Mixing
Are there algorithms for solving a sequence of sparse linear systems of the form $(x_iI+B)y_i = b$?
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I am struggling to solve a sequence of linear systems of the form
$$(x_i I + B ) y_i = b, quad i = 1,2,dotsc,$$
where $I$ is identity, $x_i$ is a real number and $B$ is sparse. Both the matrix $B$ and the right-hand side $b$ are independent of $x_i$. Both $B$ and $b$ are complex.
Iterative methods are inapplicable here because
$$||B^{-1}|| ,,,, text{sup}_i ,,, |x_{i+1} - x_i| gg 1. $$
Can I exploit the fact that all matrices have the same sparsity pattern?
numerical-methods numerical-linear-algebra
edited Jan 30 at 8:31
Carl Christian
5,9511723
asked Jan 29 at 20:35
IndianIndian
1
$endgroup$
add a comment |
$begingroup$
I am struggling to solve a sequence of linear systems of the form
$$(x_i I + B ) y_i = b, quad i = 1,2,dotsc,$$
where $I$ is identity, $x_i$ is a real number and $B$ is sparse. Both the matrix $B$ and the right-hand side $b$ are independent of $x_i$. Both $B$ and $b$ are complex.
Iterative methods are inapplicable here because
$$||B^{-1}|| ,,,, text{sup}_i ,,, |x_{i+1} - x_i| gg 1. $$
Can I exploit the fact that all matrices have the same sparsity pattern?
numerical-methods numerical-linear-algebra
edited Jan 30 at 8:31
Carl Christian
5,9511723
asked Jan 29 at 20:35
IndianIndian
1
$endgroup$
$begingroup$
A number of iterative methods are applicable to your problem. You should explain which methods are excluded by your condition.
$endgroup$
– Carl Christian
Jan 29 at 20:42
$begingroup$
I have formated your question and made some small changes. Please check carefully that nothing has been lost. You can roll back the changes.
$endgroup$
– Carl Christian
Jan 30 at 8:33
$begingroup$
Can’t you solve it as $B y_i = b - x_i$ with a direct solver? Then you’ll factor the matrix only once and then solve the problem with multiple righthand sides?
$endgroup$
– VorKir
Feb 3 at 6:02
$begingroup$
O, sorry, I misread the equation
$endgroup$
– VorKir
Feb 3 at 17:56
$begingroup$
Why do you think a Jacobi preconditioner won't work? It looks like the diagonal is much larger than the spectrum of $B$?
$endgroup$
– VorKir
Feb 3 at 21:36
add a comment |
$begingroup$
I am struggling to solve a sequence of linear systems of the form
$$(x_i I + B ) y_i = b, quad i = 1,2,dotsc,$$
where $I$ is identity, $x_i$ is a real number and $B$ is sparse. Both the matrix $B$ and the right-hand side $b$ are independent of $x_i$. Both $B$ and $b$ are complex.
Iterative methods are inapplicable here because
$$||B^{-1}|| ,,,, text{sup}_i ,,, |x_{i+1} - x_i| gg 1. $$
Can I exploit the fact that all matrices have the same sparsity pattern?
numerical-methods numerical-linear-algebra
edited Jan 30 at 8:31
Carl Christian
5,9511723
asked Jan 29 at 20:35
IndianIndian
1
$endgroup$
I am struggling to solve a sequence of linear systems of the form
$$(x_i I + B ) y_i = b, quad i = 1,2,dotsc,$$
where $I$ is identity, $x_i$ is a real number and $B$ is sparse. Both the matrix $B$ and the right-hand side $b$ are independent of $x_i$. Both $B$ and $b$ are complex.
Iterative methods are inapplicable here because
$$||B^{-1}|| ,,,, text{sup}_i ,,, |x_{i+1} - x_i| gg 1. $$
Can I exploit the fact that all matrices have the same sparsity pattern?
numerical-methods numerical-linear-algebra
numerical-methods numerical-linear-algebra
edited Jan 30 at 8:31
Carl Christian
5,9511723
asked Jan 29 at 20:35
IndianIndian
1
edited Jan 30 at 8:31
Carl Christian
5,9511723
asked Jan 29 at 20:35
IndianIndian
1
edited Jan 30 at 8:31
Carl Christian
5,9511723
edited Jan 30 at 8:31
Carl Christian
5,9511723
edited Jan 30 at 8:31
Carl Christian
5,9511723
5,9511723
asked Jan 29 at 20:35
IndianIndian
1
asked Jan 29 at 20:35
IndianIndian
1
asked Jan 29 at 20:35
IndianIndian
1
1
$begingroup$
A number of iterative methods are applicable to your problem. You should explain which methods are excluded by your condition.
