Complete Pivoting VS Partial Pivoting in Gauss Elimination
I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)
linear-algebra matrices numerical-methods gaussian-elimination
asked Nov 18 '15 at 15:51
Arbo94
234
add a comment |
I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)
linear-algebra matrices numerical-methods gaussian-elimination
asked Nov 18 '15 at 15:51
Arbo94
234
Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
– Arbo94
Nov 18 '15 at 18:03
If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
– JP McCarthy
Nov 18 '15 at 18:06
add a comment |
I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)
linear-algebra matrices numerical-methods gaussian-elimination
asked Nov 18 '15 at 15:51
Arbo94
234
I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)
linear-algebra matrices numerical-methods gaussian-elimination
linear-algebra matrices numerical-methods gaussian-elimination
asked Nov 18 '15 at 15:51
Arbo94
234
asked Nov 18 '15 at 15:51
Arbo94
234
edited Nov 18 '15 at 15:54
asked Nov 18 '15 at 15:51
Arbo94
234
asked Nov 18 '15 at 15:51
Arbo94
234
asked Nov 18 '15 at 15:51
Arbo94
234
234
Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
– Arbo94
Nov 18 '15 at 18:03
If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
– JP McCarthy
Nov 18 '15 at 18:06
add a comment |
Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
– Arbo94
Nov 18 '15 at 18:03
If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
– JP McCarthy
Nov 18 '15 at 18:06
Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
– Arbo94
Nov 18 '15 at 18:03
Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
– Arbo94
Nov 18 '15 at 18:03
If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
– JP McCarthy
Nov 18 '15 at 18:06
If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
– JP McCarthy
Nov 18 '15 at 18:06
add a comment |
2 Answers
2
active
oldest
votes
Gaussian Elimination can be used as long as you are not using decimal rounding.
If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.
I don't know without a Google when complete pivoting is necessary.
answered Nov 18 '15 at 16:36
JP McCarthy
5,61712440
Thank you, but I didn't understand what you meant by rounding?
– Arbo94
Nov 18 '15 at 16:51
If instead of $1/3$ you use $0.333$ you are rounding.
– JP McCarthy
Nov 18 '15 at 16:53
ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
– Arbo94
Nov 18 '15 at 17:04
Are YOU going to use rounding in your row operations?
– JP McCarthy
Nov 18 '15 at 17:05
No, I'm not. because I want to run a matlab code.
– Arbo94
Nov 18 '15 at 17:13
|
show 3 more comments
I have a hard time understanding that when and under what conditions
we can use Gauss elimination with complete pivoting, and when with
partial pivoting, and when with no pivoting? (I mean what is the exact
feature of a matrix that will tell us which one to choose?)
My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.
That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.
You won't be able to tell immediately but if you solve
$$ Ax=b tag{1} $$
$$ LUx =b tag{2}$$
For partial pivoting you'd have
$$ PA = LU implies P^{'}LUx b tag{3}$$
then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm
$$ | hat{x} - x | tag{4}$$
where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.
$$ frac{| hat{x} - x |}{|x|} tag{5}$$
answered Oct 20 at 1:25
Ryan Howe
2,41911323
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Gaussian Elimination can be used as long as you are not using decimal rounding.
If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.
I don't know without a Google when complete pivoting is necessary.
answered Nov 18 '15 at 16:36
JP McCarthy
5,61712440
Thank you, but I didn't understand what you meant by rounding?
– Arbo94
Nov 18 '15 at 16:51
If instead of $1/3$ you use $0.333$ you are rounding.
– JP McCarthy
Nov 18 '15 at 16:53
ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
– Arbo94
Nov 18 '15 at 17:04
Are YOU going to use rounding in your row operations?
– JP McCarthy
Nov 18 '15 at 17:05
No, I'm not. because I want to run a matlab code.
– Arbo94
Nov 18 '15 at 17:13
|
show 3 more comments
Gaussian Elimination can be used as long as you are not using decimal rounding.
If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.
I don't know without a Google when complete pivoting is necessary.
answered Nov 18 '15 at 16:36
JP McCarthy
5,61712440
Thank you, but I didn't understand what you meant by rounding?
– Arbo94
Nov 18 '15 at 16:51
If instead of $1/3$ you use $0.333$ you are rounding.
– JP McCarthy
Nov 18 '15 at 16:53
ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
– Arbo94
Nov 18 '15 at 17:04
Are YOU going to use rounding in your row operations?
– JP McCarthy
Nov 18 '15 at 17:05
No, I'm not. because I want to run a matlab code.
– Arbo94
Nov 18 '15 at 17:13
|
show 3 more comments
Gaussian Elimination can be used as long as you are not using decimal rounding.
If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.
I don't know without a Google when complete pivoting is necessary.
answered Nov 18 '15 at 16:36
JP McCarthy
5,61712440
Gaussian Elimination can be used as long as you are not using decimal rounding.
If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.
I don't know without a Google when complete pivoting is necessary.
answered Nov 18 '15 at 16:36
JP McCarthy
5,61712440
answered Nov 18 '15 at 16:36
JP McCarthy
5,61712440
answered Nov 18 '15 at 16:36
JP McCarthy
5,61712440
answered Nov 18 '15 at 16:36
JP McCarthy
5,61712440
5,61712440
Thank you, but I didn't understand what you meant by rounding?
