Gram-Schmidt orthogonalization and approximation We want to find the polynomial of degree 4 that is the best approximation to NiMtSSRjb3NHNiI2I0kieEdGJQ== on the interval NiM3JCwkSSNQaUdJKnByb3RlY3RlZEdGJiEiIkYl . To do this we first take the standard basis for polynomials, { NiM2JyIiIkkieEc2IiokRiUiIiMqJEYlIiIkKiRGJSIiJQ== } and use Gram-Schmidt to construct an orthonormal basis. While we could do this more efficiently using built-in commands of maple, we will just use maple as a calculator.
First, we define our inner product as a function on two functions. Our inner product is the usual one, that is NiMvLUkkPCw+RzYiNiRJInBHRiY2I0kicUdGJi1JJGludEdGJjYkKiYtRig2I0kieEdGJiIiIi1GKkYwRjIvRjE7LCRJI1BpR0kqcHJvdGVjdGVkR0Y4ISIiRjc= . Note that ||p-q|| is small precisely when the values of NiMtSSJwRzYiNiNJInhHRiU= are close to those of NiMtSSJxRzYiNiNJInhHRiU= for all x in the interval NiM3JCwkSSNQaUdJKnByb3RlY3RlZEdGJiEiIkYl . For simplicity, we'll operate on expressions in NiNJInhHNiI= rather than general functions. InnerProd:=(p,q) -> int(p*q, x=-Pi..Pi); NiM+SSpJbm5lclByb2RHNiJmKjYkSSJwR0YlSSJxR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSRpbnRHNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJTYkKiY5JCIiIjklRjUvSSJ4R0YlOywkSSNQaUdGMCEiIkY7RiVGJUYl We'll also want to compute the norm of our functions. However, Norm is a maple built-in function, so we'll be a little more formal and call Norm by his full name. Norman:=p->sqrt(InnerProd(p,p)); NiM+SSdOb3JtYW5HNiJmKjYjSSJwR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSVzcXJ0RzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiU2Iy1JKklubmVyUHJvZEdGJTYkOSRGNUYlRiVGJQ== Now we do Gram-Schmidt to construct an orthonormal basis (with respect to this inner product) for polynomials of degree 4. The first element of this basis is the constant function NiMvLSZJImZHNiI2IyIiITYjSSJ4R0YnIiIi . However, we need to normalize it to be of unit length in our norm. f[0] := 1/Norman(1);
NiM+JkkiZkc2IjYjIiIhLCQqJiIiIyMiIiJGK0kjUGlHSSpwcm90ZWN0ZWRHRi8jISIiRitGLA== Next, we confirm that NiMvLSZJImZHNiI2IyIiIjYjSSJ4R0YnRis= is orthogonal to NiMmSSJmRzYiNiMiIiE= . Since we want an orthonormal basis, we still want to divide NiMmSSJmRzYiNiMiIiI= by its length InnerProd(x,f[0]); NiMiIiE= f[1]:= x/Norman(x); NiM+JkkiZkc2IjYjIiIiLCQqKEkieEdGJkYoIiInI0YoIiIjSSNQaUdJKnByb3RlY3RlZEdGMCMhIiRGLkYt