$endgroup$
– Carl Christian
Jan 29 at 20:42
$begingroup$
I have formated your question and made some small changes. Please check carefully that nothing has been lost. You can roll back the changes.
$endgroup$
– Carl Christian
Jan 30 at 8:33
$begingroup$
Can’t you solve it as $B y_i = b - x_i$ with a direct solver? Then you’ll factor the matrix only once and then solve the problem with multiple righthand sides?
$endgroup$
– VorKir
Feb 3 at 6:02
$begingroup$
O, sorry, I misread the equation
$endgroup$
– VorKir
Feb 3 at 17:56
$begingroup$
Why do you think a Jacobi preconditioner won't work? It looks like the diagonal is much larger than the spectrum of $B$?
$endgroup$
– VorKir
Feb 3 at 21:36
add a comment |
$begingroup$
A number of iterative methods are applicable to your problem. You should explain which methods are excluded by your condition.
$endgroup$
– Carl Christian
Jan 29 at 20:42
$begingroup$
I have formated your question and made some small changes. Please check carefully that nothing has been lost. You can roll back the changes.
$endgroup$
– Carl Christian
Jan 30 at 8:33
$begingroup$
Can’t you solve it as $B y_i = b - x_i$ with a direct solver? Then you’ll factor the matrix only once and then solve the problem with multiple righthand sides?
$endgroup$
– VorKir
Feb 3 at 6:02
$begingroup$
O, sorry, I misread the equation
$endgroup$
– VorKir
Feb 3 at 17:56
$begingroup$
Why do you think a Jacobi preconditioner won't work? It looks like the diagonal is much larger than the spectrum of $B$?
$endgroup$
– VorKir
Feb 3 at 21:36
$begingroup$
A number of iterative methods are applicable to your problem. You should explain which methods are excluded by your condition.
$endgroup$
– Carl Christian
Jan 29 at 20:42
$begingroup$
A number of iterative methods are applicable to your problem. You should explain which methods are excluded by your condition.
$endgroup$
– Carl Christian
Jan 29 at 20:42
$begingroup$
I have formated your question and made some small changes. Please check carefully that nothing has been lost. You can roll back the changes.
$endgroup$
– Carl Christian
Jan 30 at 8:33
$begingroup$
I have formated your question and made some small changes. Please check carefully that nothing has been lost. You can roll back the changes.
$endgroup$
– Carl Christian
Jan 30 at 8:33
$begingroup$
Can’t you solve it as $B y_i = b - x_i$ with a direct solver? Then you’ll factor the matrix only once and then solve the problem with multiple righthand sides?
$endgroup$
– VorKir
Feb 3 at 6:02
$begingroup$
Can’t you solve it as $B y_i = b - x_i$ with a direct solver? Then you’ll factor the matrix only once and then solve the problem with multiple righthand sides?
$endgroup$
– VorKir
Feb 3 at 6:02
$begingroup$
O, sorry, I misread the equation
$endgroup$
– VorKir
Feb 3 at 17:56
$begingroup$
O, sorry, I misread the equation
$endgroup$
– VorKir
Feb 3 at 17:56
$begingroup$
Why do you think a Jacobi preconditioner won't work? It looks like the diagonal is much larger than the spectrum of $B$?
$endgroup$
– VorKir
Feb 3 at 21:36
$begingroup$
Why do you think a Jacobi preconditioner won't work? It looks like the diagonal is much larger than the spectrum of $B$?
$endgroup$
– VorKir
Feb 3 at 21:36
add a comment |
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$begingroup$
A number of iterative methods are applicable to your problem. You should explain which methods are excluded by your condition.
$endgroup$
– Carl Christian
Jan 29 at 20:42
$begingroup$
I have formated your question and made some small changes. Please check carefully that nothing has been lost. You can roll back the changes.
$endgroup$
– Carl Christian
Jan 30 at 8:33
$begingroup$
Can’t you solve it as $B y_i = b - x_i$ with a direct solver? Then you’ll factor the matrix only once and then solve the problem with multiple righthand sides?
$endgroup$
– VorKir
Feb 3 at 6:02
$begingroup$
O, sorry, I misread the equation
$endgroup$
– VorKir
Feb 3 at 17:56
$begingroup$
Why do you think a Jacobi preconditioner won't work? It looks like the diagonal is much larger than the spectrum of $B$?
$endgroup$
– VorKir
Feb 3 at 21:36