 1.0*i);


0.27175258532 + 1.08392332734*i


asin(1.0 + 1.0*i);


0.666239432493 + 1.06127506191*i


acos(1.0 + 1.0*i);


0.904556894302 - 1.06127506191*i


atan(1.0 + 1.0*i);


1.0172219679 + 0.402359478109*i


log(1.0 + 1.0*i);


0.34657359028 + 0.785398163397*i


asin 2;


1.57079632679 - 1.31695789692*i


sin ws;


2.0 - 1.25983163303e-15*i


acos 2;


1.31695789692*i


cos ws;


2.0


atan(1+i);


1.0172219679 + 0.402359478109*i


tan ws;


1 + i


log(2+i);


0.804718956217 + 0.463647609001*i


exp ws;


2.0 + i


e**(i*pi);


 - 1 + 1.22460635382e-16*i


e**i;


0.540302305868 + 0.841470984808*i


z := sqrt i;


z := 0.707106781187 + 0.707106781187*i


z**2;


i


off rounded;


*** Domain mode complex-rounded changed to complex 


%-----------------end of normal floating point tests--------------------

precision(prec0+6);


12
 % arbitrary precision increase -> bigfloat functions

%----------------------start of bigfloat tests--------------------------

on rounded;


*** Domain mode complex changed to complex-rounded 


(3.25 + 8.5i) + (6.75 - 8.5i);


10.0


(3.25 + 8.5i) - (6.0 - 9.5i);


 - 2.75 + 18.0*i


(1.0 + 10.0*i)*(-6.5 + 2.5*i);


 - 31.5 - 62.5*i


(1.2 - 3.4*i)*(-5.6 + 7.8*i);


19.8 + 28.4*i


(19.8 + 28.4*i)/(-5.6 + 7.8*i);


1.2 - 3.4*i


e;


2.71828182845904524


pi;


3.14159265358979324


17*i**2;


-17


(-7.0 + 24.0*i)**(1/2);


3.0 + 4.0*i


sqrt(-7.0 + 24.0*i);


3.0 + 4.0*i


sqrt(-10.12 - 8.16*i);


1.2 - 3.4*i


sin(0.0 + 0.0*i);


0


sin(1.0 + 0.0*i);


0.841470984807896507


sin(1.0 + 1.0*i);


1.29845758141597729 + 0.634963914784736108*i


cos(0.0 + 0.0*i);


1


cos(1.0 - 0.0*i);


0.540302305868139717


cos(1.0 + 1.0*i);


0.833730025131149049 - 0.988897705762865096*i


tan(0.0 + 0.0*i);


0


tan(1.0 + 0.0*i);


1.55740772465490223


tan(1.0 + 1.0*i);


0.271752585319511717 + 1.08392332733869454*i


asin(1.0 + 1.0*i);


0.666239432492515255 + 1.06127506190503565*i


acos(1.0 + 1.0*i);


0.904556894302381364 - 1.06127506190503565*i


atan(1.0 + 1.0*i);


1.01722196789785137 + 0.402359478108525094*i


log(1.0 + 1.0*i);


0.346573590279972655 + 0.78539816339744831*i


asin 2;


1.57079632679489662 - 1.31695789692481671*i


sin ws;


2.0


acos 2;


1.31695789692481671*i


cos ws;


2.0


atan(1+i);


1.01722196789785137 + 0.402359478108525094*i


tan ws;


1 + i


log(2+i);


0.804718956217050187 + 0.463647609000806116*i


exp ws;


2.0 + i


e**(i*pi);


 - 1


e**i;


0.540302305868139717 + 0.841470984807896507*i


z := sqrt i;


z := 0.707106781186547524 + 0.707106781186547524*i


z**2;


i


off rounded;


*** Domain mode complex-rounded changed to complex 


% ---------------------------------------------------------------------

% The following examples are independent of precision.

precision prec0;


18
 % restores default precision.

% on rationalize;  % no longer needed, since it doesn't affect complex.

s:= 1.1+2.3i;


      11 + 23*i
s := -----------
         10


s/4;


 11 + 23*i
-----------
    40
  % this would have had a common factor of 4.

x:= a+1.1+2.3i;


      10*a + 11 + 23*i
x := ------------------
             10


y:= b+1.2+1.3i;


      10*b + 12 + 13*i
y := ------------------
             10


z:= x/y;


      100*a*b + (120 - 130*i)*a + (110 + 230*i)*b + 431 + 133*i
z := -----------------------------------------------------------
                             2
                        100*b  + 240*b + 313


z/4;


 100*a*b + (120 - 130*i)*a + (110 + 230*i)*b + 431 + 133*i
-----------------------------------------------------------
                         2
                 4*(100*b  + 240*b + 313)
  % this would have had a common polynomial factor b^2 + ...

z*7/4;


 7*(100*a*b + (120 - 130*i)*a + (110 + 230*i)*b + 431 + 133*i)
---------------------------------------------------------------
                           2
                   4*(100*b  + 240*b + 313)


s/(c^2+c+1);


    11 + 23*i
-----------------
      2
 10*(c  + c + 1)
  % this would have had a common factor of c^2+c+1

clear x;



zz:= x^2+(1.1+2.3i)*x+1.2+1.3i;


           2
       10*x  + (11 + 23*i)*x + 12 + 13*i
zz := -----------------------------------
                      10


ss:=1.23456789x^2+1.3579i*x+5.6789;


                  2
       123456789*x  + 135790000*i*x + 567890000
ss := ------------------------------------------
                      100000000


% off rationalize;  % not needed now.

z:= x+1.1+2.3i;


      10*x + 11 + 23*i
z := ------------------
             10


on rationalize;



z;


 10*x + 11 + 23*i
------------------
        10
               % same as previous answer.

off rationalize;



1.23456789x^2+2.3456i*x+7.89;


            2
 123456789*x  + 234560000*i*x + 789000000
------------------------------------------
                100000000


on factor;



x**2+1;


(x + i)*(x - i)


x**4-1;


(x + i)*(x - i)*(x + 1)*(x - 1)


x**4+(i+2)*x**3+(2*i+5)*x**2+(2*i+6)*x+6;


  2
(x  + i*x + 3)*(x + 1 + i)*(x + 1 - i)


(2*i+3)*x**4+(3*i-2)*x**3-2*(i+1)*x**2+i*x-1;


              2        2
i*((2 - 3*i)*x  - i)*(x  + i*x - 1)


% Multivariate examples:

x**2+y**2;


 (10*b + 10*i*x + 12 + 13*i)*(10*b - 10*i*x + 12 + 13*i)
---------------------------------------------------------
                           100


off factor;



factorize(x**2+1);


{x - i,x + i}


end;
(TIME:  complex 810 850)