Even if quad precision is not directly supported by most CPUs, many Compilers (GNU, Intel) support them. Also some software packages allow to compile with quad precision, e.g. PETSc. But is there really a need for this? I asked some days ago about Krylov subspace method in finite arithmetic, see Krylov subspace iterative methods in floating point arithmetic. I read the mentioned papers, and it seems to me that these methods are well understood in floating point arithmetic. Does iterative solvers benefit from using higher precision if the matrices are well conditioned? I.e., does the iteration counter goes down with higher precision? Are there, may be, some studies on this topic? Are there problems in scientific computing, where double precision is not enough to formulate and discretize them?


I think this is actually a duplicate of this question: Higher precision floating-point arithmetic in numerical PDE

As I stated on that question, quad precision is certainly not widely used in scientific computing. Moreover, I do not believe that quad precision is actually necessary in any significant number of applications.

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    $\begingroup$ I have not seen this question before, but I don't think that this is a duplicate. The question you mentioned is just related to numerical solution of PDEs, whereas my question is not specific to any area in scientific computing. I'm mostly interested whether there are applications which lead to system not solvable in double precision. There are some publication on quad precision for Krylov subspace methods (e.g. siam.org/meetings/la03/proceedings/hhasegaw.pdf), but they usually solve the Toepliz matrix or some other artificial problems. $\endgroup$ – Thomas W. Nov 6 '12 at 6:18
  • $\begingroup$ I didn't mean the link to be specific to PDE solvers. I think it's universally true that there are not a lot of applications for quad precision (or higher). That you can't find a lot of publications fits this well. $\endgroup$ – Wolfgang Bangerth Nov 6 '12 at 13:40
  • $\begingroup$ Identification of a duplicate belongs in the comments, and as you have more than 500 rep (3k once the beta ends) you should be voting to close as such. $\endgroup$ – dmckee --- ex-moderator kitten Nov 6 '12 at 14:18
  • $\begingroup$ Ah, I learn something new everyday. Thanks @dmckee. $\endgroup$ – Wolfgang Bangerth Nov 6 '12 at 14:59

This does not answer your question, but there are people - interested in the reproducibility and reliability of large scale simulations, e.g., climate simulation - who investigate implementational details of floating-point algorithms in double precision.

Behavioural differences have been observed, e.g., when the scalar products in the CGM are computed not by naive summation, but by more sophisticated algorithm as compensated summation or summation in order of ascending magnitude. This is particular relevant for parallel summation on many cores, because the summation (say, in MPI) is usually black-boxed.

A search like ["conjugate gradient" "compensated summation"] in a well-known search engine for scientific papers will give a few (!) publications on that. As far as I remember, the iteration count might vary slightly, and might even increase when improving the numerical accuracy. It is probably less relevant on your local quad-core work station.


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