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2 i I Octave ii Octave

3 I Octave

4 Octave MATLAB... [ ; ].. > A=0:0 2. > t=0 : pi/2: 2*pi 3. > B=[ 2 3; 4 5 6] 4. > A=[B; 7 8 9] 5. > H= Hello world! x = x min : x : x max [x min, x min + x, x min + 2 x,, x max ] x > t=0 : pi/30 : 2*pi; > plot(t, sin(t))

5 sin(x) [ ] B = B 4. B 3 [7 8 9] 3 3 A 2 3 A = > A=:00; [ ] [ ] (array constractor) [ ] (concatenation)

6 3. 5j 2. A = [5], B = 5, C = + 2j 2 + 4j 2, D = 3 3j 3 5j j 5 j A =, B = [ ] 3. A = Solving our problems with Octave., B = [ I like A]. A = [5] A = 5 A = [5] B = 5 2 > A==B A = B 0 2. j i, j, I, J i 2 sqrt(2) 3. A = [2 : 2 : 0 ; 0 : 2 : 2] A = [(2 : 2 : 0) ; (0 : 2 : 2)] > [A B] > [A ; B] 4. A B I like Solving our problems with Octave.

7 Octave (array). 2 Octave NaN Not a Number 0/0 (matrix) (scalar) m n m n m n a a 2 a n a 2 a 22 a 2n A = = [a pq ] (.)... a m a m2 a mn a pq (element) (p, q) * 2 A vec = vec (A) = a. a m a 2. a m2. a mn = α α α r α r (.2) * aij i j Octave p q

8 5. Octave a = 5.5, b = 2 pi + 3j pi i, j, I, J NaN n a = [ ] m a = 2 3 m n A = A = [ ] zeros(0,4) 0 4 zeros(3,0) 3 0 eye(3) ones(3,4) zeros(3,4) 0 rand(3,4)

9 6 A (array) A a pq = α r (p, q) r r = (q ) m + p (.3) Octave a pq α r ( ) a pq A (p, q) α r A (r) 2 (.3).2.2 M > V=vander([ ]) V = V = [m pq ], p, q =, 2,, 5 m pq 2 3 > V( 2, 3 ) 4 > V( 2 )

10 7 (.3) 2 = (3 )5 + 2 > V( 2 : 5 ) > V( [ ] ) p 2 > V( 2, : ) > V( end, : ) 2 4 > V( 2 : 4, : ) > V( 4 : 2, : ) [ ](0 5) 0 5 q 3 > V( :, 3 ) > V( :, end ) 3 4 > V( :, 3 : 4 ) > V( 2, 3 : 4 ) W > W=V( 2 : 4, 3 : 4 ).3

11 8 (.) (.2) > V( : ) > size( ans ) ans= > V=vander([ ]) > V( 2, : )=[ ] 2 3 > V=vander([ ]) > V( 2 : 3, : )=[ ] 3 > V=vander([ ]) > V( :, 3 )=[ ] 2 3 > V=vander([ ]) > V( :, 2 : 3 )=[ ] > V=vander([ ]) > V( 2 : 2 : 20 )=[ ] X > X= : 0 > X( 0 : - : ) > V=vander([ ]) > V( [,5],: )=V( [5,],: )

12 9 > V=vander([ ]) > V( [2, 5,],: ) > V=vander([ ]) > W=[ ] > V(2, : )=W > V=vander([ ]) > W=[ ] > V(2, : )=W N > V=vander([ ]) > W=[ 2; 3 4] > V(2:3, 3:4 )=W N > V=vander([ ]) > W=[ 2; 3 4] > V([, 3], [3, 4] )=W [ ].2.4 > clear wild card * > clear A A

13 0.2 Octave Octave diag rot90 tril triu fliplr flipud 90

14 2 Octave Octave Octave Octave

15 a = [a r ], b = [b r ] A = [a pq ], B = [b pq ] > t=0:0 > t*5 > t.ˆ2 > t /2 > t 2. > A=[ 2 3; 4 5 6; 7 8 9] > A*2 > A.ˆ2 > A/2 > A 3. > A=[ 2 3; 4 5 6; 7 8 9] > B=2*ones(3) > A.*B > A*B > A./B > 2*A-3*B 4. > A=[ ; -] > kron(a, A) 2...ˆ > x=-5 : 0. : 5 f (x) = x 3 x 2 5x 2 > y = x.ˆ3 x.ˆ2 5 x 2

