added: SandBoxCS224 removed: - - -SandBoxCS224 removed: -\end {axiom}
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Here is a simple Axiom command:
\begin{axiom}
integrate(1/(a+z^3), z=0..1,"noPole")
\end{axiom}
axiomintegrate(1/(a+z^3), z=0..1,"noPole")
And here is a REDUCE command:
\begin{reduce}
load_package sfgamma;
load_package defint;
int(1/(a+z^3), z,0,1);
\end{reduce}
\begin{reduce}
load_package sfgamma;
load_package defint;
int(1/(a+z^3), z,0,1);
\end{reduce}
<hr />
Please review the list of [Common Mistakes]? and the list of [MathAction Problems]? if you are have never used MathAction? before. If you are learning to use Axiom and think that someone must have solved some particular problem before you, check this list of Common [Axiom Problems]?.
axiom)set output tex off )set output algebra on
axiomsolve([x^2 + y^2 - 2*(ax*x + ay*y) = l1, x^2 + y^2 - 2*(cx*x + cy*y) = l2],[x,y]) (2) [ (- 2cy + 2ay)y - l2 + l1 [x= ------------------------, 2cx - 2ax 2 2 2 2 2 (4cy - 8ay cy + 4cx - 8ax cx + 4ay + 4ax )y + 2 2 (4cy - 4ay)l2 + (- 4cy + 4ay)l1 + (8ax cx - 8ax )cy - 8ay cx + 8ax ay cx * y + 2 2 2 2 l2 + (- 2l1 + 4ax cx - 4ax )l2 + l1 + (- 4cx + 4ax cx)l1 = 0 ] ]
But fails with LaTeX?.
axiom)set output tex on )set output algebra off
0**0 depends on the type of '0':
axiom(0::Float)**(0::Float) >> Error detected within library code: 0**0 is undefined
The idea was, that defining LatexWiki Image as 1 is ok whenever there is no notion of limit. However,
axiom(0::EXPR INT)**(0::EXPR INT)
is not quite in line with this, I think. There has been some discussion on this subject on axiom-developer.
It is easy to change this behaviour, if we know better...
axiomsinCosProducts := rule sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2 cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2 sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2
sinCosProducts := rule (_ sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2; _ cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2; _ sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2)
Alternatively, using a text editor you can enter the commands into a
file called, for example sincos.input exactly as in MathActon? above
and the use the command:
)read sincos.input
axiom)lib RINTERPA RINTERP PCDEN GUESS GUESSINT GUESSP )library cannot find the file RINTERPA. )library cannot find the file RINTERP. )library cannot find the file PCDEN. )library cannot find the file GUESS. )library cannot find the file GUESSINT. )library cannot find the file GUESSP. guess(n, [1, 5, 14, 34, 69, 135, 240, 416, 686, 1106], n+->n, [guessRat], [guessSum, guessProduct, guessOne],2)$GuessInteger GuessInteger is not a valid type.
z:=sum(myfn(x),x=1..10) -- This fails, why?
The reason this fails is because Axiom tries to evaluate
myfn(x) first. But x is not yet an Integer so Axiom
cannot compute myfn(x). I guess you were expecting Axiom
to "wait" and not evaluate myfn(x) until after x has
been assigned the value 1, right? But Axiom does not work
this way.
The solution is to write myfn(x) so that is can be applied
to something symbolic like x. For example something this:
axiommyfn(i : Expression Integer) : Expression Integer == i Function declaration myfn : Expression Integer -> Expression Integer has been added to workspace.
axiommyfn(x)
Compiling function myfn with type Expression Integer -> Expression Integer
axiomz:=sum(myfn(x),x=1..10)
Thanks for your quick response. I tried to respond to this earlier, but didn't see it in the sand box, please forgive me if you get multiple copies.
