Re: [Scilab-users] improve accuracy of roots
Hello, After a quick search on the internet I have found and translated in Scilab "multroot", a Matlab Package computing polynomial roots and multiplicities. If you are interested proceed to this url: http://bugzilla.scilab.org/show_bug.cgi?id=15349#c9 Enjoy ! S. Le 14/01/2019 à 21:07, Federico Miyara a écrit : Denis, What I meant is that convergence is a limiting process. On average, as the number of iterations rises you´ll be closer to the limit, bu there is no guarantee that any single iteration will bring you any closer; it may be a question of luck. Maybe (though it would require a proof, it is not self-evident for me) in the exact case of a single multiple root as (x - a)^n the convergence process is monotonous, but what if you have (x - a1)* ... * (x - an) where ak are all very similar but not identical, say, with relative differences of the order of those reported by the application of the regular version of roots. Regards, Federico Miyara On 14/01/2019 13:47, CRETE Denis wrote: Thank you Frederico! According to the page you refer to, the method seems to converge more rapidly with this factor equal to the multiplicity of the root. About overshoot, it is well known to occur for |x|^a where a <1. But for a>1, the risk of overshoot with the Newton-Raphson method seems to be very small... Best regards Denis [@@ THALES GROUP INTERNAL @@] Unité Mixte de Physique CNRS / THALES 1 Avenue Augustin Fresnel 91767 Palaiseau CEDEx - France Tel : +33 (0)1 69 41 58 52 Fax : +33 (0)1 69 41 58 78 e-mail : denis.cr...@thalesgroup.com <mailto:%20denis.cr...@thalesgroup.com> http://www.trt.thalesgroup.com/ump-cnrs-thales http://www.research.thalesgroup.com -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : samedi 12 janvier 2019 07:52 À : Users mailing list for Scilab Objet : Re: [Scilab-users] improve accuracy of roots Denis, I've found the correction here, https://en.wikipedia.org/wiki/Newton%27s_method It is useful to accelerate convergence in case of multiple roots, but I guess it is not valid to apply it once to improve accuracy because of the risk of overshoot. Regards, Federico Miyara On 10/01/2019 10:32, CRETE Denis wrote: Hello, I tried this correction to the initial roots z: z-4*(1+z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') ans = -1. - 1.923D-13i -1. + 1.189D-12i -1. - 1.189D-12i -1. - 1.919D-13i // Evaluation of new error, (and defining Z as the intended root, i.e. here Z=-1): z2=z-4*(z-Z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') z2 - Z ans = 2.233D-08 - 1.923D-13i -2.968D-08 + 1.189D-12i -2.968D-08 - 1.189D-12i 2.131D-08 - 1.919D-13i The factor 4 in the correction is a bit obscure to me, but it seems to work also for R=(3+p)^4, again with an accuracy on the roots of a ~2E-8. HTH Denis -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : jeudi 10 janvier 2019 00:32 À :users@lists.scilab.org Objet : [Scilab-users] improve accuracy of roots Dear all, Consider this code: // Define polynomial variable p = poly(0, 'p', 'roots'); // Define fourth degree polynomial R = (1 + p)^4; // Find its roots z = roots(R) The result (Scilab 6.0.1) is z = -1.0001886 -1. + 0.0001886i -1. - 0.0001886i -0.9998114 It should be something closer to -1. -1. -1. -1. Using these roots C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4))) yield seemingly accurate coefficients C = 1. 4. 6. 4. 1. but C - [1 4 6 4 1] shows the actual error: ans = 3.775D-15 1.243D-14 1.155D-14 4.441D-15 0. This is acceptable for the coefficients, but the error in the roots is too large. Somehow the errors cancel out when assembling back the polynomial but each individual zero should be closer to the theoretical value Is there some way to improve the accuracy? Regards, Federico Miyara --- El software de antivirus Avast ha analizado este correo electrónico en busca de virus. https://www.avast.com/antivirus ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users <https://antispam.utc.fr/proxy/2/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/www.avast.com/sig-email?utm_medium=email_source=link_campaign=sig-email_content=emailclient> Libre de virus. www.avast.com <https://antispam.utc.fr/proxy/2/c
Re: [Scilab-users] improve accuracy of roots
