MathAction #326 Meaning of F0 and F1 ?

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  - #326 Meaning of F0 and F1 ? <-- You are here.

axiom

)set output tex off

axiom

)set output algebra on

axiom

integrate(1/(x^7+x^6+x^5+x^4+x^3+x^2+x+1),x)

   (1)
              +-------------------------------------+
              |        2                       2          +-+         +-+
         (- 2\|- 24%%F1  - 16%%F0 %%F1 - 24%%F0  - 1  - 4\|2 %%F1 - 4\|2 %%F0)
      *
         log
                      +-+         +-+          +-+        +-+
                ((128\|2 %%F0 + 8\|2 )%%F1 + 8\|2 %%F0 - \|2 )
             *
                 +-------------------------------------+
                 |        2                       2
                \|- 24%%F1  - 16%%F0 %%F1 - 24%%F0  - 1
            + 
                                  2             2                  2
              (- 512%%F0 - 32)%%F1  + (- 512%%F0  - 4)%%F1 - 32%%F0  - 4%%F0
            + 
              3x + 1
     + 
            +-------------------------------------+
            |        2                       2          +-+         +-+
         (2\|- 24%%F1  - 16%%F0 %%F1 - 24%%F0  - 1  - 4\|2 %%F1 - 4\|2 %%F0)
      *
         log
                        +-+         +-+          +-+        +-+
                ((- 128\|2 %%F0 - 8\|2 )%%F1 - 8\|2 %%F0 + \|2 )
             *
                 +-------------------------------------+
                 |        2                       2
                \|- 24%%F1  - 16%%F0 %%F1 - 24%%F0  - 1
            + 
                                  2             2                  2
              (- 512%%F0 - 32)%%F1  + (- 512%%F0  - 4)%%F1 - 32%%F0  - 4%%F0
            + 
              3x + 1
     + 
           +-+
         8\|2 %%F1
      *
         log
                                 2            2                    3
              (1024%%F0 + 64)%%F1  + (1024%%F0  + 8)%%F1 + 1024%%F0  + 32%%F0
            + 
              3x + 3
     + 
         +-+                   3         2                       +-+     2
       8\|2 %%F0 log(- 1024%%F0  + 64%%F0  - 24%%F0 + 3x - 5) - \|2 log(x  + 1)
     + 
         +-+               +-+
       2\|2 log(x + 1) + 2\|2 atan(x)
  /
       +-+
     8\|2

Type: Union(Expression(Integer),...)

Apparently these are "rootOf" expressions:

axiom

%%F0::InputForm

   (2)
   (rootOf
    (/ (+ (+ (+ (* 2048 (^ %%F0 4)) (* 64 (^ %%F0 2))) (* 16 %%F0)) 1) 2048)
    %%F0)

Type: InputForm

axiom

%%F1::InputForm

   (3)
   (rootOf

     (/

       (+

         (+

           (+

             (*  128

               (^

                 (rootOf

                   (/

                     (+
                      (+ (+ (* 2048 (^ %%F0 4)) (* 64 (^ %%F0 2))) (* 16 %%F0))
                      1)

                    2048)

                  %%F0)

                3)
               )

             (*  (* 128 %%F1)

               (^

                 (rootOf

                   (/

                     (+
                      (+ (+ (* 2048 (^ %%F0 4)) (* 64 (^ %%F0 2))) (* 16 %%F0))
                      1)

                    2048)

                  %%F0)

                2)
               )
             )

           (*  (+ (* 128 (^ %%F1 2)) 4)

             (rootOf

               (/
                (+ (+ (+ (* 2048 (^ %%F0 4)) (* 64 (^ %%F0 2))) (* 16 %%F0)) 1)
                2048)

              %%F0)
             )
           )

        (+ (+ (* 128 (^ %%F1 3)) (* 4 %%F1)) 1))

      128)

    %%F1)

Type: InputForm

We can in fact solve for some of these.

axiom

)set output tex on

axiom

)set output algebra off

axiom

definingPolynomial %%F0

$\label{eq1}{{{2048}\ {\%\%F 0^4}}+{{64}\ {\%\%F 0^2}}+{{16}\ \%\%F 0}+ 1}\over{2048}$

(1)

Type: Expression(Integer)

axiom

F0:=radicalSolve(%::UP('%%F0,COMPLEX FRAC INT))

$\label{eq2}\begin{array}{@{}l} \displaystyle \left[{\%\%F 0 ={-{{1 \over 2}\ {\sqrt{-{{1 \over{16}}\ i}}}}-{{1 \over 8}\ i}}}, \:{\%\%F 0 ={{{1 \over 2}\ {\sqrt{-{{1 \over{1 6}}\ i}}}}-{{1 \over 8}\ i}}}, \: \right. \ \ \displaystyle \left.{\%\%F 0 ={-{{1 \over 2}\ {\sqrt{{1 \over{16}}\ i}}}+{{1 \over 8}\ i}}}, \:{\%\%F 0 ={{{1 \over 2}\ {\sqrt{{1 \over{1 6}}\ i}}}+{{1 \over 8}\ i}}}\right]$

(2)

Type: List(Equation(Expression(Complex(Fraction(Integer)))))

axiom

definingPolynomial %%F1

$\label{eq3}{\left( \begin{array}{@{}l} \displaystyle {{128}\ {\%\%F 0^3}}+{{128}\ \%\%F 1 \ {\%\%F 0^2}}+ \ \ \displaystyle {{\left({{128}\ {\%\%F 1^2}}+ 4 \right)}\ \%\%F 0}+{{128}\ {\%\%F 1^3}}+{4 \ \%\%F 1}+ 1$

(3)

Type: Expression(Integer)

axiom

F11:=subst(%,`%%F0=rhs(F0.1))::UP('%%F1,?)

$\label{eq4}\begin{array}{@{}l} \displaystyle {\%\%F 1^3}+{{\left(-{{1 \over 2}\ {\sqrt{-{{1 \over{16}}\ i}}}}-{{1 \over 8}\ i}\right)}\ {\%\%F 1^2}}+ \ \ \displaystyle {{\left({{1 \over 8}\ i \ {\sqrt{-{{1 \over{16}}\ i}}}}+{1 \over{64}}-{{1 \over{64}}\ i}\right)}\ \%\%F 1}+ \ \ \displaystyle {{\left({1 \over{128}}+{{1 \over{128}}\ i}\right)}\ {\sqrt{-{{1 \over{16}}\ i}}}}+{1 \over{512}}-{{1 \over{512}}\ i}$

(4)

Type: UnivariatePolynomial(%%F1,Expression(Complex(Fraction(Integer))))

Why doesn't Axiom automatically provide this sort of information?

Perhaps related to #262

... --test1, Tue, 06 May 2014 17:37:13 +0000 reply

Severity: normal => wishlist

Subject: Be Bold !!

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