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Re: Big Loss of Precision

From: Thomas D. Dean
Subject: Re: Big Loss of Precision
Date: Sat, 16 Jul 2016 14:11:08 -0700
User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.8.0

On 07/16/2016 12:05 PM, Montgomery-Smith, Stephen wrote:
R is of the same order of magnitude as 1e4.  So dot(R,cross(R,V)) is of the 
same order of magnitude as 1e8.  1e8 times octave eps is about 1e-8.
From: Help-octave address@hidden on behalf of Thomas D. Dean address@hidden
Sent: Saturday, July 16, 2016 1:35 PM
To: address@hidden
Subject: Big Loss of Precision

The cross product of two vectors is orthogonal to both.  The dot product
of two orthogonal vectors is zero.

This is not an octave problem, just an ill-formed situation.

octave:1405> printf(" %.20f\n",V);

octave:1406> printf(" %.20f\n",R);

octave:1407> dot(cross(R,V),R)
ans =    1.4901e-08
octave:1408> dot(cross(R,V),V)
ans =    3.6380e-12

cross has 6 multiplications and 3 add/subtract
dot has 3 multiplications and 3 add/subtract

Octave eps is 2.2204e-16

machine eps is 1.11022e-16 (quadmath 1.925930e-34).

This case has a very serious loss of precision!

Tom Dean

A similar problem with similar magnitudes.  Very different outcome.
A = [2 4 8]; norm(A)
B = A.^4; norm(B)


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