{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

UPDATE:  compare the following output to the output generated by my more evolved and more "elegant" elegant approach, namely gauss++.
____________________________________________________________

mwh@coyote2:[~]:
---> rref
Enter the number of rows in the matrix: 3
Enter the number of columns in the matrix: 4

Enter row 1, column 1 : 4
Enter row 1, column 2 : 3
Enter row 1, column 3 : 1
Enter row 1, column 4 : 2

Enter row 2, column 1 : 2
Enter row 2, column 2 : -1
Enter row 2, column 3 : 2
Enter row 2, column 4 : 3

Enter row 3, column 1 : 4
Enter row 3, column 2 : 2
Enter row 3, column 3 : -1
Enter row 3, column 4 : -5


Original Matrix:

_________________________________________________________

   4   3   1   2
   2   -1   2   3
   4   2   -1   -5
_________________________________________________________


Dividing row 1 by 4
_________________________________________________________

   1   3/4   1/4   1/2
   2   -1   2   3
   4   2   -1   -5
_________________________________________________________


Multiplying row 1 by -2 and adding result to row 2
_________________________________________________________

   1   3/4   1/4   1/2
   0   -5/2   3/2   2
   4   2   -1   -5
_________________________________________________________


Multiplying row 1 by -4 and adding result to row 3
_________________________________________________________

   1   3/4   1/4   1/2
   0   -5/2   3/2   2
   0   -1   -2   -7
_________________________________________________________


Dividing row 2 by -5/2
_________________________________________________________

   1   3/4   1/4   1/2
   0   1   -3/5   -4/5
   0   -1   -2   -7
_________________________________________________________


Multiplying row 2 by -3/4 and adding result to row 1
_________________________________________________________

   1   0   7/10   11/10
   0   1   -3/5   -4/5
   0   -1   -2   -7
_________________________________________________________


Multiplying row 2 by 1 and adding result to row 3
_________________________________________________________

   1   0   7/10   11/10
   0   1   -3/5   -4/5
   0   0   -13/5   -39/5
_________________________________________________________


Dividing row 3 by -13/5
_________________________________________________________

   1   0   7/10   11/10
   0   1   -3/5   -4/5
   0   0   1   3
_________________________________________________________


Multiplying row 3 by -7/10 and adding result to row 1
_________________________________________________________

   1   0   0   -1
   0   1   -3/5   -4/5
   0   0   1   3
_________________________________________________________


Multiplying row 3 by 3/5 and adding result to row 2
_________________________________________________________

   1   0   0   -1
   0   1   0   1
   0   0   1   3
_________________________________________________________

************************************************************
Row reduced echelon form:
_________________________________________________________

   1   0   0   -1
   0   1   0   1
   0   0   1   3
_________________________________________________________

************************************************************
So, we see there is a unique solution for vector x:

x1 = -1
x2 = 1
x3 = 3

An example where there are no solutions:

mwh@coyote2:[~]:
---> rref
Enter the number of rows in the matrix: 3
Enter the number of columns in the matrix: 4

Enter row 1, column 1 : 1
Enter row 1, column 2 : -3
Enter row 1, column 3 : 4
Enter row 1, column 4 : 1

Enter row 2, column 1 : 2
Enter row 2, column 2 : 13
Enter row 2, column 3 : -17
Enter row 2, column 4 : 2

Enter row 3, column 1 : -2
Enter row 3, column 2 : 6
Enter row 3, column 3 : -8
Enter row 3, column 4 : 1


Original Matrix:

_________________________________________________________

   1   -3   4   1
   2   13   -17   2
   -2   6   -8   1
_________________________________________________________


Dividing row 1 by 1
_________________________________________________________

   1   -3   4   1
   2   13   -17   2
   -2   6   -8   1
_________________________________________________________


Multiplying row 1 by -2 and adding result to row 2
_________________________________________________________

   1   -3   4   1
   0   19   -25   0
   -2   6   -8   1
_________________________________________________________


Multiplying row 1 by 2 and adding result to row 3
_________________________________________________________

   1   -3   4   1
   0   19   -25   0
   0   0   0   3
_________________________________________________________


Dividing row 2 by 19
_________________________________________________________

   1   -3   4   1
   0   1   -25/19   0
   0   0   0   3
_________________________________________________________


Multiplying row 2 by 3 and adding result to row 1
_________________________________________________________

   1   0   1/19   1
   0   1   -25/19   0
   0   0   0   3
_________________________________________________________


Multiplying row 2 by 0 and adding result to row 3
_________________________________________________________

