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Aug 30, 2021
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Fix some typos in comments #582

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95b6abf
unify comments: homogenize notation for transpose (**T) and conjugate…
jip Jun 14, 2021
a0634a9
unify comments: TRANS parameter
jip Jun 14, 2021
f42f512
revert "comments fixed: expression for residual in xTBT02"
jip Jun 14, 2021
8262b4a
fix comments: expression for residual
jip Jun 14, 2021
2eeed4b
unify comments: expression for residual
jip Jun 14, 2021
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16 changes: 7 additions & 9 deletions TESTING/LIN/ctbt02.f
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Expand Up @@ -29,13 +29,11 @@
*> \verbatim
*>
*> CTBT02 computes the residual for the computed solution to a
*> triangular system of linear equations A*x = b, A**T *x = b, or
*> A**H *x = b when A is a triangular band matrix. Here A**T denotes
*> the transpose of A, A**H denotes the conjugate transpose of A, and
*> x and b are N by NRHS matrices. The test ratio is the maximum over
*> the number of right hand sides of
*> norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A, A**T, or A**H, and EPS is the machine epsilon.
*> triangular system of linear equations op(A)*X = B, when A is a
*> triangular band matrix. The test ratio is the maximum over the
*> number of right hand sides of
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
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@weslleyspereira weslleyspereira Jun 14, 2021

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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?

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@jip jip Jun 15, 2021 •
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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.

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@weslleyspereira weslleyspereira Jun 15, 2021

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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.

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@jip jip Sep 6, 2021

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(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.