Notice that while NiMtJkkiZkc2IjYjIiIiNiNJInhHRiY= is orthogonal to NiMqJEkieEc2IiIiIw== , NiMmSSJmRzYiNiMiIiE= is not. InnerProd(x^2,f[0]); NiMsJComIiIjIyIiIkYlSSNQaUdJKnByb3RlY3RlZEdGKSMiIiZGJSNGJyIiJA== InnerProd(x^2,f[1]); NiMiIiE= This means we have to use Gram-Schmidt to find a function in the span of NiM8JSZJImZHNiI2IyIiISZGJTYjIiIiKiRJInhHRiYiIiM= which is orthogonal to both NiMmSSJmRzYiNiMiIiE= and NiMmSSJmRzYiNiMiIiI= . Note that since NiMmSSJmRzYiNiMiIiE= and NiMmSSJmRzYiNiMiIiI= are already normalized, we don't have to divide by their lengths. ftemp:= x^2 - InnerProd(x^2,f[0])*f[0] - InnerProd(x^2,f[1])*f[1];
f[2] := collect(ftemp/Norman(ftemp),x); NiM+SSZmdGVtcEc2IiwmKiRJInhHRiUiIiMiIiIqJEkjUGlHSSpwcm90ZWN0ZWRHRi1GKSMhIiIiIiQ= NiM+JkkiZkc2IjYjIiIjLCYqKCIjNSMiIiJGKEkjUGlHSSpwcm90ZWN0ZWRHRi8jISImRihJInhHRiZGKCMiIiQiIiUqJkYuIyEiIkYoRitGLCNGOEY1 Now we play the same game with NiMqJEkieEc2IiIiJA== and NiMqJEkieEc2IiIiJQ== . Note that if you were working these integrals out by hand, you'd quickly realize that the even powers are orthogonal to the odd ones. ftemp:= x^3 - InnerProd(x^3,f[0])*f[0] - InnerProd(x^3,f[1])*f[1] - InnerProd(x^3,f[2])*f[2]:
f[3] := collect(ftemp/Norman(ftemp),x); NiM+JkkiZkc2IjYjIiIkLCYqKCIjOSMiIiIiIiNJI1BpR0kqcHJvdGVjdGVkR0YwIyEiKEYuSSJ4R0YmRigjIiImIiIlKihGLyMhIiRGLkYrRixGM0YtI0Y5RjY= ftemp:= x^4 - InnerProd(x^4,f[0])*f[0] - InnerProd(x^4,f[1])*f[1] - InnerProd(x^4,f[2])*f[2] - InnerProd(x^4,f[3])*f[3]:
f[4] := collect(ftemp/Norman(ftemp),x); NiM+JkkiZkc2IjYjIiIlLCgqKCIiIyMiIiJGK0kjUGlHSSpwcm90ZWN0ZWRHRi8jISIqRitJInhHRiZGKCMiJDAiIiM7KihGLiMhIiZGK0YrRixGMkYrIyEjWCIiKSomRitGLEYuIyEiIkYrIyIiKkY1 So, we now have an orthonormal basis for fourth degree polynomials on NiM3JCwkSSNQaUdJKnByb3RlY3RlZEdGJiEiIkYl . Just in case you are disturbed by all the NiNJI1BpR0kqcHJvdGVjdGVkR0Yk s and square roots, let's also write their approximate values, and also draw their graphs. OrthoBasis:=[f[0],f[1],f[2],f[3],f[4]]; NiM+SStPcnRob0Jhc2lzRzYiNycsJComIiIjIyIiIkYpSSNQaUdJKnByb3RlY3RlZEdGLSMhIiJGKUYqLCQqKEkieEdGJUYrIiInRipGLCMhIiRGKUYqLCYqKCIjNUYqRiwjISImRilGMkYpIyIiJCIiJSomRixGLkY4RiojRi9GPSwmKigiIzlGKkYsIyEiKEYpRjJGPCMiIiZGPSooRixGNEZCRipGMkYrI0Y1Rj0sKCooRilGKkYsIyEiKkYpRjJGPSMiJDAiIiM7KihGLEY5RilGKkYyRikjISNYIiIpRigjIiIqRk8= evalf(OrthoBasis,4); NiM3JyQiJSkpUiEiJSwkSSJ4RzYiJCIlKD4jRiYsJiokRigiIiMkIiVhOEYmJCElZVdGJiIiIiwmKiRGKCIiJCQiJSVcKSEiJkYoJCElTl1GJiwoKiRGKCIiJSQiJWZgRjlGLSQhJVNYRiYkIiUnWyVGJkYz plot(OrthoBasis,x=-Pi..Pi, color=[black,red,blue,green,brown],thickness=2); 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U$!3oh?'