   1   0   1/19   1
   0   1   -25/19   0
   0   0   0   3
_________________________________________________________


Dividing row 3 by 3
_________________________________________________________

   1   0   1/19   1
   0   1   -25/19   0
   0   0   0   1
_________________________________________________________


Multiplying row 3 by -1 and adding result to row 1
_________________________________________________________

   1   0   1/19   0
   0   1   -25/19   0
   0   0   0   1
_________________________________________________________


Multiplying row 3 by 0 and adding result to row 2
_________________________________________________________

   1   0   1/19   0
   0   1   -25/19   0
   0   0   0   1
_________________________________________________________

************************************************************
Row reduced echelon form:
_________________________________________________________

   1   0   1/19   0
   0   1   -25/19   0
   0   0   0   1
_________________________________________________________

************************************************************

An example where there are an infinite number of solutions (with 2 free variables).  Notice the zero rows in the row echelon form.

mwh@coyote2:[~]:
---> rref
Enter the number of rows in the matrix: 4
Enter the number of columns in the matrix: 5

Enter row 1, column 1 : 1
Enter row 1, column 2 : 2
Enter row 1, column 3 : 3
Enter row 1, column 4 : 4
Enter row 1, column 5 : 5

Enter row 2, column 1 : -1
Enter row 2, column 2 : -2
Enter row 2, column 3 : -3
Enter row 2, column 4 : -4
Enter row 2, column 5 : -5

Enter row 3, column 1 : 10
Enter row 3, column 2 : 20
Enter row 3, column 3 : 30
Enter row 3, column 4 : 40
Enter row 3, column 5 : 50

Enter row 4, column 1 : -3
Enter row 4, column 2 : -5
Enter row 4, column 3 : 32
Enter row 4, column 4 : 12
Enter row 4, column 5 : 100


Original Matrix:

_________________________________________________________

   1   2   3   4   5
   -1   -2   -3   -4   -5
   10   20   30   40   50
   -3   -5   32   12   100
_________________________________________________________


Dividing row 1 by 1
_________________________________________________________

   1   2   3   4   5
   -1   -2   -3   -4   -5
   10   20   30   40   50
   -3   -5   32   12   100
_________________________________________________________


Multiplying row 1 by 1 and adding result to row 2
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   10   20   30   40   50
   -3   -5   32   12   100
_________________________________________________________


Multiplying row 1 by -10 and adding result to row 3
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   -3   -5   32   12   100
_________________________________________________________


Multiplying row 1 by 3 and adding result to row 4
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________

Prevented division by zero
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 2 by -2 and adding result to row 1
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 2 by 0 and adding result to row 3
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 2 by -1 and adding result to row 4
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________

Prevented division by zero
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 3 by -2 and adding result to row 1
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 3 by 0 and adding result to row 2
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 3 by -1 and adding result to row 4
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Dividing row 4 by 1
_________________________________________________________

   1   2   3   4   5
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 4 by -2 and adding result to row 1
_________________________________________________________

   1   0   -79   -44   -225
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 4 by 0 and adding result to row 2
_________________________________________________________

   1   0   -79   -44   -225
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________


Multiplying row 4 by 0 and adding result to row 3
_________________________________________________________

   1   0   -79   -44   -225
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________

************************************************************
Row reduced echelon form:
_________________________________________________________

   1   0   -79   -44   -225
   0   0   0   0   0
   0   0   0   0   0
   0   1   41   24   115
_________________________________________________________

************************************************************

I have been reading the posts.  It's one of the first things I do in the morning.  Do not mistaken the absence of comments or responses as a sign that I am not interested. 

I have tragedies to report, and I am glad to read what others have to say.

I just wanted to post a few examples of the output of rref to give you an idea of the work it does.   Final analysis is left to the "user".

May our toothaches be of the minor nuisance variety and not the full blown kind that leave one at the mercy of an expensive oral surgeon who only accepts cash (cash that you already spent on food,  wires, adapters, and mysterious black boxes).

Here is the output generated by gauss++ when fed the system from .