*> where op(A) = A, A**T, or A**H, and EPS is the machine epsilon.
*> \endverbatim
*
* Arguments:
Expand Down Expand Up @@ -142,7 +140,7 @@
*> \verbatim
*> RESID is REAL
*> The maximum over the number of right hand sides of
*> norm(B - op(A)*X) / ( norm(op(A)) * norm(X) * EPS ).
*> norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*> \endverbatim
*
* Authors:
Expand Down Expand Up @@ -221,7 +219,7 @@ SUBROUTINE CTBT02( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, X,
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(B - op(A)*X) / ( norm(op(A)) * norm(X) * EPS ).
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*
RESID = ZERO
DO 10 J = 1, NRHS
Expand Down
23 changes: 10 additions & 13 deletions TESTING/LIN/ctpt02.f
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Expand Up @@ -28,14 +28,11 @@
*> \verbatim
*>
*> CTPT02 computes the residual for the computed solution to a
*> triangular system of linear equations A*x = b, A**T *x = b, or
*> A**H *x = b, when the triangular matrix A is stored in packed format.
*> Here A**T denotes the transpose of A, A**H denotes the conjugate
*> transpose of A, and x and b are N by NRHS matrices. The test ratio
*> is the maximum over the number of right hand sides of
*> triangular system of linear equations op(A)*X = B, when the
*> triangular matrix A is stored in packed format. The test ratio is
*> the maximum over the number of right hand sides of
*> norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A, A**T, or A**H, and EPS is the machine epsilon.
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
*> where op(A) = A, A**T, or A**H, and EPS is the machine epsilon.
*> \endverbatim
*
* Arguments:
Expand All @@ -53,9 +50,9 @@
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': A *x = b (No transpose)
*> = 'T': A**T *x = b (Transpose)
*> = 'C': A**H *x = b (Conjugate transpose)
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose)
*> \endverbatim
*>
*> \param[in] DIAG
Expand Down Expand Up @@ -130,7 +127,7 @@
*> \verbatim
*> RESID is REAL
*> The maximum over the number of right hand sides of
*> norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*> norm(op(A)*B - B) / ( norm(op(A)) * norm(X) * EPS ).
*> \endverbatim
*
* Authors:
Expand Down Expand Up @@ -191,7 +188,7 @@ SUBROUTINE CTPT02( UPLO, TRANS, DIAG, N, NRHS, AP, X, LDX, B, LDB,
RETURN
END IF
*
* Compute the 1-norm of A or A**H.
* Compute the 1-norm of op(A).
*
IF( LSAME( TRANS, 'N' ) ) THEN
ANORM = CLANTP( '1', UPLO, DIAG, N, AP, RWORK )
Expand All @@ -208,7 +205,7 @@ SUBROUTINE CTPT02( UPLO, TRANS, DIAG, N, NRHS, AP, X, LDX, B, LDB,
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
* norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ).
*
RESID = ZERO
DO 10 J = 1, NRHS
Expand Down
24 changes: 11 additions & 13 deletions TESTING/LIN/ctrt02.f
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Original file line number Diff line number Diff line change
Expand Up @@ -29,13 +29,11 @@
*> \verbatim
*>
*> CTRT02 computes the residual for the computed solution to a
*> triangular system of linear equations A*x = b, A**T *x = b,
*> or A**H *x = b. Here A is a triangular matrix, A**T is the transpose
*> of A, A**H is the conjugate transpose of A, and x and b are N by NRHS
*> matrices. The test ratio is the maximum over the number of right
*> hand sides of
*> norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A, A**T, or A**H, and EPS is the machine epsilon.
*> triangular system of linear equations op(A)*X = B, where A is a
*> triangular matrix. The test ratio is the maximum over the number of
*> right hand sides of
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
*> where op(A) = A, A**T, or A**H, and EPS is the machine epsilon.
*> \endverbatim
*
* Arguments:
Expand All @@ -53,9 +51,9 @@
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': A *x = b (No transpose)
*> = 'T': A**T *x = b (Transpose)
*> = 'C': A**H *x = b (Conjugate transpose)
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose)
*> \endverbatim
*>
*> \param[in] DIAG
Expand Down Expand Up @@ -138,7 +136,7 @@
*> \verbatim
*> RESID is REAL
*> The maximum over the number of right hand sides of
*> norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ).
*> \endverbatim
*
* Authors:
Expand Down Expand Up @@ -200,7 +198,7 @@ SUBROUTINE CTRT02( UPLO, TRANS, DIAG, N, NRHS, A, LDA, X, LDX, B,
RETURN
END IF
*
* Compute the 1-norm of A or A**H.
* Compute the 1-norm of op(A).
*
IF( LSAME( TRANS, 'N' ) ) THEN
ANORM = CLANTR( '1', UPLO, DIAG, N, N, A, LDA, RWORK )