16 Octave αa = [αa r ] α a or a α [ ] α a = α a r a/α or α\a αa = [αa pq ] [ ] α A = α a pq [(a pq ) n ] α A or A α A/α or α\a A.ˆn a + b = [a r + b r ] a + b a b = [a r b r ] a b A + B = [a pq + b pq ] A + B A B = [a pq b pq ] A B A B = [a pq b pq ] A. B A B = [a pq /b pq ] A./B [ ] AB = a pk b kq A B Kronecker A B = [a pq B] kron(a, B) AX = B X = A B A\B XA = B X = BA B/A A n Aˆn A t A. A t A k

17 2 4 > plot(x, y) > grid 4. Kronecker = = 2.. [ ] / x y z = [ ] [ ]

18 Octave Octave abs angle real imag conj sin cos tan atan atan2 4 exp log log0 rem sign sqrt Octave 2.2

19 2 6 A = a b c d cos cos (a) cos (A) = cos (c) cos (b) cos (d) cos * > t=0 : pi/30 : 2*pi > plot( t, sin(t)) > V=vander([ ]) > sum(v) > trace(v) > sum(v ) > sum(rot90(v)) 2 3 rank = x a y = b z c * (array) Octave (list) arrayable cos

20 Octave Octave det sum trace norm rank null Kernel eig poly expm logm sqrtm > [a b]=eig([ 2;3 4]) a = b = a b > *[ ; ]-[ 2;3 4]*[ ; ]

21 8 3 Visualization plot 3.. > t=0 : pi/30 : 2*pi ; > plot(t, sin(t)); > axis([0 2*pi - ]) 2. > t=0 : pi/30 : 2*pi ; > plot(cos(t), sin(t)) > axis([- - ]) 3. > t=0 : pi/30 : 2*pi ; > plot(t, cos(t), t, sin(t)) 4. > t=linspace(0, 2*pi, 30); > plot(t, cos(t), r, t, sin(t), b* ) plot plot(, ), plot(,, r ) r red r 3.

22 3 9 line line line line 2 line line linspace(x min, x max, ) [x min, x max ] > hold on > plot(t, sin(3*t)) > hold off 3.. y = exp( 0.3t) sin(3t), 0 < t < 0 2. y = 2 sin(t) + cos(4t), 0 < t < 2π 3. y = 2 sin(t) + cos(4t) y = 2 cos(t) + sin(3t), 0 < t < 2π 4. x = sin(t) y = cos(t) + sin(3t), 0 < t < 2π

23 m o c x x r + + g * b - w : Exercise 3.. line 0.5 voltage time > gset size square gset size nosquare > gset title Exercise 3.. > gset xlabel time > gset ylabel voltage axis([x min x max y min y max ])

24 plot(x,y) ploar(t,r) semilogx(x,y) loglog(x,y) stem(x,y) grid on/off clg clearplot 3 plot(t, y, t, y 2, t, y 3 ) > grid on grid off > clg * 3..2 polar 3.2. > t=0 : pi/50 : 2*pi ; > polar(t, exp(-0.*t)) 2. > t=0 : pi/50 : 2*pi ; > polar(cos(t), sin(t)) 3. > t=0 : pi/50 : 2*pi ; > polar(t, sin(2*t)) polar * clg clg

25 3 22 line r = sin(2θ) 20 line Bode polar( ) 3..3 semilogx

26 G (s) = s 2 + 2ζs + { ( y = 20 log G = 0 log x 2 ) } (2ζx) > x=logspace(-,, 00); > zeta=. > y=-0*log0((.0-x.ˆ2).ˆ2+(2.0*zeta*x).ˆ2); > semilogx(x, y, r ) > grid on logspace( 0 0 ) [0, 0 ] 00 semilogx zeta ϕ = tan 2ζx x 2 tan atan2 atan2( y x ) > y=-80*atan2(2.0*zeta*x,.0-x.ˆ2)/pi; 3..4 Bode subplot(m,n,p) *2 m n p *2 oneplot()

27 line line > x=logspace(-,, 00); > y=-0*log0((.0-x.ˆ2).ˆ2+(0.2*x).ˆ2); > z=-80*atan2(0.2*x,.0-x.ˆ2)/pi; > subplot(2,,); > semilogx(x, y, r );grid on > subplot(2,,2); > semilogx(x, z, b );grid on y > axis([0.,0,-80,0]) 3..5 Gnuplot Matlab Octave Gnuplot Gnuplot plot splot gplot gsplot gplot plot gsplot Matlab Octave Gnuplot Octave Octave

28 line line gplot Gnuplot Gnuplot g Octave Gnuplot set Octave gset gplot x (t) = sin t + sin(3t + π 4 ), x 2(t) =.8 cos(t) 0.3cos(2t π 5 ) > t=[0 : pi/30 : 2*pi] ; > data=[t, sin(t)+sin(3*t+pi/4),.8*cos(t)-0.3*cos(2*t-pi/5)]; > gplot [0:2*pi][-2.2:2] data with lines, data using :3 with impulses gplot 2D gplot [ ] n t, x, x2, plot gplot