 *** Hentrich tinkering with gauss++ ...
RREF: Hentrich tinkering with I/O:

Please enter dimension of square matrix A: 4

First enter Matrix A in form { {  }{  }{  } }: { { 1 2 3 4 }{ -1 -2 -3 -4 }{ 10 20 30 40 }{ -3 -5 32 12 } } 

Now enter Vector b in this form {  }: { 5 -5 50 100 }

 A =
{
{  1  2  3  4  }
{  -1  -2  -3  -4  }
{  10  20  30  40  }
{  -3  -5  32  12  }
}

 b = {  5  -5  50  100  }

The augmented matrix [ A | b ]:

[
 {  1  2  3  4  } | [ 5 ]
 {  -1  -2  -3  -4  } | [ -5 ]
 {  10  20  30  40  } | [ 50 ]
 {  -3  -5  32  12  } | [ 100 ]
]

________*** press enter ***_____________________

FLOATING-POINT ("real") representation:
________________________________

A =
{
{  1  2  3  4  }
{  -1  -2  -3  -4  }
{  10  20  30  40  }
{  -3  -5  32  12  }
}

b = {  5  -5  50  100  }
________________________________

Gaussian Elimination:
 ... Fill zeros under the diagonal of row 2 with multiplier: -1

[Gauss:]
Multiplying row 1 by 1 and adding result to row 2

M(1, 0) = -1 + (1 * 1) = 0
M(1, 1) = -2 + (1 * 2) = 0
M(1, 2) = -3 + (1 * 3) = 0
M(1, 3) = -4 + (1 * 4) = 0
b(1) = b(1) -  (-1)*b(0) = -5 - -5 = 0

[
 {  1  2  3  4  } | [ 5 ]
 {  0  0  0  0  } | [ 0 ]
 {  10  20  30  40  } | [ 50 ]
 {  -3  -5  32  12  } | [ 100 ]
]

________*** press enter ***_____________________

row 3 with multiplier: 10

[Gauss:]
Multiplying row 1 by -10 and adding result to row 3

M(2, 0) = 10 + (-10 * 1) = 0
M(2, 1) = 20 + (-10 * 2) = 0
M(2, 2) = 30 + (-10 * 3) = 0
M(2, 3) = 40 + (-10 * 4) = 0
b(2) = b(2) -  (10)*b(0) = 50 - 50 = 0

[
 {  1  2  3  4  } | [ 5 ]
 {  0  0  0  0  } | [ 0 ]
 {  0  0  0  0  } | [ 0 ]
 {  -3  -5  32  12  } | [ 100 ]
]

________*** press enter ***_____________________

row 4 with multiplier: -3

[Gauss:]
Multiplying row 1 by 3 and adding result to row 4

M(3, 0) = -3 + (3 * 1) = 0
M(3, 1) = -5 + (3 * 2) = 1
M(3, 2) = 32 + (3 * 3) = 41
M(3, 3) = 12 + (3 * 4) = 24
b(3) = b(3) -  (-3)*b(0) = 100 - -15 = 115

[
 {  1  2  3  4  } | [ 5 ]
 {  0  0  0  0  } | [ 0 ]
 {  0  0  0  0  } | [ 0 ]
 {  0  1  41  24  } | [ 115 ]
]

________*** press enter ***_____________________


[
 {  1  2  3  4  } | [ 5 ]
 {  -1  -2  -3  -4  } | [ -5 ]
 {  10  20  30  40  } | [ 50 ]
 {  -3  -5  32  12  } | [ 100 ]
]

________*** press enter ***_____________________


Elimination with Partial Pivot:
[Swapped row 1 with pivot row 3]


[
 {  10  20  30  40  } | [ 50 ]
 {  -1  -2  -3  -4  } | [ -5 ]
 {  1  2  3  4  } | [ 5 ]
 {  -3  -5  32  12  } | [ 100 ]
]

________*** press enter ***_____________________


[Partial Pivot:]
Multiplying row 1 by 1/10 and adding result to row 2

M(1, 0) = -1 + (1/10 * 10) = 0
M(1, 1) = -2 + (1/10 * 20) = 0
M(1, 2) = -3 + (1/10 * 30) = 0
M(1, 3) = -4 + (1/10 * 40) = 0
[STATE] multiplier: -1/10
 i = 1; j = 0
Then:

scale_and_add(A[0].slice(0),1/10, A[1].slice(0) ):

-5 - -1/10*50 = -5 - -5 = 0
[Partial Pivot:]
Multiplying row 1 by -1/10 and adding result to row 3

M(2, 0) = 1 + (-1/10 * 10) = 0
M(2, 1) = 2 + (-1/10 * 20) = 0
M(2, 2) = 3 + (-1/10 * 30) = 0
M(2, 3) = 4 + (-1/10 * 40) = 0
[STATE] multiplier: 1/10
 i = 2; j = 0
Then:

scale_and_add(A[0].slice(0),-1/10, A[2].slice(0) ):

5 - 1/10*50 = 5 - 5 = 0
[Partial Pivot:]
Multiplying row 1 by 3/10 and adding result to row 4

