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assume_pos-false-makelist-solve

assume_pos:false;

liste:makelist(i,i,-5...

liste^2;

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assume_pos

? assume_pos;

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assume_pos-factor-integrate-sqrt-true

assume_pos:true;

t:integrate(1/(x*sqrt...

factor(t);

Calculate

assume_pos-factor-integrate-sqrt-true

assume_pos:true;

t:integrate(1/(x*sqrt...

factor(t);

Calculate

assume_pos-makelist-true

assume_pos:true;

makelist(i,i,-5,5);

Calculate

assume_pos-factor-integrate-sqrt-true

assume_pos:true;

t:integrate(1/(x*sqrt...

factor(t);

Calculate

assume_pos-integrate-sqrt-true

assume_pos:true;

t:integrate(1/(x*sqrt...

Calculate

assume_pos-expand-factor-integrate-sqrt-true

assume_pos:true;

t:integrate(1/(x*sqrt...

expand(factor(t));

Calculate

assume_pos-integrate-true

assume_pos:true;

f[0](x):=1/(1+x)^3;

f[n](x):=integrate(f[...

Calculate

assume_pos-false-makelist-solve

assume_pos:false;

liste:makelist(i,i,-5...

liste^2;

Calculate

assume_pos

Run Example
(%i1)assume_pos : true;
(%o1)                                true
(%i2) solveradcan : true;
(%o2)                                true
(%i3) globalsolve: true;
(%o3)                                true
(%i4) backsubst : false;
(%o4)                                false
(%i5) algebraic : true;
(%o5)                                true
(%i6) keepfloat : true;
(%o6)                                true
(%i7) N : 2;
(%o7)                                  2
(%i8) C : 1;
(%o8)                                  1
(%i9) D : 2;
(%o9)                                  2
(%i10) W[m, n] := H(m, n) - E * S(m, n);
(%o10)                   W     := H(m, n) - E S(m, n)
                          m, n
(%i11) matrix_H : genmatrix(W, D, D);
                 [ H(1, 1) - S(1, 1) E  H(1, 2) - S(1, 2) E ]
(%o11)           [                                          ]
                 [ H(2, 1) - S(2, 1) E  H(2, 2) - S(2, 2) E ]
(%i12) det : factor(xthru(radcan(determinant(matrix_H))));
                        2                    2
(%o12) S(1, 1) S(2, 2) E  - S(1, 2) S(2, 1) E  - H(1, 1) S(2, 2) E
 - S(1, 1) H(2, 2) E + H(1, 2) S(2, 1) E + S(1, 2) H(2, 1) E + H(1, 1) H(2, 2)
 - H(1, 2) H(2, 1)
(%i13) solve_det : solve(det = 0, E);
                     2        2
(%o13) [E = - (sqrt(H (1, 1) S (2, 2) + (- 2 H(1, 1) S(1, 1) H(2, 2)
 - 2 H(1, 1) H(1, 2) S(2, 1) + (4 S(1, 1) H(1, 2) - 2 H(1, 1) S(1, 2)) H(2, 1))
            2        2
