Sophie: yacas-1.0.47rev7-1mdk i586

yacas-1.0.47rev7-1mdk.i586.rpm

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An assorted selection of the tests found in the Wester benchmark

1. <font color=0000ff>Compute 50!</font> : 
In&gt 50! 
<font color=ff0000> Out&gt 30414093201713378043612608166064768844377641568960512000000000000 </font>

2. <font color=0000ff>Compute the prime decomposition of 6!.</font> : 
In&gt ans:=Factors(6!) 
<font color=ff0000> Out&gt {{2,4},{5,1},{3,2}} </font>
This list contains lists of two elements. This list should
be interpreted as 
In&gt PrettyForm(FW(ans)) 
<font color=ff0000> 
 4        2
2  * 5 * 3 

Out&gt True </font>

3. <font color=0000ff>Compute 1/2 + ... + 1/10.</font> : 
In&gt  Sum(i,2,10,1/i) 
<font color=ff0000> Out&gt 4861/2520 </font>

4. <font color=0000ff>Compute a numerical approximation of e^(Pi*sqrt(163)) to 50 digits.</font> : 
In&gt Precision(50) 
<font color=ff0000></font>In&gt N(Exp(Pi*Sqrt(163))) 
<font color=ff0000> Out&gt 262537412640768743.99999999999925007259719818568885219682604177332393 </font>

5. <font color=0000ff>Compute an infinite decimal representation of 1/7</font> : 
In&gt Decimal(1/7) 
<font color=ff0000> Out&gt {0,{1,4,2,8,5,7}} </font>

6. <font color=0000ff>Compute the first terms of the continued fraction of Pi.</font> : 
In&gt PrettyForm(ContFrac(Pi())) 
<font color=ff0000> 
                 1             
3 + ---------------------------
                   1           
    7 + -----------------------
                     1         
        15 + ------------------
                       1       
             1 + --------------
                          1    
                 292 + --------
                       1 + rest

Out&gt True </font>

7. <font color=0000ff>Simplify sqrt(2*sqrt(3)+4).</font> : 
In&gt  RadSimp(Sqrt(2*Sqrt(3)+4)) 
<font color=ff0000> Out&gt 1+Sqrt(3) </font>

8. <font color=0000ff>Simplify sqrt(14+3*sqrt(3+2*sqrt(5-12*sqrt(3-2*sqrt(2))))).</font> : 
In&gt RadSimp(Sqrt(14+3*Sqrt(3+2*Sqrt(5-12*Sqrt(3-2*Sqrt(2)))))) 
<font color=ff0000> Out&gt 3+Sqrt(2) </font>

9. <font color=0000ff>Simplify 2*infinity-3.</font> : 
In&gt 2*Infinity-3 
<font color=ff0000> Out&gt Infinity </font>
Infinity is also defined for comparisons like a &lt Infinity

10. <font color=0000ff>Compute the normal form of (x^2-4)/(x^2+4x+4).</font> : 
In&gt  PrettyForm(GcdReduce((x^2-4)/(x^2+4*x+4),x)) 
<font color=ff0000> 
-2 + x
------
2 + x 

Out&gt True </font>

11. <font color=0000ff>Expand (x+1)^5, then differentiate and factorize.</font> : 
In&gt  ans:=Factors(D(x)Expand((x+1)^5)) 
<font color=ff0000></font>In&gt PrettyForm(FW(ans)) 
<font color=ff0000> 
         4    
( 1 + x )  * 5

Out&gt True </font>

12. <font color=0000ff>Simplify sqrt(997) - (997^3)^(1/6).</font> : 
In&gt  RadSimp(Sqrt(997)-997^3^(1/6)) 
<font color=ff0000> Out&gt 0 </font>

13. <font color=0000ff>Simplify sqrt(999983) - (999983^3)^(1/6).</font> : 
In&gt RadSimp(Sqrt(999983)-999983^3^(1/6)) 
<font color=ff0000> Out&gt 0 </font>

14. <font color=0000ff>Recognize that (2^(1/3)+4^(1/3))^3-6*(2^(1/3)+4^(1/3)) - 6 is 0.</font> : 
In&gt RadSimp((2^(1/3)+4^(1/3))^3-6*(2^(1/3)+4^(1/3))-6) 
<font color=ff0000> Out&gt 0 </font>

15. <font color=0000ff>Simplify log e^z into z only for -Pi < Im(z) <= Pi.</font> : 
In&gt Simplify(Ln(Exp(z))) 
<font color=ff0000> Out&gt z </font>

