---------------------------------------------------
-x_1 + x_2 + 2x_3 = 10
3x_1 - x_2 + x_3 = -20
-x_1 + 3x_2 + 4x_3 = 40
A=[-1 1 2;3 -1 1;-1 3 4]
A*[x_1;X_2;x_3]=[10;-20;40]
Check det(A) is not 0.
[x_1;X_2;x_3]=inv(A)*[10;-20;40]
b=[10;-20;40]
>x=inv(A)*b' or
>x=A\b' (Gaussian Elimenation)
>ones(3,2)
>eye(3,3)
>eye(3)
>eye(3,2)
>ones(3,2)+eye(3,2)
>diag([2 4 5])
e-values are 2 4 5.
A and b are given above.
>F=[Ab']
>G=rref(F) --->give solution
>rrefmovie(F)
e-value
det(B-\labbda*I)=0
>D=diag([1 2 3])
>eig(D)
>P=[1 2 3;4 5 6;5 7 8]
>B=inv(P)*A*P
>eig(B)
* eig(D)=eig(B)
complex nbr i,j
>angle()
>abs()
>real()
>imag(b)
>conj(b)=b'
>exp(i*pi)
>x=-3:3
>abs(x)>1
>y=x(abs(x)>1)
>x=-3:3
>find(x>2)
-----------------------------------------------------
>gcd(69,15)
>rem(7,2) --->remainder
>floor(7/2) -->quotient
----------------------------------------------
>> A=1:9
>> B=9-A
>>tf=A>A
>>tf=(A==B)
>> tf=B-(A>2)
>> x=(-3:3)/3
>>A=1:9
>> B=9-A
>> tf=A>4
>> tf=~(A>4)
>> tf=(A>2)&(A<6)
>> D=[2 1 3 4 7]
>>min(D)
>>median(D)
>>prod(D)
>>sum(D)
>>cumsum(D)
>>std(D) %std deviation
>>sort(D)
>>clf
>>hist(D,3) ---->figure1
>>figure
>>hist(D) ---->figure2
>> y=hist(D,3)
>>y
>>[y x]=hist(D,3) %x:midpoint
>>hist(D,3)
>>bar(x,y)
%Scatter plot and ...
>>tomato--->run tomato.m
>>plot(x,y,'r*')
>>[sx k]=sort(x)
>>sy=y(k)
>>plot(sx,sy)
-------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Least Square %
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (1) Y = alpha + beta*x = AC
% (2) Y = alpha + beta*x + gamma*x^2 =AC
x=[2 6 4 8 12 10 18 0 16 14]';
y=[4.70 4.84 5.02 5.38 5.47 5.61 5.95 4.01 5.90 5.80]';
[x y]
x=[2 6 4 8 12 10 18 0 16 14]';
y=[4.70 4.84 5.02 5.38 5.47 5.61 5.95 4.01 5.90 5.80]';
[x y]
ans =
2.0000 4.7000
6.0000 4.8400
4.0000 5.0200
8.0000 5.3800
12.0000 5.4700
10.0000 5.6100
18.0000 5.9500
0 4.0100
16.0000 5.9000
14.0000 5.8000
A=[ones(size(x)) x]
A =
1 2
1 6
1 4
1 8
1 12
1 10
1 18
1 0
1 16
1 14
c = A\y
c =
4.3985
0.0966
alpha =c(1)
alpha =
4.3985
beta = c(2)
beta =
0.0966
Y= alpha +beta*x
Y =
4.5918
4.9782
4.7850
5.1714
5.5578
5.3646
6.1375
4.3985
5.9442
5.7510
clc
[x y Y]
ans =
2.0000 4.7000 4.5918
6.0000 4.8400 4.9782
4.0000 5.0200 4.7850
8.0000 5.3800 5.1714
12.0000 5.4700 5.5578