 S(2, 2) + S (1, 1) H (2, 2) + ((4 H(1, 1) S(1, 2) - 2 S(1, 1) H(1, 2)) S(2, 1)
                                         2        2
 - 2 S(1, 1) S(1, 2) H(2, 1)) H(2, 2) + H (1, 2) S (2, 1)
                                        2        2
 - 2 H(1, 2) S(1, 2) H(2, 1) S(2, 1) + S (1, 2) H (2, 1)) - H(1, 1) S(2, 2)
 - S(1, 1) H(2, 2) + H(1, 2) S(2, 1) + S(1, 2) H(2, 1))
/(2 S(1, 1) S(2, 2) - 2 S(1, 2) S(2, 1)), 
           2        2
E = (sqrt(H (1, 1) S (2, 2) + (- 2 H(1, 1) S(1, 1) H(2, 2)
 - 2 H(1, 1) H(1, 2) S(2, 1) + (4 S(1, 1) H(1, 2) - 2 H(1, 1) S(1, 2)) H(2, 1))
            2        2
 S(2, 2) + S (1, 1) H (2, 2) + ((4 H(1, 1) S(1, 2) - 2 S(1, 1) H(1, 2)) S(2, 1)
                                         2        2
 - 2 S(1, 1) S(1, 2) H(2, 1)) H(2, 2) + H (1, 2) S (2, 1)
                                        2        2
 - 2 H(1, 2) S(1, 2) H(2, 1) S(2, 1) + S (1, 2) H (2, 1)) + H(1, 1) S(2, 2)
 + S(1, 1) H(2, 2) - H(1, 2) S(2, 1) - S(1, 2) H(2, 1))
/(2 S(1, 1) S(2, 2) - 2 S(1, 2) S(2, 1))]
(%i14) second(solve_det[1]);
                2        2
(%o14) - (sqrt(H (1, 1) S (2, 2) + (- 2 H(1, 1) S(1, 1) H(2, 2)
 - 2 H(1, 1) H(1, 2) S(2, 1) + (4 S(1, 1) H(1, 2) - 2 H(1, 1) S(1, 2)) H(2, 1))
            2        2
 S(2, 2) + S (1, 1) H (2, 2) + ((4 H(1, 1) S(1, 2) - 2 S(1, 1) H(1, 2)) S(2, 1)
                                         2        2
 - 2 S(1, 1) S(1, 2) H(2, 1)) H(2, 2) + H (1, 2) S (2, 1)
                                        2        2
 - 2 H(1, 2) S(1, 2) H(2, 1) S(2, 1) + S (1, 2) H (2, 1)) - H(1, 1) S(2, 2)
 - S(1, 1) H(2, 2) + H(1, 2) S(2, 1) + S(1, 2) H(2, 1))
/(2 S(1, 1) S(2, 2) - 2 S(1, 2) S(2, 1))
(%i15) second(solve_det[2]);
              2        2
(%o15) (sqrt(H (1, 1) S (2, 2) + (- 2 H(1, 1) S(1, 1) H(2, 2)
 - 2 H(1, 1) H(1, 2) S(2, 1) + (4 S(1, 1) H(1, 2) - 2 H(1, 1) S(1, 2)) H(2, 1))
            2        2
 S(2, 2) + S (1, 1) H (2, 2) + ((4 H(1, 1) S(1, 2) - 2 S(1, 1) H(1, 2)) S(2, 1)
                                         2        2
 - 2 S(1, 1) S(1, 2) H(2, 1)) H(2, 2) + H (1, 2) S (2, 1)
                                        2        2
 - 2 H(1, 2) S(1, 2) H(2, 1) S(2, 1) + S (1, 2) H (2, 1)) + H(1, 1) S(2, 2)
 + S(1, 1) H(2, 2) - H(1, 2) S(2, 1) - S(1, 2) H(2, 1))
/(2 S(1, 1) S(2, 2) - 2 S(1, 2) S(2, 1))
(%i16) display2d : false;

(%o16) false
(%i17) 
det;

(%o17) S(1,1)*S(2,2)*E^2-S(1,2)*S(2,1)*E^2-H(1,1)*S(2,2)*E-S(1,1)*H(2,2)*E
                        +H(1,2)*S(2,1)*E+S(1,2)*H(2,1)*E+H(1,1)*H(2,2)
                        -H(1,2)*H(2,1)
(%i18) 
solve_det;

