MathAction SandBox11

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(1) -> )set output tex off
 

)set output algebra on

solve([x^2 + y^2 - 2*(ax*x + ay*y) = l1, x^2 + y^2 - 2*(cx*x + cy*y) = l2],[x,y])

 (1)
 [
 (- 2 cy + 2 ay)y - l2 + l1
 [x = --------------------------,
 2 cx - 2 ax

 2 2 2 2 2
 (4 cy - 8 ay cy + 4 cx - 8 ax cx + 4 ay + 4 ax )y
 + 
 2
 (4 cy - 4 ay)l2 + (- 4 cy + 4 ay)l1 + (8 ax cx - 8 ax )cy
 + 
 2
 - 8 ay cx + 8 ax ay cx
 *
 y
 + 
 2 2 2 2
 l2 + (- 2 l1 + 4 ax cx - 4 ax )l2 + l1 + (- 4 cx + 4 ax cx)l1
 = 
 0
 ]
 ]

But fails with LaTeX.

)set output tex on
 

)set output algebra off

(0::Float)^(0::Float)

 >> Error detected within library code:
 0^0 is undefined

The idea was, that defining 0^0 as 1 is ok whenever there is no notion of limit. However,

(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...

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

 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

guess([1, 5, 14, 34, 69, 135, 240, 416, 686, 1106], [guessRat], [guessSum, guessProduct, guessOne])

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:

myfn(i : Expression Integer) : Expression Integer == i

 Function declaration myfn : Expression(Integer) -> Expression(
 Integer) has been added to workspace.

myfn(x)

Compiling function myfn with type Expression(Integer) -> Expression(
 Integer)

z:=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.

a(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.

output(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

z := 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.

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.

b(n, k, p) == binomial(n,k) * p^(k) * (1.0-p)^(n-k)

output(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

z := 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.

For univariate functions the approach

c(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

output(c(3)) -- test to see if function can be evaluated for sensible arguments

Compiling function c with type PositiveInteger -> Float 
 0.046875

z := 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.

But interestingly something like

d(k) == 1.5 * k -- coerce output to be a Float

z := sum(d(i), i=1..3) -- This works!

Compiling function d with type Variable(i) -> Polynomial(Float)

output(z)

 9.0

Bill, thanks again for your quick help, unforutnatly I lack a local Axiom expert, any ideas would really be welcome here.

z := 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.

CFM(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.

--test ComputeFundamentalMatrix
X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]

output(X)

 +0.0 0.5 0.0+
 | |
 |0.5 0.0 0.5|
 | |
 +0.0 0.5 0.0+

N := 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.

Any ideas where I'm blowing it here? I tried explicitly setting N to be a Matrix type but that failed too.

CFM(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

--test ComputeFundamentalMatrix
X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]

output(X)

 +0.0 0.5 0.0+
 | |
 |0.5 0.0 0.5|
 | |
 +0.0 0.5 0.0+

N : 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.

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.

)set output tex off
 

)set output algebra on

FunFun := x^4 - 6* x^3 + 11* x^2 + 2* x + 1

 4 3 2
 (24) x - 6 x + 11 x + 2 x + 1

radicalSolve(FunFun)

 (25)
 [
 x
 = 
 -
 ROOT
 +------------------+2 +------------------+
 | +----+ | +----+
 | | 79 | | 79
 - 3|144 |- -- + 2069 + 10 3|144 |- -- + 2069
 \| \| 3 \| \| 3
 + 
 - 169
 *
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 + 
 +------------------+
 | +----+
 | | 79
 - 48 3|144 |- -- + 2069
 \| \| 3
 /
 +------------------+
 | +----+
 | | 79
 3 3|144 |- -- + 2069
 \| \| 3
 *
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 + 
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------ + 3
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 /
 2
 ,

 x
 = 
 ROOT
 +------------------+2 +------------------+
 | +----+ | +----+
 | | 79 | | 79
 (- 3|144 |- -- + 2069 + 10 3|144 |- -- + 2069 - 169)
 \| \| 3 \| \| 3
 *
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 + 
 +------------------+
 | +----+
 | | 79
 - 48 3|144 |- -- + 2069
 \| \| 3
 /
 +------------------+
 | +----+
 | | 79
 3 3|144 |- -- + 2069
 \| \| 3
 *
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 + 
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------ + 3
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 /
 2
 ,