Expand All @@ -217,7 +215,7 @@ SUBROUTINE CTRT02( UPLO, TRANS, DIAG, N, NRHS, A, LDA, X, LDX, B,
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS )
* norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS )
*
RESID = ZERO
DO 10 J = 1, NRHS
Expand Down
15 changes: 7 additions & 8 deletions TESTING/LIN/dtbt02.f
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Original file line number Diff line number Diff line change
Expand Up @@ -28,12 +28,11 @@
*> \verbatim
*>
*> DTBT02 computes the residual for the computed solution to a
*> triangular system of linear equations A*x = b or A' *x = b when
*> A is a triangular band matrix. Here A' is the transpose of A and
*> x and b are N by NRHS matrices. The test ratio is the maximum over
*> the number of right hand sides of
*> norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A or A' and EPS is the machine epsilon.
*> triangular system of linear equations op(A)*X = B, when A is a
*> triangular band matrix. The test ratio is the maximum over the
*> number of right hand sides of
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
*> where op(A) = A, A**T, or A**H and EPS is the machine epsilon.
*> The norm used is the 1-norm.
*> \endverbatim
*
Expand Down Expand Up @@ -136,7 +135,7 @@
*> \verbatim
*> RESID is DOUBLE PRECISION
*> The maximum over the number of right hand sides of
*> norm(B - op(A)*X) / ( norm(op(A)) * norm(X) * EPS ).
*> norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*> \endverbatim
*
* Authors:
Expand Down Expand Up @@ -214,7 +213,7 @@ SUBROUTINE DTBT02( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, X,
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(B - op(A)*X) / ( norm(op(A)) * norm(X) * EPS ).
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*
RESID = ZERO
DO 10 J = 1, NRHS
Expand Down
21 changes: 10 additions & 11 deletions TESTING/LIN/dtpt02.f
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Original file line number Diff line number Diff line change
Expand Up @@ -27,12 +27,11 @@
*> \verbatim
*>
*> DTPT02 computes the residual for the computed solution to a
*> triangular system of linear equations A*x = b or A'*x = b when
*> the triangular matrix A is stored in packed format. Here A' is the
*> transpose of A and x and b are N by NRHS matrices. The test ratio is
*> triangular system of linear equations op(A)*X = B, when the
*> triangular matrix A is stored in packed format. The test ratio is
*> the maximum over the number of right hand sides of
*> norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A or A' and EPS is the machine epsilon.
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
*> where op(A) = A or A**T, and EPS is the machine epsilon.
*> The norm used is the 1-norm.
*> \endverbatim
*
Expand All @@ -51,9 +50,9 @@
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': A *x = b (No transpose)
*> = 'T': A'*x = b (Transpose)
*> = 'C': A'*x = b (Conjugate transpose = Transpose)
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
Expand Down Expand Up @@ -123,7 +122,7 @@
*> \verbatim
*> RESID is DOUBLE PRECISION
*> The maximum over the number of right hand sides of
*> norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ).
*> \endverbatim
*
* Authors:
Expand Down Expand Up @@ -183,7 +182,7 @@ SUBROUTINE DTPT02( UPLO, TRANS, DIAG, N, NRHS, AP, X, LDX, B, LDB,
RETURN
END IF
*
* Compute the 1-norm of A or A'.
* Compute the 1-norm of op(A).
*
IF( LSAME( TRANS, 'N' ) ) THEN
ANORM = DLANTP( '1', UPLO, DIAG, N, AP, WORK )
Expand All @@ -200,7 +199,7 @@ SUBROUTINE DTPT02( UPLO, TRANS, DIAG, N, NRHS, AP, X, LDX, B, LDB,
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
* norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ).
*
RESID = ZERO
DO 10 J = 1, NRHS
Expand Down
23 changes: 11 additions & 12 deletions TESTING/LIN/dtrt02.f
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Original file line number Diff line number Diff line change
Expand Up @@ -28,12 +28,11 @@
*> \verbatim
*>
*> DTRT02 computes the residual for the computed solution to a
*> triangular system of linear equations A*x = b or A'*x = b.
*> Here A is a triangular matrix, A' is the transpose of A, and x and b
*> are N by NRHS matrices. The test ratio is the maximum over the
*> number of right hand sides of
*> norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A or A' and EPS is the machine epsilon.
*> triangular system of linear equations op(A)*X = B, where A is a
*> triangular matrix. The test ratio is the maximum over the number of
*> right hand sides of
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
*> where op(A) = A or A**T, and EPS is the machine epsilon.
