厦门大学数学科学学院：《高等代数》课程教学资源（应用与实验）Key to MATLAB Ex 5 - Symbol Computation

KeytoMatlabExercise5SchoolofMathematicalSciencesXiamenUniversityhttp:/gdjpkc.xmu.edr Key to MatLAB Exercise 5-Symbol Computation >>al=1/4+1/6 >>a2=sym(1/4+1/6 >>a3=1/4+1/6 >>a4=1/4+l1/6;eval(a4) ration >>clear: cl=a+2*a: cl ??? Undefined function or variable 'a' a+2*a Conclusion. It is illegal in the statement of c1=a+2*a" of a)where " a'is not defined before we assign data to variable cl. But"c2=a+2*a' " of b)is right, due to here a' being a character > clear; c3=a+2*a'; eval(c3) 29 Error using al Undefined function or variable 'a > clear; c4=a+2*a, a=3; eval(c4) ans = s>clear; syms a; c5=a+2*a'; eval(c5) 3*a >>clear; syms a; c6=a+2 a; eval(c6) ns 3*a > clear; syms a; c7=sym(a+2*a): subs(c7, a, 2) ans s > clear; syms a; c8=sym(a+2*a'); subs(c8, 2) ans clear; c9=sym( a+2*a+b); c9 Ex5-1

Key to MATLAB Exercise 5  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex5­1  Key to MATLAB Exercise 5 – Symbol Computation  1.  >> a1=1/4+1/6  % double  >> a2=sym(1/4+1/6)  % sym >> a3='1/4+1/6'  % char >> a4='1/4+1/6'; eval(a4) % rational  2.  1) a)  >> clear; c1=a+2*a; c1  ??? Undefined function or variable 'a'. b) >> clear; c2='a+2*a'; c2  c2 =  a+2*a  Conclusion. It is illegal in the statement of “c1=a+2*a” of a) where ‘a’ is not  defined before we assign data to variable c1. But “c2=a+2*a” of b) is right, due to here ‘a’ being a character. 2) a)  >> clear; c3='a+2*a'; eval(c3)  ??? Error using ==> eval  Undefined function or variable 'a'. b) >> clear; c4='a+2*a'; a=3; eval(c4)  ans =  9  3) a)  >> clear; syms a; c5='a+2*a'; eval(c5)  ans =  3*a  b) >> clear; syms a; c6= a+2*a; eval(c6)  ans =  3*a  c)  >> clear; syms a; c7=sym('a+2*a'); subs(c7, a, 2) ans =  6  d) >> clear; syms a; c8=sym('a+2*a'); subs(c8, 2) ans =  6  4) a)  >> clear; c9=sym(' a+2*a+b'); c9

Key to MATLAB Exercise 5 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed a+2=*a+b b) n(at2*a+b); subs(c10, 1) =sym( a+2*a+b); subs(cll, a, 1) 22? Undefined function or variable 'a cI (a+2 a+b); subs(c12, a, 1) ans = 3+b e > clear; syms a b: c13=sym(a+2*a+b): subs(c13, a, 1, b, 2) ?? Error using==> sym.subs Too many input arguments f) > clear; syms a b; c14=sym( a+2*a+b); subs(c14, [a, b], [1, sym('pi)D ans >> syms x;fx3-6*x^2+11*x-6;,g=(x-1)(X-2)°(x-3),h=x*x*(x-6)+11 d(h) X^3-6*x^2+11*x-6 > f actori (X-1)+(x-2)°(x-3) orner(f) (x*(x-6)+11)6 x,f[x^(-3) (1/x^3+6Xx^2+12/x+8)^(1/3)

Key to MATLAB Exercise 5  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex5­2  c9 =  a+2*a+b  b) >> clear; c10=sym('a+2*a+b'); subs(c10, 1) ans =  3*a+1  c)  >> clear; c11=sym(' a+2*a+b'); subs(c11, a, 1) ??? Undefined function or variable 'a'. d) >> clear; syms a; c12=sym('a+2*a+b'); subs(c12,a,1) ans =  3+b  e)  >> clear; syms a b; c13=sym('a+2*a+b'); subs(c13,a,1, b,2) ??? Error using ==> sym.subs Too many input arguments. f) >> clear; syms a b; c14=sym('a+2*a+b'); subs(c14, [a,b], [1, sym('pi')]) ans =  3+pi  3.  1) >> syms x; f=x^3­6*x^2+11*x­6; g=(x­1)*(x­2)*(x­3); h=x*[x*(x­6)+11]­6;  >> expand(g) ans =  x^3­6*x^2+11*x­6  >> expand(h) ans =  x^3­6*x^2+11*x­6  2) >> f actor(f) ans =  (x­1)*(x­2)*(x­3) 3) >> horner(f) ans =  x*(x*(x­6)+11)­6  4.  1) >> syms x; f=[x^(­3)+6*x^(­2)+12*x^(­1)+8]^(1/3) f =  (1/x^3+6/x^2+12/x+8)^(1/3)  >> simple(f)