M(3, 0) = -3 + (3/10 * 10) = 0
M(3, 1) = -5 + (3/10 * 20) = 1
M(3, 2) = 32 + (3/10 * 30) = 41
M(3, 3) = 12 + (3/10 * 40) = 24
[STATE] multiplier: -3/10
 i = 3; j = 0
Then:

scale_and_add(A[0].slice(0),3/10, A[3].slice(0) ):

100 - -3/10*50 = 100 - -15 = 115
************ What is it ? ******************
[*** STATE ***]: [ A | b ] =

[
 {  10  20  30  40  } | [ 50 ]
 {  0  0  0  0  } | [ 0 ]
 {  0  0  0  0  } | [ 0 ]
 {  0  1  41  24  } | [ 115 ]
]

________*** press enter ***_____________________


[Swapped row 2 with pivot row 4]


[
 {  10  20  30  40  } | [ 50 ]
 {  0  1  41  24  } | [ 115 ]
 {  0  0  0  0  } | [ 0 ]
 {  0  0  0  0  } | [ 0 ]
]

________*** press enter ***_____________________


[Partial Pivot:]
Multiplying row 2 by 0 and adding result to row 3

M(2, 1) = 0 + (0 * 1) = 0
M(2, 2) = 0 + (0 * 41) = 0
M(2, 3) = 0 + (0 * 24) = 0
[STATE] multiplier: 0
 i = 2; j = 1
Then:

scale_and_add(A[1].slice(1),0, A[2].slice(1) ):

0 - 0*115 = 0 - 0 = 0
[Partial Pivot:]
Multiplying row 2 by 0 and adding result to row 4

M(3, 1) = 0 + (0 * 1) = 0
M(3, 2) = 0 + (0 * 41) = 0
M(3, 3) = 0 + (0 * 24) = 0
[STATE] multiplier: 0
 i = 3; j = 1
Then:

scale_and_add(A[1].slice(1),0, A[3].slice(1) ):

0 - 0*115 = 0 - 0 = 0
************ What is it ? ******************
[*** STATE ***]: [ A | b ] =

[
 {  10  20  30  40  } | [ 50 ]
 {  0  1  41  24  } | [ 115 ]
 {  0  0  0  0  } | [ 0 ]
 {  0  0  0  0  } | [ 0 ]
]

________*** press enter ***_____________________

Exception: can't solve pivot==0

Lastly, the example with fractions as input I promised.  it was important that I decreased tolerance from 0.001 to 0.0000001 for the fractions to sufficiently represent the number.
_________________________________________________________________

 *** Hentrich tinkering with gauss++ ...
RREF: Hentrich tinkering with I/O:

Please enter dimension of square matrix A: 2

First enter Matrix A in form { {  }{  }{  } }: { { -1/2 3/4 }{ 5/16 12 } }

Now enter Vector b in this form {  }: { 1 -1 }

 A =
{
{  -1/2  3/4  }
{  5/16  12  }
}

 b = {  1  -1  }

The augmented matrix [ A | b ]:

[
 {  -1/2  3/4  } | [ 1 ]
 {  5/16  12  } | [ -1 ]
]

________*** press enter ***_____________________

FLOATING-POINT ("real") representation:
________________________________

A =
{
{  -0.5  0.75  }
{  0.3125  12  }
}

b = {  1  -1  }
________________________________

Gaussian Elimination:
 ... Fill zeros under the diagonal of row 2 with multiplier: -5/8

[Gauss:]
Multiplying row 1 by 5/8 and adding result to row 2

M(1, 0) = 5/16 + (5/8 * -1/2) = 0
M(1, 1) = 12 + (5/8 * 3/4) = 399/32
b(1) = b(1) -  (-5/8)*b(0) = -1 - -0.625 = -3/8

[
 {  -1/2  3/4  } | [ 1 ]
 {  0  399/32  } | [ -3/8 ]
]

________*** press enter ***_____________________


[
 {  -1/2  3/4  } | [ 1 ]
 {  0  399/32  } | [ -3/8 ]
]

________*** press enter ***_____________________

[backsub: n = 2; Now i = 1]|--->
[Row 2; pivot value: 399/32]

b(1): -3/8;  x(1) will become [-3/8  - (0)]/pivot = (-3/8)/(399/32)
So, x(1) = (-3/8 - (0))/(399/32) = -4/133 =?= s/m : -4/133

NEW x(1) = -4/133
Peek at x so far: {  0  -4/133  }
[
 {  -1/2  3/4  } | [ 1 ]
 {  0  1  } | [ -4/133 ]
]

