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sursumCorda

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Express one (suitable) algebraic number ...

sursumCorda 1279 Maple 2025
field algebraic-numbers
November 01 2025
2 4

For instance, I would like to represent “3^(1/3)” and “4^(1/4)” respectively as elements of the field generated by “3^(1/3) + 2^(1/2) + 1^(1/1)”. 
I think the Algebraic package and the evala procedure should already offer a direct command, but I couldn't find it. The following results are computed by SymPy's `to_number_field` function: 

-48/755*(3^(1/3) + 2^(1/2) + 1^(1/1))^5 + 
 213/755*(3^(1/3) + 2^(1/2) + 1^(1/1))^4 - 
 52/755*(3^(1/3) + 2^(1/2) + 1^(1/1))^3 - 
 174/755*(3^(1/3) + 2^(1/2) + 1^(1/1))^2 - 
 232/755*(3^(1/3) + 2^(1/2) + 1^(1/1)) + 
 277/151: # originally computed by SymPy's `to_number_field`
is(3^(1/3) = %);
                              true

48/755*(3^(1/3) + 2^(1/2) + 1^(1/1))^5 - 
 213/755*(3^(1/3) + 2^(1/2) + 1^(1/1))^4 + 
 52/755*(3^(1/3) + 2^(1/2) + 1^(1/1))^3 + 
 174/755*(3^(1/3) + 2^(1/2) + 1^(1/1))^2 + 
 987/755*(3^(1/3) + 2^(1/2) + 1^(1/1)) - 
 428/151: # originally computed by SymPy's `to_number_field`
is(4^(1/4) = %);
                              true

Does there exist a direct command in Maple to find the above representations? 

Symbolic calculus w.r.t. ABSTRACT vector...

sursumCorda 1279 Maple 2025
differentiation linear-algebra symbolic
October 28 2025
0 0

Although several similar problems were asked many years ago (see, e.g., the section “Formal linear algebra” here), there appears to be no new progress so far. It is said that such functionalities exists in the Physics package, but I cannot find any corresponding examples. 
In short, can Maple at present calculate these examples in terms of symbolic array constructs completely automatically?

Why does RealDomain:-solve return an ext...

sursumCorda 1279 Maple 2025
solve realdomain
August 24 2025
1 0

As we can see, RealDomain:-solve gives an incorrect solution to the following system: 

> 

restart;
> 

sys := `~`[diff](sqrt(2*a^2-8*a+10)+sqrt(b^2-6*b+10)+sqrt(2*a^2-2*a*b+b^2), [a, b]):
> 

RealDomain:-solve(`~`[`=`](sys, 0), {a, b})

{a = 5/3, b = 5/2}, {a = a, b = 2*a/(a-1)}

 (1)
> 

plot(eval(sys, {max(2*5^(1/2), (2*a^2-8*a+10)^(1/2)+2^(1/2)*((a^2-4*a+5)/(a-1)^2)^(1/2)+2^(1/2)*(a^2*(a^2-4*a+5)/(a-1)^2)^(1/2)), min(2*5^(1/2), (2*a^2-8*a+10)^(1/2)+2^(1/2)*((a^2-4*a+5)/(a-1)^2)^(1/2)+2^(1/2)*(a^2*(a^2-4*a+5)/(a-1)^2)^(1/2))}[-1]), a = -infinity .. infinity)

 
> 

extrema(sqrt(2*a^2-8*a+10)+sqrt(b^2-6*b+10)+sqrt(2*a^2-2*a*b+b^2), {}, {a, b})

{max(2*5^(1/2), (2*a^2-8*a+10)^(1/2)+2^(1/2)*((a^2-4*a+5)/(a-1)^2)^(1/2)+2^(1/2)*(a^2*(a^2-4*a+5)/(a-1)^2)^(1/2)), min(2*5^(1/2), (2*a^2-8*a+10)^(1/2)+2^(1/2)*((a^2-4*a+5)/(a-1)^2)^(1/2)+2^(1/2)*(a^2*(a^2-4*a+5)/(a-1)^2)^(1/2))}

 (2)

