/*************** -*- Mode: MACSYMA; Package: MAXIMA -*-  ******************/
/***************************************************************************
***                                                                    *****
***     Copyright (c) 1984 by William Schelter,University of Texas     *****
***     All rights reserved                                            *****
***************************************************************************/


/* -*- Mode: MACSYMA; Package: CL-MAXIMA -*- */
block([],kill(all),%rnum:0);
0$
X^10-2*X^4+1/2;
X^10-2*X^4+1/2$
NROOTS(%,-6,9.1);
4$
REALROOTS(X^5-X-1,5.0E-6);
[X = 612003/524288]$
EV(%[1],FLOAT);
X = 1.16730309$
EV(X^5-X-1,%);
-7.396496210176906E-6;
(2*X+1)^3 = 13.5*(X^5+1);
(2*X+1)^3 = 13.5*(X^5+1)$
sort(ALLROOTS(%));
[X = -1.0157555438281209,X = 0.82967499021293611,X = 1.0,
       X = -0.96596251521963683*%I-0.40695972319240747,
       X = 0.96596251521963683*%I-0.40695972319240747];
/* lispm
[X = 0.829675,X = -1.0157557,X = 0.9659626*%I-0.4069597,
	X = -0.9659626*%I-0.4069597,X = 1.0000001]; */
/* result tops20 [X = 0.82967498,X = -1.01575564,X = 0.96596261*%I-0.406959705,
 X = -0.96596261*%I-0.406959705,X = 1.00000006]$ */
EXP1:X+Z = Y;
Z+X = Y$
EXP:2*A*X-Y = 2*A^2;
2*A*X-Y = 2*A^2$
Y-2*Z = 2;
Y-2*Z = 2$
EV(LINSOLVE([EXP,EXP1,%],[X,Y,Z]),GLOBALSOLVE);
[X = A+1,Y = 2*A,Z = A-1]$
F1:2*X*(1-L1)-2*(X-1)*L2;
2*(1-L1)*X-2*L2*(X-1)$
F2:L2-L1;
L2-L1$
F3:L1*(1-X^2-Y);
L1*(-Y-X^2+1)$
F4:L2*(Y-(X-1)^2);
L2*(Y-(X-1)^2)$
ALGSYS([F1,F2,F3,F4],[X,Y,L1,L2]);
[[X = 0,Y = %R1,L1 = 0,L2 = 0],[X = 1,Y = 0,L1 = 1,L2 = 1]]$
F1:X^2-Y^2;
X^2-Y^2$
F2:X^2-X+2*Y^2-Y-1;
2*Y^2-Y+X^2-X-1$
ALGSYS([F1,F2],[X,Y]);
[[X = -1/SQRT(3),Y = 1/SQRT(3)],[X = 1/SQRT(3),Y = -1/SQRT(3)],
 [X = -1/3,Y = -1/3],[X = 1,Y = 1]]$
SOLVE(ASIN(COS(3*X))*(F(X)-1),X);
[X = %PI/6,F(X) = 1]$
EV(SOLVE(5^F(X) = 125,F(X)),SOLVERADCAN);
[F(X) = LOG(125)/LOG(5)]$
[4*X^2-Y^2 = 12,X*Y-X = 2];
[4*X^2-Y^2 = 12,X*Y-X = 2]$
SOLVE(%,[X,Y]);
[[X = 2,Y = 2],
 [X = 0.5202594388652008*%I-0.1331240357358706,
  Y = 0.07678378523787777-3.608003221870287*%I],
 [X = -0.5202594388652008*%I-0.1331240357358706,