10.0000 5.6100 5.3646
18.0000 5.9500 6.1375
0 4.0100 4.3985
16.0000 5.9000 5.9442
14.0000 5.8000 5.7510
plot(x,y,'r*')
hold on
plot(x,Y,'g')
r= y-Y
r =
0.1082
-0.1382
0.2350
0.2086
-0.0878
0.2454
-0.1875
-0.3885
-0.0442
0.0490
R2 = r'*r
R2 =
0.3880
fschange('/home/elee/mymatlab/tomato.m');clear tomato
clc
[x y]
ans =
2.0000 4.7000
6.0000 4.8400
4.0000 5.0200
8.0000 5.3800
12.0000 5.4700
10.0000 5.6100
18.0000 5.9500
0 4.0100
16.0000 5.9000
14.0000 5.8000
A=[ones(size(x)) x x.^2]
A =
1 2 4
1 6 36
1 4 16
1 8 64
1 12 144
1 10 100
1 18 324
1 0 0
1 16 256
1 14 196
c=A\y
c =
4.1826
0.1776
-0.0045
alpha =c(1)
alpha =
4.1826
beta =c(2)
beta =
0.1776
gamma =c(3)
gamma =
-0.0045
plot(x,y, 'r*')
plot(x,y,'r*')
hold on
plot(x,Y,'g')
Y= alpha + beta*x +gamma
Y =
4.5333
5.2436
4.8884
5.5987
6.3090
5.9539
7.3744
4.1781
7.0193
6.6641
fschange('/home/elee/mymatlab/tomato.m');clear tomato
Y= alpha + beta*x +gamma*x.^2
Y =
4.5198
5.0861
4.8210
5.3153
5.6658
5.5085
5.9215
4.1826
5.8723
5.7870
plot(x,y,'r*')
plot(x,y,'r*')
hold on
plot(x,Y,'g')
---------------------------------------------------------------
%Polynomial
>> p = [1 -2 0 25 116]; --->x^4 - 2x^3 + 25x + 116
>> r = roots(p)
r =
2.9832 + 2.7986i
2.9832 - 2.7986i
-1.9832 + 1.7320i
-1.9832 - 1.7320i
>> pp = poly(r)
pp =
1.0000 -2.0000 0.0000 25.0000 116.0000
>> a=[1 2 3 4] --->a(x)=x^3 + 2x^2 + 3x +4
a =
1 2 3 4
>> b=[1 4 9 6]
b =
1 4 9 6
>> c=conv(a,b) %convolution (multiplication of two polys
c =
1 6 20 40 55 54 24 --->6 degree polynomial
>> d=a+b
d =
2 6 12 10
>> e=c+[0 0 0 d]
e =
1 6 20 42 61 66 34
polyadd(c,d)
ans =
1 6 20 52 81 96 84
[q, r] =deconv(c,b)
q =
1 2 3 4
r =
0 0 0 0 0 0 0
g=[1 6 20 48 69 72 44]
g =
1 6 20 48 69 72 44
h=polyder(g)
h =
6 30 80 144 138 72
x=linspace(-1,3)
x =
Columns 1 through 6
-1.0000 -0.9596 -0.9192 -0.8788 -0.8384 -0.7980
Columns 7 through 12
-0.7576 -0.7172 -0.6768 -0.6364 -0.5960 -0.5556
.........
Columns 91 through 96
2.6364 2.6768 2.7172 2.7576 2.7980 2.8384
Columns 97 through 100
2.8788 2.9192 2.9596 3.0000
help linspace
LINSPACE Linearly spaced vector.
LINSPACE(x1, x2) generates a row vector of 100 linearly
equally spaced points between x1 and x2.
LINSPACE(x1, x2, N) generates N points between x1 and x2.