I tried to simplify the code from my original program, and generated a univariate function, however my actual code has a multivariate function, and your excellent hint on the use of the Expression qualifier on the parameter and return type which works great for the univariate function case appears to fail for multivarite functions. Please consider the following example.
axioma(n : Expression Integer, k : Expression Integer, p : Expression Float) : Expression Float == binomial(n,k) * p**(k) * (1.0-p)**(n-k) Function declaration a : (Expression Integer,Expression Integer, Expression Float) -> Expression Float has been added to workspace.
axiomoutput(a(4,3,0.25)) -- see that the function actually evaluates for sensible values
Compiling function a with type (Expression Integer,Expression Integer,Expression Float) -> Expression Float 0.046875
axiomz := sum(a(4,i,0.25), i=1..3) --- this fails There are 6 exposed and 2 unexposed library operations named sum having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sum 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 sum with argument type(s) Expression Float SegmentBinding PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. output(z) 55
I did notice in the Axiom online book, chapter 6.6, around page 241, the recommendation to use untyped functions, which appears to allow Axiom to do inference on parameter and result type.
axiomb(n, k, p) == binomial(n,k) * p**(k) * (1.0-p)**(n-k)
axiomoutput(b(4,3,0.25)) -- see that the function actually evaluates for sensible values
Compiling function b with type (PositiveInteger,PositiveInteger, Float) -> Float 0.046875
axiomz := sum(b(4,i,0.25), i=1..3) --- this fails
Compiling function b with type (PositiveInteger,Variable i,Float) -> Expression Float There are 6 exposed and 2 unexposed library operations named sum having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sum 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 sum with argument type(s) Expression Float SegmentBinding PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. output(z) 55
For univariate functions the approach
axiomc(k) == binomial(4,k) * 0.25**k * (1.0 - 0.25)**(4-k) -- This approach is only a test, but is not suitable for my program
axiomoutput(c(3)) -- test to see if function can be evaluated for sensible arguments
Compiling function c with type PositiveInteger -> Float 0.046875
axiomz := sum(c(i), i=1..3) -- still doesn't work
Compiling function c with type Variable i -> Expression Float There are 6 exposed and 2 unexposed library operations named sum having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sum to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named sum with argument type(s) Expression Float SegmentBinding PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. output(z) 55
But interestingly something like
axiomd(k) == 1.5 * k -- coerce uotput to be a Float
axiomz := sum(d(i), i=1..3) -- This works!
Compiling function d with type Variable i -> Polynomial Float
axiomoutput(z) 9.0
Bill, thanks again for your quick help, unforutnatly I lack a local Axiom expert, any ideas would really be welcome here.
Try thisaxiomz := reduce(+,[b(4,i,0.25) for i in 1..3])
Thanks Bill Page for your help, it is much appreciated (although I used a for loop and not reduce :-)).
I'm having a bit of difficulty getting a Function returning a matrix to work as expected, perhaps it is just cockpit error, but I don't see the error of my ways.
axiomCFM(Q : Matrix(Float)): Matrix(Float) == x := nrows(Q) MyIdentityMatrix : Matrix(Float) := new(x, x, 0) for i in 1..nrows(MyIdentityMatrix) repeat MyIdnetityMatrix(i,i) := 1.0 Ninv := MyIdnetityMatrix - Q N := inverse(Ninv) N Function declaration CFM : Matrix Float -> Matrix Float has been added to workspace.
axiom--test ComputeFundamentalMatrix X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]
axiomoutput(X) +0.0 0.5 0.0+ | | |0.5 0.0 0.5| | | +0.0 0.5 0.0+
axiomN := CFM(X) The form on the left hand side of an assignment must be a single variable, a Tuple of variables or a reference to an entry in an object supporting the setelt operation. output(N) N
Any ideas where I'm blowing it here? I tried explicitly setting N to be a Matrix type but that failed too.
axiomCFM(Q : Matrix(Float)): Matrix(Float) == x := nrows(Q) MyIdentityMatrix : Matrix(Float) := new(x, x, 0) for i in 1..nrows(MyIdentityMatrix) repeat MyIdnetityMatrix(i,i) := 1.0 Ninv := MyIdnetityMatrix - Q N := inverse(Ninv) N Function declaration CFM : Matrix Float -> Matrix Float has been added to workspace. Compiled code for CFM has been cleared. 1 old definition(s) deleted for function or rule CFM
axiom--test ComputeFundamentalMatrix X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]
axiomoutput(X) +0.0 0.5 0.0+ | | |0.5 0.0 0.5| | | +0.0 0.5 0.0+
axiomN : Matrix(Float) := CFM(X) The form on the left hand side of an assignment must be a single variable, a Tuple of variables or a reference to an entry in an object supporting the setelt operation. output(N) N is declared as being in Matrix Float but has not been given a value.
Thanks again for all your help.