16. <font color=0000ff>Invert the 2x2 matrix [[a,b],[1,ab]]. </font> : 
In&gt  A:={{a,b},{1,a*b}} 
<font color=ff0000></font>In&gt ans:=Inverse(A) 
<font color=ff0000> Out&gt {{(a*b)/(b*a^2-b),(-b)/(b*a^2-b)},{-1/(b*a^2-b),a/(b*a^2-b)}} </font>
In&gt TableForm(Simplify(ans)) 
<font color=ff0000> {1/(a+ -1/a),1/(1-a^2)}
{1/(b-b*a^2),1/(b*a-b/a)}
Out&gt True </font>

17. <font color=0000ff>Find the eigenvalues of the matrix [[5, -3, -7],[-2, 1, 2],[ 2, -3, -4]].</font> : 
In&gt  A:={{5,-3,-7},{-2,1,2},{2,-3,-4}} 
<font color=ff0000></font>In&gt EigenValues(A) 
<font color=ff0000> Out&gt {1,3,-2} </font>

18. <font color=0000ff>Compute the limit of (1-cos x)/x^2 when x goes to zero. </font> : 
In&gt Limit(x,0)(1-Cos(x))/x^2 
<font color=ff0000> Out&gt 1/2 </font>

19. <font color=0000ff>Compute the derivative of |x|.</font> : 
In&gt D(x)Abs(x) 
<font color=ff0000> Out&gt Sign(x) </font>

20. <font color=0000ff>Compute an antiderivative of |x|.</font> : 
In&gt AntiDeriv(Abs(x),x) 
<font color=ff0000> Out&gt (Abs(x)*x)/2 </font>

21. <font color=0000ff>Compute the derivative of |x| (piecewise defined).</font> : 
In&gt D(x)if(x<0)(-x)else x 
<font color=ff0000> Out&gt  if(x<0)-1else1 </font>

22. <font color=0000ff>Compute the antiderivative of |x| (piecewise defined).</font> : 
In&gt AntiDeriv(if(x<0)(-x)else x,x) 
<font color=ff0000> Out&gt if(x<0)(-x^2/2)else x^2/2 </font>

23. <font color=0000ff>Compute the first terms of the Taylor expansion of 1/sqrt(1-v^2/c^2) at v=0. </font> : 
In&gt ans:=Taylor(v,0,4)Sqrt(1/(1-v^2/c^2)) 
<font color=ff0000></font>In&gt ans:=Simplify(ans) 
<font color=ff0000></font>In&gt PrettyForm(ans) 
<font color=ff0000> 
       2          4
      v      3 * v 
1 + ------ + ------
         2        4
    2 * c    8 * c 

Out&gt True </font>

24. <font color=0000ff>Compute the inverse of the square of the above expansion. </font> : 
In&gt  ans:=Taylor(v,0,4)(1/ans)^2 
<font color=ff0000></font>In&gt PrettyForm(Simplify(ans)) 
<font color=ff0000> 
     2
    v 
1 - --
     2
    c 

Out&gt True </font>

25. <font color=0000ff>Compute the Taylor expansion of tan(x) at x=0 by dividing the expansion of sin(x) by that of cos(x). </font> : 
In&gt  ans1:=Taylor(x,0,5)Sin(x)/Cos(x) 
<font color=ff0000></font>In&gt PrettyForm(ans1) 
<font color=ff0000>
     3        5
    x    2 * x 
x + -- + ------
    3      15  

</font>In&gt ans2:=Taylor(x,0,5)Tan(x) 
<font color=ff0000></font>In&gt PrettyForm(ans2) 
<font color=ff0000>
     3        5
    x    2 * x 
x + -- + ------
    3      15  

</font>In&gt ans1-ans2 
<font color=ff0000> Out&gt 0 </font>

26. <font color=0000ff>Compute the Legendre polynomials directly.</font> : 
In&gt 10#Legendre(0,_x)<--1 
<font color=ff0000></font>In&gt 20#Legendre(n_IsInteger,_x)<--[Local(result);result:=[Local(x);Expand(1/(2^n*n!)*Deriv(x,n)Expand((x^2-1)^n,x));];Eval(result);] 
<font color=ff0000></font>In&gt ForEach(item,Table(Legendre(i,x),i,0,4,1))PrettyForm(item) 
<font color=ff0000>
1