(%o18) [E = -(sqrt(H(1,1)^2*S(2,2)^2+(-2*H(1,1)*S(1,1)*H(2,2)
                                     -2*H(1,1)*H(1,2)*S(2,1)
                                     +(4*S(1,1)*H(1,2)-2*H(1,1)*S(1,2))
                                      *H(2,1))
                                     *S(2,2)+S(1,1)^2*H(2,2)^2
                                    +((4*H(1,1)*S(1,2)-2*S(1,1)*H(1,2))*S(2,1)
                                     -2*S(1,1)*S(1,2)*H(2,1))
                                     *H(2,2)+H(1,2)^2*S(2,1)^2
                                    -2*H(1,2)*S(1,2)*H(2,1)*S(2,1)
                                    +S(1,2)^2*H(2,1)^2)
          -H(1,1)*S(2,2)-S(1,1)*H(2,2)+H(1,2)*S(2,1)+S(1,2)*H(2,1))
          /(2*S(1,1)*S(2,2)-2*S(1,2)*S(2,1)),
        E = (sqrt(H(1,1)^2*S(2,2)^2+(-2*H(1,1)*S(1,1)*H(2,2)
                                    -2*H(1,1)*H(1,2)*S(2,1)
                                    +(4*S(1,1)*H(1,2)-2*H(1,1)*S(1,2))*H(2,1))
                                    *S(2,2)+S(1,1)^2*H(2,2)^2
                                   +((4*H(1,1)*S(1,2)-2*S(1,1)*H(1,2))*S(2,1)
                                    -2*S(1,1)*S(1,2)*H(2,1))
                                    *H(2,2)+H(1,2)^2*S(2,1)^2
                                   -2*H(1,2)*S(1,2)*H(2,1)*S(2,1)
                                   +S(1,2)^2*H(2,1)^2)
          +H(1,1)*S(2,2)+S(1,1)*H(2,2)-H(1,2)*S(2,1)-S(1,2)*H(2,1))
          /(2*S(1,1)*S(2,2)-2*S(1,2)*S(2,1))]
(%i19) 
display2d : true;
(%o19)                               true
(%i20) tex(det);
$$S\left(1 , 1\right)\,S\left(2 , 2\right)\,E^2-S\left(1 , 2\right)\,
 S\left(2 , 1\right)\,E^2-H\left(1 , 1\right)\,S\left(2 , 2\right)\,E
 -S\left(1 , 1\right)\,H\left(2 , 2\right)\,E+H\left(1 , 2\right)\,S
 \left(2 , 1\right)\,E+S\left(1 , 2\right)\,H\left(2 , 1\right)\,E+H
 \left(1 , 1\right)\,H\left(2 , 2\right)-H\left(1 , 2\right)\,H\left(
 2 , 1\right)$$
(%o20)                               false
(%i21) tex(solve_det);
$$\left[ E=-{{\sqrt{H^2\left(1 , 1\right)\,S^2\left(2 , 2\right)+
 \left(-2\,H\left(1 , 1\right)\,S\left(1 , 1\right)\,H\left(2 , 2
 \right)-2\,H\left(1 , 1\right)\,H\left(1 , 2\right)\,S\left(2 , 1
 \right)+\left(4\,S\left(1 , 1\right)\,H\left(1 , 2\right)-2\,H\left(
 1 , 1\right)\,S\left(1 , 2\right)\right)\,H\left(2 , 1\right)\right)
 \,S\left(2 , 2\right)+S^2\left(1 , 1\right)\,H^2\left(2 , 2\right)+
 \left(\left(4\,H\left(1 , 1\right)\,S\left(1 , 2\right)-2\,S\left(1
  , 1\right)\,H\left(1 , 2\right)\right)\,S\left(2 , 1\right)-2\,S
 \left(1 , 1\right)\,S\left(1 , 2\right)\,H\left(2 , 1\right)\right)
 \,H\left(2 , 2\right)+H^2\left(1 , 2\right)\,S^2\left(2 , 1\right)-2
 \,H\left(1 , 2\right)\,S\left(1 , 2\right)\,H\left(2 , 1\right)\,S
 \left(2 , 1\right)+S^2\left(1 , 2\right)\,H^2\left(2 , 1\right)}-H
 \left(1 , 1\right)\,S\left(2 , 2\right)-S\left(1 , 1\right)\,H\left(
 2 , 2\right)+H\left(1 , 2\right)\,S\left(2 , 1\right)+S\left(1 , 2