Regards:
Bill M. (Sorry, my unique last name attracts too much spam).
although I used a for loop and not reduce :-)
Good thinking. ;)
You have a simple typographical error. You have written both:
MyIdentityMatrix
and :
MyIdnetityMatrix
BTW, instead of the complicated construction of the identify matrix you should just write:
Ninv := 1 - Q
For matrices 1 denotes the identity.
axiom)set output tex off )set output algebra on FunFun := x**4 - 6* x**3 + 11* x*x + 2* x + 1 4 3 2 (28) x - 6x + 11x + 2x + 1
axiomradicalSolve(FunFun) (29) [ x = - ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 - 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 30 |--------------------- - 169 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 - 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 , x = ROOT +---------------------+2 +---------------------+ | +-+ +----+ | +-+ +----+ |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 - 9 |--------------------- + 30 |--------------------- 3| +-+ 3| +-+ \| 27\|3 \| 27\|3 + - 169 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 - 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 , x = - ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 - 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 30 |--------------------- - 169 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 - |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 , x = ROOT +---------------------+2 +---------------------+ | +-+ +----+ | +-+ +----+ |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 - 9 |--------------------- + 30 |--------------------- 3| +-+ 3| +-+ \| 27\|3 \| 27\|3 + - 169 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 - |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 ]
axiom)set output tex on )set output algebra off
Matthias
axiomt:=matrix ([[0,1,1],[1,-2,2],[1,2,-1]])
We cat diagonalise t by finding it's eigenvalues.
axiom)set output tex off )set output algebra on e:=radicalEigenvectors(t) (31) [ +-----------------+2 +-----------------+ | +-+ +------+ | +-+ +------+ |3\|3 + \|- 1345 |3\|3 + \|- 1345 3 |----------------- - 3 |----------------- + 7 3| +-+ 3| +-+ \| 6\|3 \| 6\|3 [radval= --------------------------------------------------, radmult= 1, +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 3 |----------------- 3| +-+ \| 6\|3 radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 12\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (60\|3 + 6\|- 1345 ) |----------------- + 205\|3 + 3\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 ] , [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 6\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (117\|3 - 3\|- 1345 ) |----------------- - 71\|3 + 9\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] , [ radval = +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- + 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 , radmult= 1, radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 60\|- 3 - 60)\|3 - 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (205\|- 3 - 205)\|3 + 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 117\|- 3 - 117)\|3 + 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 71\|- 3 + 71)\|3 + 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] , [ radval = +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- - 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 , radmult= 1, radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((60\|- 3 - 60)\|3 + 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 205\|- 3 - 205)\|3 - 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((117\|- 3 - 117)\|3 - 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (71\|- 3 + 71)\|3 - 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] ]
axiomd:=diagonalMatrix([e.1.radval,e.2.radval,e.3.radval]) Function definition for d is being overwritten. Compiled code for d has been cleared. (32) +-----------------+2 +-----------------+ | +-+ +------+ | +-+ +------+ |3\|3 + \|- 1345 |3\|3 + \|- 1345 3 |----------------- - 3 |----------------- + 7 3| +-+ 3| +-+ \| 6\|3 \| 6\|3 [[--------------------------------------------------,0,0], +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 3 |----------------- 3| +-+ \| 6\|3 [0, +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- + 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 , 0] , [0, 0, +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- - 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 ] ]
Now prove it by constructing the simularity transformation from the eigenvectors:
axiomp:=horizConcat(horizConcat(e.1.radvect.1,e.2.radvect.1),e.3.radvect.1) (33) [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 12\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (60\|3 + 6\|- 1345 ) |----------------- + 205\|3 + 3\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 , +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 60\|- 3 - 60)\|3 - 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (205\|- 3 - 205)\|3 + 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 , +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((60\|- 3 - 60)\|3 + 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 205\|- 3 - 205)\|3 - 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 6\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (117\|3 - 3\|- 1345 ) |----------------- - 71\|3 + 9\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 , +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 117\|- 3 - 117)\|3 + 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 71\|- 3 + 71)\|3 + 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 , +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((117\|- 3 - 117)\|3 - 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (71\|- 3 + 71)\|3 - 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1,1,1]]
axiomp*d*inverse(p) +0 1 1 + | | (34) |1 - 2 2 | | | +1 2 - 1+
axiom)set output tex on )set output algebra off
\end{axiom}
axiomintegrate(exp(x**4),x)
But Maple can...
axiomf(x) == (1/4)*x*(-Gamma(1/4,-x**4)*Gamma(3/4)+%pi*sqrt(2))/((-x**4)**(1/4)*Gamma(3/4))
axiomD(f(x),x)
Compiling function f with type Variable x -> Expression DoubleFloat
Gamma(x,y) is not an elementary function.
