J: Gauss-Jordan Elimination

The task of coding a method for solving Gauss-Jordan equations of a linear system of algebraic equations is the exercise I have chosen to advance in the learning process. The system is Ax = b, where A is an n-by-n matrix, b and unknown x are n-vectors. First, I started with the simplest form with control structures:

   gj0 =: dyad :0 NB. usage is same to %.
    y=.y,.b 
    for_d.i.#y do. 
     for_r.i.#y do. 
      if.r=d do.continue.end. NB. do not eliminate d'th row
      t=.%/ (<"1(r,d),:(d,d)) { y
      for_c.d}.>:i.#y do.
       y=.(((<r,c){y)-(t*(<d,c){y)) (<r,c)} y
      end.
      y=.0 (<r,d)} y NB. ensure zero
     end. 
    end. NB. now A is diagonal but not identity matrix, so:
    x=.{:"1 y NB. x = b
    for_r.i.#y do.
     x=.((r{x)%(<r,r){y) r} x NB. divide by coefficients on diagonal
    end. 
   )
   Ab =: (".;._2) 0 :0
   0.25 _0.16 _0.38  0.17
   0.19 _0.22 _0.02  0.41
   0.13  0.08 _0.08 _0.13
   0.13 _0.1  _0.32  0.65
   )
   b =: 0.37 0.01 0.01 1.51
   (,.".&.>)('A';'b';'gj0 A,.b';'b %. A')
┌────────┬──────────────────────┐
│A       │0.25 _0.16 _0.38  0.17│
│        │0.19 _0.22 _0.02  0.41│
│        │0.13  0.08 _0.08 _0.13│
│        │0.13  _0.1 _0.32  0.65│
├────────┼──────────────────────┤
│b       │0.37 0.01 0.01 1.51   │
├────────┼──────────────────────┤
│b gj0 A │_1 3 _2 2             │
├────────┼──────────────────────┤
│b %. A  │_1 3 _2 2             │
└────────┴──────────────────────┘

      

        
        
        
      

    

Right! Then I decided to get rid of as many control structures as possible:

   gj1 =:dyad :0
    y=.y,.b 
    for_d.i.#y do.
     for_r.d ({.,]}.~[:>:[) i.#y do. NB. for indices without d
      t=.%/ (<"1(r,d),:(d,d)) { y
      y=.((r{y)-(t*d{y)) r}y NB. no need to iterate for each column
      y=.0 (<r,d)} y
     end. 
    end.
    ({:"1 y)%(+/}:"1 y) NB. b divide by sum of each item of A (drop zeroes)
   )
   b gj1 A
_1 3 _2 2

      

        
        
        
      

    

OK, now I can try to translate for_r.

-loop into an unspoken form ... but it looks like it will look more cumbersome and I think I am wrong, but what research without errors? I really want to code the Gauss-Iord method tacitly:

• J-coding exercise
• see if performance is better
• try to understand the code in a few weeks :)

Please help me write it to the end or point out a better approach.