Expression for minimum iterations for SOR method
$begingroup$
I try to find an expression for the minimum iterations needed to solve a problem $Au=f$ by using the Successive Over-Relaxation (SOR) method for a certain $omega$ as a function of the size of matrix $Ainmathbb{R}^{ntimes n}$, $n$ and with an initial guess $u^{0}=emptyset$, such that the initial error is $f$.
Assuming I am able to find an expression for the rate of convergence of the corresponding Jacobi iteration matrix of matrix $A$, which does not depend on the size matrix $A$. Then I also have an expression for the rate of convergence of the SOR iteration matrix, since for a $omega_{opt}$, the optimal choice of $omega$, and $mu=rho(B_{Jac})$, we have
$$
rho(B_{SOR(omega)})=omega-1~mbox{for}~0leqomegaleqomega_{opt}\
rho(B_{SOR(omega)})=frac{1}{4}Big(omegamu+sqrt{omega^{2}mu^{2}-4(omega-1)}Big)^{2}~mbox{for}~omega_{opt}leqomegaleq2
$$
Now, using the rate of convergence of the SOR method I find the following expression for the minimal iterations needed
$$
||e^{k}||_{2}=frac{mu_{SOR}^{k}}{1-mu_{SOR}}||e^{0}||
$$
with $k$ the number of the iterations, thus for a certain stopping criterium $e^{k}leq e^{stop}$
$$
k=logBig[frac{||e^{stop}||}{||e^{0}||}(1-mu_{SOR})Big]cdot(log[mu_{SOR}])^{-1}
$$
However, the only term, which could depend on the size of matrix $A$ is $||e_{0}||=||f||$. Now, vector $f$ is very irregular and therefore hard to write into an analytical expression.
My question is: is there another way to approach the minimal amount of iterations needed by the SOR method, if necessary assuming we know the rate of convergence (of the SOR method) or is this the right way and is simply the irregularity of the vector $f$ a bottleneck in this approach?
linear-algebra matrices numerical-methods
asked Dec 24 '18 at 16:24
rs4rs35rs4rs35
237
$endgroup$
add a comment |
$begingroup$
I try to find an expression for the minimum iterations needed to solve a problem $Au=f$ by using the Successive Over-Relaxation (SOR) method for a certain $omega$ as a function of the size of matrix $Ainmathbb{R}^{ntimes n}$, $n$ and with an initial guess $u^{0}=emptyset$, such that the initial error is $f$.
Assuming I am able to find an expression for the rate of convergence of the corresponding Jacobi iteration matrix of matrix $A$, which does not depend on the size matrix $A$. Then I also have an expression for the rate of convergence of the SOR iteration matrix, since for a $omega_{opt}$, the optimal choice of $omega$, and $mu=rho(B_{Jac})$, we have
$$
rho(B_{SOR(omega)})=omega-1~mbox{for}~0leqomegaleqomega_{opt}\
rho(B_{SOR(omega)})=frac{1}{4}Big(omegamu+sqrt{omega^{2}mu^{2}-4(omega-1)}Big)^{2}~mbox{for}~omega_{opt}leqomegaleq2
$$
Now, using the rate of convergence of the SOR method I find the following expression for the minimal iterations needed
$$
||e^{k}||_{2}=frac{mu_{SOR}^{k}}{1-mu_{SOR}}||e^{0}||
$$
with $k$ the number of the iterations, thus for a certain stopping criterium $e^{k}leq e^{stop}$
$$
k=logBig[frac{||e^{stop}||}{||e^{0}||}(1-mu_{SOR})Big]cdot(log[mu_{SOR}])^{-1}
$$
However, the only term, which could depend on the size of matrix $A$ is $||e_{0}||=||f||$. Now, vector $f$ is very irregular and therefore hard to write into an analytical expression.
My question is: is there another way to approach the minimal amount of iterations needed by the SOR method, if necessary assuming we know the rate of convergence (of the SOR method) or is this the right way and is simply the irregularity of the vector $f$ a bottleneck in this approach?
linear-algebra matrices numerical-methods
asked Dec 24 '18 at 16:24
rs4rs35rs4rs35
237
$endgroup$
add a comment |
$begingroup$
I try to find an expression for the minimum iterations needed to solve a problem $Au=f$ by using the Successive Over-Relaxation (SOR) method for a certain $omega$ as a function of the size of matrix $Ainmathbb{R}^{ntimes n}$, $n$ and with an initial guess $u^{0}=emptyset$, such that the initial error is $f$.