 x
 = 
 -
 ROOT
 +------------------+2 +------------------+
 | +----+ | +----+
 | | 79 | | 79
 - 3|144 |- -- + 2069 + 10 3|144 |- -- + 2069
 \| \| 3 \| \| 3
 + 
 - 169
 *
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 + 
 +------------------+
 | +----+
 | | 79
 48 3|144 |- -- + 2069
 \| \| 3
 /
 +------------------+
 | +----+
 | | 79
 3 3|144 |- -- + 2069
 \| \| 3
 *
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 + 
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 - |------------------------------------------------------ + 3
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 /
 2
 ,

 x
 = 
 ROOT
 +------------------+2 +------------------+
 | +----+ | +----+
 | | 79 | | 79
 (- 3|144 |- -- + 2069 + 10 3|144 |- -- + 2069 - 169)
 \| \| 3 \| \| 3
 *
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 + 
 +------------------+
 | +----+
 | | 79
 48 3|144 |- -- + 2069
 \| \| 3
 /
 +------------------+
 | +----+
 | | 79
 3 3|144 |- -- + 2069
 \| \| 3
 *
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 |------------------------------------------------------
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 + 
 +------------------------------------------------------+
 | +------------------+2 +------------------+
 | | +----+ | +----+
 | | | 79 | | 79
 |3|144 |- -- + 2069 + 5 3|144 |- -- + 2069 + 169
 |\| \| 3 \| \| 3
 - |------------------------------------------------------ + 3
 | +------------------+
 | | +----+
 | | | 79
 | 3 3|144 |- -- + 2069
 \| \| \| 3
 /
 2
 ]

)set output tex on
 

)set output algebra off

Matthias

t:=matrix ([[0,1,1],[1,-2,2],[1,2,-1]])

We can diagonalise t by finding it's eigenvalues.

)set output tex off
 

)set output algebra on

e:=radicalEigenvectors(t)

 (27)
 [
 +-------------+2 +-------------+
 | +------+ | +------+
 | | 1345 | | 1345
 | |- ---- + 3 | |- ---- + 3
 |\| 3 |\| 3
 3 3|------------- - 3 3|------------- + 7
 \| 6 \| 6
 [radval = --------------------------------------------, radmult = 1,
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 3 3|-------------
 \| 6
 radvect = [matrix1]]
 ,

 [
 radval
 = 
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (- 3 \|- 3 - 3) 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (- 3 \|- 3 + 3) 3|------------- + 14
 \| 6
 /
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (3 \|- 3 - 3) 3|-------------
 \| 6
 ,
 radmult = 1, radvect = [matrix2]]
 ,

 [
 radval
 = 
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (- 3 \|- 3 + 3) 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (- 3 \|- 3 - 3) 3|------------- - 14
 \| 6
 /
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (3 \|- 3 + 3) 3|-------------
 \| 6
 ,
 radmult = 1, radvect = [matrix3]]
 ]

 where matrix1
 = 
 [
 [
 +-------------+2 +-------------+
 | +------+ | +------+
 | | 1345 | | 1345
 | |- ---- + 3 +------+ | |- ---- + 3
 |\| 3 | 1345 |\| 3
 - 12 3|------------- + (6 |- ---- + 60) 3|-------------
 \| 6 \| 3 \| 6
 + 
 +------+
 | 1345
 3 |- ---- + 205
 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 126 3|-------------
 \| 6
 ]
 ,

 [
 +-------------+2 +-------------+
 | +------+ | +------+
 | | 1345 | | 1345
 | |- ---- + 3 +------+ | |- ---- + 3
 |\| 3 | 1345 |\| 3
 6 3|------------- + (- 3 |- ---- + 117) 3|-------------
 \| 6 \| 3 \| 6
 + 
 +------+
 | 1345
 9 |- ---- - 71
 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 126 3|-------------
 \| 6
 ]
 ,
 [1]]

 and matrix2
 = 
 [
 [
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 - 24 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 +------+ +------+ | |- ---- + 3
 | 1345 +---+ | 1345 |\| 3
 ((- 6 |- ---- - 60)\|- 3 - 6 |- ---- - 60) 3|-------------
 \| 3 \| 3 \| 6
 + 
 +------+ +------+
 | 1345 +---+ | 1345
 (3 |- ---- + 205)\|- 3 - 3 |- ---- - 205
 \| 3 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 252 3|-------------
 \| 6
 ]
 ,

