# 2.7 Conversion¶

Conversion is the process of changing an object of one type into an object of another type. The syntax for conversion is: object::newType

By default, 3 has the type PositiveInteger.

3

$3$

Type: PositiveInteger

We can change this into an object of type Fraction Integer by using ::.

3 :: Fraction Integer

$3$

Type: Fraction Integer

A coercion is a special kind of conversion that FriCAS is allowed to do automatically when you enter an expression. Coercions are usually somewhat safer than more general conversions. The FriCAS library contains operations called coerce and convert. Only the coerce operations can be used by the interpreter to change an object into an object of another type unless you explicitly use a ::.

By now you will be quite familiar with what types and modes look like. It is useful to think of a type or mode as a pattern for what you want the result to be.

Let’s start with a square matrix of polynomials with complex rational number coefficients. SquareMatrix

m : SquareMatrix(2,POLY COMPLEX FRAC INT)


Type: Void

m := matrix [ [x-3/4*%i,z*y^2+1/2],[3/7*%i*y^4 - x,12-%i*9/5] ]

$\begin{split}\left[ \begin{array}{cc} {x -{{3 \over 4} \ i}} & {{{{y} ^ {2}} \ z}+{1 \over 2}} \\ {{{3 \over 7} \ i \ {{y} ^ {4}}} -x} & {{12} -{{9 \over 5} \ i}} \end{array} \right]\end{split}$

Type: SquareMatrix(2,Polynomial Complex Fraction Integer)

We first want to interchange the Complex and Fraction layers. We do the conversion by doing the interchange in the type expression.

m1 := m :: SquareMatrix(2,POLY FRAC COMPLEX INT)

$\begin{split}\left[ \begin{array}{cc} {x -{{3 \ i} \over 4}} & {{{{y} ^ {2}} \ z}+{1 \over 2}} \\ {{{{3 \ i} \over 7} \ {{y} ^ {4}}} -x} & {{{60} -{9 \ i}} \over 5} \end{array} \right]\end{split}$

Type: SquareMatrix(2,Polynomial Fraction Complex Integer)

Interchange the Polynomial and the Fraction levels.

m2 := m1 :: SquareMatrix(2,FRAC POLY COMPLEX INT)

$\begin{split}\left[ \begin{array}{cc} {{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {{y} ^ {2}} \ z}+1} \over 2} \\ {{{3 \ i \ {{y} ^ {4}}} -{7 \ x}} \over 7} & {{{60} -{9 \ i}} \over 5} \end{array} \right]\end{split}$

Type: SquareMatrix(2,Fraction Polynomial Complex Integer)

Interchange the Polynomial and the Complex levels.

m3 := m2 :: SquareMatrix(2,FRAC COMPLEX POLY INT)

$\begin{split}\left[ \begin{array}{cc} {{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {{y} ^ {2}} \ z}+1} \over 2} \\ {{-{7 \ x}+{3 \ {{y} ^ {4}} \ i}} \over 7} & {{{60} -{9 \ i}} \over 5} \end{array} \right]\end{split}$

Type: SquareMatrix(2,Fraction Complex Polynomial Integer)

All the entries have changed types, although in comparing the last two results only the entry in the lower left corner looks different. We did all the intermediate steps to show you what FriCAS can do.

In fact, we could have combined all these into one conversion.

m :: SquareMatrix(2,FRAC COMPLEX POLY INT)

$\begin{split}\left[ \begin{array}{cc} {{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {{y} ^ {2}} \ z}+1} \over 2} \\ {{-{7 \ x}+{3 \ {{y} ^ {4}} \ i}} \over 7} & {{{60} -{9 \ i}} \over 5} \end{array} \right]\end{split}$

Type: SquareMatrix(2,Fraction Complex Polynomial Integer)

There are times when FriCAS is not be able to do the conversion in one step. You may need to break up the transformation into several conversions in order to get an object of the desired type.

We cannot move either Fraction or Complex above (or to the left of, depending on how you look at it) SquareMatrix because each of these levels requires that its argument type have commutative multiplication, whereas SquareMatrix does not. That is because Fraction requires that its argument belong to the category IntegralDomain and category Complex requires that its argument belong to CommutativeRing. See ugTypesBasic for a brief discussion of categories. The Integer level did not move anywhere because it does not allow any arguments. We also did not move the SquareMatrix part anywhere, but we could have.

Recall that m looks like this.

m

$\begin{split}\left[ \begin{array}{cc} {x -{{3 \over 4} \ i}} & {{{{y} ^ {2}} \ z}+{1 \over 2}} \\ {{{3 \over 7} \ i \ {{y} ^ {4}}} -x} & {{12} -{{9 \over 5} \ i}} \end{array} \right]\end{split}$

Type: SquareMatrix(2,Polynomial Complex Fraction Integer)

If we want a polynomial with matrix coefficients rather than a matrix with polynomial entries, we can just do the conversion.

m :: POLY SquareMatrix(2,COMPLEX FRAC INT)

$\begin{split}{{\left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]} \ {{y} ^ {2}} \ z}+{{\left[ \begin{array}{cc} 0 & 0 \\ {{3 \over 7} \ i} & 0 \end{array} \right]} \ {{y} ^ {4}}}+{{\left[ \begin{array}{cc} 1 & 0 \\ -1 & 0 \end{array} \right]} \ x}+{\left[ \begin{array}{cc} -{{3 \over 4} \ i} & {1 \over 2} \\ 0 & {{12} -{{9 \over 5} \ i}} \end{array} \right]}\end{split}$

Type: Polynomial SquareMatrix(2,Complex Fraction Integer)

We have not yet used modes for any conversions. Modes are a great shorthand for indicating the type of the object you want. Instead of using the long type expression in the last example, we could have simply said this.

m :: POLY ?

$\begin{split}{{\left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]} \ {{y} ^ {2}} \ z}+{{\left[ \begin{array}{cc} 0 & 0 \\ {{3 \over 7} \ i} & 0 \end{array} \right]} \ {{y} ^ {4}}}+{{\left[ \begin{array}{cc} 1 & 0 \\ -1 & 0 \end{array} \right]} \ x}+{\left[ \begin{array}{cc} -{{3 \over 4} \ i} & {1 \over 2} \\ 0 & {{12} -{{9 \over 5} \ i}} \end{array} \right]}\end{split}$

Type: Polynomial SquareMatrix(2,Complex Fraction Integer)

We can also indicate more structure if we want the entries of the matrices to be fractions.

m :: POLY SquareMatrix(2,FRAC ?)

$\begin{split}{{\left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]} \ {{y} ^ {2}} \ z}+{{\left[ \begin{array}{cc} 0 & 0 \\ {{3 \ i} \over 7} & 0 \end{array} \right]} \ {{y} ^ {4}}}+{{\left[ \begin{array}{cc} 1 & 0 \\ -1 & 0 \end{array} \right]} \ x}+{\left[ \begin{array}{cc} -{{3 \ i} \over 4} & {1 \over 2} \\ 0 & {{{60} -{9 \ i}} \over 5} \end{array} \right]}\end{split}$

Type: Polynomial SquareMatrix(2,Fraction Complex Integer)