________*** press enter ***_____________________

[backsub: n = 2; Now i = 0]|--->
[Row 1; pivot value: -1/2]
3/4 * -4/133 = -3/133| Total Sum = -3/133 |
*

b(0): 1;  x(0) will become [1  - (-3/133)]/pivot = (136/133)/(-1/2)
So, x(0) = (1 - (-3/133))/(-1/2) = -272/133 =?= s/m : -272/133


scale_and_add(A[0].slice(1), -1, A[0].slice(1) ):


M(0, 1) = 3/4 + (-3/4)*(1) = 3/4 + -3/4 = 0
NEW x(0) = -272/133
Peek at x so far: {  -272/133  -4/133  }
[
 {  1  0  } | [ -272/133 ]
 {  0  1  } | [ -4/133 ]
]

________*** press enter ***_____________________


Gaussian elimination solution is x = {  -272/133  -4/133  }
A*x = {  -272/133  -4/133  }


FLOATING-POINT ("real") representation:
________________________________

x = {  -2.04511  -0.0300752  }
________________________________

I have been watching a python for beginners video on youtube.

Holden and Ibra,

I mainly posted this for posterity, as well as the code using Matrix.h/MatrixIO.h and fractions.h/fractions.cpp, not simply because of the flawlessness of command line input, that is matrix as object-in-its-own-right in the iostream:

{ { 1 1 1 1 1 1 }{ 2 -2 -1 1 -1 2 }{ 1 -1 3 -1 2 -1 }{ 1 3 -1 2 -1 -3 }{ 3 -1 2 -3 3 -1 }{ 1 4 -2 4 2 -3 } }

That can even contain fractions as elements:

{ { 1/2 1 1 1/6 1 1 }{ 2 -2 -1 1 -1 2 }{ 1 -1 3 -1 2 -1 }{ 1 3 -1 2 -1 -3 }{ 3 -1 2 -3 3 -1 }{ 1 4 -2 4 2 -3/10 } }

In the future, if some web-archives archeologist compiles and runs the different code, comparing rref to gauss++, he or she would immediately witness how much more elegant, and in fact, superior, the latest version I created is to the one that I haphazardly created that uses "brute force" (rref).

I sometimes do not wish to pause to upload such things here, but I feel it makes it more real for me to share breakthoughs of this kind.

I had alerted Holden that I was investigating Stroustrup's Matrix.h in chapter 24 (Numerics) of text, and - well - one thing led to another and the project merged and snowballed with my previous creation.  I had hard-coded some of the internal functions of Matrix.h, and that helped a great deal in coding the "animation" (the changing of states with an explanation of each).

The difference between the old post (rref) = BRUTE FORCE and the latest (gauss++) is very dramatic.  In fact, following the algorithms spelled out in the code, I find myself using this method by osmosis when performing by hand (smaller, 3 by 3, matrices).

I've come up with my own notation for the "Back Substitution" stage.

It may seem grotesque to delight in such secret breakthroughs, but, you have to remember that I have been improving that fraction.h class (and fraction.cpp) since its first incarnation in 1999 during a C++ course at a community college.  So, due to sentimental value [or the organic independence I experience on a subjective level, being intimate with the nature of rational numbers] I love to use it as a test type to see if a certain class, such as Matrix, will absorb it (the Fraction type).  I made the adjustments in order for them to play well together, and it felt kind of like a "mild religious experience," that is, there is a part of me working on things that my conscious mind is merely "recording/perceiving" - The Tinkering Organism-in-Itself is up to tinkering, and it is frustrated to be attached to this living animal body who is so moody and needy - it needs a tremendous amount to "get through the day," ... the coffee, the tobacco ... not to mention the need to eat, which I daily ignore until I am dizzy.

We must be brainwashed apes.   Sorry I have been absent from the board, but I do not leave computers on 24/7.  If I keep machines running compiling and debugging, when I reach a certain point, I let them (and my eyeballs) rest.  By the time I am done, my eyes are fried, and there is no "energy" left to invest in this board.  At the moment, I am trying to force myself to document a large matrix being fed through this code.

 Tomorrow morning might be very "sacred" - that is, we might have some thunder storms just prior to my nicotine stained fingers planting some seeds in the dirt.   I tracked down a few tomato and squash and cucumber seedlings to join the garlic in a patch of dirt along the New Jersey Parkway we rent for $11 a season.   I had to drive far as all local places were hit fast and hard by the local population [Raw Primates with apetites, as are we all].

