Fix some typos in comments

[jip]

These typos were overlooked while working on the PR #581. I hope they are the last.

[codecov]

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Merging #582 (2eeed4b) into master (34c8291) will not change coverage.
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@@           Coverage Diff           @@
##           master     #582   +/-   ##
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  Coverage   82.37%   82.37%           
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  Files        1894     1894           
  Lines      190679   190679           
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  Hits       157065   157065           
  Misses      33614    33614           

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[weslleyspereira]

I think this is still incorrect. The expression norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ) is a number, so it does not have right-hand sides. I suggest something like:

The test ratio is the maximum over
norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS )
for every column b of B,
where op(A) = A, A**T, or A**H, x is the solution vector, and EPS is the machine epsilon.

What do you think?

[jip]

Yes, this looks better. Two adjoining "where" sections (the first one - "for every column b of B" - is implicit, and the second one is explicit) could be merged:

The test ratio is the maximum over
  norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
where op(A) = A, A**T, or A**H, b is the column of B, x is
the solution vector, and EPS is the machine epsilon.

but this is a matter of taste.

I believe this variant is incorrect, too. The original comment states "x and b are N by NRHS matrices". So will be the residual R := b - op(A)*x. The code computes taxicab-based norm || R(j) || for each j-th column R(j) from R. Then it takes a max(j) || R(j) || which means in toto a computing of kinda 1-norm, but based on taxicab-based norm |Re(z)| + |Im(z)|, not magnitude-based norm |z|. So, the phrase "the maximum over" is excessive.

And this variant may be improved further:

1. Since x and b are matrices not vectors here, capital letters X and B are used.
2. I'm using norm(op(A)*X - B) not norm(B - op(A)*X) since the first is the exact expression which is computed. I know they give identical results. I don't insist on this summands swapping here.

So, the result may look like this:

The test ratio is set to
  norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
where op(A) = A, A**T, or A**H, B is the N by NRHS matrix of
right-hand sides, X is the N by NRHS matrix of solution vectors, and
EPS is the machine epsilon.

When "kinda 1-norms" in both norm(op(A)*X - B) and norm(X) will be replaced by "true 1-norm", then the following statement could be added: "The norm used is the 1-norm.".

I'm ready to update this PR with any of variants discussed above.

[weslleyspereira]

Yes, this looks better. Two adjoining "where" sections (the first one - "for every column b of B" - is implicit, and the second one is explicit) could be merged:

The test ratio is the maximum over
  norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
where op(A) = A, A**T, or A**H, b is the column of B, x is
the solution vector, and EPS is the machine epsilon.

but this is a matter of taste.

I would go with either this one or the version I proposed. Maybe you can modify it a bit:

The test ratio is the maximum over
  norm(b - op(A)*x) / ( ||op(A)||_1 * norm(x) * EPS ),
where op(A) = A, A**T, or A**H, b is the column of B, x is
the solution vector, and EPS is the machine epsilon.

As you pointed out, the norm operator is that "kinda 1-norm" for vectors of size N, in which |Re(z)| + |Im(z)| replaces the magnitude |z| for all complex scalars z. This is what the subroutine computes if I understand the algorithm correctly.

I believe this variant is incorrect, too. The original comment states "x and b are N by NRHS matrices". So will be the residual R := b - op(A)*x. The code computes taxicab-based norm || R(j) || for each j-th column R(j) from R. Then it takes a max(j) || R(j) || which means in toto a computing of kinda 1-norm, but based on taxicab-based norm |Re(z)| + |Im(z)|, not magnitude-based norm |z|. So, the phrase "the maximum over" is excessive.

And this variant may be improved further:

1. Since x and b are matrices not vectors here, capital letters X and B are used.

2. I'm using `norm(op(A)*X - B)` not `norm(B - op(A)*X)` since the first is the exact expression which is computed. I know they give identical results. I don't insist on this summands swapping here.

So, the result may look like this:

The test ratio is set to
  norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
where op(A) = A, A**T, or A**H, B is the N by NRHS matrix of
right-hand sides, X is the N by NRHS matrix of solution vectors, and
EPS is the machine epsilon.

I don't agree with this proposal. The algorithm does not compute norm(X), where "capital" X is the original input matrix. Maybe I am missing something, please correct me if I am wrong.

[jip]

(Sorry for delayed answer)

I think you are right. I'll use the last variant you've proposed:

I would go with either this one or the version I proposed. Maybe you can modify it a bit:

The test ratio is the maximum over
  norm(b - op(A)*x) / ( ||op(A)||_1 * norm(x) * EPS ),
where op(A) = A, A**T, or A**H, b is the column of B, x is
the solution vector, and EPS is the machine epsilon.

I'll try either to update this PR or to create a new one.

[weslleyspereira]

I just noticed another thing that is not related to your changes, but maybe you can help me. Is there a bug here?

lapack/TESTING/LIN/ctbt02.f

Lines 231 to 235 in 2eeed4b

	IF( XNORM.LE.ZERO ) THEN
	RESID = ONE / EPS
	ELSE
	RESID = MAX( RESID, ( ( BNORM / ANORM ) / XNORM ) / EPS )
	END IF

I think so. Suppose that the last column of X is full of zeros. Then, XNORM = ZERO in the last iteration ( J = NRHS ).
Then, the subroutine computes RESID = ONE / EPS. There are many files with the same pattern.

I can propose a PR to solve this (if this is actually a bug).

[jip]

I just noticed another thing that is not related to your changes, but maybe you can help me. Is there a bug here?
[...]
I can propose a PR to solve this (if this is actually a bug).

I don't see any problems here. If any column of X is full of zeros, then the RESID takes a maximal value, and the rest columns of X doesn't matter. IMHO.