 \right)\,H\left(2 , 1\right)}\over{2\,S\left(1 , 1\right)\,S\left(2
  , 2\right)-2\,S\left(1 , 2\right)\,S\left(2 , 1\right)}} , E={{
 \sqrt{H^2\left(1 , 1\right)\,S^2\left(2 , 2\right)+\left(-2\,H\left(
 1 , 1\right)\,S\left(1 , 1\right)\,H\left(2 , 2\right)-2\,H\left(1
  , 1\right)\,H\left(1 , 2\right)\,S\left(2 , 1\right)+\left(4\,S
 \left(1 , 1\right)\,H\left(1 , 2\right)-2\,H\left(1 , 1\right)\,S
 \left(1 , 2\right)\right)\,H\left(2 , 1\right)\right)\,S\left(2 , 2
 \right)+S^2\left(1 , 1\right)\,H^2\left(2 , 2\right)+\left(\left(4\,
 H\left(1 , 1\right)\,S\left(1 , 2\right)-2\,S\left(1 , 1\right)\,H
 \left(1 , 2\right)\right)\,S\left(2 , 1\right)-2\,S\left(1 , 1
 \right)\,S\left(1 , 2\right)\,H\left(2 , 1\right)\right)\,H\left(2
  , 2\right)+H^2\left(1 , 2\right)\,S^2\left(2 , 1\right)-2\,H\left(1
  , 2\right)\,S\left(1 , 2\right)\,H\left(2 , 1\right)\,S\left(2 , 1
 \right)+S^2\left(1 , 2\right)\,H^2\left(2 , 1\right)}+H\left(1 , 1
 \right)\,S\left(2 , 2\right)+S\left(1 , 1\right)\,H\left(2 , 2
 \right)-H\left(1 , 2\right)\,S\left(2 , 1\right)-S\left(1 , 2\right)
 \,H\left(2 , 1\right)}\over{2\,S\left(1 , 1\right)\,S\left(2 , 2
 \right)-2\,S\left(1 , 2\right)\,S\left(2 , 1\right)}} \right] $$
(%o21)                               false
(%i22) is(second(solve_det[1]) >
 second(solve_det[2]));
(%o22)                              unknown
(%i23) X[m, n] := H(m, n) - second(solve_det[1]) * S(m, n);
(%o23)           X     := H(m, n) - second(solve_det ) S(m, n)
                  m, n                              1
(%i24) matrix_H : genmatrix(X, D, D);
                               2        2
(%o24) matrix([(S(1, 1) (sqrt(H (1, 1) S (2, 2)
 + (- 2 H(1, 1) S(1, 1) H(2, 2) - 2 H(1, 1) H(1, 2) S(2, 1)
                                                               2        2
 + (4 S(1, 1) H(1, 2) - 2 H(1, 1) S(1, 2)) H(2, 1)) S(2, 2) + S (1, 1) H (2, 2)
 + ((4 H(1, 1) S(1, 2) - 2 S(1, 1) H(1, 2)) S(2, 1)
                                         2        2
 - 2 S(1, 1) S(1, 2) H(2, 1)) H(2, 2) + H (1, 2) S (2, 1)
                                        2        2
 - 2 H(1, 2) S(1, 2) H(2, 1) S(2, 1) + S (1, 2) H (2, 1)) - H(1, 1) S(2, 2)
 - S(1, 1) H(2, 2) + H(1, 2) S(2, 1) + S(1, 2) H(2, 1)))
/(2 S(1, 1) S(2, 2) - 2 S(1, 2) S(2, 1)) + H(1, 1), 
                2        2
(S(1, 2) (sqrt(H (1, 1) S (2, 2) + (- 2 H(1, 1) S(1, 1) H(2, 2)
 - 2 H(1, 1) H(1, 2) S(2, 1) + (4 S(1, 1) H(1, 2) - 2 H(1, 1) S(1, 2)) H(2, 1))
            2        2
 S(2, 2) + S (1, 1) H (2, 2) + ((4 H(1, 1) S(1, 2) - 2 S(1, 1) H(1, 2)) S(2, 1)
                                         2        2
 - 2 S(1, 1) S(1, 2) H(2, 1)) H(2, 2) + H (1, 2) S (2, 1)
                                        2        2
 - 2 H(1, 2) S(1, 2) H(2, 1) S(2, 1) + S (1, 2) H (2, 1)) - H(1, 1) S(2, 2)
 - S(1, 1) H(2, 2) + H(1, 2) S(2, 1) + S(1, 2) H(2, 1)))
/(2 S(1, 1) S(2, 2) - 2 S(1, 2) S(2, 1)) + H(1, 2)], 
                 2        2
