while for break.
, , x y y(k) = x(k+1)/x(k) x(k), . . , . x , , x . y x. - break:
function y=div(x)
for k=1:length(x)-1
if x(k)==0
break
end
y(k)=x(k+1)/x(k);
end
15. , . . plotyy(), , .
16. ( ), fplot(), MATLAB .
fplot() -, , : fplot(' -', [a,b]).
- myfun :
function y=myfun(x);
y=cos(10*x).*(sin(x)+0.1*sin(100*pi*x));
y=myfun(x);
fplot(myfun,[0,1])
17. , .
x(t) y(t), y(x). , :
t;
x(t) y(t);
plot() .
t=[0:0.01:2*pi];
x=sin(t);
y=0.1*cos(t);
plot(x,y)
18. :
) [x;y] . meshgrid(x,y). x y , . , meshgrid() :
) ;
) MATLAB;
) , .
14. z(x,y) = x+y3 (. 5.16), x y [-8;8] 0,5. .
>> [x,y]=meshgrid(-8:0.5:8);
>> z=x+y.^3;
>> plot3(x,y,z,k)
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.
) , z . meshgrid() .
) meshgrid() [x,y], .
) plot3() , .
) , , zOy. .
, plot3() plot() 2D .
19. MATLAB:
1. File Save As.
2. , .
:
1. , Windows, .
2. File Export.
3. , (). , Windows.
Word [ * .tif] [ * .bmp].
20. , y x . (. . ), , . comet(x,y), , , , , ( ). . , . S. comet() comet(x,y,S), S . S 0,1.
, 2D . comet3(x,y,z), x, y z. comet3() , , comet(), .
21. sym.
,
>>x=sym('x')
x =
x
'x' .
x=sym('x','real') x . x=sym('x','positive') x () , x=sym('x','unreal') (. . ).
syms.
>> syms a b c
a, b, c.
>> Pi=sym('pi');
Pi = π, π .
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, Symbolic ( ) π , , . , sin
>> S=sin(Pi/5)
S =
1/4*2^(1/2)*(5-5^(1/2))^(1/2)
syms , .
vpa:
R=vpa(S) S, D, digits. R sym;
R=vpa(S,D) S D.
digits . :
digits ( 32);
digits(D) D .
22. simplify(S) S. , .
:
>> syms a b x
>>simplify((a^2-2*a*b+b^2)/(a-b))
ans =
a-b
simple.
simple(S) S , . [R,HOW] = simple(S) . R, HOW . :
>> syms x
>> [R,HOW]=simple(cos(x)^2+sin(x)^2)
R =
HOW =
combine
>> [R,HOW]=simple(2*cos(x)^2-sin(x)^2)
R =
3*cos(x)^2-1
HOW =
simplify
>> [R,HOW]=simple(cos(x)^2-sin(x)^2)
R =
cos(2*x)
HOW =
combine
[R] = simple(S):
>> [R]=simple(cos(x)^2-sin(x)^2)
R =
cos(2*x)
. subs :
subs(S) S , MATLAB;
subs(S, NEW) S NEW;
subs(S, OLD, NEW) OLD NEW S. OLD NEW . S OLD , NEW , .
:
>>syms a b x y;
>>subs(x-y,y,1)
ans =
x-1
>>subs(sin(x)+cos(y),[x y],[a b])
ans =
sin(a)+cos(b)
, :
>> s=sym('x^(x+1)');
>> f=subs(s,'x',1.5)
f =
2.7557
vpa:
>> f=subs(s,'x','1.5')
f =
(1.5)^((1.5)+1)
>> vpa(f,40)
ans =
2.7556759606310753604719445840441
23. F(x), () , limit, :
limit(F,x,a) F x = a;
limit(F,x,a,'right') limit(F,x,a,'left') a .
S diff, diff(S, x, n). n - ( n) S x, . .
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Sn(x) =.
diff(S, x) (n = 1 ).
S , diff , , .
I = dx I = dx.
f(x) x, a b .
int(f, x) ( ) f x.
int(f, x, a, b) f x b.
f , .
f , int(f, x) , int(f, x, a, b) .
f(x) a:
f(x) =.
a = 0 :
f(x) =.
f ( ) taylor:
taylor(f) . , n - a, , taylor(f,n,x,a).
cos(x):
>> syms x
>> taylor(cos(x))
ans =
1-1/2*x^2+1/24*x^4
>> taylor(cos(x),5,x,2)
ans =
cos(2)-sin(2)*(x-2)-1/2*cos(2)*(x-2)^2+1/6*sin(2)*(x-2)^3+1/24*cos(2)*(x-2)^4
24.
solve(expr1, expr2,..., exprN, var1, var2,..., varN). varI, , exprI. , , exprI = 0.
:
;
.
solve , . , doc solve.
:
x3 - 1 = 0.