29 3 26 line D 2D 2D 3.2. gsplot 3.4 (torus) x = (r + r 2 cos ϕ) cos θ, y = (r + r 2 cos ϕ) sin θ, z = r 2 sin ϕ > t=0:pi/20:60*pi; > r=2+0.5*cos(t/30); > x=r.*cos(t) > y=r.*sin(t) > z=0.5*sin(t/30); > p=[x; y; z] ; > gsplot p gsplot 3D 3

30 3 27 line z = sin x cos x gsplot 3 x,y,z gshow view octave:> view is 60 rot_x, 30 rot_z, scale, scale_z gset view 45, π x, y π z = sin x cos y > t=-pi:pi/20:pi; > x=t; y=t; > z=sin(x )*cos(y);

31 3 28 line > mesh(x,y,z) > axis([-pi pi -pi pi - ]) > gset ticslevel > k=4;n=2ˆk-; > theta=pi*[-n:2:n]/n; > phi=(pi/2)*[-n:2:n] /n; > x=cos(phi)*cos(theta); > y=cos(phi)*sin(theta); > z=sin(phi)*ones(size(theta)); > mesh(x,y,z);

32 Octave Octave 2 Octave (script) (function) Octave sin(theta) Octave.m * Octave M-file.m myprog.m Octave > myprog * Matlab

33 4 30 File Save As > sum([:00]) % % % This is my first program : the addition from to 00 sum([:00]) wa00.m Octave > wa00 Octave C 2. % This is my second program : the addition from to 00 wa=0; for i=:00 wa=wa+i; endfor wa for endfor wafor.m wa 3. n wa3 function a=wa3(n) % This is my third program : the addition from to n a=0; for i=:n a=a+i; endfor %

34 4 3 % Of course you can write shortly as follows: % a=sum([:n]) % function a=wa3(n) a a = wa3(n) Matlab 4. (recursive programming) recwa.m function a=recwa(n) % This is my first recursive program : the addition from to n if n== a=; return; else a=n+recwa(n-); endif n= a= n=n a=n+recwa(n-) n 4.2 a=b+c;

35 for endfor while endwhile 3 (operator) >, <, <=, >= ==, = and: & or : not : 0 for for index=: 2: 00 endfor index index while while expression endwhile expression index<00 5. wa3 while function a=wa4(n) % This is my 5th program : the addition from to n a=0; i=; while i < n+ a=a+i; i=i+; endwhile

36 4 33 n+ n < <= 4.3 (flow control) if else end if endif if else 2 endif if elseif 2 2 else 3 endif A, B wa seki function C=WaorSeki(A,B,ch) % if ch== w then C=A+B and if ch== s then C=A*B. if ch== w C=A+B; elseif ch== s C=A*B; else printf("cannot calculate!!\n"); endif

37 4 34 w wa s seki Cannot calculate!! switch case endswitch > help -i switch y. function TI() % turtle initialization: save as "TI.m" global U X Path deg U=[0;]; X=[0;0]; Path=X; deg=pi/80; U y X Path deg radian (global) 2. function R(a) % rotation by a degree: save as "R.m" global U X Path deg theta=a*deg; U=[cos(theta) sin(theta); -sin(theta) cos(theta)]* U; 3.

38 4 35 function F(s) % forward s length: save as "F.m" global U X Path deg Path=Path+s*U; X=[X Path]; 4. function J(s) % jump s length: save as "J.m" global U X Path deg Path=Path+s*U; X=[X NaN Path]; 5. function ST() % show graphics:save as "ST.m" global U X Path deg gset size square; plot(x(, :), X(2, :)); 6. No. % TI; for i=:20 for j=:0 F(); R(08); endfor R(8); endfor ST; 7. No. 2 TI; for i=:0 for j=:0 F(); R(08); endfor R(36); endfor ST;

39 branch.m rectree function branch(length, level) % save as "branch.m" if level== return; else F(length); R(45); branch(length/2, level -); R(-90); branch(length/2, level -); R(45); F(-length); endif function rectree(length, level) % recursive tree:save as "rectree.m" TI; branch(length, level); ST; > rectree(0, 5) M-file subfunction function op() % % - optical art: coded by H. Kawakami Nov. 4, 2004 % - main program: This main program is replced by the problem % - you want to solve. % global U X Path deg parity TI(); for i=:6 parity=-parity; A(); B(,5); H([0;0]); if parity== R(60*i); else

