Tarmo:
Thanks for the explanation. I guess I still do not get the reliance today
on a rule of thumb method to approximate eigenvalues when there has been
technology to do it accurately for many years.
- The power method and similar iterative methods are easy to program.
- Engineers have calculated eigenvalues on computers since the 1950s.
- The routines that later were incorporated into EISPACK (and then LAPACK)
were originally written in ALGOL in 1971.
- IMSL and NAG have been going since the early 1970s.
- Maple started in 1980.
- MATLAB and Mathematica started in 1985.
Where have you guys been?
Best wishes,
John
Tarmo Veskioja wrote:
>
>
> Bill Harris wrote:
>
>>NB. Calculate the consistency ratio
>>n =: #
>>cr =: (ev - n) % <: @: n
>>
>>
> Your cr is called the absolute consistency index (CI). What Coyle refers
> to is called relative consistency ratio (CR). From Coyle's paper on page
> 5:
>
> "...the calculation of a Consistency Ratio (CR) to measure how
> consistent the judgements have been relative to large samples of purely
> random judgements. If the CR is much in excess of 0.1 the judgements are
> untrustworthy because they are too close for comfort to randomness..."
>
> So you need to do
> CR = CI % MACI, where MACI is a mean absolute consistency index of
> random matrices.
> Saaty and others have MACI precomputed for matrices up to size 10.
> NB. MACI based on 1 000 000 random matrices
> 3 4 5 6 7
> 8 9 10
> 0.524500 0.884133 1.10857 1.24902 1.34039 1.40437 1.45050 1.48607
> The 0.5245 for 3x3 matrices is exact.
>
> One can use Roger's code for generating random matrices (I had to put
> back the dot behind y, since I still use J5.04):
> st1=: 3 : 0
> i=. 1+(,~y.) [EMAIL PROTECTED] #saaty_scale
> b=. >/~i.y.
> ((b*i) + |:b*i+#saaty_scale) { 1,(,%) saaty_scale
> )
> Thanks Roger for the speedup!
>
>
> Then you can use something like
> am =: +/ % # NB. arithmetic mean
> maci =: 13 : 'am cr"2 st1"0 (x. # y.)' NB. cr =: (ev - n) % <: @: n
> maci_precomputed =: 1000 maci"0 (3+i.8) NB. larger x. will
> obviously give a more precise answer
> (cr exp) % (<:<:#exp) {maci_precomputed NB. relative consistency
> ratio for exp
>
>>NB. Results for all 4 of Coyle's 3x3 matrices
>> cr exp
>>0.014531883 NB. Coyle: 0.072
>>
>>
> I think Coyle's CR=0.072 is a typo and it should be 0.0145319 % 0.5425 =
> 0.027 (actually 0.0277062).
>
>> cr und
>>0.014531883 NB. Coyle: 0.026
>>
>>
> It should also be 0.0277062
>
>> cr rod
>>_4.4408921e_16 NB. Coyle: 0
>>
>>
> 0 considering rounding errors.
>
>> cr pod
>>0.00061677663 NB. Coyle: 0
>>
>>
> 0.00061677 % 0.5425
> 0.0011369
>
> Bill Harris wrote:
>
>>If I get a moment, would anyone object if I pick some of the best code,
>>add the eigenvalue calculation, and put it on the Wiki as a Saaty
>>framework in J?
>>
> I don't mind. I might have some ideas to add but it is past midnight
> already.
>
> Cheers,
> Tarmo
>
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