Assuming I am able to find an expression for the rate of convergence of the corresponding Jacobi iteration matrix of matrix $A$, which does not depend on the size matrix $A$. Then I also have an expression for the rate of convergence of the SOR iteration matrix, since for a $omega_{opt}$, the optimal choice of $omega$, and $mu=rho(B_{Jac})$, we have
$$
rho(B_{SOR(omega)})=omega-1~mbox{for}~0leqomegaleqomega_{opt}\
rho(B_{SOR(omega)})=frac{1}{4}Big(omegamu+sqrt{omega^{2}mu^{2}-4(omega-1)}Big)^{2}~mbox{for}~omega_{opt}leqomegaleq2
$$
Now, using the rate of convergence of the SOR method I find the following expression for the minimal iterations needed
$$
||e^{k}||_{2}=frac{mu_{SOR}^{k}}{1-mu_{SOR}}||e^{0}||
$$
with $k$ the number of the iterations, thus for a certain stopping criterium $e^{k}leq e^{stop}$
$$
k=logBig[frac{||e^{stop}||}{||e^{0}||}(1-mu_{SOR})Big]cdot(log[mu_{SOR}])^{-1}
$$
However, the only term, which could depend on the size of matrix $A$ is $||e_{0}||=||f||$. Now, vector $f$ is very irregular and therefore hard to write into an analytical expression.
My question is: is there another way to approach the minimal amount of iterations needed by the SOR method, if necessary assuming we know the rate of convergence (of the SOR method) or is this the right way and is simply the irregularity of the vector $f$ a bottleneck in this approach?
linear-algebra matrices numerical-methods
asked Dec 24 '18 at 16:24
rs4rs35rs4rs35
237
$endgroup$
I try to find an expression for the minimum iterations needed to solve a problem $Au=f$ by using the Successive Over-Relaxation (SOR) method for a certain $omega$ as a function of the size of matrix $Ainmathbb{R}^{ntimes n}$, $n$ and with an initial guess $u^{0}=emptyset$, such that the initial error is $f$.
Assuming I am able to find an expression for the rate of convergence of the corresponding Jacobi iteration matrix of matrix $A$, which does not depend on the size matrix $A$. Then I also have an expression for the rate of convergence of the SOR iteration matrix, since for a $omega_{opt}$, the optimal choice of $omega$, and $mu=rho(B_{Jac})$, we have
$$
rho(B_{SOR(omega)})=omega-1~mbox{for}~0leqomegaleqomega_{opt}\
rho(B_{SOR(omega)})=frac{1}{4}Big(omegamu+sqrt{omega^{2}mu^{2}-4(omega-1)}Big)^{2}~mbox{for}~omega_{opt}leqomegaleq2
$$
Now, using the rate of convergence of the SOR method I find the following expression for the minimal iterations needed
$$
||e^{k}||_{2}=frac{mu_{SOR}^{k}}{1-mu_{SOR}}||e^{0}||
$$
with $k$ the number of the iterations, thus for a certain stopping criterium $e^{k}leq e^{stop}$
$$
k=logBig[frac{||e^{stop}||}{||e^{0}||}(1-mu_{SOR})Big]cdot(log[mu_{SOR}])^{-1}
$$
However, the only term, which could depend on the size of matrix $A$ is $||e_{0}||=||f||$. Now, vector $f$ is very irregular and therefore hard to write into an analytical expression.
My question is: is there another way to approach the minimal amount of iterations needed by the SOR method, if necessary assuming we know the rate of convergence (of the SOR method) or is this the right way and is simply the irregularity of the vector $f$ a bottleneck in this approach?
linear-algebra matrices numerical-methods
linear-algebra matrices numerical-methods
asked Dec 24 '18 at 16:24
rs4rs35rs4rs35
237
asked Dec 24 '18 at 16:24
rs4rs35rs4rs35
237
edited Dec 25 '18 at 8:26
rs4rs35
asked Dec 24 '18 at 16:24
rs4rs35rs4rs35
237
asked Dec 24 '18 at 16:24
rs4rs35rs4rs35
237
asked Dec 24 '18 at 16:24
rs4rs35rs4rs35
237
237
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051418%2fexpression-for-minimum-iterations-for-sor-method%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051418%2fexpression-for-minimum-iterations-for-sor-method%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
