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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.
________________________________________
Sent: Saturday, July 16, 2016 1:35 PM
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);
3.99120302628332668249
-0.91842329903575536942
1.29246846883179156151

octave:1406> printf(" %.20f\n",R);
-1518.35794621729837672319
-4017.17627231122196462820
4816.75939627815478161210

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)
C=cross(A,B)
dot(C,A)
dot(C,B)

tomdean

```