See also LOGSPACE, :.
linspace(1,3,4)
ans =
1.0000 1.6667 2.3333 3.0000
x=linspace(-1,3);
p=[1 4 -7 -10];
v=polyval(p,x);
plot(x,v)
title('x^3 + 4 x^2 - 7x + 10')
--------------------------------------------------
%n(x)/d(x)
>> n=[1 -10 100]
n =
1 -10 100
>> d=[1 10 100 0]
d =
1 10 100 0
>> z = roots(n)
z =
5.0000 + 8.6603i
5.0000 - 8.6603i
>> p = roots(d)
p =
0
-5.0000 + 8.6603i
-5.0000 - 8.6603i
>> [nn, dd] = polyder(n,d)
nn =
Columns 1 through 3
-1 20 -100
Columns 4 through 5
-2000 -10000
%i.e, -x^4 + 20x^3 - 100x^2 -2000x -10000
dd =
Columns 1 through 3
1 20 300
Columns 4 through 6
2000 10000 0
Column 7
0
>> [r,p,k]=residue(n,d)
r =
0.0000 + 1.1547i
0.0000 - 1.1547i
1.0000
p =
-5.0000 + 8.6603i
-5.0000 - 8.6603i
0
k =
[]
------------------------------------------------------------
x=[2 6 4 8 12 10 18 0 16 14]';
y=[4.70 4.84 5.02 5.38 5.47 5.61 5.95 4.01 5.90 5.80]';
x
x =
2
6
4
8
12
10
18
0
16
14
y
y =
4.7000
4.8400
5.0200
5.3800
5.4700
5.6100
5.9500
4.0100
5.9000
5.8000
x=1:10
x =
1 2 3 4 5 6 7 8 9 10
y=x.^3
y =
Columns 1 through 5
1 8 27 64 125
Columns 6 through 10
216 343 512 729 1000
polyfit(x,y,1)
ans =
105.4000 -277.2000
xi=linspace(1,10);
fschange('/home/elee/mymatlab/polynomial_fit.m');clear polynomial_fit
polynomial_fit
??? Error: File: /home/elee/mymatlab/polynomial_fit.m Line: 6 Column: 7
Missing operator, comma, or semicolon.
fschange('/home/elee/mymatlab/polynomial_fit.m');clear polynomial_fit
polynomial_fit
x =
1 2 3 4 5 6 7 8 9 10
p =
105.4000 -277.2000
fschange('/home/elee/mymatlab/polynomial_fit.m');clear polynomial_fit
polynomial_fit
x =
1 2 3 4 5 6 7 8 9 10
p =
105.4000 -277.2000
p =
16.5000 -76.1000 85.8000
??? Error using ==> plot
String argument is an unknown option.
Error in ==> /home/elee/mymatlab/polynomial_fit.m
On line 19 ==> plot(x,y,'-o',xi,y_linear,'r:',y_quadratic,'m--')
fschange('/home/elee/mymatlab/polynomial_fit.m');clear polynomial_fit
polynomial_fit
x =
1 2 3 4 5 6 7 8 9 10
p =
105.4000 -277.2000
p =
16.5000 -76.1000 85.8000
----------------------------------------------
rand ---> Unif(0,1)
randn ---> Std Normal
>> rand
ans =
0.5681
>> rand(2,3)
ans =
0.3704 0.5466 0.6946
0.7027 0.4449 0.6213
>> rand(2)
ans =
0.7948 0.5226
0.9568 0.8801
>> rand('seed',13)>> b=rand(2,1)
b =
0.6678
0.9863
>> rand('seed',0)>> a=rand(3,2)
a =
0.2190 0.6793
0.0470 0.9347
0.6789 0.3835
>> rand('seed',13)>> c=rand(2,1)
c =
0.6678
0.9863
>> x=rand(1,3)
x =
0.6143 0.7985 0.6083
>> a=12;b=99;
>> y=a+(b-a)*x
y =
65.4454 81.4655 64.9230
>> ceil(1.2)
ans =
2
>> ceil(1.9)