[(S(2, 1) (sqrt(H (1, 1) S (2, 2) + (- 2 H(1, 1) S(1, 1) H(2, 2)
 - 2 H(1, 1) H(1, 2) S(2, 1) + (4 S(1, 1) H(1, 2) - 2 H(1, 1) S(1, 2)) H(2, 1))
            2        2
 S(2, 2) + S (1, 1) H (2, 2) + ((4 H(1, 1) S(1, 2) - 2 S(1, 1) H(1, 2)) S(2, 1)
                                         2        2
 - 2 S(1, 1) S(1, 2) H(2, 1)) H(2, 2) + H (1, 2) S (2, 1)
                                        2        2
 - 2 H(1, 2) S(1, 2) H(2, 1) S(2, 1) + S (1, 2) H (2, 1)) - H(1, 1) S(2, 2)
 - S(1, 1) H(2, 2) + H(1, 2) S(2, 1) + S(1, 2) H(2, 1)))
/(2 S(1, 1) S(2, 2) - 2 S(1, 2) S(2, 1)) + H(2, 1), 
                2        2
(S(2, 2) (sqrt(H (1, 1) S (2, 2) + (- 2 H(1, 1) S(1, 1) H(2, 2)
 - 2 H(1, 1) H(1, 2) S(2, 1) + (4 S(1, 1) H(1, 2) - 2 H(1, 1) S(1, 2)) H(2, 1))
            2        2
 S(2, 2) + S (1, 1) H (2, 2) + ((4 H(1, 1) S(1, 2) - 2 S(1, 1) H(1, 2)) S(2, 1)
                                         2        2
 - 2 S(1, 1) S(1, 2) H(2, 1)) H(2, 2) + H (1, 2) S (2, 1)
                                        2        2
 - 2 H(1, 2) S(1, 2) H(2, 1) S(2, 1) + S (1, 2) H (2, 1)) - H(1, 1) S(2, 2)
 - S(1, 1) H(2, 2) + H(1, 2) S(2, 1) + S(1, 2) H(2, 1)))
/(2 S(1, 1) S(2, 2) - 2 S(1, 2) S(2, 1)) + H(2, 2)])
(%i25) vector_c : makelist(c[i], i, 1, D);
(%o25)                             [c , c ]
                                     1   2
(%i26) 
Run Example
assume_pos:true;
(%o1)                                true
(%i2) solve(x^2+8*x+15=0,x);
(%o2)                         [x = - 3, x = - 5]
(%i3) 
Run Example
assume_pos:true;
(%o1)                                true
(%i2) t:integrate(1/(x*sqrt(0.7+0.3/x^3)),x);

rat: replaced -0.7 by -7/10 = -0.7
                                    0.3
                            20 sqrt(--- + 0.7) - 2 sqrt(70)
                                     3
                                    x
                     10 log(-------------------------------)
                                    0.3
                            20 sqrt(--- + 0.7) + 2 sqrt(70)
                                     3
                                    x
(%o2)              - ---------------------------------------
                                   3 sqrt(70)
(%i3) expand(factor(t));

rat: replaced 0.7 by 7/10 = 0.7

rat: replaced 0.3 by 3/10 = 0.3
                                       3
                            10 sqrt(7 x  + 3)
(%o3) - (10 log(------------------------------------------
                           3                           3/2
                10 sqrt(7 x  + 3) + sqrt(10) sqrt(70) x
                                                   3/2
                                sqrt(10) sqrt(70) x
                    - ------------------------------------------))/(3 sqrt(70))
                                 3                           3/2
                      10 sqrt(7 x  + 3) + sqrt(10) sqrt(70) x
(%i4) 

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