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Right-preconditioning and fixed point linear iterations

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Given a linear system $Atextbfx=textbfb$, we can express it into the easier-to-solve right-preconditioned form:

$$ AM^-1textbfy=textbfb, quad textbfy= M^-1textbfx $$

On the other hand, the left-preconditioned system is:

$$ M^-1Atextbfx=M^-1textbfb $$

From the literature we know it's easy to show that the left-preconditioned linear system above is equivalent to a fixed-point linear iteration as:

$$ textbfx^(k+1)=(I-M^-1A)textbfx^(k)+M^-1textbfb $$

Or equivalently:

$$ textbfx^(k+1)=textbfx^(k) + M^-1textbfr^(k) $$

My question is: how can I get an update scheme like those starting from the right-preconditioning formulation?

fixed-point-theorems

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asked 16 hours ago

S. G. RinaldoS. G. Rinaldo

1

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S. G. Rinaldo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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0

$begingroup$

Given a linear system $Atextbfx=textbfb$, we can express it into the easier-to-solve right-preconditioned form:

$$ AM^-1textbfy=textbfb, quad textbfy= M^-1textbfx $$

On the other hand, the left-preconditioned system is:

$$ M^-1Atextbfx=M^-1textbfb $$

From the literature we know it's easy to show that the left-preconditioned linear system above is equivalent to a fixed-point linear iteration as:

$$ textbfx^(k+1)=(I-M^-1A)textbfx^(k)+M^-1textbfb $$

Or equivalently:

$$ textbfx^(k+1)=textbfx^(k) + M^-1textbfr^(k) $$

My question is: how can I get an update scheme like those starting from the right-preconditioning formulation?

fixed-point-theorems

share|cite|improve this question

asked 16 hours ago

S. G. RinaldoS. G. Rinaldo

1

New contributor

S. G. Rinaldo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

Given a linear system $Atextbfx=textbfb$, we can express it into the easier-to-solve right-preconditioned form:

$$ AM^-1textbfy=textbfb, quad textbfy= M^-1textbfx $$

On the other hand, the left-preconditioned system is:

$$ M^-1Atextbfx=M^-1textbfb $$

From the literature we know it's easy to show that the left-preconditioned linear system above is equivalent to a fixed-point linear iteration as:

$$ textbfx^(k+1)=(I-M^-1A)textbfx^(k)+M^-1textbfb $$

Or equivalently:

$$ textbfx^(k+1)=textbfx^(k) + M^-1textbfr^(k) $$

My question is: how can I get an update scheme like those starting from the right-preconditioning formulation?

fixed-point-theorems

share|cite|improve this question

asked 16 hours ago

S. G. RinaldoS. G. Rinaldo

1

New contributor

S. G. Rinaldo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked 16 hours ago

S. G. RinaldoS. G. Rinaldo

1

New contributor

S. G. Rinaldo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked 16 hours ago

S. G. RinaldoS. G. Rinaldo

1

New contributor

S. G. Rinaldo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 16 hours ago

S. G. RinaldoS. G. Rinaldo

1

New contributor

S. G. Rinaldo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 16 hours ago

S. G. RinaldoS. G. Rinaldo

1