MATLABSymbolic Math Toolbox

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2019 6 4

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Symbolic Math Toolbox

• MATLAB•

•

•–

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sym

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•

•• sym( )

•

x=sym(‘x’); x

y=sym(‘y’); y

syms a b; a,b

a = sym(1/3);b = sym(1/7);

a 1/3, b 1/7

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•

+ a+b;sym(1/3) + sym(1/7); 10/21

- a-b;sym(1/3) – sym(1/7); 4/21

* a*b;sym(1/3) * sym(1/7); 1/21

/ a / b;sym(7) / sym(3); 7/36

x=sym(‘x’);c = sin(x)/x^2;

xb

a; b; c; a b c

a * b; !×#

clear all;

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^ sym(1/3)^sym(3) 1/27

sqrt sqrt(sym(2)) 2^(1/2)

sym( ,’d’)10 )

sym(1/3,’d’); 2.333333333

ii

sym(2 + 3*i);sym(i)^2;

2+3i-1

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a=sym(pi);b=sym(pi,’r’);

π

b=sym(pi,’d’);

b=sym(pi/5) + sym(pi/3)

clear

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•sin( )

Ang = 60; 60

sin(Ang); NGsin(Ang*pi/180);

cos(pi/4);

tan(pi/6);

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asin(1/2); arcsin(1/2)acos(1/2); arccos(1/2)

atan(sqrt(3)); arctan( 3)

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sin(sym(pi));cos(sym(pi/2));

atan(sym(sqrt(3)));syms x y

simplify(sin(x)^2+cos(x)^2);eq = expand(cos(x+y));

combine(eq, ’sincos’);

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• exp

exp(x) e x

exp(2); !"

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• log log(x) log x log2(x) 2log10(x):

log(2); log 2log2(3); log% 3

log10(1020); log'( 102014

syms x y z; x, y, z exp(1); exp(sym(1));

exp(sym(2))*exp(sym(3));exp(x)+exp(y);

!"!#×!%!& + !(

log(x);log2(sym(8));log10(sym(x));

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S = log(sym(2)) + log(sym(3));combine(S, ‘log’)

clear 15

(1)

x = 1:10; 1 10

y1 = 0:0.2:1y2 = 1:-0.2:0

n=10; x1=-5; x2=5;y3 = linspace(x1,x2,n);

x1 x2 n

y4 = linspace(x1, x2); n 100

clear

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(2)

n=5; z1 = zeros(1, n);

n

z2 = ones(1, n); 1z3=rand(1,n);t=[1 4 3 8 2];

t(2); t 2t(2:4); t 2 4clear

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• a,b,c,d 1 4a

•• q

(1)

syms a b c dp=[a b c d];

p=[a, b, c, d]

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• c1…c4 14 c

•• d

(2)

c=sym(‘c’, [1 4]); c=[c1, c2, c3, c4]

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syms ee ff gg hh; e=[ee ff gg hh];

1 4 e

et=e.’;et=transpose(e);

e

c + e

c – e

c * et ( )

* csym(2) * c

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n=3; nA=zeros(n);B=ones(n); 1C=rand(n);

D=[1 2 3; 4 5 6; 7 8 9];

A(2,3) 2 3A(:, 1) 1A(3, :) 3clear 21

• A•• B

(1)

syms d e f; A=[d e f; f d e];

A = [d, e, f][f, d, e]

A(2,3) ans = e

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• 3 4C

• Ci_j

•• D

(2)

C=sym(‘C’, [3 4]); C = [ C1_1, C1_2, C1_3, C1_4][ C2_1, C2_2, C2_3, C2_4][ C3_1, C3_2, C3_3, C3_4]

C(2,3) ans = C2_3

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syms j,k,m; E=[j k m; m j k; k m j];

3 3 E

A + E

A – E

A * E

* Asym(2) * A

inv(A)clear

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h = inline('3*x^3 + 2*x^2 +1');h(3)

h(x)h(3)

syms f(x)f(x) = x^4-2*x^3+4*x^2-5*x+6;

f(x)

f(-5) f(-5)

syms g(s, t)g(s, t) = s + 2*t

2 g(s,t)

g(1,2) g(1,2)

clear25

•

[n,d] = numden(sym(4/5));

syms x y[n,d] = numden(x/y + y/x) syms a b[n,d] = numden([a/b, 1/b; 1/a, 1/(a*b)]

quorem(sym(7), sym(3));

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syms xeq1 = (x^3+x+1)/(x^2 + x + 1);

[eq2, eq3] = numden(eq1);[q, r] = quorem(eq2, eq3, x);

clear

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• i i

real z) z

imag(z) z

abs(z)

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i^2; -1

eq1 = 4 – 5i;eq2 = sym(4- 5i);

real(eq1); real(eq2);imag(eq1); imag(eq2);

abs(eq1); abs(eq2);

clear

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• max min

• abs

A=[3 5 12 6 29 3];M1=max A)

A

M2=min(A) Aabs(-5.0)

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B=[sym(1/3) sym(2/7) sym(11/9)];M1=max(B)M2=min(B)

C = [2 8 4; 7 3 9]M3=max(C)

M4=max(C, [], 2)

clear

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factorial(n) n factorial(5)

syms nf = n^2+1;

fFac = factorial(f)round(a) a round(sym(2/3))

mod(a, b) a m mod(23, 3)

gcd(a,b) a b gcd(123, 45)

lcm(a,b) a b lcm(6, 8)

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•

factor n)( )

nthprime(n) n

isprime(n) n10

nextprime(n)prevprime(n)