Download solve_returns_an_unsatisfiable_real_solution.mw

This appears to be a bug; is it possible to fix it? 
Text: 

sys := diff~(sqrt(2*a^2 - 8*a + 10) + sqrt(b^2 - 6*b + 10) + sqrt(2*a^2 - 2*a*b + b^2), [a, b]):
RealDomain:-solve(sys =~ 0, {a, b});

About the robustness of Student:-Numeric...

sursumCorda 1279 Maple 2025
linear-algebra factorization decomposition
July 31 2025
1 0

As the following worksheet shows, Student:-NumericalAnalysis:-MatrixDecomposition cannot factorize the input matrix  and throws an error, but if we simply reorder or exchange the elements of , no error will be raised. (The reason for setting  is that LinearAlgebra:-LUDecomposition can be used for other methods.) 
 

> 

restart
> 

with(Student:-NumericalAnalysis, MatrixDecomposition)
> 

m := Matrix([[3*(sqrt(3)+1)/8,-1/2,1/2,-(sqrt(3)+1)/8,-1/2,1/2,-(sqrt(3)+1)/8,-1/2,1/2,-(sqrt(3)+1)/8],
> 

             [-1/2,sqrt(3)-1,-(sqrt(3)-1),-1/2,0,0,1/2,0,0,1/2],
> 

             [1/2,-(sqrt(3)-1),sqrt(3)-1,1/2,0,0,-1/2,0,0,-1/2],
> 

             [-(sqrt(3)+1)/8,-1/2,1/2,3*(sqrt(3)+1)/8,1/2,-1/2,-(sqrt(3)+1)/8,1/2,-1/2,-(sqrt(3)+1)/8],
> 

             [-1/2,0,0,1/2,sqrt(3)-1,-(sqrt(3)-1),-1/2,0,0,1/2],
> 

             [1/2,0,0,-1/2,-(sqrt(3)-1),sqrt(3)-1,1/2,0,0,-1/2],
> 

             [-(sqrt(3)+1)/8,1/2,-1/2,-(sqrt(3)+1)/8,-1/2,1/2,3*(sqrt(3)+1)/8,1/2,-1/2,-(sqrt(3)+1)/8],
> 

             [-1/2,0,0,1/2,0,0,1/2,sqrt(3)-1,-(sqrt(3)-1),-1/2],
> 

             [1/2,0,0,-1/2,0,0,-1/2,-(sqrt(3)-1),sqrt(3)-1,1/2],
> 

             [-(sqrt(3)+1)/8,1/2,-1/2,-(sqrt(3)+1)/8,1/2,-1/2,-(sqrt(3)+1)/8,-1/2,1/2,3*(sqrt(3)+1)/8]],

> 

            'shape'='symmetric'):
> 

MatrixDecomposition(m, 'method' = 'LDLt'): # this does not work 

Error, (in Student:-NumericalAnalysis:-MatrixDecomposition) a pivot element 0 is encountered, and the entries below it are not all 0; the factorization cannot continue

  
> 

MatrixDecomposition(m([1, 4, 7, 10, 2, 5, 8, 3, 6, 9] $ 2), 'method' = 'LDLt'): # yet this works 
> 

MatrixDecomposition(m([2, 3, 5, 6, 8, 9, 1, 4, 7, 10] $ 2), 'method' = 'LDLt'): # this also works 
> 

randomize(5):
> 

k := 0:
to 1e3 do
        try
                MatrixDecomposition(m(combinat:-randperm(10) $ 2), 'method' = 'LDLt')
        catch :
                k++
        end
od:
k/1e3;

.469

 (1)


 

Download LDL_factorization_robustness.mw

The last instance above suggests that it only works on about half of the inputs (that are equivalent to each other). Although I tried changing the value of Digits, the failure rate remained high. Is Student:-NumericalAnalysis:-MatrixDecomposition not robust enough? 

How to `Describe` “Iterator:-Product”? ...

sursumCorda 1279 Maple 2024
March 03 2025
0 2

I would like to generate a brief description of the object Iterator:-Product but I get the following error:  

Describe(Iterator:-Product);

object Product :: Class<<36893490916968945900>>:

    ModuleApply( )

    ModuleCopy( self::_Product, proto::_Product, 
Error, (in Describe) `proto` does not evaluate to a module

How do I get rid of this message? 

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