:
>> syms x,y=x^3-1;S=solve(y)
S =
[ 1]
[ -1/2+1/2*i*3^(1/2)]
[ -1/2-1/2*i*3^(1/2)]
x1 = 1, x2 =, x3 =, S(1), S(2), S(3) S.
( ) MATLAB
r = dsolve('eq1','eq2',...,'cond1','cond2',...,'v').
eq1, eq2,..., v , () cond1, cond2,.... t, . eqI (condI) , , eqI (condI) = 0.
D , d/dt, D2 d2/dt2 . . D.
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'y(a) = b' 'Dy(a) = b', , a b , . . , C1, C2 . . , solve. dsolve , doc dsolve.
:
1) x'' = -2x', 2) y'' = -ax+y', y(0) = b, 3) y(4) - y = 5exsinx+x4, 4) y''+4y'+3y = cost, y(0) = 1, y'(0) = 0.
3 - 4 - .
:
>> dsolve('D2x=-2*x')
ans =
C1*cos(2^(1/2)*t)+C2*sin(2^(1/2)*t)
>> dsolve('D2y=-a*x+y','y(0)=b','x')
ans =
a*x+C1*sinh(x)+b*cosh(x)
25. , (, plot), Symbolic ezplot:
ezplot(f) f(x) x [- 2*pi;2*pi];
ezplot(f,xmin,xmax) , x xmin xmax;
ezplot(f, [xmin, xmax, ymin, ymax]) f(x,) = 0 xmin < < xmax, ymin < y < ymax.
f(t) ezpolar:
ezpolar(f) f(t) t 0 2π;
ezpolar(f,[a b]) f(t) t a b.
, ezcontour f(x,y). ezcontourf . ezplot3. ezsurf, ezsurfc, ezmesh, ezmeshc , f(x,y). doc < >.
26. f(t) t
L(s) = f(t)e-stdt.
f(t) , L(s) . f(t) :
) f(t) t, . ( , . );
) f(t) = 0 t < 0;
) M > 0 p ≥ 0 , t │ f(t) │ < Mept (p │ f(t) │).
) 7.1.
7.1.
f(t) | L(s) = f(t)e-stdt. |
1 | s-1 |
e-at | (s+a)-1 |
sinat | a(s2+a2)-1 |
tn | n!s-n-1 |
e-atcoswt | |
tne-at |
MATLAB f(t) laplace(F,t,s).
7.1 f(t):
>> syms a t w s
>> n=sym('n','positive');
>> laplace(1,t,s)
ans =
1/s
>> laplace(exp(-a*t),t,s)
ans =
1/(s+a)
>> laplace(sin(a*t),t,s)
ans =
a/(s^2+a^2)
>> laplace(t^n,t,s)
ans =
s^(-n-1)*gamma(n+1)
>> laplace(exp(-a*t)*cos(w*t),t,s)
ans =
(s+a)/((s+a)^2+w^2)
>> laplace(t^n*exp(-a*t),t,s)
ans =
gamma(n+1)*(s+a)^(-n-1)
laplace ( doc laplace).
f(t) = L(s)estds.
MATLAB L(s) ilaplace(L,s,t). 7.1 L(s):
>> syms a t w s
>> n=sym('n','positive');
>> ilaplace(1/s,s,t)
ans =
>> ilaplace(1/(s+a),s,t)
ans =
exp(-a*t)
>> ilaplace(a/(s^2+a^2),s,t)
ans =
a/(a^2)^(1/2)*sin((a^2)^(1/2)*t)
>> ilaplace(s^(-n-1)*gamma(n+1),s,t)
ans =
t^n
>> ilaplace((s+a)/((s+a)^2+w^2),s,t)
ans =
exp(-a*t)*cos(w*t)
>> ilaplace(gamma(n+1)*(s+a)^(-n-1),s,t)
ans =
exp(-a*t)*t^n
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27. Maple MATLAB , () .
Maple, LAB, maple.
Symbolic MATLAB Maple Waterloo, MATLAB Maple. Maple 3000 . MATLAB Symbolic, , Maple . , Maple, ( ), Maple.
Maple solve.
:
> 2.
:
>> maple('solve((x-2)/(x+3)>2,{x})')
ans =
{-8 < x, x < -3}
, - 8 < x < - 3 .
:
:
>> maple('solve({(x-2)/(x+3)<=51,sqrt(x)*(sqrt(x)-1)<10,10*x^2+4*x>=69},x)')
ans =
{RootOf(-69+10*_Z^2+4*_Z,2.4343879744638981326796776750121) <= x, x < RootOf(-21*_Z+_Z^2+100,13.701562118716424343244108837311)}
>> vpa(ans,4)
ans =
{2.434 <= x, x < 13.70}
, 2,434 ≤ x < 13,70 ( 4 - ).