Gaussian elimination with partial pivoting
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GEPP( Gaussian elimination with partial pivoting) is a method used to solve Ax=b.
Let us assume that A is non singular matrix and GEPP is backward stable.
Let x* be the approximate solution computed with GEPP and x is the exact solution.
I want to give a bound for the error in the solution.
Here is my attempt, since GEPP is backward stable, then we have:
||x*-x||/||x|| = O(eps machine), then ||x*-x||=O(||x|| eps machine).
Since x=inv(A)b, then ||x-x||=O(||inv(A)*b|| eps machine) < O(||b||/sigma m eps machine)
where sigma m is the smallest singular value of A.
Does his seem correct bound ?
numerical-linear-algebra
asked Aug 8 at 19:22
Mike
414
add a comment |Â
up vote
0
down vote
favorite
GEPP( Gaussian elimination with partial pivoting) is a method used to solve Ax=b.
Let us assume that A is non singular matrix and GEPP is backward stable.
Let x* be the approximate solution computed with GEPP and x is the exact solution.
I want to give a bound for the error in the solution.
Here is my attempt, since GEPP is backward stable, then we have:
||x*-x||/||x|| = O(eps machine), then ||x*-x||=O(||x|| eps machine).
Since x=inv(A)b, then ||x-x||=O(||inv(A)*b|| eps machine) < O(||b||/sigma m eps machine)
where sigma m is the smallest singular value of A.
Does his seem correct bound ?
numerical-linear-algebra
asked Aug 8 at 19:22
Mike
414
An approximate solution $x^*$ of $Ax=b$ is backward stable, if $(A+E)x^*=b+f$ such that $|E|/|A|=mathcalO(epsilon_mathrmmach)$ and $|f|/|b|=mathcalO(epsilon_mathrmmach)$ in some suitable norms.
– Algebraic Pavel
Aug 10 at 11:25
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
GEPP( Gaussian elimination with partial pivoting) is a method used to solve Ax=b.
Let us assume that A is non singular matrix and GEPP is backward stable.
Let x* be the approximate solution computed with GEPP and x is the exact solution.
I want to give a bound for the error in the solution.
Here is my attempt, since GEPP is backward stable, then we have:
||x*-x||/||x|| = O(eps machine), then ||x*-x||=O(||x|| eps machine).
Since x=inv(A)b, then ||x-x||=O(||inv(A)*b|| eps machine) < O(||b||/sigma m eps machine)
where sigma m is the smallest singular value of A.
Does his seem correct bound ?
numerical-linear-algebra
asked Aug 8 at 19:22
Mike
414
GEPP( Gaussian elimination with partial pivoting) is a method used to solve Ax=b.
Let us assume that A is non singular matrix and GEPP is backward stable.
Let x* be the approximate solution computed with GEPP and x is the exact solution.
I want to give a bound for the error in the solution.
Here is my attempt, since GEPP is backward stable, then we have:
||x*-x||/||x|| = O(eps machine), then ||x*-x||=O(||x|| eps machine).
Since x=inv(A)b, then ||x-x||=O(||inv(A)*b|| eps machine) < O(||b||/sigma m eps machine)
where sigma m is the smallest singular value of A.
Does his seem correct bound ?
numerical-linear-algebra
asked Aug 8 at 19:22
Mike
414
asked Aug 8 at 19:22
Mike
414
asked Aug 8 at 19:22
Mike
414
asked Aug 8 at 19:22
Mike
414
414
An approximate solution $x^*$ of $Ax=b$ is backward stable, if $(A+E)x^*=b+f$ such that $|E|/|A|=mathcalO(epsilon_mathrmmach)$ and $|f|/|b|=mathcalO(epsilon_mathrmmach)$ in some suitable norms.
– Algebraic Pavel
Aug 10 at 11:25
add a comment |Â
An approximate solution $x^*$ of $Ax=b$ is backward stable, if $(A+E)x^*=b+f$ such that $|E|/|A|=mathcalO(epsilon_mathrmmach)$ and $|f|/|b|=mathcalO(epsilon_mathrmmach)$ in some suitable norms.
– Algebraic Pavel
Aug 10 at 11:25
An approximate solution $x^*$ of $Ax=b$ is backward stable, if $(A+E)x^*=b+f$ such that $|E|/|A|=mathcalO(epsilon_mathrmmach)$ and $|f|/|b|=mathcalO(epsilon_mathrmmach)$ in some suitable norms.
– Algebraic Pavel
Aug 10 at 11:25
An approximate solution $x^*$ of $Ax=b$ is backward stable, if $(A+E)x^*=b+f$ such that $|E|/|A|=mathcalO(epsilon_mathrmmach)$ and $|f|/|b|=mathcalO(epsilon_mathrmmach)$ in some suitable norms.
– Algebraic Pavel
Aug 10 at 11:25
add a comment |Â
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An approximate solution $x^*$ of $Ax=b$ is backward stable, if $(A+E)x^*=b+f$ such that $|E|/|A|=mathcalO(epsilon_mathrmmach)$ and $|f|/|b|=mathcalO(epsilon_mathrmmach)$ in some suitable norms.
– Algebraic Pavel
Aug 10 at 11:25