 [
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 12 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 +------+ +------+ | |- ---- + 3
 | 1345 +---+ | 1345 |\| 3
 ((3 |- ---- - 117)\|- 3 + 3 |- ---- - 117) 3|-------------
 \| 3 \| 3 \| 6
 + 
 +------+ +------+
 | 1345 +---+ | 1345
 (9 |- ---- - 71)\|- 3 - 9 |- ---- + 71
 \| 3 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 252 3|-------------
 \| 6
 ]
 ,
 [1]]

 and matrix3
 = 
 [
 [
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 - 24 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 +------+ +------+ | |- ---- + 3
 | 1345 +---+ | 1345 |\| 3
 ((6 |- ---- + 60)\|- 3 - 6 |- ---- - 60) 3|-------------
 \| 3 \| 3 \| 6
 + 
 +------+ +------+
 | 1345 +---+ | 1345
 (- 3 |- ---- - 205)\|- 3 - 3 |- ---- - 205
 \| 3 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 252 3|-------------
 \| 6
 ]
 ,

 [
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 12 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 +------+ +------+ | |- ---- + 3
 | 1345 +---+ | 1345 |\| 3
 ((- 3 |- ---- + 117)\|- 3 + 3 |- ---- - 117) 3|-------------
 \| 3 \| 3 \| 6
 + 
 +------+ +------+
 | 1345 +---+ | 1345
 (- 9 |- ---- + 71)\|- 3 - 9 |- ---- + 71
 \| 3 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 252 3|-------------
 \| 6
 ]
 ,
 [1]]

d:=diagonalMatrix([e.1.radval,e.2.radval,e.3.radval])

 Function definition for d is being overwritten.
 Compiled code for d has been cleared.

 (28)
 +-------------+2 +-------------+
 | +------+ | +------+
 | | 1345 | | 1345
 | |- ---- + 3 | |- ---- + 3
 |\| 3 |\| 3
 3 3|------------- - 3 3|------------- + 7
 \| 6 \| 6
 [[--------------------------------------------, 0, 0],
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 3 3|-------------
 \| 6

 [0,

 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (- 3 \|- 3 - 3) 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (- 3 \|- 3 + 3) 3|------------- + 14
 \| 6
 /
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (3 \|- 3 - 3) 3|-------------
 \| 6
 ,
 0]
 ,

 [0, 0,

 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (- 3 \|- 3 + 3) 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (- 3 \|- 3 - 3) 3|------------- - 14
 \| 6
 /
 +-------------+
 | +------+
 | | 1345
 | |- ---- + 3
 +---+ |\| 3
 (3 \|- 3 + 3) 3|-------------
 \| 6
 ]
 ]

Now prove it by constructing the simularity transformation from the eigenvectors:

p:=horizConcat(horizConcat(e.1.radvect.1,e.2.radvect.1),e.3.radvect.1)

 (29)
 [
 [
 +-------------+2 +-------------+
 | +------+ | +------+
 | | 1345 | | 1345
 | |- ---- + 3 +------+ | |- ---- + 3
 |\| 3 | 1345 |\| 3
 - 12 3|------------- + (6 |- ---- + 60) 3|-------------
 \| 6 \| 3 \| 6
 + 
 +------+
 | 1345
 3 |- ---- + 205
 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 126 3|-------------
 \| 6
 ,

 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 - 24 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 +------+ +------+ | |- ---- + 3
 | 1345 +---+ | 1345 |\| 3
 ((- 6 |- ---- - 60)\|- 3 - 6 |- ---- - 60) 3|-------------
 \| 3 \| 3 \| 6
 + 
 +------+ +------+
 | 1345 +---+ | 1345
 (3 |- ---- + 205)\|- 3 - 3 |- ---- - 205
 \| 3 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 252 3|-------------
 \| 6
 ,

 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 - 24 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 +------+ +------+ | |- ---- + 3
 | 1345 +---+ | 1345 |\| 3
 ((6 |- ---- + 60)\|- 3 - 6 |- ---- - 60) 3|-------------
 \| 3 \| 3 \| 6
 + 
 +------+ +------+
 | 1345 +---+ | 1345
 (- 3 |- ---- - 205)\|- 3 - 3 |- ---- - 205
 \| 3 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 252 3|-------------
 \| 6
 ]
 ,

 [
 +-------------+2 +-------------+
 | +------+ | +------+
 | | 1345 | | 1345
 | |- ---- + 3 +------+ | |- ---- + 3
 |\| 3 | 1345 |\| 3
 6 3|------------- + (- 3 |- ---- + 117) 3|-------------
 \| 6 \| 3 \| 6
 + 
 +------+
 | 1345
 9 |- ---- - 71
 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 126 3|-------------
 \| 6
 ,

 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 12 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 +------+ +------+ | |- ---- + 3
 | 1345 +---+ | 1345 |\| 3
 ((3 |- ---- - 117)\|- 3 + 3 |- ---- - 117) 3|-------------
 \| 3 \| 3 \| 6
 + 
 +------+ +------+
 | 1345 +---+ | 1345
 (9 |- ---- - 71)\|- 3 - 9 |- ---- + 71
 \| 3 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 252 3|-------------
 \| 6
 ,

