KeytoMatlabExercise6SchoolofMathematicalSciencesXiamenUniversityhttp:/gdjpkc.xmu.edr Key to MATLAB Exercise 6-Polynomial a=[10003-4 >>a2=[1000 0 0 >r[01-3; poly(r) ans >>a=1-2;-3-4;poly(a) 3-10 >> a=1 23; poly(a ans >> syms x;f=(x-1)*(x-2)(X-3) f (x-1)*(x-2)*(x-3) >horner(f) (x-1)*(x-2)*(x-3) >>subs(f, 4 ans >subs(f,1:10) 062460120210336504 Ex
Key to MATLAB Exercise 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Key to Ex61 Key to MATLAB Exercise 6 – Polynomial 1. 1) >> a=[1 0 0 0 3 4] a = 1 0 0 0 3 4 2) >> a2=[1 0 0 0] a2 = 1 0 0 0 3) >> r=[0 1 3]; poly(r) ans = 1 2 3 0 4) >> a=[1 2; 3 4]; poly(a) ans = 1 3 10 2. 1) >> a=[1 2 3]; poly(a) ans = 1 6 11 6 2) >> syms x; f=(x1)*(x2)*(x3) f = (x1)*(x2)*(x3) 3) >> pretty(f) (x 1) (x 2) (x 3) 4) >> horner(f) ans = (x1)*(x2)*(x3) 3. 1) >> subs(f, 4) ans = 6 2) >> subs(f, 1:10) ans = 0 0 0 6 24 60 120 210 336 504
Key to MATLAB Exercise 6 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed syms x, mat=eye(2); sym pol a=x2+1; subs(sym pol a, mat) ans Let mat t12 It generate a matrix, the(i,)) element of which is obtained by substituting 2122 2+1+1_(12+102+1 the x in x+I with the (i, )element of mat, that is,21+1 22+10+1 12+1 >>clear; mat=eye(2); pol a=[10 1; polyvalm(pol a, mat) ans 02 The answer is the result of 01 > syms x; mat =eye(2): pol a=x2+1; polyvalm(pol a, mat) ??? Inputs to polyvalm must be floats, namely single or double Error in==> polyvalm at 27 Y=diag(p(1)*ones(m, I, superiorfloat(p, X))) 02 2+x-4:g=2*x^2+1:fpl f plus_g sym 3*x^2+x-3 > clear; f0=[11-4: g0=20 1; f plus g vector=f0+g0 in vector form f plus g vector= > clear; f0=[l 1-4: g0=20]; f plus g sym=poly 2sym(f0)+poly2sym(g0) 3*x^2+x-3 2) >clear; syms x; fx2+x-4: g=2*x2+1; f multiply g sym=fg
Key to MATLAB Exercise 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Key to Ex62 4. 1) >> syms x; mat = eye (2); sym_pol_a = x^2+1; subs(sym_pol_a, mat) ans = 2 1 1 2 Let 11 12 21 22 t t mat t t Ê ˆ = Á ˜ Ë ¯ , It generate a matrix, the (i, j) element of which is obtained by substituting the x in 2 x +1 with the (i, j) element of mat, that is, 2 2 2 2 11 12 2 2 2 2 21 22 1 1 1 1 0 1 . 1 1 0 1 1 1 t t t t Ê + + ˆ Ê + + ˆ Á ˜ = Á ˜ + + + + Ë ¯ Ë ¯ 2) >> clear; mat = eye (2); pol_a = [1 0 1]; polyvalm(pol_a, mat) ans = 2 0 0 2 The answer is the result of 2 1 0 1 0 0 1 0 1 Ê ˆ Ê ˆ Á ˜ + Á ˜ Ë ¯ Ë ¯ . 3) >> syms x; mat = eye (2); pol_a =x^2+1; polyvalm(pol_a, mat) ??? Inputs to polyvalm must be floats, namely single or double. Error in ==> polyvalm at 27 Y = diag(p(1) * ones(m,1,superiorfloat(p,X))); >> clear;syms x; mat = eye (2); pol_a =x^2+1; polyvalm(sym2poly(pol_a), mat) ans = 2 0 0 2 5. 1) >> clear; syms x; f=x^2+x4; g=2*x^2+1; f_plus_g_sym=f+g % in sym form f_plus_g_sym = 3*x^2+x3 Or >> clear; f0=[1 1 4]; g0=[2 0 1]; f_plus_g_vector=f0+g0 % in vector form f_plus_g_vector = 3 1 3 Or >> clear; f0=[1 1 4]; g0=[2 0 1]; f_plus_g_sym=poly2sym(f0)+poly2sym(g0) f_plus_g_sym = 3*x^2+x3 2) >> clear; syms x; f=x^2+x4; g=2*x^2+1; f_multiply_g_sym=f*g
Key to MATLAB Exercise 6 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed f multiply g sym (x^2+x-4)°(2*x^2+1 s> collect(f multiply g sym) ans 2*x^4+2*x^3-7*x^2+x4 > clear; f0=[1 1-4:g0=20 1; f multiply g vector-conv(f0, go) f multiply g vector 2 > poly 2sym(f multiply g vector) ans 2*x^4-7*x^2+2*x^3+x4 >> clear; syms x; f-x2+x-4: g=2*x2+1; subs(f, g) ans (2*x^2+1)2+2*x^2-3 clear; 0[1 1-4; g0=[20 1]; [q, r]=decon(f0, g0 0.5000 0 > clear; syms x, fx2:g=3x5+l; f plus g sym=f+g f pl x^2+3*x^5+1 >clear; f0=10001001: g0=300001; f plus g vectorf0+g0 >>f plus g sym=poly2sym(f plus g vector) f plus g sym= x^2+3*x^5+1 s>clear; syms x; fx2: g3x 5+1; f multiply g sym=fg f multiply g sym poly2sym(sym2 poly(f multiply g sym)) ans 3*x^7+x^2 > clear; f0=1000100; g0=300001; f multiply g vector=conv(f0, g0) f multiply g vector
