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### [Lapack] Fwd: mistake in BLAS/scabs1.f

 ```From: "Vitaly N. Golovach" > Date: December 13, 2011 7:25:22 AM MST To: "Langou, Julien" > Subject: Re: [Lapack] mistake in BLAS/scabs1.f Hi Julien, Thank you for the detailed answer! I enjoyed reading the references that you sent me. I agree, it makes sense to define a norm like | Re( z ) | + | Im ( z ) |, because it is so easy to compute and that such a norm can be useful for certain routines where it is not crucial to have the exact Euclidean norm. I imagine that ZGETF2 is such a routine, although I am not familiar with the algorithm (partial pivoting with row interchanges), so it is difficult to judge whether replacing | Re( z ) | + | Im ( z ) | by a properly coded ( | Re( z ) |^2 + | Im ( z ) |^2 )^(1/2) should improve the results. The two norms differ only by a factor of 2^(1/2), at most. However, "absolute value" is not a good name for this measure of length when you refer to complex numbers. Most people will only read the description of the routine IZAMAX and trust that it "finds the index of element having max. absolute value". If you have the power to do it, you might want to consider adding another sentence to the description saying what is actually meant by "absolute value". I also liked the term "pseudo 1-norm" used at the end of the paragraph in Nick Higham's book, p.500. But making a too drastic change is also not good, because the people in your field are, probably, used to seeing "absolute value" in that place. A simple example where IZAMAX would give different results depending on which norm is used is as follows. Consider an array consisting of two complex numbers z_1 = (6,0) and z_2 = (3,4). The Euclidean version would compare 6>5, whereas the "pseudo 1-norm" would compare 6<7. Thanks again for your explanation and wish you good luck! Best, Vitaly ******************************* Vitaly N. Golovach LPMMC, Maison des Magist?res CNRS 25 avenue des Martyrs, BP166 38042 Grenoble Cedex, FRANCE Tel.: +33 4 76 88 79 82 Fax.: +33 4 76 88 79 83 Email: vitaly.n.golovach@Domain.Removed ******************************* On Mon, Dec 12, 2011 at 9:39 PM, Langou, Julien > wrote: Hi Vitaly, The definition of absolute value, abs( z ), of a complex number, z, in the BLAS has always been abs( z ) = | Re( z ) | + | Im ( z ) |. ( Always means since the seventies. ) I guess the reason is that a reliable computation of the modulus: | z | = ( | Re( z ) |^2 + | Im ( z ) |^2 )^(1/2) is complicated (see for example dlapy2 to do this operation without unneccessary overflows or underflows) and since it is complicate, it consumes time. There is a nice paragraph in Nick Higham's book "Accuracy and Stability of Numerical Algorithms", 2nd edition p.500, Sec 27.8. If you look at the original Level 1 BLAS paper from 1979 form Lawson, Hanson, Kincaid, and Krogh. ( available from Kincaid webpage http://www.cs.utexas.edu/users/kincaid/blas.pdf ) ( page 311) you would read some justification on this choice. So * a subroutine like ICAMAX would look like the largest element in absolute value with the definition: abs( z ) = | Re( z ) | + | Im ( z ) |. * a subroutine like ICAMAX does not compute the infinite norm of a complex vector. * a subroutine like ICASUM does not compute the 1-norm of a complex vector. Such routine do exist in the new BLAS standard. http://www.netlib.org/blas/blast-forum/chapter2.pdf page 19, table 2.1, under NORM. Anyway, you do not think "absolute value" is appropriate to reference: abs( z ) = | Re( z ) | + | Im ( z ) |. Fair enough. Suggestions? "taxi modulus"? One thing I have always wondered is whether GETRF which uses GETF2 which uses ICAMAX is more or less stable than one who would use | z | = ( | Re( z ) |^2 + | Im ( z ) |^2 )^(1/2) .... Cheers, Julien. On Dec 12, 2011, at 8:40 AM, Vitaly N. Golovach wrote: Dear LPACK development team, There is a mistake in the function SCABS1 in BLAS, see file BLAS/scabs1.f. The absolute value of a complex number is not the sum of the absolute values of its real and imaginary parts. Either the documentation is wrong or the code is wrong. The function SCABS1 is used only in BLAS/icamax.f. Actually, it is also used in BLAS/caxpy.f, but in a trivial way, where it cannot cause errors. Best, Vitaly ******************************* Vitaly N. Golovach LPMMC, Maison des Magist?res CNRS 25 avenue des Martyrs, BP166 38042 Grenoble Cedex, FRANCE Tel.: +33 4 76 88 79 82 Fax.: +33 4 76 88 79 83 Email: vitaly.n.golovach@Domain.Removed ******************************* REAL FUNCTION SCABS1(Z) * .. Scalar Arguments .. COMPLEX Z * .. * * Purpose * ======= * * SCABS1 computes absolute value of a complex number * * ===================================================================== * * .. Intrinsic Functions .. INTRINSIC ABS,AIMAG,REAL * .. SCABS1 = ABS(REAL(Z)) + ABS(AIMAG(Z)) RETURN END _______________________________________________ Lapack mailing list Lapack@Domain.Removed http://lists.eecs.utk.edu/mailman/listinfo/lapack -------------- next part -------------- An HTML attachment was scrubbed... URL: http://lists.eecs.utk.edu/mailman/private/lapack/attachments/20111213/e28ce26e/attachment-0001.html ```
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