[Scilab-users] partial fractions

[Federico Miyara]

Samuel,

Thanks for the tip!

Any possibility that partfrac() ever become part of Scilab? I think it 
should actually replace pfss(), since it seems to be exactly what a user 
expects from a partial fraction function.

Regards,

Federico Miyara


On 29/12/2021 17:32, Samuel Gougeon wrote:
> Hello Federico,
>
> Le 16/07/2021 à 21:11, Federico Miyara a écrit :
>>
>> Dear All,
>>
>> I'm wonderig whether there is a standard method to obtain the partial 
>> fraction expansion of a rational function.
>>
>> Searching, I've found the function pfss(), which purportedly is meant 
>> for that, but I find it doesn't give a complete solution. For instance:
>>
>> // Poles and zeros
>> p = [-8:-1];
>> z = [0 0 0 0];
>>
>> // Numerator and denominator
>> N = prod(%s - z)
>> D = prod(%s - p)
>>
>> // Rational function
>> H = N/D
>>
>> // Partial fractions
>> HH = pfss(H)
>>
>> The result is
>>
>> HH  =
>>        HH(1)
>>
>>    -8 -5.7428571s -3.9515873s² +0.2960317s³ -0.0946429s⁴ -0.006746s⁵ 
>> -0.0001984s⁶
>> ------------------------------------------------------------------------------ 
>>
>>               40320 +69264s +48860s² +18424s³ +4025s⁴ +511s⁵ +35s⁶ +s⁷
>>
>>        HH(2)
>>
>>    0.0001984
>>    ---------
>>      1 +s
>>
>> The second component in the list is indeed one of the partial 
>> fractions, but the first one is not an irreductible version. 
>
>
> The pfss() rmax option must be used, and set to a big value, for 
> instance 1/%eps.
> https://help.scilab.org/docs/6.1.0/en_US/pfss.html
> Then:
>
> --> D = list2vec(pfss(H,1/%eps))'
>  D  =
>
>    -0.8126984  3.3347222  -5.4  4.3402778  -1.7777778  0.3375 
> -0.0222222  0.0001984
>    ----------  ---------  ----  ---------  ----------  ------ 
> ----------  ---------
>       8 +s       7 +s     6 +s    5 +s        4 +s      3 +s      2 
> +s       1 +s
>
> --> clean(sum(D)-H)
>  ans  =
>
>    0
>    -
>    1
>
>> Computing HH(1) + HH(2) yields H with good approximation.
>>
>> I guess I could proceed iteratively with HH(1), but it would be nice 
>> to have a function performing this automatically. I couldn't find 
>> such a function
>
>
> In addition, partfrac() is available on FileExchange since 2015 @ 
> https://fileexchange.scilab.org/toolboxes/451000
> It manages poles multiplicity. With it, your example (with only simple 
> poles) gives the following:
>
> --> H = N/D
>  H  =
> s⁴
> -------------------------------------------------------------------
>    40320 +109584s +118124s² +67284s³ +22449s⁴ +4536s⁵ +546s⁶ +36s⁷ +s⁸
>
> --> [I, R, F, T] = partfrac(H);
>
> --> I  // integer part of the fraction
>  I  =
>    0.
>
> --> R   // Undecomposed remainder of the fraction
>  R  =
> s⁴
> -------------------------------------------------------------------
>    40320 +109584s +118124s² +67284s³ +22449s⁴ +4536s⁵ +546s⁶ +36s⁷ +s⁸
>
> --> F   // Decomposition: row #1 = numerators ; row #2 = poles value ; 
> row #3 = poles multiplicity
>  F  =
>
>   -0.8126984   3.3347222  -5.4   4.3402778  -1.7777778 0.3375  
> -0.0222222   0.0001984
>   -8.         -7.         -6.   -5.         -4. -3.      -2.         -1.
>    1.          1.          1.    1.          1. 1.       1.          1.
>
> --> T  // Decomposition output as text
>  T  =
>   "    -0.8126984   3.3347222   -5.4   4.3402778 -1.7777778   0.3375   
> -0.0222222   0.0001984"
>   "0 + ---------- + --------- + ---- + --------- + ---------- + ------ 
> + ---------- + ---------"
>   "       8+%s        7+%s      6+%s     5+%s 4+%s       3+%s       
> 2+%s        1+%s   "
>
>
> --> D = F(1,:)./((%s-F(2,:)).^F(3,:))
>  D  =
>
>    -0.8126984  3.3347222  -5.4 4.3402778  -1.7777778  0.3375  
> -0.0222222  0.0001984
>    ----------  ---------  ----  ---------  ----------  ------ 
> ----------  ---------
>       8 +s       7 +s     6 +s    5 +s        4 +s      3 +s      2 
> +s       1 +s
>
> -->  clean(R-sum(D))
>  ans  =
>
>    0
>    -
>    1
>
> Regards
> Samuel
>
>
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