Lecture 22: Gauss, Prince of Mathematics

1. Reading: TSS, chapter 11

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2. Solving a system of equations:

   2 * z	+	2 * x	+	3 * y	=	10
   2 * z	+	5 * x	+	12 * y	=	31
   4 * z	+	1 * x	+	-2 * y	=	1

   
   A solution is a substitution of one number per variable such that the equations work out.
   Here is a solution for the above set of equations: z = 1, x = 1, y = 2. Task: check that it is a solution.
   
   
3. Gauss suggested a method for solving a system of equations. It consists of two phases: triangulation and then the solution process proper.
   First, we need a representation for equations.

   (list
   	(list	2	2	3	10)
   	(list	2	5	12	31)
   	(list	4	1	-2	1)	)

   
   Second, we need to see what it means to manipulate (representations of) these equations:

   (list (list 2 2 3 10) (list 2 5 12 31) (list 4 1 -2 1) )

   (list (list 2 2 3 10) (list 3 9 21) (list -3 -8 -19) )

   (list (list 2 2 3 10) (list 3 9 21) (list 1 2) )

   
   This last set of equations has triangular form and it is easy to solve.
   Task: Solve the corresponding system of equations systematically.
   
   
4. Time to design a Scheme program for triangulation:
   Data Definition:

   A matrix is a (listof (listof num)).
   
     Constraints:
      1. Each item on a matrix has the same length.
      2. The first item on a matrix is by one item longer than the matrix itself.
   
   A trimat is a (listof (listof num)).
   
     Constraints:
      1. Each item on a trimat is by one item longer than the following one.
      2. The first item on a trimat is by one item longer than the trimat itself.
      3. The last one contains two items (nums). 
   

   
   Contract, header, and purpose:

     triangulate : matrix -> trimat
     (define (triangulate a-matrix) ...)
   
     Purpose: compute a triangular matrix from a-matrix (with the same solution)
   

   
   Example: see above
   Template:
   1. trivial problem: matrix with one row
   2. trivial solution: the matrix
   3. generative step:
      subtract the first row from the remaining rows an appropriate number of times from the remaining ones, eliminate the leading 0
      then process the remaining equations
      combine the resulting trimat with the first row
   
   Definition:

   (define (triangulate mat)
     (cond
       ((empty? (rest mat)) (first mat))
       (else (local ((define row1 (first mat)))
   	    (cons row1 (triangulate (subtract-row1-from-all-rows row1 (rest mat))))))))
   

   
   does every recursive application use triangulate on a matrix? YES!
   Define auxiliary function:

   #|
     subtract-row1-from-all-rows : (listof num) (listof (listof num)) -> (listof (listof num))
   
     (define (subtract-row1-from-all-rows row1 other-rows) ...)   
   
      Purpose: subtract enough multiples of row1 from all rows in other-rows
        so that first number of each new one is 0. Eliminate the 0, i.e., 
        all rows in result are one shorter than row1
   
      Example: 
       (subtract-row1-from-all-rows (list 2 3 4)
   				 (list (list 2 3 4)
   				       (list 4 4 4)))
       = 
       (list 
        (list  0  0)
        (list -2 -4))
   |#