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POWrite this Scala Matrix multiplication in Haskell
 

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    <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="https://stackoverflow.com/questions/8308015/can-you-overload-in-haskell">Can you overload + in haskell?</a> </p> </blockquote> <p>Can you implement a Matrix class and an * operator that will work on two matrices?:</p> <pre><code>scala&gt; val x = Matrix(3, 1,2,3,4,5,6) x: Matrix = [1.0, 2.0, 3.0] [4.0, 5.0, 6.0] scala&gt; x*x.transpose res0: Matrix = [14.0, 32.0] [32.0, 77.0] </code></pre> <p>and just so people don't say that it's hard, here is the Scala implementation (courtesy of <a href="http://jsmerritt.blogspot.com/2008/07/matrix-multiplication-in-scala.html" rel="nofollow noreferrer">Jonathan Merritt</a>):</p> <pre><code>class Matrix(els: List[List[Double]]) { /** elements of the matrix, stored as a list of its rows */ val elements: List[List[Double]] = els def nRows: Int = elements.length def nCols: Int = if (elements.isEmpty) 0 else elements.head.length /** all rows of the matrix must have the same number of columns */ require(elements.forall(_.length == nCols)) /* Add to each elem of matrix */ private def addRows(a: List[Double], b: List[Double]): List[Double] = List.map2(a,b)(_+_) private def subRows(a: List[Double], b: List[Double]):List[Double] = List.map2(a,b)(_-_) def +(other: Matrix): Matrix = { require((other.nRows == nRows) &amp;&amp; (other.nCols == nCols)) new Matrix( List.map2(elements, other.elements) (addRows(_,_)) ) } def -(other: Matrix): Matrix = { require((other.nRows == nRows) &amp;&amp; (other.nCols == nCols)) new Matrix( List.map2(elements, other.elements) (subRows(_,_)) ) } def transpose(): Matrix = new Matrix(List.transpose(elements)) private def dotVectors(a: List[Double], b: List[Double]): Double = { val multipliedElements = List.map2(a,b)(_*_) (0.0 /: multipliedElements)(_+_) } def *(other: Matrix): Matrix = { require(nCols == other.nRows) val t = other.transpose() new Matrix( for (row &lt;- elements) yield { for (otherCol &lt;- t.elements) yield dotVectors(row, otherCol) } ) override def toString(): String = { val rowStrings = for (row &lt;- elements) yield row.mkString("[", ", ", "]") rowStrings.mkString("", "\n", "\n") } } /* Matrix constructor from a bunch of numbers */ object Matrix { def apply(nCols: Int, els: Double*):Matrix = { def splitRowsWorker( inList: List[Double], working: List[List[Double]]): List[List[Double]] = if (inList.isEmpty) working else { val (a, b) = inList.splitAt(nCols) splitRowsWorker(b, working + a) } def splitRows(inList: List[Double]) = splitRowsWorker(inList, List[List[Double]]()) val rows: List[List[Double]] = splitRows(els.toList) new Matrix(rows) } } </code></pre> <p><strong>EDIT</strong> I understood that strictly speaking the answer is No: overloading * is not possible without side-effects of defining also a + and others or special tricks. The <a href="http://hackage.haskell.org/package/numeric-prelude" rel="nofollow noreferrer">numeric-prelude package</a> describes it best:</p> <blockquote> <p>In some cases, the hierarchy is not finely-grained enough: Operations that are often defined independently are lumped together. For instance, in a financial application one might want a type "Dollar", or in a graphics application one might want a type "Vector". It is reasonable to add two Vectors or Dollars, but not, in general, reasonable to multiply them. But the programmer is currently forced to define a method for '(*)' when she defines a method for '(+)'.</p> </blockquote>
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    1. COThe answers to your previous question showed it exactly. I don't see how they are wishy-washy.
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    2. CONot really. It left out implementation of a lot of other Num operations undefined or something like that. I want to know if there is "correct" way of doing this, or if it's IMPOSSIBLE in haskell (maybe people don't want to admit it?). In Scala - there is no question that this is OK! In any case - one way to settle this for sure is to implement the example above in Haskell.
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    3. COThere are 7 `Num` operations: `(+)`, `(*)`, `(-)`, `negate`, `signum`, `abs`, `fromInteger`. The first four, no problem; `signum` I suppose should return a unit matrix that is a scalar multiple of the input, `abs` should return a scalar (and it can't, because the typeclass insists it must return another matrix), and `fromInteger` doesn't make sense. OK, fine, make `abs` and `fromInteger` return errors. That's not perfect but it'll do --- you're already resigned to getting runtime errors when you add two matrices of different sizes. Could someone else expand on the Numeric Prelude, please?
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