x


          2
-1 + 3 * x 
-----------
     2     


              3
-3 * x + 5 * x 
---------------
       2       


           2         4
3   -15 * x    35 * x 
- + -------- + -------
8      4          8   

</font>
27. <font color=0000ff>Compute the Legendre polynomials recursively, using their recurrence of order 2. </font> : 
In&gt 10#LegendreRecursive(0,_x)<--1 
<font color=ff0000></font>In&gt 20#LegendreRecursive(1,_x)<--x 
<font color=ff0000></font>In&gt 30#LegendreRecursive(n_IsPositiveInteger,_x)<--Expand(((2*n-1)*x*LegendreRecursive(n-1,x)-(n-1)*LegendreRecursive(n-2,x))/n) 
<font color=ff0000></font>In&gt ForEach(item,Table(LegendreRecursive(i,x),i,0,4,1))PrettyForm(item) 
<font color=ff0000>
1


x


          2
-1 + 3 * x 
-----------
     2     


              3
-3 * x + 5 * x 
---------------
       2       


           2         4
3   -15 * x    35 * x 
- + -------- + -------
8      4          8   

</font>
28. <font color=0000ff>Evaluate the fourth Legendre polynomial at 1. </font> : 
In&gt Legendre(4,1) 
<font color=ff0000> Out&gt 1 </font>

29. <font color=0000ff>Define the polynomial p = sum( i=1..5, ai*x^i ). </font> : 
In&gt ans:=Sum(MakeVector(a,5)*FillList(x,5)^(1..5)) 
<font color=ff0000></font>In&gt PrettyForm(ans) 
<font color=ff0000>
               2         3         4         5
a1 * x + a2 * x  + a3 * x  + a4 * x  + a5 * x 

</font>
30. <font color=0000ff>Apply Horner's rule to the above polynomial.</font> : 
In&gt ans:=Sum(MakeVector(a,5)*FillList(x,5)^(1..5)) 
<font color=ff0000></font>In&gt PrettyForm(Horner(ans,x)) 
<font color=ff0000>
( ( ( ( a5 * x + a4 ) * x + a3 ) * x + a2 ) * x + a1 ) * x

</font>
31. <font color=0000ff>Compute the first terms of the continued fraction of Pi. </font> : 
In&gt pi:=N(Pi,20) 
<font color=ff0000></font>In&gt a:=ContFrac(pi,6) 
<font color=ff0000></font>In&gt PrettyForm(a) 
<font color=ff0000>
                 1             
3 + ---------------------------
                   1           
    7 + -----------------------
                     1         
        15 + ------------------
                       1       
             1 + --------------
                          1    
                 292 + --------
                       1 + rest

</font>
32. <font color=0000ff>Compute an infinite decimal representation of 1/7. </font> : 
In&gt Decimal(1/7) 
<font color=ff0000> Out&gt {0,{1,4,2,8,5,7}} </font>
This result means that the decimal expansion of 1/7 is
0.142857142857142....

33. <font color=0000ff>Evaluate TRUE and FALSE. </font> : 
In&gt True And False 
<font color=ff0000> Out&gt  False </font>

34. <font color=0000ff>Solve the equation tan(x) = 1. </font> : 
In&gt  Solve(Tan(x)==1,x) 
<font color=ff0000> Out&gt Pi/4 </font>

35. <font color=0000ff>Revert the Taylor expansion of sin(y) + cos(y) at y=0. </font> : 
In&gt t:=InverseTaylor(y,0,6)Sin(y)+Cos(y) 
<font color=ff0000></font>In&gt PrettyForm(t) 
<font color=ff0000>
                 2                3                
        ( y - 1 )    2 * ( y - 1 )             4   
y - 1 + ---------- + -------------- + ( y - 1 )  + 
            2              3                       

              5
17 * ( y - 1 ) 
---------------
      10       

</font>And check that it is in fact the inverse up to degree 5:
In&gt s:=Taylor(y,0,6)Sin(y)+Cos(y) 
<font color=ff0000></font>In&gt BigOh(Subst(y,s)t,y,6) 
<font color=ff0000> Out&gt y </font>

36. <font color=0000ff>Solve the linear (dependent) system x+y+z=6,2x+y+2z=10,x+3y+z=10.</font> : 
In&gt  Solve({x+y+z==6,2*x+y+2*z==10,x+3*y+z==10},{x,y,z}) 
<font color=ff0000> Out&gt {{4-z,2,z}} </font>

37. <font color=0000ff>Evaluate True And False</font> : 
In&gt True And False 
<font color=ff0000> Out&gt  False </font>
In&gt  CanProve(True And False) 
<font color=ff0000> Out&gt False </font>

38. <font color=0000ff>Simplify x or (not x)</font> : 
In&gt  CanProve(x Or Not x) 
<font color=ff0000> Out&gt True </font>

39. <font color=0000ff>Simplify x or y or (x and y)</font> : 
In&gt  CanProve(x Or y Or x And y) 
<font color=ff0000> Out&gt x Or y </font>
39  examples done
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