# Simplification of Rational Polynomials **URL:** \ **Category:** General usage\ **Created:** [June 22, 2023, 9:01am UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105 "2023-06-22T09:01:24Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![Thaisa](https://avatars.discourse-cdn.com/v4/letter/t/c5a1d2/32.png) [@Thaisa](https://scilab.discourse.group/u/Thaisa)\ **Post date:** [June 22, 2023, 9:01am UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/1 "2023-06-22T09:01:24Z")
Hi, everyone! Please, I have ZBBE (end of this message), an array of rational polynomials to be simplified. I’ve tried to use commands “simp\_mode (%T)” before the equation that defines ZBBE and “simp(ZBBE)”, also separating numerators and denominators, and ZBBE has kept the same. So I’ve tried to use “gcd” command, with the following script: ```auto mdc1=gcd([ZBBE(k,1).num ZBBE(k,1).den]); if mdc1~=1 simp_num=ZBBE(k,1).num./mdc1; simp_den=ZBBE(k,1).den./mdc1; ZBE(k,1)=simp_num/simp_den; end ``` Now I have ZBE (also at the end of this message). Some irreducible rational polynomials have kept the same as I expected. Others, like ZBBE(1,1) was properly simplified. But I have a problem with the last one, ZBBE(23,1), which became of even grater degree. Please, could someone solve my problem or at least explain to me what is going wrong? I thank you a lot. ```auto -> ZBBE ZBBE = 0.0000124 +1.262D-09s +8.496D-12s² +4.316D-16s³ +1.453D-18s⁴ ------------------------------------------------------------ 4.234D-08 +2.052D-12s +2.896D-14s² +7.018D-19s³ +4.952D-21s⁴ 3.135D+08 +200000s ------------------------- 353004.51 +1941.1718s +s² 0.0000124 +1.284D-09s +8.645D-12s² +4.469D-16s³ +1.504D-18s⁴ ------------------------------------------------------------ 4.234D-08 +2.088D-12s +2.947D-14s² +7.266D-19s³ +5.127D-21s⁴ 1.818D-13 +2.236D-17s +1.505D-19s² +9.255D-24s³ +3.115D-26s⁴ ------------------------------------------------------------ 3.240D-16 +1.636D-18s +4.695D-22s² +1.355D-24s³ +1.388D-28s⁴ +2.804D-31s⁵ 6.614D+11 +2.556D+08s +166666.67s² ---------------------------------- 6613756.6s +2556.2372s² +s³ 0.0000008 +6.435D-10s +3.922D-13s² +1.046D-16s³ ------------------------------------------------- 8.984D-12s +7.078D-15s² +2.987D-18s³ +6.274D-22s⁴ 0.014861s +0.0000095s² -------------------------------- 2.972205 +0.001893s +7.431D-08s² 1.135D+08 +166666.67s ------------------------- 96457.316 +951.74433s +s² 7.375D+08 +583333.33s ------------------------- 1839060.2 +4404.5491s +s² 1.601D-13 +2.236D-17s +1.325D-19s² +9.255D-24s³ +2.743D-26s⁴ ------------------------------------------------------------ 3.240D-16 +1.441D-18s +4.695D-22s² +1.193D-24s³ +1.388D-28s⁴ +2.469D-31s⁵ 489383.56s +845s² ------------------------- 9411222.3 +16592.466s +s² 2.987D+11 +1.439D+08s +111111.11s² ---------------------------------- 3882915.2s +1871.042s² +s³ 1.632D-17s +2.601D-19s² +8.777D-22s³ ------------------------------------------------------------ 4.410D-16 +7.029D-18s +2.387D-20s² +1.425D-24s³ +3.456D-27s⁴ 0.1760929s +0.0002869s² --------------------------------- 8.804645 +0.0143449s +0.0000009s² 1.0174757s +0.0023123s² ---------------------------------- 14.535367 +0.0330322s +0.0000071s² 1.125D-17s +1.176D-19s² +2.382D-22s³ ------------------------------------------------------------ 2.250D-16 +2.351D-18s +4.866D-21s² +5.569D-25s³ +7.657D-28s⁴ 2.644D-13 +6.668D-19s² +3.806D-25s⁴ ------------------------------------------------------------ 6.480D-16 +2.115D-18s +1.634D-21s² +3.693D-24s³ +9.328D-28s⁴ +1.612D-30s⁵ 449410.95s +816s² ----------------------- 7022046.1 +13096.5s +s² 0.0001426 +0.0000005s ----------------------- 2.139D-09s +3.060D-12s² 1.192D-17s +1.253D-19s² +2.553D-22s³ ------------------------------------------------------------ 2.250D-16 +2.364D-18s +4.924D-21s² +5.935D-25s³ +8.205D-28s⁴ 0.1937022s +0.0003156s² -------------------------------- 8.804645 +0.0143449s +0.000001s² 1.417D+11 +23864070s +111111.11s² -------------------------------------- 2.435D+08 +1275510.2s +405.68919s² +s³ 2.744D-18s +2.898D-20s² +6.638D-23s³ ------------------------------------------------------------ 3.920D-16 +4.140D-18s +9.505D-21s² +1.475D-25s³ +2.478D-28s⁴ --> ZBE ZBE = 4.248D-12 +4.316D-16s +1.453D-18s² ---------------------------------- 1.448D-14 +7.018D-19s +4.952D-21s² 3.135D+08 +200000s ------------------------- 353004.51 +1941.1718s +s² 4.322D-12 +4.469D-16s +1.504D-18s² ---------------------------------- 1.473D-14 +7.266D-19s +5.127D-21s² 1.818D-13 +2.236D-17s +1.505D-19s² +9.255D-24s³ +3.115D-26s⁴ ------------------------------------------------------------ 3.240D-16 +1.636D-18s +4.695D-22s² +1.355D-24s³ +1.388D-28s⁴ +2.804D-31s⁵ 6.614D+11 +2.556D+08s +166666.67s² ---------------------------------- 6613756.6s +2556.2372s² +s³ 3.218D-10 +1.267D-13s +1.046D-16s² ------------------------------------ 3.539D-15s +1.394D-18s² +6.274D-22s³ 0.014861s +0.0000095s² -------------------------------- 2.972205 +0.001893s +7.431D-08s² 1.135D+08 +166666.67s ------------------------- 96457.316 +951.74433s +s² 7.375D+08 +583333.33s ------------------------- 1839060.2 +4404.5491s +s² 1.601D-13 +2.236D-17s +1.325D-19s² +9.255D-24s³ +2.743D-26s⁴ ------------------------------------------------------------ 3.240D-16 +1.441D-18s +4.695D-22s² +1.193D-24s³ +1.388D-28s⁴ +2.469D-31s⁵ 489383.56s +845s² ------------------------- 9411222.3 +16592.466s +s² 2.987D+11 +1.439D+08s +111111.11s² ---------------------------------- 3882915.2s +1871.042s² +s³ 1.632D-17s +2.601D-19s² +8.777D-22s³ ------------------------------------------------------------ 4.410D-16 +7.029D-18s +2.387D-20s² +1.425D-24s³ +3.456D-27s⁴ 0.1760929s +0.0002869s² --------------------------------- 8.804645 +0.0143449s +0.0000009s² 1.0174757s +0.0023123s² ---------------------------------- 14.535367 +0.0330322s +0.0000071s² 3.094D-20s +2.382D-22s² ----------------------------------------------- 6.188D-19 +4.764D-21s +2.784D-25s² +7.657D-28s³ 2.308D-19 -4.793D-37s +3.806D-25s² ----------------------------------------------- 5.657D-22 +1.846D-24s +9.328D-28s² +1.612D-30s³ 449410.95s +816s² ----------------------- 7022046.1 +13096.5s +s² 0.0001426 +0.0000005s ----------------------- 2.139D-09s +3.060D-12s² 1.192D-17s +1.253D-19s² +2.553D-22s³ ------------------------------------------------------------ 2.250D-16 +2.364D-18s +4.924D-21s² +5.935D-25s³ +8.205D-28s⁴ 0.1937022s +0.0003156s² -------------------------------- 8.804645 +0.0143449s +0.000001s² 1.417D+11 +23864070s +111111.11s² -------------------------------------- 2.435D+08 +1275510.2s +405.68919s² +s³ 3.813D-16s +6.771D-18s² +3.820D-20s³ +6.638D-23s⁴ ------------------------------------------------------------ 5.448D-14 +9.673D-16s +5.460D-18s² +9.526D-21s³ +1.820D-25s⁴ +2.478D-28s⁵ ``` ---