Key to MATLAB Exercise 5 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed (2*x+1)3/x3)^(1/3) (1+6*x+12*x22+8*x3)x^3)^(1/3) factor (2*x+1)3/x^3)(1/3) (1/x^3+6/Xx^2+12/x+8)(1/3) combine (1/x^3+6/Xx^2+12/x+8)(1/3) (1/x^3+6/Xx^2+12/x+8)^(1/3) convert(sincos) (1/x^3+6/Xx^2+12/x+8)^(1/3) (1/x^3+6x^2+12/x+8)^(1/3) (1/x^3+6Xx^2+12/x+8)^(1/3) (1/x^3+6Xx^2+12/x+8)^(1/3) (2 y+1)X (2*x+1)^3/x^3)(1/3) etty(f) /1612\1/3 32x XX 12 +8 Omitted 2)to 6) >>clear; syms a x y; s=a*sin(x )+exp(y) ms t; subs(s, y, log(t))

Key to MATLAB Exercise 5  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex5­3  simplify:  ((2*x+1)^3/x^3)^(1/3)  radsimp:  (2*x+1)/x  combine(trig):  ((1+6*x+12*x^2+8*x^3)/x^3)^(1/3)  factor:  ((2*x+1)^3/x^3)^(1/3)  expand:  (1/x^3+6/x^2+12/x+8)^(1/3)  combine:  (1/x^3+6/x^2+12/x+8)^(1/3)  convert(exp):  (1/x^3+6/x^2+12/x+8)^(1/3)  convert(sincos):  (1/x^3+6/x^2+12/x+8)^(1/3)  convert(tan):  (1/x^3+6/x^2+12/x+8)^(1/3)  collect(x):  (1/x^3+6/x^2+12/x+8)^(1/3)  mwcos2sin:  (1/x^3+6/x^2+12/x+8)^(1/3)  ans =  (2*x+1)/x  >> simplify(f) ans =  ((2*x+1)^3/x^3)^(1/3)  >> pretty(f) / 1 6  12  \1/3  |­­­­ + ­­­­ + ­­­­ + 8 | | 3  2 x  |  \ x x  /  It implies that the result is 1  3  3 2  1 6 12  8 x x x Ê ˆ Á + + + ˜ Ë ¯ .  Omitted 2) to 6).  5.  >>clear; syms a x y; s=a*sin(x)+exp(y) s =  a*sin(x)+exp(y) 1) >> syms t; subs(s, y, log(t)) ans =

Key to MATLAB Exercise 5 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed a°sin(x)+t subs(s,x, yi, (pi/3, 0)) 1/2*a*3~(1/2)+1 > clear; syms a x y; p=a+x y+2 y2+3x2y a+x*y+2*x*y^2+3*x^2y ans 3*x^2*y+(y+2*y^2)*x+a >collect(p, y) 2*x*y^2+(x+3*x^2)y+a >>A=randn(2): syms x:s=3* x+5 > polyvalm(sym2 poly(s), A) 8.743113.0285 Notice: subs(s, A)cannt be used here, because it equals to 3. *A *A+5. [1, 1; 1, 1-2 * > clear; syms x; f-exp(-x) f >subs(f, 0 ans ans 1.0000 >a=[1: 1: 20]; format short; subs(f, a) Columns 1 through 9 36790.1353 0.0498 0.0183 0.0067 0.0025 0.0009 0.00030.0001 Columns 10 through 18 0.0000 0.0000 0.0000 0.0000 0.0000