– Arbo94
Nov 18 '15 at 16:51
If instead of $1/3$ you use $0.333$ you are rounding.
– JP McCarthy
Nov 18 '15 at 16:53
ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
– Arbo94
Nov 18 '15 at 17:04
Are YOU going to use rounding in your row operations?
– JP McCarthy
Nov 18 '15 at 17:05
No, I'm not. because I want to run a matlab code.
– Arbo94
Nov 18 '15 at 17:13
|
show 3 more comments
Thank you, but I didn't understand what you meant by rounding?
– Arbo94
Nov 18 '15 at 16:51
If instead of $1/3$ you use $0.333$ you are rounding.
– JP McCarthy
Nov 18 '15 at 16:53
ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
– Arbo94
Nov 18 '15 at 17:04
Are YOU going to use rounding in your row operations?
– JP McCarthy
Nov 18 '15 at 17:05
No, I'm not. because I want to run a matlab code.
– Arbo94
Nov 18 '15 at 17:13
Thank you, but I didn't understand what you meant by rounding?
– Arbo94
Nov 18 '15 at 16:51
Thank you, but I didn't understand what you meant by rounding?
– Arbo94
Nov 18 '15 at 16:51
If instead of $1/3$ you use $0.333$ you are rounding.
– JP McCarthy
Nov 18 '15 at 16:53
If instead of $1/3$ you use $0.333$ you are rounding.
– JP McCarthy
Nov 18 '15 at 16:53
ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
– Arbo94
Nov 18 '15 at 17:04
ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
– Arbo94
Nov 18 '15 at 17:04
Are YOU going to use rounding in your row operations?
– JP McCarthy
Nov 18 '15 at 17:05
Are YOU going to use rounding in your row operations?
– JP McCarthy
Nov 18 '15 at 17:05
No, I'm not. because I want to run a matlab code.
– Arbo94
Nov 18 '15 at 17:13
No, I'm not. because I want to run a matlab code.
– Arbo94
Nov 18 '15 at 17:13
|
show 3 more comments
I have a hard time understanding that when and under what conditions
we can use Gauss elimination with complete pivoting, and when with
partial pivoting, and when with no pivoting? (I mean what is the exact
feature of a matrix that will tell us which one to choose?)
My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.
That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.
You won't be able to tell immediately but if you solve
$$ Ax=b tag{1} $$
$$ LUx =b tag{2}$$
For partial pivoting you'd have
$$ PA = LU implies P^{'}LUx b tag{3}$$
then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm
$$ | hat{x} - x | tag{4}$$
where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.
$$ frac{| hat{x} - x |}{|x|} tag{5}$$
answered Oct 20 at 1:25
Ryan Howe
2,41911323
add a comment |
I have a hard time understanding that when and under what conditions
we can use Gauss elimination with complete pivoting, and when with
partial pivoting, and when with no pivoting? (I mean what is the exact
feature of a matrix that will tell us which one to choose?)
My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.
That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.
You won't be able to tell immediately but if you solve
$$ Ax=b tag{1} $$
$$ LUx =b tag{2}$$
For partial pivoting you'd have
$$ PA = LU implies P^{'}LUx b tag{3}$$
then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm
$$ | hat{x} - x | tag{4}$$
where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.
$$ frac{| hat{x} - x |}{|x|} tag{5}$$
answered Oct 20 at 1:25
Ryan Howe
2,41911323
add a comment |
I have a hard time understanding that when and under what conditions
we can use Gauss elimination with complete pivoting, and when with
partial pivoting, and when with no pivoting? (I mean what is the exact
feature of a matrix that will tell us which one to choose?)
My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.
That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.
You won't be able to tell immediately but if you solve
$$ Ax=b tag{1} $$
$$ LUx =b tag{2}$$
For partial pivoting you'd have
$$ PA = LU implies P^{'}LUx b tag{3}$$
then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm
$$ | hat{x} - x | tag{4}$$
where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.
$$ frac{| hat{x} - x |}{|x|} tag{5}$$
answered Oct 20 at 1:25
Ryan Howe
2,41911323
I have a hard time understanding that when and under what conditions
we can use Gauss elimination with complete pivoting, and when with
partial pivoting, and when with no pivoting? (I mean what is the exact
feature of a matrix that will tell us which one to choose?)
My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.
That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.
You won't be able to tell immediately but if you solve
$$ Ax=b tag{1} $$
$$ LUx =b tag{2}$$
For partial pivoting you'd have
$$ PA = LU implies P^{'}LUx b tag{3}$$
then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm
$$ | hat{x} - x | tag{4}$$
where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.
$$ frac{| hat{x} - x |}{|x|} tag{5}$$
answered Oct 20 at 1:25
Ryan Howe
2,41911323
edited Oct 20 at 2:57
answered Oct 20 at 1:25
Ryan Howe
2,41911323
answered Oct 20 at 1:25
Ryan Howe
2,41911323
answered Oct 20 at 1:25
Ryan Howe
2,41911323
2,41911323
add a comment |
add a comment |
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Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
– Arbo94
Nov 18 '15 at 18:03
If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
– JP McCarthy
Nov 18 '15 at 18:06