Convergence of an Iterative Method for Linear Systems
Consider the linear system $A bf{x} = bf{b}$, where $A$ is a symmetric matrix. Suppose that $M - N$ is a splitting of $A$ (i.e. $A = M - N$), where $M$ is symmetric positive definite and $N$ is symmetric. Let $lambda_{min}(M)$ be the eigenvalue of $M$ with the smallest magnitude and let $rho(N)$ be the spectral radius of $N$. Show that if $lambda_{min}(M) > rho(N)$, then the iterative scheme $M bf{x}_{k+1} = N bf{x}_{k} + bf {b}$ converges to $bf{x}$ for any initial guess $bf {x}_0$.
I've tried showing that the iterative matrix $T = I - M^{-1}A$, where $I$ is the identity matrix, has a spectral radius less than 1 in this case and so the method converges, but I'm having trouble taking advantage of the symmetry conditions around $M$ and $N$. Any hints or pointers would be appreciated.
real-analysis linear-algebra matrices symmetric-matrices
asked Nov 30 '18 at 2:28
rw435rw435
11
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Consider the linear system $A bf{x} = bf{b}$, where $A$ is a symmetric matrix. Suppose that $M - N$ is a splitting of $A$ (i.e. $A = M - N$), where $M$ is symmetric positive definite and $N$ is symmetric. Let $lambda_{min}(M)$ be the eigenvalue of $M$ with the smallest magnitude and let $rho(N)$ be the spectral radius of $N$. Show that if $lambda_{min}(M) > rho(N)$, then the iterative scheme $M bf{x}_{k+1} = N bf{x}_{k} + bf {b}$ converges to $bf{x}$ for any initial guess $bf {x}_0$.
I've tried showing that the iterative matrix $T = I - M^{-1}A$, where $I$ is the identity matrix, has a spectral radius less than 1 in this case and so the method converges, but I'm having trouble taking advantage of the symmetry conditions around $M$ and $N$. Any hints or pointers would be appreciated.
real-analysis linear-algebra matrices symmetric-matrices
asked Nov 30 '18 at 2:28
rw435rw435
11
add a comment |
Consider the linear system $A bf{x} = bf{b}$, where $A$ is a symmetric matrix. Suppose that $M - N$ is a splitting of $A$ (i.e. $A = M - N$), where $M$ is symmetric positive definite and $N$ is symmetric. Let $lambda_{min}(M)$ be the eigenvalue of $M$ with the smallest magnitude and let $rho(N)$ be the spectral radius of $N$. Show that if $lambda_{min}(M) > rho(N)$, then the iterative scheme $M bf{x}_{k+1} = N bf{x}_{k} + bf {b}$ converges to $bf{x}$ for any initial guess $bf {x}_0$.
I've tried showing that the iterative matrix $T = I - M^{-1}A$, where $I$ is the identity matrix, has a spectral radius less than 1 in this case and so the method converges, but I'm having trouble taking advantage of the symmetry conditions around $M$ and $N$. Any hints or pointers would be appreciated.
real-analysis linear-algebra matrices symmetric-matrices
asked Nov 30 '18 at 2:28
rw435rw435
11
Consider the linear system $A bf{x} = bf{b}$, where $A$ is a symmetric matrix. Suppose that $M - N$ is a splitting of $A$ (i.e. $A = M - N$), where $M$ is symmetric positive definite and $N$ is symmetric. Let $lambda_{min}(M)$ be the eigenvalue of $M$ with the smallest magnitude and let $rho(N)$ be the spectral radius of $N$. Show that if $lambda_{min}(M) > rho(N)$, then the iterative scheme $M bf{x}_{k+1} = N bf{x}_{k} + bf {b}$ converges to $bf{x}$ for any initial guess $bf {x}_0$.
I've tried showing that the iterative matrix $T = I - M^{-1}A$, where $I$ is the identity matrix, has a spectral radius less than 1 in this case and so the method converges, but I'm having trouble taking advantage of the symmetry conditions around $M$ and $N$. Any hints or pointers would be appreciated.
real-analysis linear-algebra matrices symmetric-matrices
real-analysis linear-algebra matrices symmetric-matrices
asked Nov 30 '18 at 2:28
rw435rw435
11
asked Nov 30 '18 at 2:28
rw435rw435
11
asked Nov 30 '18 at 2:28
rw435rw435
11
asked Nov 30 '18 at 2:28
rw435rw435
11
asked Nov 30 '18 at 2:28
rw435rw435
11
11
add a comment |
add a comment |
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