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zFPIf5jf5BphW}`=NO=qb7m3V1XqN?*m@5$9Ttec6zn4nD>K!C85sK2Y$nzv&igR125nk$8-$7{k^J}=H-KZ@ScaxPf&izjtH!(2mys%*-9v7VNy9{scAc|FFLSr7( zM(e<;1yPgeE>D_15XKKtEr*=>?~~h7=rPN!Pk^HJc8p8tvAW%f$n=8_W$w>un*{SSRCQy#>13`!b z71M1pc$dX9qy%%s2_~ZqMacXiqExGWilKM=K|ff&bq-d_)#H~P4VbODp84&YyRfY$ zH=VgIbXeP@+MHi9m&ws*DEP1##OJX^R3U_hS)$<|MA>MIGxdUF<2UP-C4oZI*9oUh-*!OssBvEKL!W5dGk2)$I%Oo3AygfJAVt7A{dYy8STi52?u$ESe zd;NsNyNA7K%xx!@7}m9Kf8Oa6Z#k(D_UesU*T-rcrjibQ-mjG;O9WJwV+gFRAugXt z45r%ykeeqYMjrnEEgq=gO;-8WH#ZPmoIxskpmR!8)SA&?HIq*O3fyRf=N_OH_#meX z=f!#S=ehiS;!T{#*U~&St})lc%&tHdip~;Gsbl*HFC5gJcqm^-BIM3k4&2!U@ns0> z2FdU{EBRrwqJb^74z`7kRX0vpCZkx>qmqQ}i5_=!U%SvU{9ZGM^7|CFJLc_}#Y66Z zPk-Jy@4f}>GcHJEljfOY*;~8m?{-aDu<`$_=lPHA{hz=3|I{M==Z|Y$!c|(LD~l?{ zA8*7-2n(>cW2{e{{-YG|AfUI(npuTeBuX7U##j#x>~Gvu(tj^G>GqMZF5)20g72!#F97@=)B`B>;cR*uxj3JuO@^!g&Hi&=PNk%dtx5nWV z0yee7kRW!mKA_i&>9RMRA`{V2Cp0#MBCzunJ?GU`@8r{plIVNKn5 z?Sxb?d$DyR^>XMT%9l float: """ return sum([t**i for i in range(n + 1)]) +def question_3d(): + Ns = [25, 50, 100] + ns = Ns + t = np.linspace(-5, 5, 1000) + f_t = f1(t) + plt.figure(figsize=(15, 10)) + for i in range(len(Ns)): + N = Ns[i] + n = ns[i] + A = np.zeros((N + 1, n + 1), dtype=float) + A, b = build_A(N, n, f1) + x = householder_lstsq(A, b) + p_t = sum([x[j] * t**j for j in range(n + 1)]) + plt.subplot(1, 3, i + 1) + plt.plot(t, f_t, label='f(t)', color='blue') + plt.plot(t, p_t, label='p(t)', color='red', linestyle='--') + plt.title(f'N={N}, n={n}') + plt.xlabel('t') + plt.ylabel('y') + plt.legend() + plt.grid(True, alpha=0.3) + plt.suptitle('Least Squares Polynomial Approximation using Householder QR') + plt.tight_layout(rect=[0, 0.03, 1, 0.95]) + plt.savefig('Project1_A3d_least_squares_approximation.png') + #plt.show() + def question_3e(): # n = 4, 6, 8, 10, ..., 20 ns = list(range(4, 21, 2)) @@ -161,6 +187,7 @@ def question_3e(): if __name__ == "__main__": question_3c() + question_3d() question_3e()