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Talk:Reduced row echelon form (view source)

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=="Break"SWAP vs. "returnRows" bug==
 
The algorithm has a Bug.<br>
It does NOT do the SWAP. No conditions are checked
 
It does NOT workdo forthe thisSWAP. exampleNo matricconditions are checked<br>
 
Solve the following system of equations.
3x+y− 4z=−1
x +10z= 5
4x+y+ 6z= 1
Solution. The corresponding augmented matrix is
3 1 −4 −1
1 0 10 5
4 1 6 1
Create the first leading one by interchanging rows 1 and 2
1 0 10 5
3 1 −4 −1
4 1 6 1
Now subtract 3 times row 1 from row 2, and subtract 4 times row 1 from row 3. The result is
1 0 10 5
0 1 −34 −16
0 1 −34 −19
Now subtract row 2 from row 3 to obtain
1 0 10 5
0 1 −34 −16
0 0 0 −3
This means that the following reduced system of equations
x +10z= 5
y−34z=−16
0= −3
 
It does NOT work for this example matrix<br>
is equivalent to the original system. In other words, the two have the same solutions. But this last
system clearly has no solution (the last equation requires that x, y and z satisfy 0x+0y+0z = −3,
and no such numbers exist). Hence the original system has no solution.
 
 
Solve the following system of equations.<br>
 
3x+y− 4z=−1<br>
 
x +10z= 5<br>
 
4x+y+ 6z= 1<br>
 
Solution. The corresponding augmented matrix is<br>
 
3 1 −4 −1<br>
 
1 0 10 5<br>
 
4 1 6 1<br>
 
Create the first leading one by interchanging rows 1 and 2<br>
 
1 0 10 5<br>
 
3 1 −4 −1<br>
 
4 1 6 1<br>
 
Now subtract 3 times row 1 from row 2, and subtract 4 times row 1 from row 3. The result is<br>
 
1 0 10 5<br>
 
0 1 −34 −16<br>
 
0 1 −34 −19<br>
 
Now subtract row 2 from row 3 to obtain<br>
 
1 0 10 5<br>
 
0 1 −34 −16<br>
 
0 0 0 −3<br>
 
This means that the following reduced system of equations<br>
 
x +10z= 5<br>
 
y−34z=−16<br>
 
0= −3<br>
 
 
is equivalent to the original system. In other words, the two have the same solutions. But this last<br>
 
system clearly has no solution (the last equation requires that x, y and z satisfy 0x+0y+0z = −3,<br>
 
and no such numbers exist). Hence the original system has no solution.<br> [[User:Umariani|Umariani]]
 
:The algorithm does not have a bug. Failure to check conditions is not part of the algorithm, it is part of the implementation. That is just a degenerate case.
 
:A few points:
:# The task does not require checking for degenerate cases.
:# Even if it did, you gave no indication of which implementation fails on degenerate cases.
:# What should the implementation do in the case of a "failure"? Error message? Warning? Or just precede as far as possible? (Which is what the [[Reduced_row_echelon_form#Raku|Raku]] implementation does.)
 
: ''By the way, please sign your discussion page edits.'' --[[User:Thundergnat|Thundergnat]] ([[User talk:Thundergnat|talk]]) 12:38, 20 July 2023 (UTC)
 
:: The algorithm is useful for solving a system of linear equations, but a reduced row echelon form can be made even if there is no such solution! The reduced form of the system
<pre>
3 1 -4 -1
1 0 10 5
4 1 6 1
 
is
 
1 0 10 0
0 1 -34 0
0 0 0 1
</pre>
--[[User:Wherrera|Wherrera]] ([[User talk:Wherrera|talk]]) 20:14, 20 July 2023 (UTC)
 
=="Break" vs. "return" bug==
Wherrera
4,105

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