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Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the LU decomposition of $B$ by reusing $L$ and $U$, avoiding the cost of a new decomposition?

If the cost of reusing $L$ and $U$ becomes too similar to the cost of recomputing the entire decomposition, would there be any other decomposition more suitable in this case?

Edit: I have found my question to be very similar to solving series of linear systems with diagonal perturbations. I am also trying to solve a series of linear systems where every system is in the form $(A+\lambda I)x=b$ where $\lambda$ is a real number and all $b$s are the same. However, unlike the aforementioned question, my matrices are always dense.

Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the LU decomposition of $B$ by reusing $L$ and $U$, avoiding the cost of a new decomposition?

If the cost of reusing $L$ and $U$ becomes too similar to the cost of recomputing the entire decomposition, would there be any other decomposition more suitable in this case?

Edit: I have found my question to be very similar to solving series of linear systems with diagonal perturbations. I am also trying to solve a series of linear systems where every system is in the form $(A+\lambda I)x=b$ where $\lambda$ is a real number and all $b$s are the same. However, unlike the aforementioned question, my matrices are always dense.

Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the LU decomposition of $B$ by reusing $L$ and $U$, avoiding the cost of a new decomposition?

If the cost of reusing $L$ and $U$ becomes too similar to the cost of recomputing the entire decomposition, would there be any other decomposition more suitable in this case?

Edit: I have found my question to be very similar to solving series of linear systems with diagonal perturbations. I am also trying to solve a series of linear systems where every system is in the form $(A+\lambda I)x=b$ where $\lambda$ is a real number. However, unlike the aforementioned question, my matrices are always dense.

Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the LU decomposition of $B$ by reusing $L$ and $U$, avoiding the cost of a new decomposition?

If the cost of reusing $L$ and $U$ becomes too similar to the cost of recomputing the entire decomposition, would there be any other decomposition more suitable in this case?

Edit: I have found my question to be very similar to solving series of linear systems with diagonal perturbations. I am also trying to solve a series of linear systems where every system is in the form $(A+\lambda I)x=b$ where $\lambda$ is a real number and all $b$s are the same. However, unlike the aforementioned question, my matrices are always dense.

Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the LU decomposition of $B$ by reusing $L$ and $U$, avoiding the cost of a new decomposition?

If the cost of reusing $L$ and $U$ becomes too similar to the cost of recomputing the entire decomposition, would there be any other decomposition more suitable in this case?

Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the LU decomposition of $B$ by reusing $L$ and $U$, avoiding the cost of a new decomposition?

If the cost of reusing $L$ and $U$ becomes too similar to the cost of recomputing the entire decomposition, would there be any other decomposition more suitable in this case?

Edit: I have found my question to be very similar to solving series of linear systems with diagonal perturbations. I am also trying to solve a series of linear systems where every system is in the form $(A+\lambda I)x=b$ where $\lambda$ is a real number. However, unlike the aforementioned question, my matrices are always dense.