=1`2f#F/7$FX$!3>\m2D>%Ho"F/7$Fen$!3[!o*p1oU'H)Fgy7$Fhn$"3*[*>(z&)=D#=Fgy7$F[o$"3qKj@$)e`35F/7$F^o$"3C3a-2GK.>F/7$Fao$"3*y(oao<,8EF/7$Fdo$"3M"fK0Xk4G$F/7$Fgo$"3!=v!))e:b#z$F/7$Fjo$"3VRyG4I!H=%F/7$$!3T"4gMblG)>F/$"3%pEmg3R,J%F/7$F]p$"3F*4\;Y,OS%F/7$$!3b.@Z/"\$ypFgy$"3<%fe@yofY%F/7$F`p$"3@j-0<p0)[%F/7$$"3p0t!3)GF;mFgy$"3$\imcp/#oWF/7$Fdp$"3J*f)eX.)[S%F/7$$"312**yIK@d>F/$"3&zs%>lIp9VF/7$Fgp$"3kO^c&yaB>%F/7$Fjp$"32![+"oYi@QF/7$F]q$"3Cwk'>j=UH$F/7$F`q$"3(GZ,fp%z\EF/7$Fcq$"3oWlcz-n@>F/7$Ffq$"38^'3.g&=;5F/7$Fiq$"3x*[6Mb@<S"Fgy7$F\r$!3gSwS]D(*)H)Fgy7$F_r$!3X$y62h50r"F/7$Fbr$!3@Ze<1**eQEF/7$Fer$!3&QdQbgY8W$F/7$Fhr$!3]'R9*\3%R;%F/7$F[s$!3xf'f;&)4#4ZF/7$$"3/UPl\c"o*=F,$!3eFdgMdg9\F/7$F^s$!3cesd&Q)Qb]F/7$$"3i0+\jM/(*>F,$!3k^Z27!py4&F/7$$"3H>eDi5iH?F,$!3FpbX7")pA^F/7$$"3'Hj@5m)>i?F,$!3=sei]w(*G^F/7$Fas$!3[D!G%f]z:^F/7$$"3LPB")f:4G@F,$!3Xdo+C'>73&F/7$$"3-Gs$)foSh@F,$!36U/K5QJC]F/7$$"3s=@')f@s%>#F,$!3J@90$\aS%\F/7$Fds$!3ZCDCaESR[F/7$$"35:t#\K;TH#F,$!3='H]yR,bb%F/7$Fgs$!3#f2!oHv0jTF/7$$"3M?UD/["4U#F,$!3[$o73O@$*p$F/7$Fjs$!3+/FRW%>(HJF/7$$"31?#=([fA^DF,$!3Qq6'))4avL#F/7$F]t$!3sa^&RJzgQ"F/7$$"36TH4t41$o#F,$!3%4C<YG">4RFgy7$F`t$"3/@'3c)QD*[(Fgy7$F^bm$"3%=*[g(p<J8#F/7$Fct$"3g.:[+e'=q$F/7$$"3D#)G=JKz4HF,$"3IPiIYD.@XF/7$Ffbm$"3ZyMD#*f6'Q&F/7$$"3B43mqDJtHF,$"3k<")\t_K)H'F/7$Fft$"3#3a#*4l#))esF/7$$"3;HB"HM-#RIF,$"3Xn!=k(*RkM)F/7$F^cm$"3-#)>jJ(3G\*F/7$$"3#HWPzaiu5$F,$"32nS0[c&*p5F,7$Fit$"3?&)GCHo#o>"F,-F\u6&F^u$"1_MmX%)eqk!#;$"2wmoV()eqk"F,F`in-%+AXESLABELSG6$Q"x6"Q!Ffin-%%VIEWG6$;$!+aEfTJ!"*$"+aEfTJF^jn;$!2)**=JE'[05"F_in$"2Gy-oSt=C"F_in-%*THICKNESSG6#""# The orthonormal basis we have constructed is closely related to the Legendre polynomials , one of several families of orthogonal polynomials . Now we can turn to our minimization problem, which becomes easy with an orthonormal basis. Recall that the minimum with respect to our norm is just the projection of the target function onto the subspace spanned by the basis. With an orthogonal basis, this is trivial since it is just a sum of inner products times each basis element.