Key to MATLAB Exercise 5  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex5­4  a*sin(x)+ t  2) >> subs(s,{x,y},{pi/3,0})  ans =  1/2*a*3^(1/2)+1  6.  1) >> clear; syms a x y; p=a+x*y+2*x*y^2+3*x^2*y  p =  a+x*y+2*x*y^2+3*x^2*y  >> collect(p, x) ans =  3*x^2*y+(y+2*y^2)*x+a 2) >> collect(p, y) ans =  2*x*y^2+(x+3*x^2)*y+a 7.  >> A=randn(2); syms x; s=3*x^2­2*x+5;  >> polyvalm(sym2poly(s), A) ans =  5.2187  4.6712  8.7431  13.0285  Notice: subs(s, A) cann’t be used here, because it equals to 3.*A.*A+5.*[1, 1; 1, 1]­2.*A.  8.  >> clear; syms x; f=exp(­x) f =  exp(­x) 1) >> subs(f, 0) ans =  1  2) >> subs(f, eps)  ans =  1.0000  3) >> a=[1:1:20]; format short; subs(f, a) ans =  Columns 1 through 9  0.3679 0.1353 0.0498 0.0183 0.0067  0.0025  0.0009  0.0003  0.0001  Columns 10 through 18  0.0000 0.0000 0.0000  0.0000 0.0000  0.0000  0.0000

Key to MATLAB Exercise 5 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed 0.00000.0000 amns 19 through 20 0.00000.0000 syms a bcduvst; s a=la b; cd; s b=luv s t s ats b atu, b+v] [ c+s, d+t] 2) [2*a32*b [2*c,2°d [a*u+b°s,a*v+b Is b [(-b*s+u*d)(a*d-cb),-(b°t-v*d/(a*d-c*b) 1/a*d-c*b)*(c*u-a*s),(a*t-c*v)(a*d-c*b)] [(-s*b+a*t)/(utv*s),(-V°a+ub)(u°t-v*s) [-(s*d-c*t)(u*t-v*s),(u*d-v*c(u*t-V*s) >>s c=s a+i*s b [ c+i*s, d+i*t] I conj(a+i*u), conj(c+i*s)I

Key to MATLAB Exercise 5  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex5­5  0.0000  0.0000  Columns 19 through 20  0.0000  0.0000  9.  >> syms a b c d u v s t; s_a=[a b;c d]; s_b=[u v;s t];  1) >> s_a+s_b  ans =  [ a+u, b+v] [ c+s, d+t]  2) >> 2 * s_a ans =  [ 2*a, 2*b] [ 2*c, 2*d] 3) >> s_a * s_b  ans =  [ a*u+b*s, a*v+b*t]  [ c*u+d*s, c*v+d*t]  4) >> s_a .* s_b  ans =  [ a*u, b*v] [ c*s, d*t]  5) >> s_a\s_b  ans =  [ (­b*s+u*d)/(a*d­c*b),  ­(b*t­v*d)/(a*d­c*b)] [ ­1/(a*d­c*b)*(c*u­a*s),  (a*t­c*v)/(a*d­c*b)] 6) >> s_a/s_b  ans =  [ (­s*b+a*t)/(u*t­v*s), (­v*a+u*b)/(u*t­v*s)] [ ­(s*d­c*t)/(u*t­v*s),  (u*d­v*c)/(u*t­v*s)] 7) >> s_c=s_a+i*s_b  s_c =  [ a+i*u, b+i*v] [ c+i*s, d+i*t]  8) >> s_c'  ans =  [ conj(a+i*u), conj(c+i*s)]

Key to MATLAB Exercise 5 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed [conj(b+i°v),conj(d+i°t) ans [b+i*v, d+i*t >> det(s a) a*d-c*b 11 ans c/(a*d-c*b), a/(ad-c"b) 10 1) syms x; b=ll; x > a=[1 2: 34; syms y, y=inv(a) *b 1/2*x Omitted 2)to 6) >>a=[1234];[vdl=eig(a) -0.8246-0.4160 0.5658-0.9094 -0.3723 0 05.3723 > a=sym(1) sym(2); sym(3) sym(4) [3,4 >v, d=eig(a) ans [3/4+14*33^(1/2),314-14*33^(1/2) [5/2+1/2833^(1/2 Ex5-6