ans =
2
>> ceil(0.9)
ans =
1
>> floor(1.2)
ans =
1
>> floor(1.9)
ans =
1
>> floor(0.6)
ans =
0
>> round(1.2)
ans =
1
>> round(1.5)
ans =
2
>> round(1.9)
ans =
2
>>
>>
>> x=rand(3,1)
x =
0.8651
0.4474
0.6907
**** Create random integer
>> floor(6*x)
ans =
5
2
4
>> floor(9.98)
ans =
9
>> ceil(6*x)
ans =
6
3
5
>> x=rand(1,10)
x =
Columns 1 through 7
0.3742 0.2525 0.1195 0.1134 0.4568 0.1440 0.0305
Columns 8 through 10
0.1401 0.9484 0.2173
>> floor(6*x)
ans =
2 1 0 0 2 0 0 0 5 1
>> ceil(6*x)
ans =
3 2 1 1 3 1 1 1 6 2
>> r=randperm(9)
r =
5 8 4 3 1 7 2 6 9
--------------------------------
exprand
% exprand.m
% Task: Compute a matrix of size n x m, exponentially distributed with mean = mu
% Usage: V = exprand(mu) % for 1 x 1
% V = exprand(mu,n) % for n x n
% V = exprand(mu,n,m) % for n x m
exprand(2)
ans =
3.8706
exprand(2,4,5)
ans =
0.9987 0.6653 0.7397 0.1090 0.5346
0.6908 0.4728 5.4059 0.9735 4.5242
2.0226 1.1213 3.2577 0.0133 4.0404
1.2826 2.7186 0.6252 1.0982 4.0458
fschange('/home/elee/mymatlab/compare_distribution.m');clear compare_distribution
fschange('/home/elee/mymatlab/compare_distribution.m');clear compare_distribution
compare_distribution
fschange('/home/elee/mymatlab/compare_distribution.m');clear compare_distribution
fschange('/home/elee/mymatlab/compare_distribution.m');clear compare_distribution
compare_distribution
fschange('/home/elee/mymatlab/compare_distribution.m');clear compare_distribution
fschange('/home/elee/mymatlab/compare_distribution.m');clear compare_distribution
fschange('/home/elee/mymatlab/compare_distribution.m');clear compare_distribution
compare_distribution
clear
m=180/131;
rand('seed',0)z=exprand(m,50,1)
z =
0.3396
0.0662
1.5608
1.5626
3.7493
0.6646
1.0068
2.4426
0.0483
0.0755
1.0366
1.5281
0.0106
0.6644
0.0951
0.7425
1.5950
1.2217
3.6625
2.5721
1.0285
0.1326
1.4580
0.7390
1.6598
3.3135
1.9736
0.4183
0.0668
1.8304
0.5467
1.3760
1.9405
6.4782
0.6247
0.3899
5.5627
1.7622
1.9234
1.4485
0.1037
1.3722
2.9683
0.4375
0.7879
1.9986
0.8925
0.3731
0.4417
0.6116
t=cumsum(z)
t =
0.3396
0.4058
1.9665
3.5292
7.2785
7.9431
8.9499
11.3925
11.4408
11.5163
12.5529
14.0810
14.0917
14.7561
14.8512
15.5937
17.1887
18.4104
22.0729
24.6450
25.6735
25.8060
27.2640
28.0030
29.6628
32.9764
34.9499
35.3682
35.4350
37.2654
37.8121
39.1881
41.1286
47.6068
48.2315
48.6214
54.1841
55.9464
57.8698
59.3182
59.4219
60.7942
63.7625
64.2000
64.9879
66.9866
67.8791
68.2522
68.6939
69.3055
---------------------------------
x=linspace(0,2*pi,30)
y=sin(x)
z=cos(x)
plot(x,y,x,z)
grid on
grid off