While we know that the inner product of NiMtSSRjb3NHNiI2I0kieEdGJQ== with NiMmSSJmRzYiNiMiIiE= , NiMmSSJmRzYiNiMiIiI= , and NiMmSSJmRzYiNiMiIiQ= is zero, we write them anyway. approx := collect(
InnerProd(cos(x),f[0])*f[0] + InnerProd(cos(x),f[1])*f[1] + InnerProd(cos(x),f[2])*f[2] + InnerProd(cos(x),f[3])*f[3] + InnerProd(cos(x),f[4])*f[4], x);
evalf(approx); NiM+SSdhcHByb3hHNiIsKiooLCYqJEkjUGlHSSpwcm90ZWN0ZWRHRisiIiNGLCEjQCIiIkYuRiohIilJInhHRiUiIiUjISV2OiIiKSomLCYqJEYqISIlIyEjWEYsKiZGKEYuRiohIicjIiR2J0YxRi5GMEYsRi4qJEYqISIjIyIjOkYsKiZGKEYuRipGOCMhJE4iRjQ= NiMsKCokSSJ4RzYiIiIlJCIrPz8pZmgjISM2KiRGJSIiIyQhKyI0eUdfJSEjNSQiKyMqUUUkeSpGLyIiIg== The fit is pretty good, differing by less than .07 on the whole interval. plot([cos(x),approx], x=-Pi..Pi, color=[red, green], thickness=2); LSUlUExPVEc2Jy0lJ0NVUlZFU0c2JDdlbjckJCEzKioqKio0dGsjZlRKISM8JCEiIiIiITckJCEzNCdcOCFvWzZ0SUYsJCEzRGRSI1w3amwoKiohIz03JCQhM3ciKnByKTNQWSskRiwkITNuZ0BwbEJPMSoqRjU3JCQhMykqKnoqNFJcMFhIRiwkITN5PyRHPV8mWzIpKkY1NyQkITNAM0VbKnlzYSlHRiwkITMuLz8oUjMtUW4qRjU3JCQhM2ZYJikpPjJnOXYjRiwkITNrcmpSZGtlWyMqRjU3JCQhMyk0NTc3LmZsaCNGLCQhM20iKSkzOyozMmAnKUY1NyQkITN2IlFNOjoqSCNbI0YsJCEzWHQlKik0K19VIXpGNTckJCEzeVxBaGNJI3lOI0YsJCEzWmRWS0xUZCMzKEY1NyQkITMmZT1kLUZOKkdBRiwkITNnel5TM2RYO2hGNTckJCEzdyUqR0JQJFJjNCNGLCQhM0hCUnI7Ong1XUY1NyQkITNtQ2omZikzeGk+RiwkITNXQi9SLydSLCNRRjU3JCQhMydvcjRBTio0RT1GLCQhM0U1TD9PaVFEREY1NyQkITNRUVEqMyoqPWRxIkYsJCEzSlhqZCR5T15NIkY1NyQkITNkIT5qZ0AqPnE6RiwkIjMpPjpPQSdwNXJmISNANyQkITNgO20sJSlIN005RiwkIjNnKillKVFSI1tpOEY1NyQkITN2KDRGLEspKUhJIkYsJCIzWyI0Pz89Iz1ZRUY1NyQkITNTKjQhUiNbMFI9IkYsJCIzJVJmdCdmKjNKeCRGNTckJCEzcFlAUXFXSVU1RiwkIjMjW1oyJClIOkIvJkY1NyQkITNXcDN3JFIpXEIjKkY1JCIzSS0yZShcKltSZ0Y1NyQkITNFSmM4bzM5R3lGNSQiM0ZWPSVSTzskKjMoRjU3JCQhMyc9W0JQLThJZidGNSQiMzpQIVIjPlU/L3pGNTckJCEzQ0IzPEA/KXlCJkY1JCIzYmUyKSlcbklmJylGNTckJCEzYCopUjxZb1paUkY1JCIzYV0zMiFHTzRCKkY1NyQkITM4VHFYPVcyLEVGNSQiMzpVcFo4TmlqJypGNTckJCEzVCI0Z01ibEcpPkY1JCIzdzVwIyp5YTAvKSpGNTckJCEzcVRKWSlvY1lPIkY1JCIzeF90SEArLjIqKkY1NyQkITNiLkBaLyJcJHlwISM+JCIzWU5JRCw3bXYqKkY1NyQkITM4LHo3VktKLEohIz8kIjNnJlE+JDQ+JioqKioqRjU3JCQiM3AwdCEzKUdGO21GYXQkIjNdai4jKVwvN3kqKkY1NyQkIjM6U0ZmM3hFYThGNSQiMyR5O2J1LFElMyoqRjU3JCQiMzEyKip5SUtAZD5GNSQiM1BxN2xnazI0KSpGNTckJCIzJ1IyKClIdmUsYyNGNSQiM24sS1BsYTF1JypGNTckJCIzSWN3UjxUYmlRRjUkIjNwOmEuNkFFaiMqRjU3JCQiMyo0Kj1Kb2YwM19GNSQiMzFYMmJJaD11JylGNTckJCIzcF0xWElxT0NsRjUkIjMmSDglcHF4MVl6RjU3JCQiMyVIb0JsbWx6eihGNSQiMzliQih5JGZjNXJGNTckJCIzXnUnZiM0KXo/QCpGNSQiMzFkdyRHVidlW2dGNTckJCIzOUh2PEVDRls1RiwkIjNrLiUzSVUpbyEqXEY1NyQkIjNObGolRyUzJVI9IkYsJCIzMHUhXCkpWyJ5c1BGNTckJCIzXC0pUkVmd29JIkYsJCIzKUhWZDFmbCczRUY1NyQkIjNjTTJ2UXlGVDlGLCQiM19mRy4oeW06SCJGNTckJCIzeigzMnMkKlF4YyJGLCQiMyMpZmdlQ1tRZElGZ3Q3JCQiMzcrSz9CcyMqKnAiRiwkITMqXHVmd0JCeEciRjU3JCQiMzwjSCRlTWE7SD1GLCQhMzg5YzRHOC9iREY1NyQkIjMlPj5DWidlWWs+RiwkITNfLT1xbSQpek5RRjU3JCQiM2tZdXlmaXglNCNGLCQhMyNvSCFlYzBJLl1GNTckJCIzVDRxKSlmdS5HQUYsJCEzJD40Qip6Lk40aEY1NyQkIjN6P3cnKio9Jj5nQkYsJCEzIXlzWHZPKyQqNChGNTckJCIzISo+M2E9V2oiWyNGLCQhM1l5bC53JHksIXpGNTckJCIzQT9jKil5dSIzaSNGLCQhMzMzKkcyXVBWbilGNTckJCIzLGktSG5XSVhGRiwkITNHOkRVUzQrRCMqRjU3JCQiM3Y9UldoTi55R0YsJCEzdjhbLWcpKW9hJypGNTckJCIzdVg9IzQhSGJUSEYsJCEzbiE9NFZ6J2UrKSpGNTckJCIzc3MoKlJTQTIwSUYsJCEzKFFueTNgYnAhKipGNTckJCIzLycpW1VYQ0x0SUYsJCEzXXVlYms8cncqKkY1NyQkIjMhKSoqKlwvbCNmVEpGLEYtLSUmQ09MT1JHNiYlJFJHQkckIiM1Ri4kRi9GLkZqXWwtRiY2JDdcbzckRiokITNIKEh1SGE1UFAqRjU3JEYxJCEzVSdlIUdHWDEqZipGNTckRjckITMnUnBUMCNlIXlzKkY1NyQkITNsJlIzUixZWyhIRiwkITNhcSlSK0c4X3YqRjU3JEY8JCEzTTF3OzVHKmZ3KkY1NyQkITNLLzdIa1FFOkhGLCQhMyVRRmknZWRqZygqRjU3JEZBJCEzKVtbJCl6X0cnUigqRjU3JCQhMyFwZE4yVm0lPUdGLCQhM29sIzNFenZ3aipGNTckRkYkITM9Kj1OQCJRT2slKkY1NyQkITNHQi5nXiY0U28jRiwkITNPUWBNYHA6QiMqRjU3JEZLJCEzUyVHYyNmej8/KilGNTckRlAkITNsbEwmXE46TjopRjU3JEZVJCEzJypmYVQsUyJmRihGNTckRlokITNTdUBdOWk8SWlGNTckRmluJCEzJz5oXSJ5clRNXUY1NyRGXm8kITMhKTQueHozWmVQRjU3JEZjbyQhMzpzMixtcigqKlEjRjU3JEZobyQhMyFwYHNSPi46OyJGNTckRl1wJCIzKTRLYDokcCo+QSNGYXQ3JEZjcCQiM0xiIjQtcSVmKGUiRjU3JEZocCQiMzA2IjNicGklZUdGNTckRl1xJCIzdT43Iz0yInlkUkY1NyRGYnEkIjMqcGApM0YpcCR5XkY