Denis, What I meant is that convergence is a limiting process. On average, as the number of iterations rises you´ll be closer to the limit, bu there is no guarantee that any single iteration will bring you any closer; it may be a question of luck. Maybe (though it would require a proof, it is not self-evident for me) in the exact case of a single multiple root as (x - a)^n the convergence process is monotonous, but what if you have (x - a1)* ... * (x - an) where ak are all very similar but not identical, say, with relative differences of the order of those reported by the application of the regular version of roots. Regards, Federico Miyara On 14/01/2019 13:47, CRETE Denis wrote: Thank you Frederico! According to the page you refer to, the method seems to converge more rapidly with this factor equal to the multiplicity of the root. About overshoot, it is well known to occur for |x|^a where a <1. But for a>1, the risk of overshoot with the Newton-Raphson method seems to be very small... Best regards Denis [@@ THALES GROUP INTERNAL @@] Unité Mixte de Physique CNRS / THALES 1 Avenue Augustin Fresnel 91767 Palaiseau CEDEx - France Tel : +33 (0)1 69 41 58 52 Fax : +33 (0)1 69 41 58 78 e-mail : denis.cr...@thalesgroup.com <mailto:%20denis.cr...@thalesgroup.com> http://www.trt.thalesgroup.com/ump-cnrs-thales http://www.research.thalesgroup.com -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : samedi 12 janvier 2019 07:52 À : Users mailing list for Scilab Objet : Re: [Scilab-users] improve accuracy of roots Denis, I've found the correction here, https://en.wikipedia.org/wiki/Newton%27s_method It is useful to accelerate convergence in case of multiple roots, but I guess it is not valid to apply it once to improve accuracy because of the risk of overshoot. Regards, Federico Miyara On 10/01/2019 10:32, CRETE Denis wrote: Hello, I tried this correction to the initial roots z: z-4*(1+z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') ans = -1. - 1.923D-13i -1. + 1.189D-12i -1. - 1.189D-12i -1. - 1.919D-13i // Evaluation of new error, (and defining Z as the intended root, i.e. here Z=-1): z2=z-4*(z-Z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') z2 - Z ans = 2.233D-08 - 1.923D-13i -2.968D-08 + 1.189D-12i -2.968D-08 - 1.189D-12i 2.131D-08 - 1.919D-13i The factor 4 in the correction is a bit obscure to me, but it seems to work also for R=(3+p)^4, again with an accuracy on the roots of a ~2E-8. HTH Denis -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : jeudi 10 janvier 2019 00:32 À : users@lists.scilab.org Objet : [Scilab-users] improve accuracy of roots Dear all, Consider this code: // Define polynomial variable p = poly(0, 'p', 'roots'); // Define fourth degree polynomial R = (1 + p)^4; // Find its roots z = roots(R) The result (Scilab 6.0.1) is z = -1.0001886 -1. + 0.0001886i -1. - 0.0001886i -0.9998114 It should be something closer to -1. -1. -1. -1. Using these roots C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4))) yield seemingly accurate coefficients C = 1. 4. 6. 4. 1. but C - [1 4 6 4 1] shows the actual error: ans = 3.775D-15 1.243D-14 1.155D-14 4.441D-15 0. This is acceptable for the coefficients, but the error in the roots is too large. Somehow the errors cancel out when assembling back the polynomial but each individual zero should be closer to the theoretical value Is there some way to improve the accuracy? Regards, Federico Miyara --- El software de antivirus Avast ha analizado este correo electrónico en busca de virus. https://www.avast.com/antivirus ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users
Re: [Scilab-users] improve accuracy of roots
