How to solvle this 4th system of equations.

ArthurHuang · 06-10-2006, 01:31 PM

hi guys, I am solving a 4 system of equations like this.

-36 4 4 4 X1 -40
36 32 4 4 X2 -4
4 32 -28 4 X3 = -4
4 4 28 -24 X4 -4

I search but don't find calculator online. Is there any
body know any sites can calculate it ?

Thanks a lot!

Nylex · 06-10-2006, 01:34 PM

Do it by hand? You can use the normal way of solving simultaneous equations, but there's also a way to do it using matrices (though I can't remember what it is right now).

Why is this in programming? :/

ArthurHuang · 06-10-2006, 01:40 PM

-;( do you have a calculator?..

ppanyam · 06-11-2006, 06:26 AM

The equations do look weird. Where is the "equality" symbol ie '=' in the above equations? I found only one.. What calculator are you talking about?

ArthurHuang · 06-11-2006, 08:18 PM

It hase been solved. I think some calculators has functions of solving SOE.

adilturbo · 06-11-2006, 09:18 PM

try to clarify ur system.

u can solve it by the GAUSS elemination method

QMario · 06-12-2006, 12:31 PM

Quote:

Originally Posted by ArthurHuang

hi guys, I am solving a 4 system of equations like this.

-36 4 4 4 X1 -40
36 32 4 4 X2 -4
4 32 -28 4 X3 = -4
4 4 28 -24 X4 -4

I search but don't find calculator online. Is there any
body know any sites can calculate it ?

Thanks a lot!

Use augmented matrices.

[-36 4 4 4 X1 |-40]
[ 36 32 4 4 X2 |-4 ]
[ 4 32 -28 4 X3 |-4 ]
[ 4 4 28 -24 X4 |-4 ]

Then use the RREF function on the matrix.
RREF - reduced row echelon form

Nylex · 06-12-2006, 12:40 PM

You can also use Cramer's rule and do it with determinants. I've never learnt about augmented matrices, so don't know about those.

elyk1212 · 06-12-2006, 01:19 PM

Yeah, use augmented matrices. You just reduce to reduced Row echelon from. A graphing calculator can also help solve this equation for you, but keeping track of matrix variables (prevents mistakes) and eases doing row operations.

Look up 'linear programming' on Google, and 'solving systems of linear equations'.

Quote:

Why is this in programming? :/

Likely because it is considered linear programming, and they did not see the context, but I do not know for sure.

Quote:

You can also use Cramer's rule and do it with determinants. I've never learnt about augmented matrices, so don't know about those.

An augmented matrix, is one that is just separated, typically to represent equality on one side. Cramer's rule and determinants of matrices are actually more advanced topics than augmented matrices

By the way, I am also curious where the equality symbols are here (only one as mentioned). Am I to assume the others are equal to zero, or that the last number in each equation is the Right Hand Side?

Code:

-36 4 4 4 X1 -40
36 32 4 4 X2 -4
4 32 -28 4 X3 = -4
4 4 28 -24 X4 -4

You should place the entire system of equations on here, as is.

elyk1212 · 06-12-2006, 01:34 PM

Reduced Row Echelon example (giving answers). Say each row is X1,X2,X3, then given answer matrix:

Code:

[1 0 0 | A]
[0 1 0 | B]
[0 0 1 | C]

X1 = A
X2 = B
X3 = C

But in your example, with 5 variables, and 4 equations, you will not get full answers, you will get answers in terms of themselves, as yours in an underdetermined system. Try it out and see!

osor · 06-12-2006, 02:02 PM

Quote:

Originally Posted by Nylex

You can also use Cramer's rule and do it with determinants. I've never learnt about augmented matrices, so don't know about those.

Augmented matrices is just a shorthand way of using Gaussian elimination (which, by the way, is much less computationally intensive than Cramer's rule).

adilturbo · 06-12-2006, 05:30 PM

i want just to know if ur system like this A*x=b

Code:


	Quote:
	
	
		
			
				
        -36  4  4   4           |-40]     |x1]
     A=  36 32  4   4        b= | -4]  x= |x2]
          4 32 -28  4           | -4]     |x3]
          4  4  28 -24          | -4]     |x4]

if it is the case so:

now the new A'=A|b
will be :

Code:


	Quote:
	
	
		
			
				
         36  4  4   4 |-40
     A'= 36 32  4   4 | -4
          4 32 -28  4 | -4
          4  4  28 -24| -4

ok try to use that algo :
note: it is an algo without pivoting!!!!!!!!

