Spring 2010 Instructor: Michele Merler.

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1 Spring 200 Instructr: Michele Merler

2 . Generate the pythagric table (d nt insert the values manually)

3 . Generate the pythagric table (d nt insert the values manually) r = :2; PT = r'*r;

4 2. Cmpute the series Using 0 elements (S 0 ) Using 00 elements (S 00 ) S 4 3 Cmpare the S 0 and S 00 with the value f the limit. Which ne is clser t the limit? Display the answer t cmmand windw 5 7 9

5 2. Cmpute the series Using 0 elements (S 0 ) N = 9 Using 00 elements (S 00 ) N = 99 Cmpare the S 0 and S 00 with the value f the limit. Which ne is clser t the limit? Display the answer t cmmand windw N n n n S S

6 2. Cmpute the series S 4 Using 0 elements (S 0 ) N = 9 Using 00 elements (S 00 ) N = 99 Cmpare the S 0 and S 00 with the value f the limit. Which ne is clser t the limit? Display the answer t cmmand windw n N n S n fracvec = nes(,00)./[:2:99]; signs =(-).^[0:99]; S0 = 4 * sum(fracvec(:0).*signs(:0)); diff0 = pi - S0; S00 = 4 * sum(fracvec.*signs); diff00 = pi - S00; Ff ff

7 2. Cmpute the series S 4 Using 0 elements (S 0 ) N = 9 Using 00 elements (S 00 ) N = 99 Cmpare the S 0 and S 00 with the value f the limit. Which ne is clser t the limit? Display the answer t cmmand windw n N n S n fracvec = nes(,00)./[:2:99]; signs =(-).^[0:99]; S0 = 4 * sum(fracvec(:0).*signs(:0)); diff0 = pi - S0; S00 = 4 * sum(fracvec.*signs); diff00 = pi - S00; Ff ff

8 2. Cmpute the series S 4 Using 0 elements (S 0 ) N = 9 Using 00 elements (S 00 ) N = 99 Cmpare the S 0 and S 00 with the value f the limit. Which ne is clser t the limit? Display the answer t cmmand windw fracvec = nes(,00)./[:2:99]; signs =(-).^[0:99]; S0 = 4 * sum(fracvec(:0).*signs(:0)); diff0 = pi - S0; S00 = 4 * sum(fracvec.*signs); diff00 = pi - S00; if(diff0>diff00) disp('s00 is clser t the limit') else disp('s00 is clser t the limit') end n N n S n

9 2. Cmpute the series S 4 Using 0 elements (S 0 ) N = 9 Using 00 elements (S 00 ) N = 99 Cmpare the S 0 and S 00 with the value f the limit. Which ne is clser t the limit? Display the answer t cmmand windw fracvec = nes(,00)./[:2:99]; signs =(-).^[0:99]; S0 = 4 * sum(fracvec(:0).*signs(:0)); diff0 = pi - S0; S00 = 4 * sum(fracvec.*signs); diff00 = pi - S00; if(diff0>diff00) disp('s00 is clser t the limit') else disp('s00 is clser t the limit') end n N n S n

10 2. Cmpute the series Using 0 elements (S 0 ) Using 00 elements (S 00 ) S 4 3 Cmpare the S 0 and S 00 with the value f the limit. Which ne is clser t the limit? Display the answer t cmmand windw varsign = -; S0 = 4; fr in=3:2:9 S0 = S0 + 4 * varsign * /in; varsign = - * varsign; end

11 2. Cmpute the series Using 0 elements (S 0 ) Using 00 elements (S 00 ) S 4 3 Cmpare the S 0 and S 00 with the value f the limit. Which ne is clser t the limit? Display the answer t cmmand windw varsign = -; S00 = 4; fr in=3:2:99 S00 = S * varsign * /in; varsign = - * varsign; end

12 3. Cmpute sin(x) and cs(x) in the interval x = [0, 2π] (chse the number f elements in x s that the functins can be pltted smthly). Yu have t plt 3 graphs in the same figure: Plt sin(x) in blue in the specified interval, label the axes, assign a title t the figure Plt cs(x) in red in the specified interval, label the axes, assign a title t the figure and display the legend Cmpute the maximum between the tw functins at each pint in x, then plt the functin called maxsincs(x), with line width 2 and clr magenta, tgether with sin(x) and cs(x) in the specified interval. Label the axes, assign a title t the figure and display the legend

13 3. Cmpute sin(x) and cs(x) in the interval x = [0, 2π] (chse the number f elements in x s that the functins can be pltted smthly). Yu have t plt 3 graphs in the same figure: Plt sin(x) in blue in the specified interval, label the axes, assign a title t the figure Plt cs(x) in red in the specified interval, label the axes, assign a title t the figure and display the legend Cmpute the maximum between the tw functins at each pint in x, then plt the functin called maxsincs(x), with line width 2 and clr magenta, tgether with sin(x) and cs(x) in the specified interval. Label the axes, assign a title t the figure and display the legend x = [0:0.:2*pi]; sx = sin(x); cx = cs(x); figure subplt(3,,) plt(x,sx); xlim([0 2*pi]) title('sin(x)'); xlabel('x'); ylabel('f(x)') subplt(3,,2) plt(x,cx,'r'); xlim([0 2*pi]) title('cs(x)'); xlabel('x'); ylabel('f(x)'); legend('cs(x) ); maxsincsx = max(sin(x),cs(x)); subplt(3,,3) plt(x,sx); hld n plt(x,cx,'r'); plt(x,maxsincsx,'m','linewidth',2); hld ff xlim([0 2*pi]) title('maxsincs(x)'); xlabel('x'); ylabel('f(x)') legend('sin(x)','cs(x)','maxsincs(x)')

14 3. Output

15 4. Playing with the why functin. Open the why functin, cpy the full cde int a file named why2.m and save it. Mdify line f why2.m s that the functin returns the variable a. Write functin called cuntwhy that takes a filename.txt as input. In the functin, implement a lp which invkes the why2 cmmand at every iteratin. Stp the lp when the message returned by why2 is a repetitin f a message already seen. Write t filename.txt all the why2 messages seen, and display t cmmand windw the number f iteratins achieved.

