• In section 1.1 x0 and xi are represented using subscript, in section 1.2 they're represented using superscript. Unless there's a specific reason I'm not aware of, this should probably be changed so it's consistent
• At the end of the first paragraph in 1.3 the opening quotes are backwards, in latex use `` for opening quotes.
• Section 1.3 second paragraph, missing a space before (FP16)
• 1.3 third paragraph second sentence should be changed to either "...calculation, it may utilize..." or "...calculation, higher precision may be utilized..."
• Section 1.4.2: "It's experimented that" isn't a phrase that makes sense. It should be replaced with something like "It has been experimentally proven that..." or "Experiments have shown that..."
• The graphs and code haven't been updated to the latest versions

• We should be consistent about whether the acronym or the full name goes in the parenthesis the first time it's used. My vote is that the acronym should be in parenthesis, but either is fine as long as we're consistent.
• First sentence is kind of wordy and should probably be broken up to flow better. Maybe: "Generalized Minimal Residual Method (GMRES) and its variants are advanced numerical techniques for solving non-symmetric linear systems based on iterative refinement (IR). This method is particularly useful for..."
• Second sentence doesn't add much, it can be easily absorbed into a different sentence like I did above
• Are there other steps in a GMRES iteration? If not "include" should be changed to "are".
  As a side note this is somewhere where using a numbered list might be good, it would make it clear what the different steps are. But this is also a matter of opinion, either is probably fine.
• I think you missed a word, it should be "...using the Arnoldi process"
• Is there supposed to be a sub heading here? Right now it's kind of an abrupt transition from talking about the steps of a GMRES iteration to describing a linear system.
• You have a comma at the end of this line that I think should be deleted.
• On this line there a missing space that should be between A and r0, and again I don't think a comma is necessary at the end of the line.
• This line is missing a word and has an extra capital letter: "and is unique, though it can be computed..."
• Missing space here between "basis, called" and Krylov doesn't have an e on the end
• Be consistent about punctuation. Either this line shouldn't start with a capital or the line above should have a period at the end. Also consistency about using capitals for the vectors in V1...Vn+1
• That line also has a sentence fragment. "The initial vector s..." what about that vector?
• Typo, "approximately". maybe run it through a spellchecker if wherever you're writing doesn't have one built in.
• This line should have "and" instead of a comma between the residual types. Second sentence should end with a comma and connect to the next line, currently it's a fragment.
• Another typo, whilce -> while
• "GMRES has played a significant role" in what? Should probably expand on this.
• These two sentences say almost the same thing, the second one should probably be deleted.

The text between functions isn't quite right.
Between code block 2 and 3, "And then,Several testing matrices were taken as bellow to do the experiment:" should be: "Several testing matrices were used to do the following experiments:"
The others just need to have lowercase "r" in "results"

• #11
• "Time vs. Condition Number" is missing a space
• I'm not sure how accurate the Iterations vs Condition Number section is, 30 is what I set as the max number of iterations so GMRES just took the full number of iterations each time. The only time when it was lower is when the matrix size was 10, as apparently the matlab gmres function bounds the number of iterations by the matrix size.
• Large Matrix, LUIR has a bullet point about overhead that seems a little lost? I'm not sure where that's supposed to be
• High condition number, GMRES, Time vs Matrix size has an extra space
• Maybe add some overall conclusions at the bottom, saying something about overall trends and when it's better to use one method over the other?

• This section says basically the same thing as the last few lines of the introduction. If you want to keep it, you should add more detail like which sections are critical and which aren't.
• I don't think "(i.e., faster)" is necessary here. I think most people would get that easier = faster without needing to be told

• Maybe add a bit more detail about the LU algorithm, just enough so that someone who's never heard of it can understand what's going on. Especially what u, ul, and ur represent since those are used a bunch later.
• flop -> flops
• About the speedup, did you run some code to get that 2.7 speedup or was it in one of the papers? If it was in a paper you need to cite it, and if you got that number yourself you should say that.
• This line, delete "also" in "also the condition of A"
• Missing word: "...the unit round-off is..." (again in next sentence)
• Missing word: "...giving an even tighter bound of..."
• Is "strict and picky" a quote from the cited paper? If so it needs to be in quotes.
• This line has an extra space after "line 4"
• Here "Let" should be "and let"
• This line third sentence isn't making much sense, I'm really not sure what you're trying to say.
• This line, third sentence: "Here the trade-off is the longer the Krylov subspace that is constructed..."
• Same line, fifth sentence: delete "the" in "the returning to the outer loop"
• Sixth sentence: delete "a" in "a low or no preconditioning" and "case" should be "cases"
• This line fourth sentence: comma at the end should be a period.
• Fifth sentence: "restrict" should be either "strict" or "restrictive"

• Listing 1: should indicate that that function specifically generates matrices with the given condition number. Partly my fault, I should have specified in a comment
• "bellow" -> "below"
• The "results as below" blurbs should be plural. Ex: "With matrix with high condition number, we got the results below" should be: "With matrices with high condition numbers we got the results below:"

Hi fellows, @vgvreug @Gudauu
Here is the CAS708 report draft.
Please review it and make comments to correct it at your convenience.
Thanks.