box on
box off
gtext('cos(x)')text(2.5,0.7,'sin(x)')
legend('sin(x)','cos(x)')axis([1 6.3 -0.8 0.8])
axis square
axis ij
axis auto
axis equal
a=[0.5 1.0 1.6 1.2 0.8 2.1]
pie(a)
pie(a,a==max(a))
pareto(a)
x=-2*pi:pi/10:2*pi
y=sin(x);
z=2*cos(x);
subplot(2,1,1)
plot(x,y,x,z)
title('two plots on the same scale')subplot(2,1,2)
plotyy(x,y,x,z)
title('two plots on the different scale')
t=linspace(0,10*pi);
x=sin(t);
y=cos(t);
plot3(x,y,t)
---------------------------------------------
sym(1/3,'d') -->d: decimal
x=sym('x')>> diff(cos(x))
ans =
-sin(x)
>> int(cos(x))
ans =
sin(x)
>> syms('a','b','c','d')>> M=[a b ;c d]
M =
[ a, b]
[ c, d]
>> det(M)
ans =
a*d-b*c
>> syms a s t u omega ij
>> findsym(a*t + s/(u+3),1)
ans =
u ---->matlab think of u as a variable(default)
>> findsym(sin(a+omega),1)
ans =
omega
>> syms a b c d x
>> f=a*x^2 + b*x +c
f =
a*x^2+b*x+c
>> g_prime=diff(sqrt(3*x^2 + 2*x +5))
g_prime =
1/2/(3*x^2+2*x+5)^(1/2)*(6*x+2)
>> sym s
ans =
s
>> h = (3*s^2 + 4*s + 2)/(5*s+7)
h =
(3*s^2+4*s+2)/(5*s+7)
>> int(h)
ans =
3/10*s^2-1/25*s+57/125*log(125*s+175)
>>x=sym('x')
>> M=[3*sin(t) -cos(t^2);4*t+1 exp(-2*t)]
M =
[ 3*sin(t), -cos(t^2)]
[ 4*t+1, exp(-2*t)]
>> det(M)
ans =
3*sin(t)*exp(-2*t)+4*cos(t^2)*t+cos(t^2)
>> syms a b x
>> m=x^2
m =
x^2
>> [n,d]=numden(m)----> n:numerator, d:denominator
n =
x^2
d =
1
>> h=(x^2 +3)/(2*x-1) +x/(x-1)
h =
(x^2+3)/(2*x-1)+x/(x-1)
>> [n,d]=numden(h)
n =
x^3+x^2+2*x-3
d =
(2*x-1)*(x-1)
>> M=[3/2,(2*x+1)/3;4/x^2, 3*x+1]
M =
[ 3/2, 2/3*x+1/3]
[ 4/x^2, 3*x+1]
>> [n,d]=numden(M)
n =
[ 3, 2*x+1]
[ 4, 3*x+1]
d =
[ 2, 3]
[ x^2, 1]
M=n/d or n./d
>> syms x u v
>> f=1/(1+x^2)
f =
1/(1+x^2)
>> g=sin(x)
g =
sin(x)
>> compose(f,g)
ans =
1/(1+sin(x)^2)
>> syms x a b
>> finverse(x^2)
Warning: finverse(x^2) is not unique.
> In /usr/local/share/matlab60/toolbox/symbolic/@sym/finverse.m at line 43
ans =
x^(1/2)
>> syms c d
>> f=a*b + c*d -a*c
f =
a*b+c*d-a*c
>> finverse(f,a)
ans =
(-c*d+a)/(b-c)
>> diff(f,a)
ans =
b-c
>> syms x n
>> symsum((2*n-1)^2,1,n)---->sum of 2*n-1)^2 running 1 to n
>> factor(ans)
ans =
1/3*n*(2*n-1)*(2*n+1)
>> phi=sym('(1+sqrt(5))/2')
phi =
(1+sqrt(5))/2
>> double(phi)
and =
1.6180
>> x=sym('x');>> f=x^3 + 2*x^2 - 3*x +5
f =
x^3+2*x^2-3*x+5
>> n=sym2poly(f)
n =
1 2 -3 5
>> poly2sym(n)
ans =
x^3+2*x^2-3*x+5
>> syms a alpha b c s x
>> f=a*x^2 + b*x + c
f =
a*x^2+b*x+c
>> subs(f,x,s)
ans =
a*s^2+b*s+c
>> subs(f,a,[alpha;s])
ans =
[ alpha*x^2+b*x+c]
[ s*x^2+b*x+c]
>> g=3*x^2 + 5*x - 4
g =
3*x^2+5*x-4
>> h=subs(g,x,2) !!!!!!!!!!!!!!!!!!!!!!!!!!!!