Yes, a terrifying predicament we have been born into.  I am trying to stare down the Stomach as Slave Master.  Often my intellect rebels and the stomach becomes more tame.  Each creature is really in a bind.

I try to be a good sport, but ... It is as though one must detach entirely into oneself when engaging with something that no one in your immediate "monkey sphere" has any interest in.

. I think the tinkering style of Mr Hentrich is the way to deepen the understanding of mathematics.

So much of the code I write mutates into "educational style" step by step explanatory phenomenon.  This not only has value while creating it, and, then again, not only for potential users who might use it, but for a future version of myself who just might happen to live through whatever crazy chaos comes ... each day ... or not.   The process itself may be rewarding to engage in as it forces brutal self-honesty while translating mathematical phenomena into structures and algorithms.

I may be some kind of dork - that is, most all I program has to do with "mathematics" --- or even science/numerics; and I am anti-smart-apps.  That is, I am a command-line fanatic.  I prophesize that it is such command-line programs which will endure far longer than specialized GUI applications software, since the "Qualitates Occultae" of the mathematical phemomena is, especially using the highly abstract phenomena created with generic (STL/modern) C++, where the code may be made to look very similar to the mathematical notation in a textbook - that is, the command line programs will have a definite elegance to the programmer reading the code.   It will be reflected in how the program handles data fed into it.

So, the breakthrough seems to be about the I/O : { {}{}{} }, but the logic used is superior, and the code far easier to read with the details hidden in the Matrix and Fraction files.

This particular "animated version" is like forcing the "Ghosts of the Thoughts" behind the algorithm to show their "incarnations in a particular numerical form."

I hope all are coming to terms with mortality and keeping the demons (of hunger and need and want) that taunt us at bay.

Peace brothers.

(Hello to Raul and Silenus as well)
 ... I am sometimes able to print what is posted and then read when I am "away from distractions," so, when I get into the proper frame of mind, I may comment, but otherwise will mostly be lurking during this transitional phase where the Scholar and the Hungry Primate argue in the morning ... The body will trump the "silly rabbit" between my ears and will be on its knees praying for plants to grow - especially the squash - it goes down easy when I have no appetite for anything.

I have to say, a pessimist who plants cucumbers is an odd animal to be.  I hope to be shocked as usual one summer morning sucking the water out of that vegetable.

Now, if I could just grow tobacco and other smokable plants - I would immediately be swooped up and dragged away, forced to live on two slices of white bread, yellow cheese and bolonga ... New York State still goes after the Natives daring to trade tobacco with heavy military-grade armed force.

How can anyone ever aspire to be a writer!  I am always silencing myself.  Aspiring to be a writer would be like aspiring to have some kind of mental illness. 

_____________________
As for Makefiles, I usually compile directly on the command line, such as:

g++ -g -Wall fraction.cpp ch24_ex08_animated++.cpp -o gauss++

In any Windows environment, even with MinGW64/MSYS2 gcc installed, attach the .exe to the executable:

g++ -g -Wall fraction.cpp ch24_ex08_animated++.cpp -o gauss++.exe

_____________________________________________________

  I did use a Makefile for a project depending on several libraries of some graphics library.  I really do stubbornly prefer more fundamental and useful-to-me programs with, as Ibra suggests, as few dependencies as possible.  Also, I am a very selfish programmer with little interest in being useful to others - unless, of course, those others are trying to understand and appreciate what is hiding in the code, which is its "spirit" (?) its "inner nature," I really want to say, it's Qualitas Occulta -- or just potential for elegant approximations of numerical truth - or not.

  What is the nature of the code?  I mean, it is physical matter solidifying thought processes, so - quite philosophically heavy stuff taking place - which I guess we just take for granted.  Forgive me, I am primarily a madman, and only momentarily a coherent social organism communicating with this very precious (to Holden and I) old-school message board.

I like that Matrix.h is only like 8 pages, and MatrixIO.h only two.  Small with bare minimum in order to serve the purposes, no bloat, close to the bone.

Lean and mean, I suppose.   
I still have yet to add definitions for matrix multiplication to Matrix.h (something Stroustrup left out), but this is a "fun" part where one must refer to mathematics textbooks rather than programming textbooks - in order to define the operation, that is, what it means to multiply one matrix by another.  Before one can "program" that definition, one has to reflect upon it in one's own head first, preferably with the assistance of pen and scrap paper.     

