✎ pp:l04 [books]

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====== 4. Working with matrices and images using higher-order functions ======

===== Introduction - displaying a matrix =====

We shall represent matrices as //lists of lists//, i.e. values of type ''[ [Integer ] ]''. Each element in the outer list represents a line of the matrix. 
Hence, the matrix

$math[ \displaystyle \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right)] 

will be represented by the list ''[ [1,2,3],[4,5,6],[7,8,9] ]''.

To make signatures more legible, add the //type alias// to your code:
<code haskell> type Matrix = [[Integer]] </code>
which makes the type-name ''Matrix'' stand for ''[ [Integer] ]''.

1. Write a function ''parsem'' which takes a string and parses it to a ''Matrix''. In the string, the columns are separated by whitespace and lines - by '\n'. 
     * (Hint: You need to perform two types of very similar splitting operations: one into lines, and another into column values, for each line)

<code haskell>
parsem :: String -> Matrix
parsem = 
</code>

2. Write a function that converts a matrix to a string encoded as illustrated in the previous exercise. 
     * (Hint: use folds)
     * (Hint: test the function ''show'' on several different values)

<code haskell>
toString :: Matrix -> String
toString = 
</code>

3. Add the following to your code. Test the function ''displaymat''.
<code haskell>
displaymat = putStrLn . toString
</cod
e>

===== Matrix operations =====
4. Write a function that computes the scalar product with an integer:

$math[ \displaystyle 2 * \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 1 & 4 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \\ \end{array}\right)] 

<code haskell>
vprod :: Integer -> Matrix -> Matrix
vprod v = 
</code>

5. Write a function which adjoins two matrices by extending rows:

$math[ \displaystyle \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\\end{array}\right)  hjoin \left(\begin{array}{cc} 5 & 6 \\ 7 & 8\\\end{array}\right) = \left(\begin{array}{cc} 1 & 2 & 5 & 6 \\ 3 & 4 & 7 & 8\\\end{array}\right) ]

<code haskell>
hjoin :: Matrix -> Matrix -> Matrix
hjoin = 
</code>

6. Write a function which adjoins two matrices by adding new rows:

$math[ \displaystyle \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\\end{array}\right)  vjoin \left(\begin{array}{cc} 5 & 6 \\ 7 & 8\\\end{array}\right) = \left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6\\ 7 & 8\\ \end{array}\right) ]

<code haskell>
vjoin :: Matrix -> Matrix -> Matrix
vjoin = 
</code>

7. Write a function which adds two matrices.

<code haskell>
msum :: Matrix -> Matrix -> Matrix
msum = 
</code>

8. Write a function which computes the transposition of a matrix:

$math[ tr \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 1 & 4 & 6 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \\ \end{array}\right) ]

<code haskell>
tr :: [[a]] -> [[a]]
tr ([]:_) =
tr m =
</code>

9. Write a function which computes the vectorial product of two matrices.
  * (Hint: start by writing a function which computes $math[a_{ij}] for a given line $math[i] and column $math[j] (both represented as lists))
  * (Hint: write a function which takes a line of matrix m1 and the matrix m2 and computes the respective line from the product)

<code haskell>
mprod :: Matrix -> Matrix -> Matrix
mprod m1 m2 = 
</code>

===== Image operations =====

We can represent images as matrices of pixels. In our example a pixel will be represented as a ''Char'', which can take the values: '*' (signifying that the respective pixel is //on//) and ' ' (the pixel is off). 

<code haskell> type Image = [String] </code>

For instance, a rectangle could be represented as:

<code>
********
*      *
********
</code>

more precisely, as the string: 
<code haskell>
"********\n*      *\n********\n"
</code>

10. Implement an image-displaying function. Use the image from the Appendix as a test.
<code haskell>
toStringImg :: Image -> String
toStringImg = 

</code>

Add the following function in order to view images in a nicer format:
<code haskell>
displaym = putStrLn . toStringImg
</code>

11. Implement a function which flips an image horizontally:
<code haskell>
flipH :: Image -> Image 
flipH = 
</code>

12. Implement a function which flips an image vertically:
<code haskell>
flipV :: Image -> Image
flipV = 
</code>

13. Implement a function which rotates an image 90grd clockwise
<code haskell>
rotate90r :: Image -> Image
rotate90r =
</code>

14. Implement a function which rotates an image -90grd clockwise
<code haskell>
rotate90l :: Image -> Image
rotate90l = 
</code>

15. Implement a function which returns a diamond of a specified //height//.  Example:
<code>
                          *
                         ***
   diamond 3 =          *****
                         ***
                          *

                         *
   diamond 2 =          ***
                         *
</code>
   
<code haskell>
diamond :: Integer -> Image 
</code>

16. Implement a function with takes two images with the same dimensions. The second image is a mask. The output will be another image in which all pixels from the first image overlaying with a '*'-pixel from the mask will be displayed. All others will be deleted (made equal to ' '). Example:

<code>  
   
   ******                ****       ****
   **  **    overlay    ******  =  **  ** 
   **  **                ****       *  *
   ******                 **         **
</code>

<code haskell>
overlay :: Image -> Image -> Image
</code>

===== More matrices =====

17. Implement a function which places zeros **above** the diagonal of a square matrix. Use folds.

18. Implement a function which places zeros **below** the diagonal of a square matrix. Use folds.

19. Implement either of 17 or 18 using the function(s) ''take'' and ''drop''. Check the types of these and test them.

20. Implement a function which computes the **diagonal** matrix of a square matrix.

21. Implement a function which computes **recursively** the determinant of a matrix. What additional functions would be necessary?

===== Appendix =====

<code haskell>
l1="        ***** **            ***** **    "
l2="     ******  ****        ******  ****   "
l3="    **   *  *  ***      **   *  *  ***  "
l4="   *    *  *    ***    *    *  *    *** "
l5="       *  *      **        *  *      ** "
l6="      ** **      **       ** **      ** "
l7="      ** **      **       ** **      ** "
l8="    **** **      *      **** **      *  "
l9="   * *** **     *      * *** **     *   "
l10="      ** *******          ** *******    "
l11="      ** ******           ** ******     "
l12="      ** **               ** **         "
l13="      ** **               ** **         "
l14="      ** **               ** **         "
l15=" **   ** **          **   ** **         "
l16="***   *  *          ***   *  *          "
l17=" ***    *            ***    *           "
l18="  ******              ******            "
l19="    ***                 ***             "

logo = [l1,l2,l3,l4,l5,l6,l7,l8,l9,l10,l11,l12,l13,l14,l15,l16,l17,l18,l19]
</code>