*> The norm used is the 1-norm.
*> \endverbatim
*
Expand All @@ -52,9 +51,9 @@
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': A *x = b (No transpose)
*> = 'T': A'*x = b (Transpose)
*> = 'C': A'*x = b (Conjugate transpose = Transpose)
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
Expand Down Expand Up @@ -132,7 +131,7 @@
*> \verbatim
*> RESID is DOUBLE PRECISION
*> The maximum over the number of right hand sides of
*> norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ).
*> \endverbatim
*
* Authors:
Expand Down Expand Up @@ -193,7 +192,7 @@ SUBROUTINE DTRT02( UPLO, TRANS, DIAG, N, NRHS, A, LDA, X, LDX, B,
RETURN
END IF
*
* Compute the 1-norm of A or A'.
* Compute the 1-norm of op(A).
*
IF( LSAME( TRANS, 'N' ) ) THEN
ANORM = DLANTR( '1', UPLO, DIAG, N, N, A, LDA, WORK )
Expand All @@ -210,7 +209,7 @@ SUBROUTINE DTRT02( UPLO, TRANS, DIAG, N, NRHS, A, LDA, X, LDX, B,
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS )
* norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS )
*
RESID = ZERO
DO 10 J = 1, NRHS
Expand Down
12 changes: 6 additions & 6 deletions TESTING/LIN/stbt02.f
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Original file line number Diff line number Diff line change
Expand Up @@ -28,11 +28,11 @@
*> \verbatim
*>
*> STBT02 computes the residual for the computed solution to a
*> triangular system of linear equations op(A)*X = B when A is a
*> triangular band matrix. The test ratio is the maximum over the
*> triangular system of linear equations op(A)*X = B, when A is a
*> triangular band matrix. The test ratio is the maximum over the
*> number of right hand sides of
*> norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A or A' and EPS is the machine epsilon.
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
*> where op(A) = A or A**T, and EPS is the machine epsilon.
*> The norm used is the 1-norm.
*> \endverbatim
*
Expand Down Expand Up @@ -135,7 +135,7 @@
*> \verbatim
*> RESID is REAL
*> The maximum over the number of right hand sides of
*> norm(B - op(A)*X) / ( norm(op(A)) * norm(X) * EPS ).
*> norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*> \endverbatim
*
* Authors:
Expand Down Expand Up @@ -213,7 +213,7 @@ SUBROUTINE STBT02( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, X,
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(B - op(A)*X) / ( norm(op(A)) * norm(X) * EPS ).
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*
RESID = ZERO
DO 10 J = 1, NRHS
Expand Down
21 changes: 10 additions & 11 deletions TESTING/LIN/stpt02.f
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Original file line number Diff line number Diff line change
Expand Up @@ -27,12 +27,11 @@
*> \verbatim
*>
*> STPT02 computes the residual for the computed solution to a
*> triangular system of linear equations A*x = b or A'*x = b when
*> the triangular matrix A is stored in packed format. Here A' is the
*> transpose of A and x and b are N by NRHS matrices. The test ratio is
*> triangular system of linear equations op(A)*X = B, when the
*> triangular matrix A is stored in packed format. The test ratio is
*> the maximum over the number of right hand sides of
*> norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A or A' and EPS is the machine epsilon.
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ),
*> where op(A) = A or A**T, and EPS is the machine epsilon.
*> The norm used is the 1-norm.
*> \endverbatim
*
Expand All @@ -51,9 +50,9 @@
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the operation applied to A.
*> = 'N': A *x = b (No transpose)
*> = 'T': A'*x = b (Transpose)
*> = 'C': A'*x = b (Conjugate transpose = Transpose)
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
Expand Down Expand Up @@ -123,7 +122,7 @@
*> \verbatim
*> RESID is REAL
*> The maximum over the number of right hand sides of
*> norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*> norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ).
*> \endverbatim
*
* Authors:
Expand Down Expand Up @@ -183,7 +182,7 @@ SUBROUTINE STPT02( UPLO, TRANS, DIAG, N, NRHS, AP, X, LDX, B, LDB,
RETURN
END IF
*
* Compute the 1-norm of A or A'.
* Compute the 1-norm of op(A).
*
IF( LSAME( TRANS, 'N' ) ) THEN
ANORM = SLANTP( '1', UPLO, DIAG, N, AP, WORK )
Expand All @@ -200,7 +199,7 @@ SUBROUTINE STPT02( UPLO, TRANS, DIAG, N, NRHS, AP, X, LDX, B, LDB,
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
* norm(op(A)*X - B) / ( norm(op(A)) * norm(X) * EPS ).
*
RESID = ZERO
DO 10 J = 1, NRHS
Expand Down
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