Edit detail for SandBox Solve revision 3 of 7

Solving Equations

What method is used to solve equation in Axiom?

(1) -> solve(sin(x)=4/5,x)

(1)

solve([a=4,sin(x)=a/5],[a,x])

(2)

In the following, a workaround is necessary because of bug #128:

)set output algebra on
 

)set output tex off

solve([V_q*U_q+V_l*U_l+V_d*U_d+V_a*U_a=U_ma,               _
       V_q*rho_q+V_l*rho_l+ V_d*rho_d+V_a*rho_a=rho_ma,i   _
       V_q*t_q+V_l*t_l+V_d*t_d+V_a*t_a=t_ma,               _
       V_q+V_l+V_d+V_a=1],                                 _
      [V_q,V_l,V_d,V_a] )

   There are no library operations named i 
      Use HyperDoc Browse or issue
                                 )what op i
      to learn if there is any operation containing " i " in its name.

   Cannot find a definition or applicable library operation named i 
      with argument type(s) 
                                Variable(V_q)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

solve(v^4+b*v^3+c*v^2+d=0,v)

          4      3      2
   (3)  [v  + b v  + c v  + d = 0]

This didn't work since solve returns solutions expressible as members of the ground field only. Above, the ground field of defaults to Fraction Polynomial Integer...

Thus, the proper call is

solve((v^4+b*v^3+c*v^2+d)::EXPR INT=0,v)

   (4)
   [v = %v0, v = %v1,

       v
     = 
            +-----------------------------------------------------------+
            |       2                             2                    2
           \|- 3 %v1  + (- 2 %v0 - 2 b)%v1 - 3 %v0  - 2 b %v0 - 4 c + b   - %v1
         + 
           - %v0 - b
      /
         2
     ,

       v
     = 
              +-----------------------------------------------------------+
              |       2                             2                    2
           - \|- 3 %v1  + (- 2 %v0 - 2 b)%v1 - 3 %v0  - 2 b %v0 - 4 c + b
         + 
           - %v1 - %v0 - b
      /
         2
     ]

But really, you should use 'zerosOf':

zerosOf(v^4+b*v^3+c*v^2+d, v)

   (5)
   [%v0, %v1,

          +-----------------------------------------------------------+
          |       2                             2                    2
         \|- 3 %v1  + (- 2 %v0 - 2 b)%v1 - 3 %v0  - 2 b %v0 - 4 c + b   - %v1
       + 
         - %v0 - b
    /
       2
     ,

            +-----------------------------------------------------------+
            |       2                             2                    2
         - \|- 3 %v1  + (- 2 %v0 - 2 b)%v1 - 3 %v0  - 2 b %v0 - 4 c + b   - %v1
       + 
         - %v0 - b
    /
       2
     ]

In the following, we have to do three things:

• convert the list of Fraction Polynomial Float to a list of Fraction Polynomial Integer, since solve can only handle the latter,
• ask for an approximate solution and
• set precision to a lower value then 68, since it would take too much time otherwise

Furthermore, it's %pi, not %PI. Finally, the underscore is the escape character (like \) and is also used as the the line continuation character when it occurs at the end of th line. So K_sc is equivalent to just Ksc, but K__sc is actually K_sc.

Here is one way to use macros and _ to define more complex names that print nicely in LaTeX? form:

K_sc  ==> K___{sc_}

mu_sc ==> _\mu___{sc_}

digits(7);

l:=  [0.01*(2.25-K_sc)*K_sc/(0.01*%pi*mu_sc*(mu_sc+3.0*K_sc)/                         _
      (4.0*mu_sc+3.0*K_sc)+2.25)+0.7*(37.0-K_sc)*(4.0*mu_sc/3.0+K_sc)/                _
      (4.0*mu_sc/3.0+37.0)+0.29*(2.25-K_sc)*(4.0*mu_sc/3.0+K_sc)/                     _
      (4.0*mu_sc/3.0+2.25),                                                           _
      -0.002*mu_sc*(2.0*(2.0*mu_sc/3.0+2.25)/                                         _
      (0.01*%pi*mu_sc*(mu_sc+3.0*K_sc)/                                               _
      (4.0*mu_sc+3.0*K_sc)+2.25)+800.0*mu_sc/                                         _
      (%pi*(2.0*mu_sc*(mu_sc+3.0*K_sc)/                                               _
      (4.0*mu_sc+3.0*K_sc)+mu_sc))+1.0)+0.7*(44.0-mu_sc)*(mu_sc*(8.0*mu_sc+9.0*K_sc)/ _
      (6.0*(2.0*mu_sc+K_sc))+mu_sc)/                                                  _
      (mu_sc*(8.0*mu_sc+9.0*K_sc)/                                                    _
      (6.0*(2.0*mu_sc+K_sc))+44.0)-1.74*(2.0*mu_sc+K_sc)*(mu_sc*(8.0*mu_sc+9.0*K_sc)/ _
      (6.0*(2.0*mu_sc+K_sc))+mu_sc)/(8.0*mu_sc+9.0*K_sc)]