Code:


	Quote:
	
	
		
			
				     function A=MEG(A,b)
       [n,n] = size(A);
         for k=1:n-1
               for i=k+1:n
                    m=A(i,k)/A(k,k) ;           
                    for j = k :n
                         A(i,j)=A(i,j)-m *A(k,j);
                         b(i)=b(i)-m*b(k);
                    end
                end
          end

when ur A will be U which is an upper triangular matrice use that algo to find ur x vector:

Code:


	Quote:
	
	
		
			
				    function[x]=subs_retro(A,b) 
        [n,n]=size(A);
         x=zeros(n,1);
         x(n)=b(n)/A(n,n);//b here is the new b 
         for i=n-1:-1:1
              x(i)=(b(i)-A(i,i+1:n)*x(i+1:n))/A(i,i);//b here is the new b 
         end

firstfire · 06-17-2006, 02:49 PM

You can use `axiom':
-> a := matrix[[-36,4,4,4],[36,32,4,4],[4,32,-28,4],[4,4,28,-24]]
-> b := [-40,-4,-4,-4]
-> solve(a, b)
...

result: x = [5/6, 6/7, 5/6, 17/21]

ArthurHuang · 06-18-2006, 12:23 PM

This axiom looks really cool. Now I have a 12X8 matrix of SOE, can I use it ?
Btw: is it a software or library?

Quote:

Originally Posted by firstfire

You can use `axiom':
-> a := matrix[[-36,4,4,4],[36,32,4,4],[4,32,-28,4],[4,4,28,-24]]
-> b := [-40,-4,-4,-4]
-> solve(a, b)
...

result: x = [5/6, 6/7, 5/6, 17/21]

firstfire · 06-19-2006, 02:00 AM

Hi all.
Yes, you can use it for such SOE. If I understand correctly, 12x8 SOE have an infinite set of solutions -- linear space. Axiom give you the basis of this space and a partial solution. Any linear function of basis-vectors will be a solution of uniform equation A*x=0. Any solution of nonuniform SOE A*x=b looks like x = x_part + c*x_base.
Example:

Code:

-> a:= matrix [[1/(i+j) for i in 1..12] for j in 1..8]
(4) -> 
        +1  1   1   1   1   1   1   1    1   1   1   1+
        |-  -   -   -   -   -   -   -   --  --  --  --|
        |2  3   4   5   6   7   8   9   10  11  12  13|
        |                                              |
        |1  1   1   1   1   1   1    1   1   1   1   1|
        |-  -   -   -   -   -   -   --  --  --  --  --|
        |3  4   5   6   7   8   9   10  11  12  13  14|
        |                                              |
        |1  1   1   1   1   1    1   1   1   1   1   1|
        |-  -   -   -   -   -   --  --  --  --  --  --|
        |4  5   6   7   8   9   10  11  12  13  14  15|
        |                                              |
        |1  1   1   1   1    1   1   1   1   1   1   1|
        |-  -   -   -   -   --  --  --  --  --  --  --|
        |5  6   7   8   9   10  11  12  13  14  15  16|
   (4)  |                                              |
        |1  1   1   1    1   1   1   1   1   1   1   1|
        |-  -   -   -   --  --  --  --  --  --  --  --|
        |6  7   8   9   10  11  12  13  14  15  16  17|
        |                                              |
        |1  1   1    1   1   1   1   1   1   1   1   1|
        |-  -   -   --  --  --  --  --  --  --  --  --|
        |7  8   9   10  11  12  13  14  15  16  17  18|
        |                                              |
        |1  1    1   1   1   1   1   1   1   1   1   1|
        |-  -   --  --  --  --  --  --  --  --  --  --|
        |8  9   10  11  12  13  14  15  16  17  18  19|
        |                                              |
        |1   1   1   1   1   1   1   1   1   1   1   1|
        |-  --  --  --  --  --  --  --  --  --  --  --|
        +9  10  11  12  13  14  15  16  17  18  19  20+
                                                Type: Matrix Fraction Integer

(7) -> b:= [i for i in 1..8]     
(7) -> 
   (7)  [1,2,3,4,5,6,7,8]
                                                   Type: List PositiveInteger
(8) -> s:= solve (a,b)          
(8) -> 
   (8)
   [
     particular =
       [- 5112, 176400, - 1912680, 9424800, - 24144120, 33297264, - 23423400,
        6589440, 0, 0, 0, 0]
     ,

     basis =
           9      36   42   252 63   588 126   72
       [[-----,- ----,---,- ---,--,- ---,---,- --,1,0,0,0],
         24310   2431 221   221 17    85  17   17
           4     315 180   1050 252   441 420   180
        [----,- ----,---,- ----,---,- ---,---,- ---,0,1,0,0],
         2431   4862 221    221  17    17  17    17
          18     700 17325   49500 11550   19404 17325   6600
        [----,- ----,-----,- -----,-----,- -----,-----,- ----,0,0,1,0],
         4199   4199  8398    4199  323     323   323     323
          36    1386 16940   190575 21780   35574 30492   10890
        [----,- ----,-----,- ------,-----,- -----,-----,- -----,0,0,0,1]]
         4199   4199  4199    8398   323     323   323     323
     ]
Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer)

(16) -> -- test (comment)
(16) -> a*s.particular
(16) -> 
   (16)  [1,2,3,4,5,6,7,8]
                                                Type: Vector Fraction Integer

(17) -> a*s.basis.1   
(17) -> 
   (17)  [0,0,0,0,0,0,0,0]
                                                Type: Vector Fraction Integer

(18) -> a*(s.particular + 100 * s.basis.1)
(18) -> 
   (18)  [1,2,3,4,5,6,7,8]
                                                Type: Vector Fraction Integer

I think you interested in partial solution. Axiom is a free (at this moment, I meant it was proprietary, but released as free) software, not library.