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RE: Dependence of precision on execution speed of Inverse

Yep I agree Daniel; that's the crucial question. My implicit assumption has
been that the underlying arbitrary precision arithmetic isn't the main
contributor to the factor of ~150, but perhaps it is.

-----Original Message-----
From: Daniel Lichtblau [mailto:danl at] 
Sent: Tuesday, 2 October 2007 2:54 AM
To: Andrew Moylan
Cc: Mathgroup
Subject: [mg81715] Re: Dependence of precision on execution speed of Inverse

Andrew Moylan wrote:
> [...]
> Ah I see, thanks for your reply Daniel. So the arbitrary precision 
> numbers in Mathematica are each something like a struct with {a 
> pointer to the arbitrary precision mantissa data, the exponent, the 
> length of the mantissa data};


> and because each element in a matrix could in principle have a 
> *different* arbitrary precision, there's no way to pack the array into 
> a contiguous lump of memory. So there's no way around de-referencing a 
> lot of pointers.

Also correct.

> But Daniel, would you agree that for (hypothetical) *fixed* precision 
> (across the whole matrix) non-machine-precision matrices of numbers, 
> it
> *would* be possible to create the analogue of packed arrays and 
> therefore make optimised routines to run on them, analogous to the 
> routines that currently operate on packed arrays of machine-precision 
> numbers?

Possibly. Except the optimality improvements might or might not actually
materialize. More below.

 > I guess one can write a code (in C++ or such) that can invert
> non-machine-precision matrices (and do other operations for which 
> Mathematica employs packed arrays only for machine precision numbers) 
> tens of times faster than Mathematica can, by combining something like 
> the GNU multiple precision library ( with BLAS-like 
> linear algebra code.
> I don't know how whether the efficiency of linear algebra at 
> higher-than-machine-precision affects many users, but it has come up in 
> my application so I am quite interested in the possibilities!

The question amounts to this. How much of the relative time difference 
(vs. machine arithmetic linear algebra using Lapack/BLAS) is due to 
locality of reference for, say, dot products, and how much to software 
implementation of the underlying arithmetic? I do not know the answer. 
If a large part is from the arithmetic, then you gain but little from 
emulation of packing for bignums.

Daniel Lichtblau
Wolfram Research

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