h =
18
>>sym a x b c d
>>t = solve(a*x^3 + b*x^2 + c*x + d)
>> r=subexpr(t(2))
>> syms a b c d x s
>> f=a*x^3 + x^2 - b*x -c
f =
a*x^3+x^2-b*x-c
>> diff(f)
ans =
3*a*x^2+2*x-b
>> diff(f,a)
ans =
x^3
>> diff(f,2) ----->2nd derivative
ans =
6*a*x+2
>> diff(f,a,2)
ans =
0
F=[a*x,b*x^2;c*x^3,d*s]
F =
[ a*x, b*x^2]
[ c*x^3, d*s]
diff(F)
ans =
[ a, 2*b*x]
[ 3*c*x^2, 0]
M=[(1:8).^2]
M =
1 4 9 16 25 36 49 64
diff(M)
ans =
3 5 7 9 11 13 15
x=sym('x')
x =
x
p=int(log(x)/exp(x^2))
Warning: Explicit integral could not be found.
> In /usr/local/share/matlab60/toolbox/symbolic/@sym/int.m at line 58
p =
int(log(x)/exp(x^2),x)
F=[1 2]
F =
1 2
diff(F)
ans =
1
syms x s m n
f=sin(s+2*x)
f =
sin(s+2*x)
int(f)
ans =
-1/2*cos(s+2*x)
int(f,s)
ans =
-cos(s+2*x)
int(f,pi/2,pi)
ans =
-cos(s)
int(f,s,pi/2,pi)
ans =
2*cos(x)^2-1-2*sin(x)*cos(x)
F=[1 2 4]
F =
1 2 4
diff(F)
ans =
1 2
clear
t=sym('t')
t =
t
digits(5)
a = sym('-9.7536')
a =
-9.7536
v=int(a,t)
v =
-9.7536*t
v=v+20
v =
-9.7536*t+20
y=int(v,t)
y =
-4.8768*t^2+20.*t
y=y+30
y =
-4.8768*t^2+20.*t+30
y0=subs(y,t,30)
y0 =
-3.7591e+03
y0=subs(y,t,0)
y0 =
30
t_top=solve(v)
t_top =
2.0505
subs(y,t,t_top)
ans =
50.505
t_splat=solve(y)
t_splat =
[ -1.1676]
[ 5.2686]
t_bunk=solve(y-1.7)
t_bunk =
[ -1.1130]
[ 5.2140]
----------------------------------------------------
>> syms t
>> v=int(a,t)
v =
-9.7536000000000000000000000000000*t
>> t=sym('t')
t =
t
>> a=sym('-9.7536')
a =
-9.7536
>> v=int(a,t)
v =
-9.7536000000000000000000000000000*t
>> v=v+20
v =
-9.7536000000000000000000000000000*t+20
>> h=int(v,t)
h =
-4.8768000000000000000000000000000*t^2+20.*t
>> ezplot(h,[0,6])
%%%% Taylor series(help taylor) %%%%%%
>> x=sym('x');>> f=taylor(log(x+1)/(x-5))
f =
-1/5*x+3/50*x^2-41/750*x^3+293/7500*x^4-1207/37500*x^5
>> pretty(f)
2 41 3 293 4 1207 5
- 1/5 x + 3/50 x - --- x + ---- x - ----- x
750 7500 37500
>>
>> clear f
>> f=(x^2 - 1)*(x-2)*(x-3)
f =