Key to MATLAB Exercise 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Key to Ex63 f_multiply_g_sym = (x^2+x4)*(2*x^2+1) >> collect(f_multiply_g_sym) ans = 2*x^4+2*x^37*x^2+x4 Or >> clear; f0=[1 1 4]; g0=[2 0 1]; f_multiply_g_vector=conv(f0, g0) f_multiply_g_vector = 2 2 7 1 4 >> poly2sym(f_multiply_g_vector) ans = 2*x^47*x^2+2*x^3+x4 3) >> clear; syms x; f=x^2+x4; g=2*x^2+1; subs(f, g) ans = (2*x^2+1)^2+2*x^23 4) >> clear; f0=[1 1 4]; g0=[2 0 1]; [q, r]=deconv(f0, g0) q = 0.5000 r = 0 1.0000 4.5000 6. 1) >> clear; syms x; f=x^2; g=3*x^5+1; f_plus_g_sym=f+g f_plus_g_sym = x^2+3*x^5+1 Or >> clear; f0=[0 0 0 1 0 0]; g0=[3 0 0 0 0 1]; f_plus_g_vector=f0+g0 f_plus_g_vector = 3 0 0 1 0 1 >> f_plus_g_sym =poly2sym(f_plus_g_vector) f_plus_g_sym = x^2+3*x^5+1 2) >> clear; syms x; f=x^2; g=3*x^5+1; f_multiply_g_sym=f*g f_multiply_g_sym = x^2*(3*x^5+1) >> poly2sym(sym2poly(f_multiply_g_sym)) ans = 3*x^7+x^2 Or >> clear; f0=[0 0 0 1 0 0]; g0=[3 0 0 0 0 1]; f_multiply_g_vector=conv(f0, g0) f_multiply_g_vector =
KeytoMatlabexErcise6SchoolofMathematicalSciencesXiamenUniversityhttpgdjpkc.xmuedu.cr 00 0 00 0 >f multiply g sym=poly 2sym(f multiply g vector) f multiply g sym 3*x^7+x^2 7. for example 6.2) > clear; syms x; f-x 2: g=3 x5+1; f multiply g sym=f g f multiply_g sym x^2*(3*x^5+1) s> collect(f multiply g sym) 3*x^7+x^2 clear;f0=0001001g0-=]; f multiply_g vector=conv(f0, g0) f multiply g vector= >>f multiply_g sym-poly2sym(f multiply g vector f multiply 3*x^7+x^2 2) > clear; syms x; fx2: g=3* 5+1; f multiply g sym=fg f multiply_g sym x^2°(3*x^5+1) sym2poly(f multiply g sym 1) >> A=diag(1: 3), Pfix(10*rand(3)); B-=p*A*p B= 71135175 171175329 >>B=B 11 > roots(poly (B)) 513.6451 494764 0.8785
Key to MATLAB Exercise 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Key to Ex64 0 0 0 3 0 0 0 0 1 0 0 >> f_multiply_g_sym =poly2sym(f_multiply_g_vector) f_multiply_g_sym = 3*x^7+x^2 7. for example 6.2) 1) >> clear; syms x; f=x^2; g=3*x^5+1; f_multiply_g_sym=f*g f_multiply_g_sym = x^2*(3*x^5+1) >> collect(f_multiply_g_sym) ans = 3*x^7+x^2 Or clear; f0=[0 0 0 1 0 0]; g0=[3 0 0 0 0 1]; f_multiply_g_vector=conv(f0, g0) f_multiply_g_vector = 0 0 0 3 0 0 0 0 1 0 0 >> f_multiply_g_sym =poly2sym(f_multiply_g_vector) f_multiply_g_sym = 3*x^7+x^2 2) >> clear; syms x; f=x^2; g=3*x^5+1; f_multiply_g_sym=f*g f_multiply_g_sym = x^2*(3*x^5+1) >> sym2poly(f_multiply_g_sym) ans = 3 0 0 0 0 1 0 0 8. 1) >> A=diag(1:3); p=fix(10*rand(3)); B=p'*A*p; B = 100 71 171 71 135 175 171 175 329 >> B==B' ans = 1 1 1 1 1 1 1 1 1 >> roots(poly(B)) ans = 513.6451 49.4764 0.8785 2)
Key to matlaB Exercise 6 School of Mathematical Sciences Xiamen University h http:/edjpkc.xmu.ed >A0=diag(0: 2): P=fix(10 rand(3); B0-p"A'p B 14415244 15216450 5017 >B=B roots(poly(B)) ans 320.9116 4.0884 0.0000 >>f[12-3-1-23l;g=[11-5-60; >>[qrldeconv(f g) 3 > poly2sym(g) poly2sym(r) x^3+10*x^2+4*x+3 10. Omitted
Key to MATLAB Exercise 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Key to Ex65 >> A0=diag(0:2); p=fix(10*rand(3)); B0=p'*A*p; B = 144 152 44 152 164 50 44 50 17 >> B==B' ans = 1 1 1 1 1 1 1 1 1 >> roots(poly(B)) ans = 320.9116 4.0884 0.0000 9. >> f=[1 2 3 1 2 3]; g=[1 1 5 6 0]; >> [q,r]=deconv(f,g) q = 1 1 r = 0 0 1 10 4 3 >> poly2sym(q) ans = 1+x >> poly2sym(r) ans = x^3+10*x^2+4*x+3 10. Omitted