40 R(60*i+60); endif endfor ST(); % % - subfunctions A(a) and B(a, b) are used % - in the main program op() % function A(a) % draw triangle global U X Path deg parity for i=:3 F(a); R(parity*20); endfor function B(a,b) % draw small triangles global U X Path deg parity c=a; for i=:b R(parity*25); c=0.956*c; F(c); R(parity*20); c=0.9*c; F(c); R(parity*20); F(c); endfor % % - The following subfunctions are commonly used % - for turtle graphics % function TI(q)

41 4 38 % turtle initialization global U X Path deg parity U=[0;]; X=[0;0]; Path=X; deg=pi/80; parity=; function ST(q) % show turtle global U X Path deg parity gset size square; plot(x(,:),x(2,:)); function R(r) % rotation by r degree global U X Path deg theta=r*deg; U=[cos(theta) sin(theta);-sin(theta) cos(theta)]*u; function F(s) % forward s length global U X Path deg Path=Path+s*U; X=[X Path]; function H(h) % return to home position global U X Path deg U=[0;]; Path=h; X=[X [NaN; NaN] Path]; % (van der Pol) d 2 x dt 2 ε ( x 2) dx dt + x = 0 (4.) van der Pol 3 pp (4.)

42 line (4.) dx dt dy dt = y = ε ( x 2) y x (4.2) 2 (x, y) x y (x 0, y 0 ) Octave lsode % van der Pol equation: the first essay.

43 4 40 % save as vdp.m t0=[0 : pi/20 : 20*pi]; x0=[0.; 0.0]; x=lsode("myvdp", x0, t0); gset size square plot(x(:, ), x(:, 2)); % end of program myvdp function dx=myvdp(x, t) % van der Pol equation % save as myvdp.m dx=[x(2); 0.5*(-x()*x())*x(2)-x()]; % end of function > vdp function vdp2() % van der Pol equation: the second essay. % save as vdp2.m t0=[0 : pi/20 : 20*pi]; x0=[0.; 0.0]; x=lsode("myvdp", x0, t0); % wave forms of x(t) and x-dot(t) subplot(2,,); axis([0, 20*pi, -2.5, 2.5]); plot(t,x(:, ), r ); subplot(2,,2); axis([0, 20*pi, -2.5, 2.5]); plot(t,x(:, 2), r ); % end of program (4.2) ε myvdp 0.5 ε ε myvdp ε eps eps myeps

44 4 4 2 line line Rössler 3 Rössler dx dt dy dt dz dt = x z = x + ay = bx + xz cz (4.3) xz Rössler band *2 (4.3) a = 0.35, b = 0.4, c = 4.5 x 0 = z 0 = 0.0, y 0 = 3.0 *2

45 x 0, 8 y 8, 0 z 8 function Rossler() % main program t=0; tmax=200*pi; h=0.; x=[ ]; y=x; while t<tmax [t,x]=rk(t,x,h); y=[y; x]; endwhile gset data style line; gset para; gset size square; gset ticslevel 0; gsplot y; %end of main % Rossler equation function xdot=fn(t,x) xdot=zeros(,3); xdot()=-(x(2)+x(3)); xdot(2)=x()+0.35*x(2); xdot(3)=0.4*x()+x()*x(3)-4.5*x(3); % Runge Kutta method function [t, x]=rk(t,x,h) f=fn(t,x); f2=fn(t+h/2,x+h*f/2); f3=fn(t+h/2,x+h*f2/2); f4=fn(t+h,x+h*f3); t=t+h; x=x+h*(f+2*(f2+f3)+f4)/6; Runge-Kutta gplot m n

46 Octave 2 for endfor 2 Octave i=0; for t=0:0.0:0 i=i+; y(i)=sin(t); endfor t=0:0.0:0; y=sin(t); 2 y=zeros(,00); for i=:00 y(i)=det(x^i); endfor y Mathematica C Octave

2 1 Octave Octave Window M m.m Octave Window 1.2 octave:1> a = 1 a = 1 octave:2> b = 1.23 b = octave:3> c = 3; ; % octave:4> x = pi x =

2 1 Octave Octave Window M m.m Octave Window 1.2 octave:1> a = 1 a = 1 octave:2> b = 1.23 b = octave:3> c = 3; ; % octave:4> x = pi x = 1 1 Octave GNU Octave Matlab John W. Eaton 1992 2.0.16 2.1.35 Octave Matlab gnuplot Matlab Octave MATLAB [1] Octave [1] 2.7 Octave Matlab Octave Octave 2.1.35 2.5 2.0.16 Octave 1.1 Octave octave Octave