**Author:** ![mottelet](https://yyz2.discourse-cdn.com/free1/user_avatar/scilab.discourse.group/mottelet/32/13_2.png) [@mottelet](https://scilab.discourse.group/u/mottelet)\ **Post date:** [June 22, 2023, 11:28am UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/2 "2023-06-22T11:28:50Z")
Hello, In Scilab polynomial and rational algebra is not exact (coefficients are encoded real number with the same finite precision as elsewhere). Hence, simplification does not necessary occur as it may occur with a symbolic math package (which Scilab does not have). Could you save `ZBEE` in a Scilab binary data file with `save("data.sod","ZBEE")`and attach the file to your post (please edit your original post) ? S. ---
**Author:** ![mottelet](https://yyz2.discourse-cdn.com/free1/user_avatar/scilab.discourse.group/mottelet/32/13_2.png) [@mottelet](https://scilab.discourse.group/u/mottelet)\ **Post date:** [June 22, 2023, 11:38am UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/3 "2023-06-22T11:38:43Z")
> [@mottelet](#): > > Could you save `ZBEE` in a Scilab binary data file with `save("data.sod","ZBEE")`and attach the file to your post (please edit your original post) ? Save `ZBEE` before any simplification please. ---
**Author:** ![Thaisa](https://avatars.discourse-cdn.com/v4/letter/t/c5a1d2/32.png) [@Thaisa](https://scilab.discourse.group/u/Thaisa)\ **Post date:** [June 22, 2023, 5:27pm UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/4 "2023-06-22T17:27:58Z")
Thank you very much for your answer, @mottelet! I’ve saved the “data.sod”, but I’ve not found where I could attach the file to my post. ---
**Author:** ![mottelet](https://yyz2.discourse-cdn.com/free1/user_avatar/scilab.discourse.group/mottelet/32/13_2.png) [@mottelet](https://scilab.discourse.group/u/mottelet)\ **Post date:** [June 23, 2023, 9:21am UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/5 "2023-06-23T09:21:45Z")
Just drag and drop the file when you are editing your post. ---
**Author:** ![Thaisa](https://avatars.discourse-cdn.com/v4/letter/t/c5a1d2/32.png) [@Thaisa](https://scilab.discourse.group/u/Thaisa)\ **Post date:** [June 23, 2023, 3:41pm UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/8 "2023-06-23T15:41:44Z")
[ZBBE\_data.sod](https://scilab.discourse.group/uploads/short-url/6SA1H2NJNPEhpw5nP2jFrkjm60t.sod) (30.2 KB) ---
**Author:** ![mottelet](https://yyz2.discourse-cdn.com/free1/user_avatar/scilab.discourse.group/mottelet/32/13_2.png) [@mottelet](https://scilab.discourse.group/u/mottelet)\ **Post date:** [June 23, 2023, 4:26pm UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/9 "2023-06-23T16:26:24Z")
You saved `ZBBE` after simplification, I have the same as above. ---
**Author:** ![Thaisa](https://avatars.discourse-cdn.com/v4/letter/t/c5a1d2/32.png) [@Thaisa](https://scilab.discourse.group/u/Thaisa)\ **Post date:** [June 24, 2023, 1:38pm UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/10 "2023-06-24T13:38:59Z")
No, I really think this is ZBBE before simplification. But I will send you now ZBBE (before simplification) and ZBE = ZBBE (after simplification), exactly as they are above. [ZBBE\_data.sod](https://scilab.discourse.group/uploads/short-url/6SA1H2NJNPEhpw5nP2jFrkjm60t.sod) (30.2 KB) [ZBE\_data.sod](https://scilab.discourse.group/uploads/short-url/dkUtjlk6OaYJ7TcvY4rTCyiIqO4.sod) (30.2 KB) Thank you very much. ---