Key to MATLAB Exercise 5  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex5­6  [ conj(b+i*v), conj(d+i*t)  9) >> s_c.'  ans =  [ a+i*u, c+i*s]  [ b+i*v, d+i*t]  10) >> det(s_a) ans =  a*d­c*b  11) >> inv(s_a) ans =  [ d/(a*d­c*b), ­b/(a*d­c*b)] [ ­c/(a*d­c*b),  a/(a*d­c*b)] 10.  1) >> syms x ; b=[1;x];  >> a=[1 2; 3 4]; syms y; y=inv(a)*b  y =  ­2+x  3/2­1/2*x  Omitted 2) to 6).  11.  1) >> a=[1 2;3 4]; [v,d]=eig(a) v =  ­0.8246  ­0.4160  0.5658  ­0.9094  d =  ­0.3723  0  0  5.3723  2) >> a=[sym(1) sym(2); sym(3) sym(4)] a =  [ 1, 2] [ 3, 4] >> [v, d]=eig(a) ans =  v =  [ 1,  1] [ 3/4+1/4*33^(1/2), 3/4­1/4*33^(1/2)] d =  [ 5/2+1/2*33^(1/2),  0]

Key to MATLAB Exercise 5 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed 0,52-1/2*33(1/2) 3) syms x; a=sym(1) sym(x), sym(3)sym(4)) [3,4] >v, d=eig(a) -1/2+1/6(9+12*x)^(1/2,2-12-16*(9+12*x)(1/2 d [5/2+1/2*(9+12*x)(1/2) 0,52-12*(9+12*x)^(1/2 syms stuv, a=[s t; u v]; [ v, d]=eig(a) -(-1/2*s+1/2*v-1/2°(s^2-2*s*v+v2+4*t市u)(1/2) -(-1/2s (1/2)/u d l/2*s+1/2*v+1/2*(S^2-2*s*v+v2+4*t°u)(1/2) 0] 1/2*s+1/2*v-1/2*(s^2-2*s*v+V2+4*t*u)^(1/2) 2. For example 2) > ut=schur(a) ?? Function'schur' is not defined for values of class'sym Error in ==> schur at 30 Ivarargout( 1: nargout)]=builtin(,, varargin( 1); [,u=lu(a) ?? Function lu is not defined for values of class'sym Error in ==> lu at 54 Ivarargout( 1: nargout)=builtin('lu, varargin( ) >u, v, t=svd(a) -40455358483375693164244872262782,-91451429567730445267917697381021] -.91451429567730445267917697381021,40455358483375693164244872262780 [54649857042190426504511884932842 0,36596619062625782042296438426142 -57604843676632079133109858194273,81741556047036327308865238846391] -.81741556047036327308865238846391,-57604843676632079133109858194273] Key to Ex5-7

Key to MATLAB Exercise 5  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex5­7  [ 0, 5/2­1/2*33^(1/2)] 3) >> syms x ; a=[sym(1) sym(x);sym(3) sym(4)] a =  [ 1, x] [ 3, 4] >> [v, d]=eig(a) v =  [ ­1/2+1/6*(9+12*x)^(1/2), ­1/2­1/6*(9+12*x)^(1/2)] [ 1,  1] d =  [ 5/2+1/2*(9+12*x)^(1/2),  0] [ 0, 5/2­1/2*(9+12*x)^(1/2)] 4) >> syms s t u v; a=[s t;u v]; [v,d]=eig(a) v =  [ ­(­1/2*s+1/2*v­1/2*(s^2­2*s*v+v^2+4*t*u)^(1/2))/u,  ­(­1/2*s+1/2*v+1/2*(s^2­2*s*v+v^2+4*t*u)^(1/2))/u] [ 1,  1] d =  [ 1/2*s+1/2*v+1/2*(s^2­2*s*v+v^2+4*t*u)^(1/2),  0] [ 0,  1/2*s+1/2*v­1/2*(s^2­2*s*v+v^2+4*t*u)^(1/2)] 12.  For example 2) >> [u,t]=schur(a) ??? Function 'schur' is not defined for values of class 'sym'.  Error in ==> schur at 30  [varargout{1:nargout}] = builtin('schur', varargin{:});  >> [l,u]=lu(a) ??? Function 'lu' is not defined for values of class 'sym'.  Error in ==> lu at 54  [varargout{1:nargout}] = builtin('lu', varargin{:});  >> [u,v,t]=svd(a) u =  [ ­.40455358483375693164244872262782, ­.91451429567730445267917697381021] [ ­.91451429567730445267917697381021,  .40455358483375693164244872262780] v =  [ 5.4649857042190426504511884932842,  0] [ 0, .36596619062625782042296438426142] t = [ ­.57604843676632079133109858194273,  .81741556047036327308865238846391] [ ­.81741556047036327308865238846391, ­.57604843676632079133109858194273]