% This is a showcase for the Yacas interface in Euler. For more
% information about Yacas look at the HTML documentation of Euler.
% 
% At first, Yacas provides infinite integer computation.
% 
% Note that the yacas function takes and returns a string.
>yacas("30!")
% Alternatively, you can use >... to enter a Yacas expression.
% The result will be formatted by Yacas in this case.
>>30!
% Factor it.
>>Factor(30!)
% The next thing takes a while. On my Notebook it takes 10 seconds.
>yacas("NextPrime(30!)")
% To evaluate a result and import it into Euler variables you
% can use evaluate(string), or the function yeval.
>longestformat; y=yeval("30!"), log(y)
% Since Euler can use expression with variable x in many places,
% you can directly use the output of yacas, if Euler knows how
% to interpret it.
>e=yacas("Expand((1+x)^5)"), fplot(e,-2,0);
% Since we want to use e below, it is not wise to use >... But
% we can.
>>e:=Expand((1+x)^5)
% Differentiate that.
>>D(x) e
% You can also evaluate an expression after it went through yacas,
% and have some variables set to certain values. This is done
% with named parameters.
% 
% In this example, we integrate from 0 to a and evaluate at a=1.
>yeval("Integrate(x,0,a) e",a=1)
% Compare to the result of the Romberg method.
>romberg(e,0,1)
% Here is method you can use.
% 
% The string % uses the most recent result from Yacas.
>>e:=Expand((1+x)^3)
>>Factor(e)
% Here is a first example on how to use Yacas for something useful.
% 
% We plot the sin functions and its taylor series.
>setplot(0,2*pi,-2,2); fplot("sin(x)",0,2*pi);
>hold; fplot(yacas("Taylor(x,0,9) Sin(x)"),0,2*pi); hold;
% Let's determine the sum of n^2, n=1 to a. There is a simple
% formula for this.
>>e:=Sum(n,1,a,n^2)
>>Factor(e)
% To evalute in a=100, you can use the substitution machanism
% of Yacas.
>yeval("Eliminate(a,100,e)")
% Yacas does also know about complex numbers.
% 
% We use the % notation to access the recent result from Yacas.
% This does not work with >... however.
>yacas("Exp(Complex(1,1))"), yeval("%"), exp(1+1i)
% Here are some variations of the factorial.
>yeval("(44 *** 49) / 6!"), yeval("Bin(49,6)"), bin(49,6),
% We compute the Newton iterator to solve x^x=2 with Yacas.
>e=yacas("x - (x^x-a) / (D(x) x^x)"),
>a=2; xs=iterate(e,1,10),
% The newton() and inewton() functions of Euler must get a derivative.
% So we use Yacas to compute it.
>inewton("x^x-a",yacas("D(x) x^x"),1)
% Here is a continued fraction.
>>e:=ContFrac(N(Sqrt(2)),10)
% We evaluate with rest=0.
>yacas("Simplify(e /: {rest <- 0})"), yeval("%")-sqrt(2)
% Yacas has a (yet) not perfect solver. It may return a list of
% solutions. You get the first solution by indexing. The right
% hand side of the equation is also obtained by indexing.
>>e:=Solve(x^2+a*x==2,x)
>>e[2][2]
% To create an Euler vector from a Yacas list, you can use the
% y2vector() function.
% 
% The following example creates a list of 100 random numbers in
% Yacas, turns that into an Euler vector then plots the cumulative
% sum.
>xplot(cumsum(y2vector("Table(Random(),n,1,100,1)")));
% To demonstrate the inverse transformation, we interpolate with
% Yacas.
>x=1:5; y=sin(x); setplot(0,6,-2,2); xmark(x,y);
% The transformation is done by vector2y(). Yacas uses the LagrangeInterpolant
% to compute the interpolation. We expand it into normal form.
>pol=yacas("Expand(LagrangeInterpolant("|converty(x)|","|converty(y)|",x))"),
>hold; fplot(pol,0,6); hold;
% With Yacas, it is possible to write advanced functions like
% the Newton method without computing the derivative function.
% This is done in the function itself.
>ynewton("x^x-2",1)
>A=[1,1;1,2]; yacas("EigenValues("|converty(A)|")"), yeval("%[1]")
% This ends our introduction into Yacas. For more information,
% read the manuals.
>