 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 12 3|-------------
 \| 6
 + 
 +-------------+
 | +------+
 | | 1345
 +------+ +------+ | |- ---- + 3
 | 1345 +---+ | 1345 |\| 3
 ((- 3 |- ---- + 117)\|- 3 + 3 |- ---- - 117) 3|-------------
 \| 3 \| 3 \| 6
 + 
 +------+ +------+
 | 1345 +---+ | 1345
 (- 9 |- ---- + 71)\|- 3 - 9 |- ---- + 71
 \| 3 \| 3
 /
 +-------------+2
 | +------+
 | | 1345
 | |- ---- + 3
 |\| 3
 252 3|-------------
 \| 6
 ]
 ,
 [1, 1, 1]]

p*d*inverse(p)

 +0 1 1 +
 | |
 (30) |1 - 2 2 |
 | |
 +1 2 - 1+

)set output tex on
 

)set output algebra off

\end{axiom}

integrate(exp(x^4),x)

But Maple can...

f(x) == (1/4)*x*(-Gamma(1/4,-x^4)*Gamma(3/4)+%pi*sqrt(2))/((-x^4)^(1/4)*Gamma(3/4))

D(f(x),x)

Compiling function f with type Variable(x) -> Expression(Integer)

Martin

integrate(1/(1+x^4),x=%minusInfinity..%plusInfinity)

numeric(integrate(1/(1+x^4),x=0..1))

)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

y : FARRAY INT := new(n,1)

n0 := n

n1 := sum(x^1, x=0..n-1)

n2 := sum(x^2, x=0..n-1)

n3 := sum(x^3, x=0..n-1)

n4 := sum(x^4, x=0..n-1)

A := matrix([[n4, n3, n2],_
 [n3, n2, n1],_
 [n2, n1, n0]])

X := vector([x1, x2, x3])

B := vector([sum(x^2* u, x=0..n-1),_
 sum(x* v, x=0..n-1),_
 sum( w, x=0..n-1)])

solve([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.

integrate(1/((x+t)*sqrt(1+(x*t)^2)),t=0..%plusInfinity,"noPole")

subst(%,x=1)

integrate(1/((1+t)*sqrt(1+(1*t)^2)),t=0..%plusInfinity,"noPole")

simplify(%-subst((asinh(x^2)+asinh(1/x^2))/sqrt(1+x^4),x=1))

%::Expression Float

a := 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]])

determinant(a)

inverse(a)

As := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])

A := subMatrix(As, 2,4,2,4)

ob := orthonormalBasis(A)

P : Matrix(Expression Integer) := new(3,3,0)

setsubMatrix!(P,1,1,ob.3)

setsubMatrix!(P,1,2,ob.1)

setsubMatrix!(P,1,3,ob.2)

Pt := transpose(P)

Ps : Matrix(Expression Integer) := new(4,4,0)

Ps(1,1) := 1

setsubMatrix!(Ps,2,2,P)

PsT := transpose(Ps)

PsTAsPs := PsT * As * Ps

b1 := PsTAsPs(2,1)

l1 := PsTAsPs(2,2)

Us : Matrix(Expression Integer) := new(4,4,0)

Us(1,1) := 1

Us(2,2) := 1

Us(3,3) := 1

Us(4,4) := 1

Us(2,1) := -b1 / l1

PsUs := Ps * Us

PsUsT := transpose(PsUs)

PsUsTAsPsUs := PsUsT * As * PsUs

C := inverse(PsUs)

c := PsUsTAsPsUs(1,1)

gQ := PsUsTAsPsUs / c

x1 := transpose(matrix([[1,2,3,4]]))

v1 := transpose(x1) * As * x1

x2 := C * x1

v2 := transpose(x2) * PsUsTAsPsUs * x2

)clear all

 All user variables and function definitions have been cleared.
draw(y^2/2+(x^2-1)^2/4-1=0, x,y, range ==[-2..2, -1..1])

 Graph data being transmitted to the viewport manager...
 FriCAS2D data being transmitted to the viewport manager...

f1 := taylor(1 - x^2,x = 0)

asin f1

sin %

SandboxMSkuce?

1+1

integrate((x-1)/log(x), x)

integrate(x*exp(x)*sin(x),x)

[p for p in primes(2,1000)|(p rem 16)=1]

[p^2+1 for p in primes(2,100)]

integrate (2*x^2 + 2*x, x)

radix(36,37)

Is it error?

integrate(log(log(x)),x)