Thank you Frederico! According to the page you refer to, the method seems to converge more rapidly with this factor equal to the multiplicity of the root. About overshoot, it is well known to occur for |x|^a where a <1. But for a>1, the risk of overshoot with the Newton-Raphson method seems to be very small... Best regards Denis [@@ THALES GROUP INTERNAL @@] Unité Mixte de Physique CNRS / THALES 1 Avenue Augustin Fresnel 91767 Palaiseau CEDEx - France Tel : +33 (0)1 69 41 58 52 Fax : +33 (0)1 69 41 58 78 e-mail : denis.cr...@thalesgroup.com <mailto:%20denis.cr...@thalesgroup.com> http://www.trt.thalesgroup.com/ump-cnrs-thales http://www.research.thalesgroup.com -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : samedi 12 janvier 2019 07:52 À : Users mailing list for Scilab Objet : Re: [Scilab-users] improve accuracy of roots Denis, I've found the correction here, https://en.wikipedia.org/wiki/Newton%27s_method It is useful to accelerate convergence in case of multiple roots, but I guess it is not valid to apply it once to improve accuracy because of the risk of overshoot. Regards, Federico Miyara On 10/01/2019 10:32, CRETE Denis wrote: > Hello, > I tried this correction to the initial roots z: > > z-4*(1+z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') > ans = > >-1. - 1.923D-13i >-1. + 1.189D-12i >-1. - 1.189D-12i >-1. - 1.919D-13i > > // Evaluation of new error, (and defining Z as the intended root, i.e. here > Z=-1): > z2=z-4*(z-Z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') > z2 - Z > ans = > > 2.233D-08 - 1.923D-13i >-2.968D-08 + 1.189D-12i >-2.968D-08 - 1.189D-12i > 2.131D-08 - 1.919D-13i > > The factor 4 in the correction is a bit obscure to me, but it seems to work > also for R=(3+p)^4, again with an accuracy on the roots of a ~2E-8. > > HTH > Denis > > -Message d'origine- > De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico > Miyara > Envoyé : jeudi 10 janvier 2019 00:32 > À : users@lists.scilab.org > Objet : [Scilab-users] improve accuracy of roots > > > Dear all, > > Consider this code: > > // Define polynomial variable > p = poly(0, 'p', 'roots'); > > // Define fourth degree polynomial > R = (1 + p)^4; > > // Find its roots > z = roots(R) > > The result (Scilab 6.0.1) is > >z = > > -1.0001886 > -1. + 0.0001886i > -1. - 0.0001886i > -0.9998114 > > It should be something closer to > > -1. > -1. > -1. > -1. > > Using these roots > > C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4))) > > yield seemingly accurate coefficients >C = > > 1. 4. 6. 4. 1. > > > but > > C - [1 4 6 4 1] > > shows the actual error: > > ans = > > 3.775D-15 1.243D-14 1.155D-14 4.441D-15 0. > > This is acceptable for the coefficients, but the error in the roots is > too large. Somehow the errors cancel out when assembling back the > polynomial but each individual zero should be closer to the theoretical > value > > Is there some way to improve the accuracy? > > Regards, > > Federico Miyara > > > > > --- > El software de antivirus Avast ha analizado este correo electrónico en busca > de virus. > https://www.avast.com/antivirus > > ___ > users mailing list > users@lists.scilab.org > http://lists.scilab.org/mailman/listinfo/users > ___ > users mailing list > users@lists.scilab.org > http://lists.scilab.org/mailman/listinfo/users > > ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users
Re: [Scilab-users] improve accuracy of roots
Denis, I've found the correction here, https://en.wikipedia.org/wiki/Newton%27s_method It is useful to accelerate convergence in case of multiple roots, but I guess it is not valid to apply it once to improve accuracy because of the risk of overshoot. Regards, Federico Miyara On 10/01/2019 10:32, CRETE Denis wrote: Hello, I tried this correction to the initial roots z: z-4*(1+z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') ans = -1. - 1.923D-13i -1. + 1.189D-12i -1. - 1.189D-12i -1. - 1.919D-13i // Evaluation of new error, (and defining Z as the intended root, i.e. here Z=-1): z2=z-4*(z-Z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') z2 - Z ans = 2.233D-08 - 1.923D-13i -2.968D-08 + 1.189D-12i -2.968D-08 - 1.189D-12i 2.131D-08 - 1.919D-13i The factor 4 in the correction is a bit obscure to me, but it seems to work also for R=(3+p)^4, again with an accuracy on the roots of a ~2E-8. HTH Denis -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : jeudi 10 janvier 2019 00:32 À : users@lists.scilab.org Objet : [Scilab-users] improve accuracy of roots Dear all, Consider this code: // Define polynomial variable p = poly(0, 'p', 'roots'); // Define fourth degree polynomial R = (1 + p)^4; // Find its roots z = roots(R) The result (Scilab 6.0.1) is z = -1.0001886 -1. + 0.0001886i -1. - 0.0001886i -0.9998114 It should be something closer to -1. -1. -1. -1. Using these roots C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4))) yield seemingly accurate coefficients C = 1. 4. 6. 4. 1. but C - [1 4 6 4 1] shows the actual error: ans = 3.775D-15 1.243D-14 1.155D-14 4.441D-15 0. This is acceptable for the coefficients, but the error in the roots is too large. Somehow the errors cancel out when assembling back the polynomial but each individual zero should be closer to the theoretical value Is there some way to improve the accuracy? Regards, Federico Miyara --- El software de antivirus Avast ha analizado este correo electrónico en busca de virus. https://www.avast.com/antivirus ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users