(x^2-1)*(x-2)*(x-3)
>> collect(f)
ans =
x^4-5*x^3+5*x^2+5*x-6
>> horner(ans)
ans =
-6+(5+(5+(x-5)*x)*x)*x
>> factor(f)
ans =
(x-1)*(x+1)*(x-2)*(x-3)
>> expand(f)
ans =
x^4-5*x^3+5*x^2+5*x-6
>> clear all
>> syms x y z a
>> simplify(log(2*x/y))
ans =
log(2)+log(x/y)
>> simplify((-a^2+1)/(1-a))
ans =
a+1
>> clear
>>
>> x=sym('x');>> f=(1/x^3+6/x^2+12/x+8)^(1/3)
f =
(1/x^3+6/x^2+12/x+8)^(1/3)
>> g=simple(f)
g =
(2*x+1)/x
>> h=simple(g)
h =
2+1/x
>> clear
>> s=sym('s')
s =
s
>> Y=(10*s^2+40*s+30)/(s^2+6*s+8)
Y =
(10*s^2+40*s+30)/(s^2+6*s+8)
>> diff(int(Y))
ans =
10-15/(s+4)-5/(s+2)
>> pretty(ans)
15 5
10 - ----- - -----
s + 4 s + 2
>> x=sym('x')
x =
x
>> g=(x^3+5)/(x^2-1)
g =
(x^3+5)/(x^2-1)
>> diff(int(g))
ans =
x+3/(x-1)-2/(x+1)
>> pretty(ans)
3 2
x + ----- - -----
x - 1 x + 1
>>
>> format long
>> pi
ans =
3.14159265358979
>> digits
Digits = 32
>> vpa(pi)
ans =
3.1415926535897932384626433832795
>> vpa(pi,5)
ans =
3.1416
>> A=[sym('1/4'),sym('sqrt(2)');sym('exp(1)'),sym('3/7')]
A =
[ 1/4, sqrt(2)]
[ exp(1), 3/7]
>> vpa(A,5)
ans =
[ .25000, 1.4142]
[ 2.7183, .42857]
>> vpa(pi*sqrt(163),20)
ans =
>> syms a b c x
>> solve(a*x^2 + b*x +c)
ans =
[ 1/2/a*(-b+(b^2-4*a*c)^(1/2))]
[ 1/2/a*(-b-(b^2-4*a*c)^(1/2))]
>> pretty(ans)
[ 2 1/2]
[ -b + (b - 4 a c) ]
[1/2 --------------------]
[ a ]
[ ]
[ 2 1/2]
[ -b - (b - 4 a c) ]
[1/2 --------------------]
[ a ]
>> solve('a*x^2 + b*x = -c')
ans =
[ 1/2/a*(-b+(b^2-4*a*c)^(1/2))]
[ 1/2/a*(-b-(b^2-4*a*c)^(1/2))]
>> solve(a*x^2 + b*x +c,b)
ans =
-(a*x^2+c)/x
>> pretty(ans)
2
a x + c
- --------
x
>>
>> solve(x+1)
ans =
-1
>> solve(cos(x)-sin(x))
ans =
1/4*pi
>> double(ans)
ans =
0.78539816339745
>> clear all
>> syms x y z
>> [a1 a2]=solve(x^2+x*y_y-3,x^2-4*x+3)
??? Undefined function or variable 'y_y'.
>> [a1 a2]=solve(x^2+x*y+y-3,x^2-4*x+3)
a1 =
[ 1]
[ 3]
a2 =
[ 1]
[ -3/2]
>> S=solve(x^2+x*y_y-3,x^2-4*x+3)
??? Undefined function or variable 'y_y'.