   (9)
   [
                                          4
         (- 0.99 K_{sc} + 26.5525)\mu_{sc}
       + 
                          2                                      3
         (- 4.98574 K_{sc}  - 190.4062 K_{sc} + 7668.544)\mu_{sc}
       + 
                            3                  2
           (- 3.18243 K_{sc}  - 435.6183 K_{sc}  + 9082.386 K_{sc} + 17708.16)
        *
                   2
           \mu_{sc}
       + 
                           3                 2
         (- 208.4084 K_{sc}  + 455.729 K_{sc}  + 26696.39 K_{sc})\mu_{sc}
       + 
                          3                  2
         - 1531.879 K_{sc}  + 10061.45 K_{sc}
    /
                 4                                  3
         \mu_{sc}  + (3.0 K_{sc} + 315.9164)\mu_{sc}
       + 
                                             2
         (303.1717 K_{sc} + 8480.051)\mu_{sc}
       + 
         (6465.401 K_{sc} + 13415.27)\mu_{sc} + 10061.45 K_{sc}
     ,

                            7                                        6
         - 3.156061 \mu_{sc}  + (- 19.3567 K_{sc} - 823.8225)\mu_{sc}
       + 
                           2                                    5
         (- 40.72939 K_{sc}  - 3289.616 K_{sc} + 1817.7)\mu_{sc}
       + 
                           3                  2                          4
         (- 35.46834 K_{sc}  - 4668.297 K_{sc}  + 22513.6 K_{sc})\mu_{sc}
       + 
                           4                  3                  2         3
         (- 10.85896 K_{sc}  - 2766.375 K_{sc}  + 53897.04 K_{sc} )\mu_{sc}
       + 
                           4                  3         2
         (- 575.9874 K_{sc}  + 45929.19 K_{sc} )\mu_{sc}
       + 
                       4
         13052.6 K_{sc} \mu_{sc}
    /
                 6                                   5
         \mu_{sc}  + (6.75 K_{sc} + 352.4789)\mu_{sc}
       + 
                         2                                      4
         (15.89063 K_{sc}  + 1693.405 K_{sc} + 18907.61)\mu_{sc}
       + 
                         3                  2                           3
         (15.82031 K_{sc}  + 2951.913 K_{sc}  + 73266.98 K_{sc})\mu_{sc}
       + 
                         4                  3                  2         2
         (5.695313 K_{sc}  + 2190.631 K_{sc}  + 101037.5 K_{sc} )\mu_{sc}
       + 
                         4                  3                           4
         (574.9593 K_{sc}  + 58495.41 K_{sc} )\mu_{sc} + 11964.97 K_{sc}
     ]

-- solve exactly for fractions
a:=solve (l::LIST FRAC POLY FRAC INT::LIST FRAC POLY INT);

-- Number of results:
#a

   (11)  2

-- But only the first one is of interest.
-- Display it as a floating point result
a.1::List Equation Fraction POLY FLOAT

   (12)  [\mu_{sc} = 0.0, K_{sc} = 6.568047]

-- Now check it
map(x+->subst(x,(a.1)::List Equation FRAC POLY FLOAT),l)

   (13)  [0.0, 0.0]

)clear all

   All user variables and function definitions have been cleared.

solve((x=-1+x^2)::EQ EXPR INT,x)

              +-+             +-+
             \|5  + 1      - \|5  + 1
   (1)  [x = --------, x = ----------]
                 2              2

quadratic equation

solve(a*x^2+b*x+c=0,x)

            2
   (2)  [a x  + b x + c = 0]

solve(a*x+b=0,x)

               b
   (3)  [x = - -]
               a

quadratic equation

solve(x**2+x-1,x)

   There are no library operations named ** 
      Use HyperDoc Browse or issue
                                 )what op **
      to learn if there is any operation containing " ** " in its name.