1NyRGZ3EkIjNbWVgjb1hbWzcnRjU3JEZcciQiM0JQN1I5JSopKTRyRjU3JEZhciQiMywiSHE5MiZwbXlGNTckRmZyJCIzcWlFdiNRJTNpJilGNTckRltzJCIzL2RXQFdeJFszKkY1NyRGYHMkIjN5by93QT9ZeSUqRjU3JEZlcyQiMyg0c3FMeVJlZypGNTckRmpzJCIzWUk/XkJjNypwKkY1NyRGX3QkIjMjMzs3PipbQ2goKkY1NyRGZXQkIjMpcChmRDIvQSR5KkY1NyRGW3UkIjNtdkFuIyoqcE13KkY1NyRGYHUkIjMxN20hR1krL3EqRjU3JEZldSQiMygqW1BTSTFSNScqRjU3JEZqdSQiM0BdMyQqUidSelsqRjU3JEZfdiQiM05bXiE+KFFJOSIqRjU3JEZkdiQiM2c+U29LN3R2JilGNTckRml2JCIzJyo+I0dnJlxSMHpGNTckRl53JCIzd1MhW2p5NChIckY1NyRGY3ckIjNvNV8ob25MTTgnRjU3JEZodyQiM18lKnpyO0UySF5GNTckRl14JCIzKnlDLT15anUmUkY1NyRGYngkIjNEKDRBYDgxOyNHRjU3JEZneCQiMyZveFl5I1F6OzpGNTckRlx5JCIzJ3pdMylRNnVyQ0ZhdDckRmF5JCEzSXJBSmVXQC02RjU3JEZmeSQhM2Q5Im9yaSEzQENGNTckRlt6JCEzYmFLci8jXF54JEY1NyRGYHokITNML25EWGhPRV1GNTckRmV6JCEzXiNlKWZ1W1pBaUY1NyRGanokITMpXDBUT2laUkgoRjU3JEZfW2wkITN3V3Q2cmdBXCIpRjU3JEZkW2wkITMpKjN3WCFHaTUlKilGNTckRmlbbCQhM056VFhXJVtdVypGNTckJCIzUSE0blYscDsiR0YsJCEzViJmdz1jZEtpKkY1NyRGXlxsJCEzcCMqWzk2cyo+dCpGNTckJCIzRCMpRz1KS3o0SEYsJCEzdCZlS0UhcCR6dipGNTckRmNcbCQhM1k0KG8kKWUhPm0oKkY1NyQkIjNCNDNtcURKdEhGLCQhM18lXChvSilwaHYqRjU3JEZoXGwkITNuLFE5Myl6c3MqRjU3JEZdXWwkITNHemxOIXkvJilmKkY1NyRGYl1sJCEzNSFlSS5gNVBQKkY1LUZlXWw2JkZnXWxGal1sRmhdbEZqXWwtJStBWEVTTEFCRUxTRzYkUSJ4NiJRIUZiW20tJSVWSUVXRzYkOyQhK2FFZlRKISIqJCIrYUVmVEpGalttOyQhMilRJz1RISoqKipSNSEjOyQiMnVkXVo0JioqUjVGYFxtLSUqVEhJQ0tORVNTRzYjIiIj For comparison, let's look at the Taylor polynomial for the cosine on the same interval. Note that while the Taylor polynomial fits much better for small values, over the whole interval it is much worse. tailor:=convert(taylor(cos(x),x,5),polynom); plot([cos(x), tailor], x=-Pi..Pi, color=[red, blue], thickness=2); NiM+SSd0YWlsb3JHNiIsKCIiIkYnKiRJInhHRiUiIiMjISIiRioqJEYpIiIlI0YnIiND 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 More explictly, we can compare the norms of the differences, and see the dramatic difference. We also show a plot of the two differences on the same axes. evalf(Norman(cos(x)-approx));