Re: [Scilab-users] improve accuracy of roots
Denis, Thank you. If this were really a general solution it would be great, since it improves the root accuracy by several orders, but I don't fully get the rationale behind this method. It seems you are trying to apply a variant of the Raphson-Newton method, aren't you? However, in cases like this, in which there are repeated roots, the derivative approaches zero as you get closer to the root, but the polynomial goes to zero faster. In that case the factor 4 may imlpy that the next approximation gets closer to the actual root. But this is speculation. It could also overshoot. Regards, Federico Miyara On 10/01/2019 10:32, CRETE Denis wrote: Hello, I tried this correction to the initial roots z: z-4*(1+z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') ans = -1. - 1.923D-13i -1. + 1.189D-12i -1. - 1.189D-12i -1. - 1.919D-13i // Evaluation of new error, (and defining Z as the intended root, i.e. here Z=-1): z2=z-4*(z-Z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') z2 - Z ans = 2.233D-08 - 1.923D-13i -2.968D-08 + 1.189D-12i -2.968D-08 - 1.189D-12i 2.131D-08 - 1.919D-13i The factor 4 in the correction is a bit obscure to me, but it seems to work also for R=(3+p)^4, again with an accuracy on the roots of a ~2E-8. HTH Denis -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : jeudi 10 janvier 2019 00:32 À :users@lists.scilab.org Objet : [Scilab-users] improve accuracy of roots Dear all, Consider this code: // Define polynomial variable p = poly(0, 'p', 'roots'); // Define fourth degree polynomial R = (1 + p)^4; // Find its roots z = roots(R) The result (Scilab 6.0.1) is z = -1.0001886 -1. + 0.0001886i -1. - 0.0001886i -0.9998114 It should be something closer to -1. -1. -1. -1. Using these roots C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4))) yield seemingly accurate coefficients C = 1. 4. 6. 4. 1. but C - [1 4 6 4 1] shows the actual error: ans = 3.775D-15 1.243D-14 1.155D-14 4.441D-15 0. This is acceptable for the coefficients, but the error in the roots is too large. Somehow the errors cancel out when assembling back the polynomial but each individual zero should be closer to the theoretical value Is there some way to improve the accuracy? Regards, Federico Miyara --- El software de antivirus Avast ha analizado este correo electrónico en busca de virus. https://www.avast.com/antivirus ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users
Re: [Scilab-users] improve accuracy of roots
Hi, This behavior has already been discussed here : http://bugzilla.scilab.org/show_bug.cgi?id=15349 S. Le 10/01/2019 à 14:32, CRETE Denis a écrit : Hello, I tried this correction to the initial roots z: z-4*(1+z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') ans = -1. - 1.923D-13i -1. + 1.189D-12i -1. - 1.189D-12i -1. - 1.919D-13i // Evaluation of new error, (and defining Z as the intended root, i.e. here Z=-1): z2=z-4*(z-Z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') z2 - Z ans = 2.233D-08 - 1.923D-13i -2.968D-08 + 1.189D-12i -2.968D-08 - 1.189D-12i 2.131D-08 - 1.919D-13i The factor 4 in the correction is a bit obscure to me, but it seems to work also for R=(3+p)^4, again with an accuracy on the roots of a ~2E-8. HTH Denis -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : jeudi 10 janvier 2019 00:32 À : users@lists.scilab.org Objet : [Scilab-users] improve accuracy of roots Dear all, Consider this code: // Define polynomial variable p = poly(0, 'p', 'roots'); // Define fourth degree polynomial R = (1 + p)^4; // Find its roots z = roots(R) The result (Scilab 6.0.1) is z = -1.0001886 -1. + 0.0001886i -1. - 0.0001886i -0.9998114 It should be something closer to -1. -1. -1. -1. Using these roots C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4))) yield seemingly accurate coefficients C = 1. 