  Y = 3.608003221870287*%I+0.07678378523787777],
 [X = -1.733751846381093,Y = -0.1535675710019696]];
/* lispm
[[X = 2,Y = 2],[X = 0.52025807*%I-0.13312346,Y = 0.07678269-3.6080136*%I],
	[X = -0.52025807*%I-0.13312346,Y = 3.6080136*%I+0.07678269],
 [X = -1.7337531,Y = -0.1535668]]$ */
/* tops 20 :[[X = 2,Y = 2],[X = 0.52025944*%I-0.133124037,Y = 0.076783786-3.6080032*%I],
 [X = -0.52025944*%I-0.133124037,Y = 3.6080032*%I+0.076783786],
 [X = -1.73375185,Y = -0.153567577]]$ */
SOLVE(X^3+A*X+1,X);
[X = (-SQRT(3)*%I/2-1/2)*(SQRT(4*A^3+27)/(6*SQRT(3))-1/2)^(1/3)
   -(SQRT(3)*%I/2-1/2)*A/(3*(SQRT(4*A^3+27)/(6*SQRT(3))-1/2)^(1/3)),
 X = (SQRT(3)*%I/2-1/2)*(SQRT(4*A^3+27)/(6*SQRT(3))-1/2)^(1/3)
   -(-SQRT(3)*%I/2-1/2)*A/(3*(SQRT(4*A^3+27)/(6*SQRT(3))-1/2)^(1/3)),
 X = (SQRT(4*A^3+27)/(6*SQRT(3))-1/2)^(1/3)
   -A/(3*(SQRT(4*A^3+27)/(6*SQRT(3))-1/2)^(1/3))]$
SOLVE(X^3-1);
[X = (SQRT(3)*%I-1)/2,X = -(SQRT(3)*%I+1)/2,X = 1]$
SOLVE(X^6-1);
[X = (SQRT(3)*%I+1)/2,X = (SQRT(3)*%I-1)/2,X = -1,X = -(SQRT(3)*%I+1)/2,
 X = -(SQRT(3)*%I-1)/2,X = 1]$
EV(X^6-1,%[1]);
(SQRT(3)*%I+1)^6/64-1$
EXPAND(%);
0$
EXP:X^2-1;
X^2-1$
SOLVE(%,X);
[X = -1,X = 1]$
EV(EXP,%[1]);
0$
H[I,J]:=1/(I+J-1);
H[I,J]:=1/(I+J-1)$
GENMATRIX(H,3,3);
MATRIX([1,1/2,1/3],[1/2,1/3,1/4],[1/3,1/4,1/5])$
[2*X-(A-1)*Y = 5*B,A*X+B*Y+C = 0];
[2*X-(A-1)*Y = 5*B,B*Y+A*X+C = 0]$
AUGCOEFMATRIX(%,[X,Y]);
MATRIX([2,1-A,-5*B],[A,B,C])$
MATRIX([2,1-A,-5*B],[A,B,C]);
MATRIX([2,1-A,-5*B],[A,B,C])$
ECHELON(%);
MATRIX([1,-(A-1)/2,-5*B/2],[0,1,(2*C+5*A*B)/(2*B+A^2-A)])$
MATRIX([2,1-A,-5*B],[A,B,C]);
MATRIX([2,1-A,-5*B],[A,B,C])$
TRIANGULARIZE(%);
MATRIX([2,1-A,-5*B],[0,2*B+A^2-A,2*C+5*A*B])$
MATRIX([2,1-A,-5*B],[A,B,C]);
MATRIX([2,1-A,-5*B],[A,B,C])$
RANK(%);
2$
A:MATRIX([3,1],[2,4]);
MATRIX([3,1],[2,4])$
EXPAND(CHARPOLY(A,LAMBDA));
LAMBDA^2-7*LAMBDA+10$
EXP:(PROGRAMMODE:TRUE,SOLVE(%));
[LAMBDA = 5,LAMBDA = 2]$
MATRIX([X1],[X2]);
MATRIX([X1],[X2])$
EV(A . %-LAMBDA*%,EXP[1]);
MATRIX([X2-2*X1],[2*X1-X2])$
EXP:%[1,1] = 0;
X2-2*X1 = 0$
X1^2+X2^2 = 1;
X2^2+X1^2 = 1$
SOLVE([EXP,%],[X1,X2]);
[[X1 = -1/SQRT(5),X2 = -2/SQRT(5)],[X1 = 1/SQRT(5),X2 = 2/SQRT(5)]]$