PART II output from gauss++
__________________________________


[
 {  1  2  -1  -2  } | [ 3 ]
 {  0  0  4  1  } | [ 5 ]
 {  2  -1  0  1  } | [ 0 ]
 {  1  1  1  0  } | [ 2 ]
]

________*** press enter ***_____________________


Elimination with Partial Pivot:
[Swapped row 1 with pivot row 3]


[
 {  2  -1  0  1  } | [ 0 ]
 {  0  0  4  1  } | [ 5 ]
 {  1  2  -1  -2  } | [ 3 ]
 {  1  1  1  0  } | [ 2 ]
]

________*** press enter ***_____________________


[Partial Pivot:]
Multiplying row 1 by 0 and adding result to row 2

M(1, 0) = 0 + (0 * 2) = 0
M(1, 1) = 0 + (0 * -1) = 0
M(1, 2) = 4 + (0 * 0) = 4
M(1, 3) = 1 + (0 * 1) = 1
[STATE] multiplier: 0
 i = 1; j = 0
Then:

scale_and_add(A[0].slice(0),0, A[1].slice(0) ):

5 - 0*0 = 5 - 0 = 5
[Partial Pivot:]
Multiplying row 1 by -1/2 and adding result to row 3

M(2, 0) = 1 + (-1/2 * 2) = 0
M(2, 1) = 2 + (-1/2 * -1) = 5/2
M(2, 2) = -1 + (-1/2 * 0) = -1
M(2, 3) = -2 + (-1/2 * 1) = -5/2
[STATE] multiplier: 1/2
 i = 2; j = 0
Then:

scale_and_add(A[0].slice(0),-1/2, A[2].slice(0) ):

3 - 1/2*0 = 3 - 0 = 3
[Partial Pivot:]
Multiplying row 1 by -1/2 and adding result to row 4

M(3, 0) = 1 + (-1/2 * 2) = 0
M(3, 1) = 1 + (-1/2 * -1) = 3/2
M(3, 2) = 1 + (-1/2 * 0) = 1
M(3, 3) = 0 + (-1/2 * 1) = -1/2
[STATE] multiplier: 1/2
 i = 3; j = 0
Then:

scale_and_add(A[0].slice(0),-1/2, A[3].slice(0) ):

2 - 1/2*0 = 2 - 0 = 2
************ What is it ? ******************
[*** STATE ***]: [ A | b ] =

[
 {  2  -1  0  1  } | [ 0 ]
 {  0  0  4  1  } | [ 5 ]
 {  0  5/2  -1  -5/2  } | [ 3 ]
 {  0  3/2  1  -1/2  } | [ 2 ]
]

________*** press enter ***_____________________


[Swapped row 2 with pivot row 3]


[
 {  2  -1  0  1  } | [ 0 ]
 {  0  5/2  -1  -5/2  } | [ 3 ]
 {  0  0  4  1  } | [ 5 ]
 {  0  3/2  1  -1/2  } | [ 2 ]
]

________*** press enter ***_____________________


[Partial Pivot:]
Multiplying row 2 by 0 and adding result to row 3

M(2, 1) = 0 + (0 * 5/2) = 0
M(2, 2) = 4 + (0 * -1) = 4
M(2, 3) = 1 + (0 * -5/2) = 1
[STATE] multiplier: 0
 i = 2; j = 1
Then:

scale_and_add(A[1].slice(1),0, A[2].slice(1) ):

5 - 0*3 = 5 - 0 = 5
[Partial Pivot:]
Multiplying row 2 by -3/5 and adding result to row 4

M(3, 1) = 3/2 + (-3/5 * 5/2) = 0
M(3, 2) = 1 + (-3/5 * -1) = 8/5
M(3, 3) = -1/2 + (-3/5 * -5/2) = 1
[STATE] multiplier: 3/5
 i = 3; j = 1
Then:

scale_and_add(A[1].slice(1),-3/5, A[3].slice(1) ):

2 - 3/5*3 = 2 - 1.8 = 1/5
************ What is it ? ******************
[*** STATE ***]: [ A | b ] =

[
 {  2  -1  0  1  } | [ 0 ]
 {  0  5/2  -1  -5/2  } | [ 3 ]
 {  0  0  4  1  } | [ 5 ]
 {  0  0  8/5  1  } | [ 1/5 ]
]

________*** press enter ***_____________________


[Partial Pivot:]
Multiplying row 3 by -2/5 and adding result to row 4

M(3, 2) = 8/5 + (-2/5 * 4) = 0
M(3, 3) = 1 + (-2/5 * 1) = 3/5
[STATE] multiplier: 2/5
 i = 3; j = 2
Then:

scale_and_add(A[2].slice(2),-2/5, A[3].slice(2) ):

1/5 - 2/5*5 = 1/5 - 2 = -9/5
************ What is it ? ******************
[*** STATE ***]: [ A | b ] =

