Given the Linear system   $$Ax = b$$  where $A$ is an $s$-sparse ($s$ is the maximum number of non-zero entries in $A$), $k$-conditioned ($k$ is the ratio between the highest and the smallest eigenvalue) matrix of size $N$, how can I express the time complexity of CG method based on those three parameters?  I have found out different questions in Stack exchange (,), but none of them considers all three parameters.
I have been trying to figure out the time complexity of the conjugate gradient method.  I have to solve a system of linear equations given by  $$Ax=b$$  where $A$ is a sparse, symmetric, positive definite matrix.  What would be the time complexity of the conjugate gradient method?
Compare classifiers based on AUROC or accuracy? How do I catch a PHP fatal (`E_ERROR`) error? I can use set_error_handler() to catch most PHP errors, but it doesn't work for fatal (E_ERROR) errors, such as calling a function that doesn't exist. Is there another way to catch these errors?  I am trying to call mail() for all errors and am running PHP 5.2.3. How to clip an Openlayers OSM Background to a Vector Boundary? table of contents: adding a vertical space In a table of contents that has been generated with LaTeX's \tableofcontents command, I would like to add a vertical space before a Chapter item in the table of contents. One way that I have in my notes is to use  \addtocontents{toc}{\protect\vspace{10pt}}   at a desired location in the document so that this gets processed by \tableofcontents to include an appropriate vertical space in the table of contents. I have two questions:   Is the above the correct way to get a vertical space in a table of contents? Why is it necessary to use the \protect command here?   Thank you... Is there any direct relation among first, third, fourth order moments? Complete classification of the groups for which converse of Lagrange's Theorem holds Using a singular or plural verb after "and/or" If the subject of a sentence is separated by "and/or", should the verb be pluralized, as with "and", or agree with the rightmost subject, as with "or"? For example:     His co-workers and/or his boss was at fault.   vs.     His co-workers and/or his boss were at fault. boolean variable values in PHP to javascript implementation how can i forward requests to my own IP address to a local IP address in my network (possibly using named)? My setup  I have a linux server running in my basement. I currently have my router looking to that server for DNS lookup, so I can use named to spoof a bunch of domain names to point to my local server's local address, 192.168.0.111. This works great.  I also am running Apache and PHP on this server, and have forwarded port 80 on my router to this machine, so my public IP xx.xxx.x.xx is sent to the basement server.  This is a great development set up for me, because I can develop websites in my local network and pull them up on numerous devices with a *.mylocal.net address. I can also quickly share a website on my public IP with others that are outside of my network.  I have a static IP, so that makes this even easier.  The problem  Some applications need to have an absolute URL. My spoofed DNS doesn't work outside my network, and neither does my internal IP address.  I need to be able to access the sites at the same IP that they are accessed publicly, which is my public IP: xx.xxx.x.xx    For some reason, I can't access my own public IP address from within my own network.  My router's public IP is xx.xxx.x.xx, and when I hit xx.xxx.x.xx from outside my network I see exactly what I want -- my development website. But, when I hit xx.xxx.x.xx from inside my network, the connection times out.    I assume my ISP has a firewall that blocks requests from xx.xxx.x.xx to xx.xxx.x.xx where the origin and the destination are the same IP (or maybe this is impossible on any network?)  So how can I set up my network to forward the internal request to my public IP to the local IP of my basement server?   I would like to do this within the local DNS server I set up, or in my router, so the change can be applied to all devices in my network, and not just the basement server or the windows tablet/laptop, etc individually. I am trying to use Stack Exchange Andoid App to use my accounts in Stack Overflow &amp; Area 51.   For the Area 51, there is only Area 51 Discussion. Even Area 51 Discussion isn't working in the App!   VERSION 1.0.51  When adding this site, it's showing the correct full name as in this screen-shot:     However, when I pin this site to my fav sites in Stack Exchange, it is showing: Unknown Site. When clicking it, it shows black screen then just quit the App!  Here is the screen-shot:       Is the site not yet capable with Android? or is it a bug in the App itself?  PS: this question isn't duplicate:   In the previous versions of the App, the site was opening but with Unknown Site name. However, in this version, it quits the app when pressing the area site! It isn't opening at all! This question is also to track this bug in the App. As it's about the site name issue &amp; the content as well. Unable to get a generic ResponseEntity where T is a generic class "SomeClass" Are the rules regarding absolute modifiers too absolute? Introduction to AdS/CFT Best Place to put Feature Requests? Why should I install hardware drivers if my computer is working fine? I've just installed Windows 7 on my PC.  I went online to look for drivers. For , Foxconn offers LAN drivers, on-board audio drivers, and chipset drivers. And for , Nvidia offers a graphics card driver, obviously.  But then I stopped and wondered... what benefit would I gain from installing them?   The LAN cable is plugged in and I'm online; no problems there.  The on-board audio is working fine.  A quick Google tells me that a chipset is responsible for moving data around my motherboard – which seems to be happening OK!  And my graphics look perfect.    So why should I install the drivers? What would they give me, other than badly designed "management" interfaces, and pointless "notifications" whenever I plug in an audio jack?  If there are different considerations for each of the components – chipset, audio, LAN, and graphics (and any others you think I should be aware of) – please break it down in your answer.     Update: I've had a couple of answers already, saying that I'll get better stability, bugfixes, etc.   Can I slightly expand my question... Can I get all these same benefits if I install the drivers via Windows Update? And if I do this, would I avoid having to install the vendor-specific UIs, which I find so annoying? How to make a half cylinder? I am having a tough time building a half cylinder and have not found any tutorial showing me how to do that. How to model effectively using exact measurements? I always struggle when trying to create a floor plan or furniture. How do I model using exact measurements. I never hear "scissor" or "pant" or "jean". It's always "scissors", "pants", and "jeans", respectively. Are they considered plural? I have found a few mods for minecraft I like a lot. I know you cant use them all at the same time. I know that some go together well, so how do I find out what goes with what?  Is there a way to load some for now, and unload them later so that I can change them out? I got a patch device but I don't think it works that well.  I tried to play with this stuff on my own but I got things so messed up that I had to delete everything and start over. So what am I doing wrong?  KH Number of surjections from $\{1,...,m\}$ to $\{1,...,n\}$ Let $m,n$ be two integers such that, $m\ge n$. Compute the number of surjections from $\{1,...,m\}$ to $\{1,...,n\}$   There are $n^m$ functions (total).  we subtract from $n^m$ the number of non-surjective functions.  There are $\binom{n}{1}(n-1)^m$ functions missing one element.  There are $\binom{n}{2}(n-2)^m$ functions missing two elements, but how many times did we count this in the previous count ?  Then we have to add this again by inclusion-exclusion but why is the difference always 1?