   Cannot find a definition or applicable library operation named ** 
      with argument type(s) 
                                 Variable(x)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

radicalSolve(a*x**2+b*x+c=0,x)

   There are no library operations named ** 
      Use HyperDoc Browse or issue
                                 )what op **
      to learn if there is any operation containing " ** " in its name.

   Cannot find a definition or applicable library operation named ** 
      with argument type(s) 
                                 Variable(x)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

This doesn't work on mine michen:

L := [ A = 2*P1+P2, B = 2*P2+P1, C = 2*Q1+Q2, D = 2*Q2+Q1]

   (4)  [A = P2 + 2 P1, B = 2 P2 + P1, C = Q2 + 2 Q1, D = 2 Q2 + Q1]

solve(L, [P1,P2])

   (5)  []

But it should, observe this:

solve([L.1,L.2],[P1,P2])

               - B + 2 A       2 B - A
   (6)  [[P1 = ---------, P2 = -------]]
                   3              3

solve([L.3,L.4],[Q1,Q2])

               - D + 2 C       2 D - C
   (7)  [[Q1 = ---------, Q2 = -------]]
                   3              3

First two equationa do not depend on , the later two don't depend on .

)set output algebra on
 

)set output tex off

radicalSolve(a*x**3+b*x**2+c*x+d=0,x)

   There are no library operations named ** 
      Use HyperDoc Browse or issue
                                 )what op **
      to learn if there is any operation containing " ** " in its name.

   Cannot find a definition or applicable library operation named ** 
      with argument type(s) 
                                 Variable(x)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

solve(sin(x)=4/5,x)

                  4
   (8)  [x = asin(-)]
                  5

solve(x^2=y,x)

              2
   (9)  [y - x  = 0]

Use braces { } not parenthesis ( )

solve(x3=x0+(x1-x0)*t + (x2-x0) *u,u)

              x3 - t x1 + (t - 1)x0
   (10)  [u = ---------------------]
                     x2 - x0

solve([x+y=3,x-y=1],[x,y])

   (11)  [[x = 2, y = 1]]

An error in the way MathAction? folds the LaTeX? output from Axiom prevents this expression from displaying properly. As a work-a-round it is necessary to disable the LaTeX? output and replace it with a ASCII text equivalent.

)set output tex off
 

)set output algebra on

A:=i=(a*x+c*z+e)/(z+g)

             c z + a x + e
   (12)  i = -------------
                 z + g

B:=j=(b*x+d*z+f)/(z+g)

             d z + b x + f
   (13)  j = -------------
                 z + g

solve([A,B],[x,z])

   (14)
         (c g - e)j + (- d g + f)i - c f + d e      - a g j + b g i + a f - b e
   [[x = -------------------------------------, z = ---------------------------]
                 a j - b i - a d + b c                 a j - b i - a d + b c
     ]

)set output tex on
 

)set output algebra off

solve(sin(x)=4/5,x)

(3)

L := [ A = 2*P1+P2, B = 2*P2+P1, C = 2*Q1+Q2, D = 2*Q2+Q1]

   There are 3 exposed and 0 unexposed library operations named 
      equation having 2 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                            )display op equation
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      equation with argument type(s) 
                   Equation(Fraction(Polynomial(Integer)))
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

solve(14=x*1.1^x,x)

   There are 20 exposed and 3 unexposed library operations named solve 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op solve
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named solve
      with argument type(s) 
                         Equation(Expression(Float))
                                 Variable(x)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

)set output algebra on
 

)set output tex off

zerosOf((1-a)*x^4+x^3+x^2+x+1,x)

   (16)
   [%x10, %x11,

         -
            ROOT
                       2               2
                 (- 3 a  + 6 a - 3)%x11
               + 
                        2
                 ((- 2 a  + 4 a - 2)%x10 + 2 a - 2)%x11
               + 
                       2               2
                 (- 3 a  + 6 a - 3)%x10  + (2 a - 2)%x10 + 4 a - 3
       + 
         (- a + 1)%x11 + (- a + 1)%x10 + 1
    /
       2 a - 2
     ,