[
 {  2  -1  0  1  } | [ 0 ]
 {  0  5/2  -1  -5/2  } | [ 3 ]
 {  0  0  4  1  } | [ 5 ]
 {  0  0  0  3/5  } | [ -9/5 ]
]

________*** press enter ***_____________________


************ What is it ? ******************
[*** STATE ***]: [ A | b ] =

[
 {  2  -1  0  1  } | [ 0 ]
 {  0  5/2  -1  -5/2  } | [ 3 ]
 {  0  0  4  1  } | [ 5 ]
 {  0  0  0  3/5  } | [ -9/5 ]
]

________*** press enter ***_____________________

[backsub: n = 4; Now i = 3]|--->
[Row 4; pivot value: 3/5]

b(3): -9/5;  x(3) will become [-9/5  - (0)]/pivot = (-9/5)/(3/5)
So, x(3) = (-9/5 - (0))/(3/5) = -3 =?= s/m : -3

NEW x(3) = -3
Peek at x so far: {  0  0  0  -3  }
[
 {  2  -1  0  1  } | [ 0 ]
 {  0  5/2  -1  -5/2  } | [ 3 ]
 {  0  0  4  1  } | [ 5 ]
 {  0  0  0  1  } | [ -3 ]
]

________*** press enter ***_____________________

[backsub: n = 4; Now i = 2]|--->
[Row 3; pivot value: 4]
1 * -3 = -3| Total Sum = -3 |
*

b(2): 5;  x(2) will become [5  - (-3)]/pivot = (/(4)
So, x(2) = (5 - (-3))/(4) = 2 =?= s/m : 2


scale_and_add(A[2].slice(3), -1, A[2].slice(3) ):


M(2, 3) = 1 + (-1)*(1) = 1 + -1 = 0
NEW x(2) = 2
Peek at x so far: {  0  0  2  -3  }
[
 {  2  -1  0  1  } | [ 0 ]
 {  0  5/2  -1  -5/2  } | [ 3 ]
 {  0  0  1  0  } | [ 2 ]
 {  0  0  0  1  } | [ -3 ]
]

________*** press enter ***_____________________

[backsub: n = 4; Now i = 1]|--->
[Row 2; pivot value: 5/2]
-1 * 2 = -2| Total Sum = -2 |
*
-5/2 * -3 = 15/2| Total Sum = 11/2 |
*

b(1): 3;  x(1) will become [3  - (11/2)]/pivot = (-5/2)/(5/2)
So, x(1) = (3 - (11/2))/(5/2) = -1 =?= s/m : -1


scale_and_add(A[1].slice(2), -1, A[1].slice(2) ):


M(1, 2) = -1 + (1)*(1) = -1 + 1 = 0
M(1, 3) = -5/2 + (5/2)*(1) = -5/2 + 5/2 = 0
NEW x(1) = -1
Peek at x so far: {  0  -1  2  -3  }
[
 {  2  -1  0  1  } | [ 0 ]
 {  0  1  0  0  } | [ -1 ]
 {  0  0  1  0  } | [ 2 ]
 {  0  0  0  1  } | [ -3 ]
]

________*** press enter ***_____________________

[backsub: n = 4; Now i = 0]|--->
[Row 1; pivot value: 2]
-1 * -1 = 1| Total Sum = 1 |
*
0 * 2 = 0| Total Sum = 1 |
*
1 * -3 = -3| Total Sum = -2 |
*

b(0): 0;  x(0) will become [0  - (-2)]/pivot = (2)/(2)
So, x(0) = (0 - (-2))/(2) = 1 =?= s/m : 1


scale_and_add(A[0].slice(1), -1, A[0].slice(1) ):


M(0, 1) = -1 + (1)*(1) = -1 + 1 = 0
M(0, 2) = 0 + (0)*(1) = 0 + 0 = 0
M(0, 3) = 1 + (-1)*(1) = 1 + -1 = 0
NEW x(0) = 1
Peek at x so far: {  1  -1  2  -3  }
[
 {  1  0  0  0  } | [ 1 ]
 {  0  1  0  0  } | [ -1 ]
 {  0  0  1  0  } | [ 2 ]
 {  0  0  0  1  } | [ -3 ]
]

________*** press enter ***_____________________


Pivot elimination solution is x = {  1  -1  2  -3  }
A*x = {  3  5  0  2  }


FLOATING-POINT ("real") representation:
________________________________

x = {  1  -1  2  -3  }
________________________________
PS C:\Users\mwh>