>> S=solve(x^2+x*y+y-3,x^2-4*x+3)
S =
x: [2x1 sym]
y: [2x1 sym]
>> S.y
ans =
[ 1]
[ -3/2]
>> S.x
ans =
[ 1]
[ 3]
>> S=solve(sin(x+y)-exp(x)*y,x^2-y-2)
S =
x: [1x1 sym]
y: [1x1 sym]
>> S.x
ans =
-6.0173272500593065641097297117905
>> S.y
ans =
34.208227234306296508646214438330
>> clear a
>> clear
>> syms d n p q
>> eq1=d+n/2+p/2-q;
>> eq2=n+d+q-p-10;
>> eq3=q+d-n/d-p;
>> eq4=q+p-n-8*d-1;
>> S=solve(eq1,eq2,eq3,eq4)
S =
d: [2x1 sym]
n: [2x1 sym]
p: [2x1 sym]
q: [2x1 sym]
>> S.d
ans =
[ 4+i*15^(1/2)]
[ 4-i*15^(1/2)]
>> S.n
ans =
[ 35/4+1/4*i*15^(1/2)]
[ 35/4-1/4*i*15^(1/2)]
--------------------------------------------
>> syms y
>> Dy=1+y^2
Dy =
1+y^2
>> dsolve('Dy=1+y^2')
ans =
tan(t+C1)
>> dsolve('Dy=1+y^2,y(0)=1')
ans =
tan(t+1/4*pi)
>> dsolve('D2y-2*Dy-3*y=0','y(0)=0','y(1)=1')
ans =
1/(exp(3)-exp(-1))*exp(3*t)-1/(exp(3)-exp(-1))*exp(-t)
>> y=simple(ans)
y =
(exp(3*t)-exp(-t))/(exp(3)-exp(-1))
>> pretty(y)
exp(3 t) - exp(-t)
------------------
exp(3) - exp(-1)
>>ezplot(y,[-6,2])
>> [t g]=dsolve('Df=3*f+4*g','Dg=-4f+3*g')
t =
-16/9+exp(3*t)*(C1+4*t*C2)
g =
exp(3*t)*C2+4/3
>> [t g]=dsolve('Df=3*f+4*g','Dg=-4f+3*g','f(0)=0','g(0)=1')
t =
-16/9+exp(3*t)*(16/9-4/3*t)
g =
-1/3*exp(3*t)+4/3
---------------------------------------------------
Maple using matlab
>> maple('f:=x+2*t*y-t-2*t^3;')
ans =
f := x+2*t*y-t-2*t^3
%g=dt/dt
>> maple('g:=diff(f,t)')
ans =
g := 2*y-1-6*t^2
>> maple('g:=diff(f,y)')
ans =
g := 2*t
>> maple('g:=diff(f,x)')
ans =
g := 1
>> maple('g:=diff(f,t)')
ans =
g := 2*y-1-6*t^2
>> maple('h:=solve(g,y)')
ans =
h := 1/2+3*t^2
>> clear
>> syms x t y
>> f = x+2*t*y-t-2*t^3
f =
x+2*t*y-t-2*t^3
%Solve g=0 for y
>> g=diff(f,t),h=solve(g,y)
g =
2*y-1-6*t^2
h =
1/2+3*t^2
%\dx/dt = x/t - x
>> maple('a:=diff(x(t),t)=x(t)/t-x(t)')
ans =
a := diff(x(t),t) = x(t)/t-x(t)
>> maple('b:=dsolve(a,x(t))')
ans =
b := x(t) = _C1*t*exp(-t)
%\int_{y_0}^{y_1} 1/\sqrt(c-y)>> maple('c:=2;a:=int(1/sqrt(c-y),y=y_0..y_1)')
ans =
c := 2a := -2*(2-y_1)^(1/2)+2*(2-y_0)^(1/2)
>> maple('A:=matrix([[1, 1, 0, 3],[2, 1, -1, 1],[3, -1, -1, 2],[-1 ,2, 3, -1]])')
ans =
A := matrix([[1, 1, 0, 3], [2, 1, -1, 1], [3, -1, -1, 2], [-1, 2, 3, -1]])
>> maple('x:=''x'' ')
ans =
x := 'x'
>> maple('b:=array(1..4,[4,1,-3,4])')
ans =
b := vector([4, 1, -3, 4])
>> maple('x:=linsolve(A,b)')
ans =
x := vector([-1, 2, 0, 1])