Notations

M ==> SquareMatrix

I ==> Integer

N ==> NonNegativeInteger

-- setting up a 2x2 matrix with entries in the ring of integers
m0 : M(2,I) := matrix [[1, 2], [3, 4]]

(1)

-- and a 3x3 matrix with entries in the ring of  2x2 matrices of integers
mm0 : M(3,M(2,I)):= matrix [[m0, m0^2, m0^3], [m0^4,m0^5,m0^6],[m0^7,m0^8,m0^9]]

(2)

PROBLEM: (?Unlike Aldor) SPAD/FriCAS 1.3.3 does NOT support types that depend on preceding formal parameters in function declarations:

fun( ndim:N, m:M(ndim,I) ):M(ndim, I) == m-m^3

   Cannot convert the value from type Symbol to NonNegativeInteger .

Let us try to find out some alternative approaches...

Approach: multiple function definitions, each with a different value of n: WORKS, but with unpleasant code repetition

fun2(m:M(2,I)):M(2, I) == m-m^3

   Function declaration fun2 : SquareMatrix(2,Integer) -> SquareMatrix(
      2,Integer) has been added to workspace.

fun3(m:M(3, M(2, I))):M(3, M(2, I)) == m-m^3

   Function declaration fun3 : SquareMatrix(3,SquareMatrix(2,Integer))
       -> SquareMatrix(3,SquareMatrix(2,Integer)) has been added to 
      workspace.

fun2(m0)

Compiling function fun2 with type SquareMatrix(2,Integer) -> 
      SquareMatrix(2,Integer)

(3)

fun3(mm0)

Compiling function fun3 with type SquareMatrix(3,SquareMatrix(2,
      Integer)) -> SquareMatrix(3,SquareMatrix(2,Integer))

(4)

Approach: multiple specializations of the same function fun with different values of n. DOES NOT WORK

fun(2:N, m:M(2,I)):M(2, I) == m-m^3

   Function declaration fun : (NonNegativeInteger, SquareMatrix(2,
      Integer)) -> SquareMatrix(2,Integer) has been added to workspace.

fun(3:N, m:M(3, M(2, I))):M(3, M(2, I)) == m-m^3

   Function declaration fun : (NonNegativeInteger, SquareMatrix(3,
      SquareMatrix(2,Integer))) -> SquareMatrix(3,SquareMatrix(2,
      Integer)) has been added to workspace.

fun(2,m0) -- "WRONG" RESULT: definition of fun(2,...) has been overwritten by fun(3,...) and m0 has been promoted to a diagonal 3x3 matrix.

Compiling function fun with type (NonNegativeInteger, SquareMatrix(3
      ,SquareMatrix(2,Integer))) -> SquareMatrix(3,SquareMatrix(2,
      Integer))

(5)

fun(3,mm0)

(6)

Approach: funMaker, a function that returns an anonymous function. ERROR because a value-dependent return type for funMaker is needed

funMaker(n:N,R:Ring):(M(n,R)->M(n,R)) == (m:M(n,R)):M(n,R) +-> m-m^3

   Cannot convert the value from type Symbol to NonNegativeInteger .

Approach: %funMaker, a macro that returns an anonymous function: WORKS and requires minimal code writing.

macro %funMaker(n,R) == (m:M(n,R)):M(n,R) +-> m-m^3

ff2 := %funMaker(2, I)

(7)

ff3 := %funMaker(3, M(2,I))

(8)

ff2(m0)

(9)

ff3(mm0)

(10)