Martin
axiomintegrate(1/(1+x^4),x=%minusInfinity..%plusInfinity)
axiomnumeric(integrate(1/(1+x^4),x=0..1))
axiom)clear co All user variables and function definitions have been cleared. All )browse facility databases have been cleared. Internally cached functions and constructors have been cleared. )clear completely is finished. n := 32
axiomy : FARRAY INT := new(n,1)
axiomn0 := n
axiomn1 := sum(x^1, x=0..n-1)
axiomn2 := sum(x^2, x=0..n-1)
axiomn3 := sum(x^3, x=0..n-1)
axiomn4 := sum(x^4, x=0..n-1)
axiomA := matrix([[n4, n3, n2],_ [n3, n2, n1],_ [n2, n1, n0]])
axiomX := vector([x1, x2, x3])
axiomB := vector([sum(x^2* u, x=0..n-1),_ sum(x* v, x=0..n-1),_ sum( w, x=0..n-1)])
axiomsolve([A * X = B], [x1, x2, x3]) 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) List Equation Vector Fraction Polynomial Integer List OrderedVariableList [x1,x2,x3] Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
axiomintegrate(1/((x+t)*sqrt(1+(x*t)**2)),t=0..%plusInfinity,"noPole")
axiomsubst(%,x=1)
axiomintegrate(1/((1+t)*sqrt(1+(1*t)**2)),t=0..%plusInfinity,"noPole")
axiomsimplify(%-subst((asinh(x^2)+asinh(1/x^2))/sqrt(1+x^4),x=1))
axiom%::Expression Float
axioma := matrix([ [-1,0,0,0,1,0], [0,1,0,0,0,0], [0,0,2,0,0,-2], [0,0,0,4,0,0], [0,0,0,0,3,0], [0,0,-3,0,0,3]])
axiomdeterminant(a)
axiominverse(a)
axiomAs := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])
axiomA := subMatrix(As, 2,4,2,4)
axiomob := orthonormalBasis(A)
axiomP : Matrix(Expression Integer) := new(3,3,0)
axiomsetsubMatrix!(P,1,1,ob.3)
axiomsetsubMatrix!(P,1,2,ob.1)
axiomsetsubMatrix!(P,1,3,ob.2)
axiomPt := transpose(P)
axiomPs : Matrix(Expression Integer) := new(4,4,0)
axiomPs(1,1) := 1
axiomsetsubMatrix!(Ps,2,2,P)
axiomPsT := transpose(Ps)
axiomPsTAsPs := PsT * As * Ps
axiomb1 := PsTAsPs(2,1)
axioml1 := PsTAsPs(2,2)
axiomUs : Matrix(Expression Integer) := new(4,4,0)
axiomUs(1,1) := 1
axiomUs(2,2) := 1
axiomUs(3,3) := 1
axiomUs(4,4) := 1
axiomUs(2,1) := -b1 / l1
axiomPsUs := Ps * Us
axiomPsUsT := transpose(PsUs)
axiomPsUsTAsPsUs := PsUsT * As * PsUs
axiomC := inverse(PsUs)
axiomc := PsUsTAsPsUs(1,1)
axiomgQ := PsUsTAsPsUs / c
axiomx1 := transpose(matrix([[1,2,3,4]]))
axiomv1 := transpose(x1) * As * x1
axiomx2 := C * x1
axiomv2 := transpose(x2) * PsUsTAsPsUs * x2
axiomdraw(y**2/2+(x**2-1)**2/4-1=0, x,y, range ==[-2..2, -1..1]) There are 20 exposed and 18 unexposed library operations named ** having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op ** 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 ** with argument type(s) FlexibleArray Integer PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
axiomf1 := taylor(1 - x**2,x = 0)
axiomasin f1
axiomsin %
SandboxMSkuce?
axiom1+1
axiomintegrate((x-1)/log(x), x)
axiomintegrate(x*exp(x)*sin(x),x)
axiom[p for p in primes(2,1000)|(p rem 16)=1]
axiom[p**2+1 for p in primes(2,100)]
axiomintegrate (2x^2 + 2x, x) Cannot find a definition or applicable library operation named 2 with argument type(s) Variable x Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
axiomradix(36,37) >> Error detected within library code: index out of range
Is it error?
axiomintegrate(log(log(x)),x)