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factor(sym(112/81));

syms x yF=factor(y^6 - x^6)

nthprime(200) 200isprime(37);

nextprime(100); 100

prevprime(100); 100

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round(n) round(sym(1/3));

ceil(n) ceil(sym(1/3));

floor(n) floor(sym(1/3));

fix(n) fix(sym(1/3));fix(sym(-1/3));

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• sort

A=[1 3 21 6 12 9 4];sort(A);

syms a b c dB=[b c a d];

sort(B)sort(A, ‘descend’)sort(B, ‘descend’)

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• sum

syms a b cA= [a b b c];

S=sum(A)A= [a b c; b c a; a c b];

S=sum(A)S=sum(A,2)

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• prod

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syms a b cA= [a b b c];S=prod(A)

A= [a b c; b c a; a c b];S=prod(A)

S=prod(A,2)

• expand• factor

= expand( );factor( );

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•

syms xp = (x-2) * (x-4);

q=expand(p)(x-2)(x-4)

factor(q); q

r = (x+1)/(x+2);expand(r)

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simplify( ) syms xeq1 = 1/(1+1/(1+1/(1+x)));

simplify(eq1);sign( ) a = sym(-5); b = sym(1/3);

sign(a)sign(b)

coeffs ) eq2 = 6*x^3 - 5*x^2 + 2*x -3*x + 4;

c=coeffs(eq2)coeffs ,

)eq3 = 6*x^3 - 5*y^2 + 1;

cx=coeffs(eq3,x)cy=coeffs(eq3,y)

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• subs

syms xy = x^2;x=2; y x 2 y=4

x=2;subs(y) subs x

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syms a bsubs(a+b, a, 4);

a 4

subs(a*b^2, a+b, 5) a+b 5

clear

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• subs(s, old, new) s oldnew

• Solve

syms xsolve(x^3 - 6*x^2 == 6 - 11*x)

x^3 - 6*x^2 = 6 - 11*x

solve(x^3 - 6*x^2 + 11*x - 6) x^3 - 6*x^2 + 11*x – 6 = 0= 0

syms x y solve(6*x^2 - 6*x^2*y + x*y^2

- x*y + y^3 - y^2 == 0, y)

6*x^2 - 6*x^2*y + x*y^2 - x*y + y^3 - y^2 = 0y

syms x y z[x, y, z] = solve(z == 4*x, x == y,

z == x^2 + y^2)

z = 4*x, x = y, z = x^2 + y^2x, y, z

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syms x y zeq1 = 2*x + y + z ==2;eq2 = -x + y – z == 3;

eq3 = x + 2*y + 3*z == -10;[x y z] = solve([eq1, eq2, eq3) (1) solve

syms y1 y2y1=x+3;y2=3*x;

solve(y1 == y2)

y1 y2

clear45

• plot

plot(x, y)• x1≦x≦x2)

ezplot(‘ (x)’, [x1 x2])•

hold on (hold off )•

→Save as

(1)

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(2)

x=linspace(0, 2*pi, 100);plot(x, sin(x));

y = sin(x) (0≦x≦2π)

y = cos(x) (0≦x≦2π)

ezplot(’sin(x)’, [0, 3]); y = sin(x) (0≦x≦3)

hold onezplot(’sin(x)’, [0, 6.28]);ezplot(’cos(x)’, [0, 6.28]);hold off 47

2

syms xfplot([sin(x),cos(x)], [-2*pi, 2*pi]);

sin(x) cos(x)

ezplotjpg

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3 (1)

t=linspace(0, 10, 40); (1) t

[x, y] = meshgrid(t, t); (2) t meshgrid xy

z = sin(x) + cos(y/2); (3)

mesh(x, y, z); (4) mesh 3

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3 (2)

ezplot3(‘sin(t)’, ‘cos(t)’, ‘t’)50

• ezplot3 3

ezplot3(x, y, z, [min, max])• min≦ t ≦max x= x(t), y = y(t), z = z(t)

0≦ t ≦2π )

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syms tx = (1-t)*sin(100*t);y = (1-t)*cos(100*t);z = sqrt(1 – x^2 – y^2);ezplot3(x, y, z, [0 1])

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• diffdiff( , )

(1)

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syms f(x)f(x) = sin(x^2);df = diff(f, x)

f(x) = sin(x^2)

df2 = df(2); x = 2

double(df2) (double)

• diff( , )

• diff( , , n) n

(2)

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syms x tdiff(sin(x*t^2), t)

t

syms sd = diff(t^6, 4)

!" 45 6

syms x ydiff(x*cos(x*y), y, 2)

y 2

• int

int( , )

int( , , [ ])

(1)

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(2)

syms xf = x^2;int(f, x);clear

f x

syms xf = x*log(1+x);int(f, x, [0 1]);clear

f 0 1

syms x tint(2*x, x, [sin(t) 1])

sin(t) 1

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• limit

limit( , , , ) ( ) ( )( )

Inf’right’ ’left’

(1)

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(2)

syms xf = sin(x)/x;limit(f, x, 0);

lim$→&'()($)$

syms xf2 = (2*x^2+x-3)/(x^2-2*x+1);limit(f2, x, Inf);

lim$→, -$./$01

$.0-$/2

limit(1/x, x, 0, right);

limit(1/x, x, 0, left);

clear

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• taylor

taylor( , , ) =

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taylor( , , ‘Order’, )

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(2)

syms xf = sin(x)/x;t1=taylor(f, x)

sin(x)

t2=taylor(f, x, ‘Order’, 8); 8

fplot([t1 t2 f])xlim([-4 4])grid on

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• diff

diff(y, x) == y !"!# = y

• dsolve( , )

•

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syms a y(t)eqn = diff(y,t) == a*y;

!"!# = %&

dsolve(eqn); C2cond = y(0) == 5; Y(0) = 5 Ysol = dsolve(eqn, cond)

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