4. 6. 4. 1. but C - [1 4 6 4 1] shows the actual error: ans = 3.775D-15 1.243D-14 1.155D-14 4.441D-15 0. This is acceptable for the coefficients, but the error in the roots is too large. Somehow the errors cancel out when assembling back the polynomial but each individual zero should be closer to the theoretical value Is there some way to improve the accuracy? Regards, Federico Miyara --- El software de antivirus Avast ha analizado este correo electrónico en busca de virus. https://antispam.utc.fr/proxy/2/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/www.avast.com/antivirus ___ users mailing list users@lists.scilab.org https://antispam.utc.fr/proxy/1/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org https://antispam.utc.fr/proxy/1/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/lists.scilab.org/mailman/listinfo/users -- Stéphane Mottelet Ingénieur de recherche EA 4297 Transformations Intégrées de la Matière Renouvelable Département Génie des Procédés Industriels Sorbonne Universités - Université de Technologie de Compiègne CS 60319, 60203 Compiègne cedex Tel : +33(0)344234688 http://www.utc.fr/~mottelet ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users
Re: [Scilab-users] improve accuracy of roots
Hello, I tried this correction to the initial roots z: z-4*(1+z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') ans = -1. - 1.923D-13i -1. + 1.189D-12i -1. - 1.189D-12i -1. - 1.919D-13i // Evaluation of new error, (and defining Z as the intended root, i.e. here Z=-1): z2=z-4*(z-Z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))') z2 - Z ans = 2.233D-08 - 1.923D-13i -2.968D-08 + 1.189D-12i -2.968D-08 - 1.189D-12i 2.131D-08 - 1.919D-13i The factor 4 in the correction is a bit obscure to me, but it seems to work also for R=(3+p)^4, again with an accuracy on the roots of a ~2E-8. HTH Denis -Message d'origine- De : users [mailto:users-boun...@lists.scilab.org] De la part de Federico Miyara Envoyé : jeudi 10 janvier 2019 00:32 À : users@lists.scilab.org Objet : [Scilab-users] improve accuracy of roots Dear all, Consider this code: // Define polynomial variable p = poly(0, 'p', 'roots'); // Define fourth degree polynomial R = (1 + p)^4; // Find its roots z = roots(R) The result (Scilab 6.0.1) is z = -1.0001886 -1. + 0.0001886i -1. - 0.0001886i -0.9998114 It should be something closer to -1. -1. -1. -1. Using these roots C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4))) yield seemingly accurate coefficients C = 1. 4. 6. 4. 1. but C - [1 4 6 4 1] shows the actual error: ans = 3.775D-15 1.243D-14 1.155D-14 4.441D-15 0. This is acceptable for the coefficients, but the error in the roots is too large. Somehow the errors cancel out when assembling back the polynomial but each individual zero should be closer to the theoretical value Is there some way to improve the accuracy? Regards, Federico Miyara --- El software de antivirus Avast ha analizado este correo electrónico en busca de virus. https://www.avast.com/antivirus ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users
[Scilab-users] improve accuracy of roots
Dear all, Consider this code: // Define polynomial variable p = poly(0, 'p', 'roots'); // Define fourth degree polynomial R = (1 + p)^4; // Find its roots z = roots(R) The result (Scilab 6.0.1) is z = -1.0001886 -1. + 0.0001886i -1. - 0.0001886i -0.9998114 It should be something closer to -1. -1. -1. -1. Using these roots C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4))) yield seemingly accurate coefficients C = 1. 4. 6. 4. 1. but C - [1 4 6 4 1] shows the actual error: ans = 3.775D-15 1.243D-14 1.155D-14 4.441D-15 0. This is acceptable for the coefficients, but the error in the roots is too large. Somehow the errors cancel out when assembling back the polynomial but each individual zero should be closer to the theoretical value Is there some way to improve the accuracy? Regards, Federico Miyara --- El software de antivirus Avast ha analizado este correo electrónico en busca de virus. https://www.avast.com/antivirus ___ users mailing list users@lists.scilab.org http://lists.scilab.org/mailman/listinfo/users