         ROOT
                    2               2          2
              (- 3 a  + 6 a - 3)%x11  + ((- 2 a  + 4 a - 2)%x10 + 2 a - 2)%x11
            + 
                    2               2
              (- 3 a  + 6 a - 3)%x10  + (2 a - 2)%x10 + 4 a - 3
       + 
         (- a + 1)%x11 + (- a + 1)%x10 + 1
    /
       2 a - 2
     ]

)set output algebra off
 

)set output tex on

solve( dx_p = ( m / b ) * ( c0 - b * dv_p + c0 * log(c0) -     _
              c0 * log( c0 - b * dv_p ) + v0 * log(c0) -       _
              v0 * log( c0 - b * dv_p ) - ( c0 / b ) ), dv_p )

(4)

solve(((1+sqrt(5))^n-(1-sqrt(5))^n)/(sqrt(5)*2^n)=10^1001, n)

(5)

)set output tex off
 

)set output algebra on

solve([nnH+2*niH+nnCs+2*niCs = n, (niH+niCs)*niH/nnH = SH, (niH+niCs)*niCs/nnCs = SCs, (niH+nnH)/(niCs+nnCs) = alpha],[nnH, niH, nnCs, niCs])

   (19)
   [
     [
         nnH
       = 
                                   2                                        2
             ((- 2 SH + 2 SCs)alpha  + (- 3 SH + 3 SCs)alpha - SH + SCs)niCs
           + 
                              2
                 (- 2 SH alpha  + (- SH - SCs)alpha)n
               + 
                                    2      2                      2
                 (- 4 SCs SH + 2 SCs )alpha  + (- 6 SCs SH + 4 SCs )alpha
               + 
                                   2
                 - 2 SCs SH + 2 SCs
            *
               niCs
           + 
                        2                   2                     2
             SCs alpha n  + ((2 SCs SH - SCs )alpha + SCs SH - SCs )n
        /
                                   2           2
           (SCs alpha + SCs)n + SCs alpha + SCs
       ,

         niH
       = 
                                             2
             ((SH - SCs)alpha + SH - SCs)niCs
           + 
                                          2                         2
             (SH alpha n + (2 SCs SH - SCs )alpha + 2 SCs SH - 2 SCs )niCs
           + 
                            2
             (- SCs SH + SCs )n
        /
                      2
           SCs n + SCs
       ,

         nnCs
       = 
                                               2
             ((- SH + SCs)alpha - SH + SCs)niCs
           + 
             (((- SH - SCs)alpha - 2 SCs)n - 2 SCs SH alpha - 2 SCs SH)niCs
           + 
                  2
             SCs n  + SCs SH n
        /
                                   2           2
           (SCs alpha + SCs)n + SCs alpha + SCs
       ,

                           2                                      3
           ((SH - SCs)alpha  + (2 SH - 2 SCs)alpha + SH - SCs)niCs
         + 
                        2                                            2      2
               (SH alpha  + (SH + SCs)alpha + SCs)n + (2 SCs SH - SCs )alpha
             + 
                                2                         2
               (5 SCs SH - 3 SCs )alpha + 3 SCs SH - 2 SCs
          *
                 2
             niCs
         + 
                  2                      2          2                2
           ((2 SCs alpha - SCs SH + 3 SCs )n + 2 SCs SH alpha + 2 SCs SH)niCs
         + 
                2 2      2
           - SCs n  - SCs SH n
       = 
         0
       ]
     ]

solve((-v^3 cos(x) sin(x)^2 / sqrt(v^2 cos(x)^2 + 2ah) - v^2 sin(x)^2 + v cos(x) sqrt(v^2 cos(x)^2 + 2 a h) + v^2 cos(x)^2) / a

solve((-v^3 * cos(x) * sin(x)^2 / sqrt(v^2 * cos(x)^2 + 2*a*h) - v^2 * sin(x)^2 + v * cos(x) * sqrt(v^2 * cos(x)^2 + 2 * a * h) + v^2 * cos(x)^2) / a

  Line   1: solve((-v^3 * cos(x) * sin(x)^2 / sqrt(v^2 * cos(x)^2 + 2*a*h) - v^2 * sin(x)^2 + v * cos(x) * sqrt(v^2 * cos(x)^2 + 2 * a * h) + v^2 * cos(x)^2) / a
           .....A..............................................................................................................................................B
  Error  A: Missing mate.
  Error  B: syntax error at top level
  Error  B: Possibly missing a ) 
   3 error(s) parsing