**Author:** ![mottelet](https://yyz2.discourse-cdn.com/free1/user_avatar/scilab.discourse.group/mottelet/32/13_2.png) [@mottelet](https://scilab.discourse.group/u/mottelet)\ **Post date:** [June 26, 2023, 6:28am UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/11 "2023-06-26T06:28:16Z")
I don’t see any simplification occuring in `ZBBE(23,1)`: ```auto --> simp_mode ans = T --> ZBBE(23,1) ans = 2.744D-18s +2.898D-20s^2 +6.638D-23s^3 --------------------------------------------------------------- 3.920D-16 +4.140D-18s +9.505D-21s^2 +1.475D-25s^3 +2.478D-28s^4 --> simp(ZBBE(23,1)) ans = 2.744D-18s +2.898D-20s^2 +6.638D-23s^3 --------------------------------------------------------------- 3.920D-16 +4.140D-18s +9.505D-21s^2 +1.475D-25s^3 +2.478D-28s^4 ``` ---
**Author:** ![Thaisa](https://avatars.discourse-cdn.com/v4/letter/t/c5a1d2/32.png) [@Thaisa](https://scilab.discourse.group/u/Thaisa)\ **Post date:** [June 26, 2023, 2:35pm UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/12 "2023-06-26T14:35:27Z")
That is exactly my problem. The command “simp” has made no simplification for me. Because of that, I’ve used gcd. I’ve got some simplifications with that, but the last one was worse. ---
**Author:** ![mottelet](https://yyz2.discourse-cdn.com/free1/user_avatar/scilab.discourse.group/mottelet/32/13_2.png) [@mottelet](https://scilab.discourse.group/u/mottelet)\ **Post date:** [June 26, 2023, 4:08pm UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/13 "2023-06-26T16:08:50Z")
Be warned that the relative magnitude of your coefficients in the denominator is 10^12 which seems to be the source of the problem. Moreover, the common divisor computation seems to also depend on the absolute magnitude of coefficients. By renormalizing (without loss of precision) the numerator and denominator you obtain the following results: ```scilab r = ZBBE(23,1); rs = r.num*1e23/(r.den*1e23) roots(r.num),roots(r.den) --> r=ZBBE(23,1); --> rsc=r.num*1e23/(r.den*1e23) rsc = 274400s +2897.664s^2 +6.6382848s^3 ---------------------------------------------------------- 39200000 +413952s +950.5216s^2 +0.0147517s^3 +0.0000248s^4 --> roots(r.num),roots(r.den) ans = -297.61905 + 0.i -138.88889 + 0.i 0. + 0.i ans = -79.325044 + 6183.6046i -79.325044 - 6183.6046i -297.61905 + 0.i -138.96896 + 0.i ``` although the common root -297.61905 seems evident, the second one is not (-138.8888 vs. -138.96896). This is confirmed by ```scilab d = bezout(rsc.num,rsc.den) --> d=bezout(rsc.num,rsc.den) d = 297.61905 +s ``` You can then divide the numerator and denominator: ```scilab [r1,q1]=pdiv(rsc.num,d) [r2,q2]=pdiv(rsc.den,d) rsimp=q1/q2 --> [r1,q1]=pdiv(rsc.num,d) r1 = -0.0004253 q1 = 0.0000014 +921.984s +6.6382848s^2 --> [r2,q2]=pdiv(rsc.den,d) r2 = 0.0002042 q2 = 131712 +948.3264s +0.0073759s^2 +0.0000248s^3 --> rsimp=q1/q2 rsimp = 0.0000014 +921.984s +6.6382848s^2 --------------------------------------------- 131712 +948.3264s +0.0073759s^2 +0.0000248s^3 ``` ---
**Author:** ![Thaisa](https://avatars.discourse-cdn.com/v4/letter/t/c5a1d2/32.png) [@Thaisa](https://scilab.discourse.group/u/Thaisa)\ **Post date:** [June 26, 2023, 5:38pm UTC](https://scilab.discourse.group/t/simplification-of-rational-polynomials/105/14 "2023-06-26T17:38:49Z")
Thank you so much for your help, @mottelet ! I will implement your scripts, and now I see my problem be solved.