evalf(Norman(cos(x)-tailor)); NiMkIisnKVJoLFkhIzY= NiMkIisnUiUzJip6ISM1 plot([cos(x)-approx, cos(x)-tailor], x=-Pi..Pi, color=[green,blue], thickness=2); 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 Just to emphasize the ease of solving the minimization problem once the basis is in hand, here are a couple more examples. Naturally, since our basis was constructed specifically for the interval NiM3JCwkSSNQaUdJKnByb3RlY3RlZEdGJiEiIkYl , we must work there. However, it would be not hard to adapt this same basis to work on any interval. In order to avoid lots of typing, we write a Maple function that computes the inner product for us, and another that shows us the graphs. MakeApprox:=(f,basis)->collect(add(InnerProd(f,basis[i])*basis[i], i=1..nops(basis)),x):
ShowApprox:=proc(f,basis)
local a;
a:=MakeApprox(f,basis);
print(a);
print(evalf(a), " norm of difference=",evalf(Norman(a-f)));
plot([f, a], x=-Pi..Pi, color=[red, green], thickness=2);
end: First, just to check, we apply it to the same problem we just solved. ShowApprox(cos(x),OrthoBasis); NiMsKiooLCYqJEkjUGlHSSpwcm90ZWN0ZWRHRigiIiNGKSEjQCIiIkYrRichIilJInhHNiIiIiUjISV2OiIiKSomLCYqJEYnISIlIyEjWEYpKiZGJUYrRichIicjIiR2J0YvRitGLUYpRisqJEYnISIjIyIjOkYpKiZGJUYrRidGNiMhJE4iRjI= NiUsKCokSSJ4RzYiIiIlJCIrPz8pZmgjISM2KiRGJSIiIyQhKyI0eUdfJSEjNSQiKyMqUUUkeSpGLyIiIlE1fm5vcm1+b2Z+ZGlmZmVyZW5jZT1GJiQiK01ZZyxZRio= 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 Now we will try several other functions, just to show off. First, sin(x), which doesn't do as well (we'd need to add a fifth degree polynomial to our basis for a good fit.) ShowApprox(sin(x),OrthoBasis); NiMsJiooLCYhIzoiIiIqJEkjUGlHSSpwcm90ZWN0ZWRHRioiIiNGJ0YnRikhIidJInhHNiIiIiQjIiNORisqJiwmKiRGKSEiI0YvKiZGJUYnRikhIiUjISNARitGJ0YtRidGJw== NiUsJiokSSJ4RzYiIiIkJCErOyhwKFEkKiEjNkYlJCIrckskKXAmKSEjNVE1fm5vcm1+b2Z+ZGlmZmVyZW5jZT1GJiQiKywyJTNtIkYt 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 ShowApprox(cos(1+x)-x,OrthoBasis); 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