4. Playing with the why functin. Open the why functin, cpy the full cde int a file named why2.m and save it. Mdify line f why2.m s that the functin returns the variable a.

16 4. Playing with the why functin. Open the why functin, cpy the full cde int a file named why2.m and save it. Mdify line f why2.m s that the functin returns the variable a. Write functin called cuntwhy that takes a filename.txt as input. In the functin, implement a lp which invkes the why2 cmmand at every iteratin. Stp the lp when the message returned by why2 is a repetitin f a message already seen. Write t filename.txt all the why2 messages seen, and display t cmmand windw the number f iteratins achieved.

17 Write functin called cuntwhy that takes a filename.txt as input. In the functin, implement a lp which invkes the why2 cmmand at every iteratin. Stp the lp when the message returned by why2 is a repetitin f a message already seen. Write t filename.txt all the why2 messages seen, and display t cmmand windw the number f iteratins achieved. numtimes = cuntwhy('whyanswers.txt'); disp(numtimes)

18 functin [cunt] = cuntwhy(filename) fid = fpen(filename,'w'); cunt = 0; in = ; cntrl = 0; while v{in} = why2; fprintf(fid,'%s\n',v{in}); cntrl = 0; if(in>) fr in2 = :in- if(strcmp(v{in},v{in2})) cntrl=; break end end end if(cntrl==) break else in = in+; end end fclse(fid) cunt = in

Handle = anther type f variables in MATLAB An identifier fr a functin h = @sin; handlename = @functinname; x = h(pi/2); x =

19 Handle = anther type f variables in MATLAB An identifier fr a functin h handlename x = h(pi/2); x = sin(pi/2); Can be inserted in structs and cells, nt arrays S.a S.b S.c C A

20 Handle t annymus functin h x + y; Example h x + y; x = h(,2); Example 2 h fun(x) + y; val = h(@sin, pi/2, 3); Example 3 functin [val] = myfun(fun,x,y) val = fun(x) + y; myfun(@sin, pi/2, 3); handlename ) bdy;

Handle t annymus functin h = @(x,y) x + y; handlename = @(v,v2, )

nt have a specified name, but it is identified nly trugh the functin

Example 2 h = @(fun,x,y) fun(x) + y; val = h(@sin, pi/2, 3); In

21 Handle t annymus functin h x + y; handlename ) bdy; Example h x + y; x = h(,2); Nte that the functin des nt have a specified name, but it is identified nly trugh the functin handle h! Example 2 h fun(x) + y; val = h(@sin, pi/2, 3); In these cases the variable fun is a functin handle itself! Example 3 functin [val] = myfun(fun,x,y) val = fun(x) + y; myfun(@sin, pi/2, 3);

22 fzer() finds a zer f a functin f(x) the clsest pssible t a specified pint x xz = fzer(@sin,x); res = fzer(funhandle,startpint); Example: find the zer f the functin sin(x) clsest t π/3 x = [-pi:0.:pi]; x = pi/3; xzer = fzer(@sin,x); plt(x, sin(x)); hld n plt(x, 0.*x,'k'); plt(x, sin(x),'g*'); plt(xzer, sin(xzer),'rd'); hld ff xlim([-pi pi]);

23 fzer() finds a zer f a functin f(x) inside a specified range x xz = fzer(@sin,x); res = fzer(funhandle,range); Example2: find the zer f the functin sin(x) between -π/3 and π/3 x = [-pi:0.:pi]; x = [-pi/3 pi/3]; xzer = fzer(@sin,x); plt(x, sin(x)); hld n plt(x, 0.*x,'k'); plt(x, sin(x),'g*'); plt(xzer, sin(xzer),'rd'); hld ff xlim([-pi pi]);

:pi]; If x is a range, the functin fun() x = [-pi/3 pi/3]; referenced by funhandle MUST change xzer = fzer(@sin,x);

24 fzer() finds a zer f a functin f(x) inside a specified range x xz = fzer(@sin,x); res = fzer(funhandle,range); Example2: find the zer f the functin sin(x) between -π/3 and π/3 x = [-pi:0.:pi]; If x is a range, the functin fun() x = [-pi/3 pi/3]; referenced by funhandle MUST change xzer = fzer(@sin,x); sign between x() and x(2)! plt(x, sin(x)); hld n plt(x, 0.*x,'k'); Sign(fun(x())) ~= Sign(fun(x(2)))!!! plt(x, sin(x),'g*'); plt(xzer, sin(xzer),'rd'); hld ff xlim([-pi pi]);

25 rts() - returns a vectr whse elements are the rts f a plynmial Example p = [ -5 6]; res = rts(p); [3;2] 2 x 5x 6 0 ( x 3)( x 2) 0 res = rts(vec);

26 Suppse we have the fllwing system f linear equatins: x + y 2z = 4 3x + 5y + z = -2-2x + 3y 0z = 7 We want t slve it and find the values f x, y and z We can stre the equatins in the fllwing way: AX = b, with A b z y x X

27 Suppse we have the fllwing system f linear equatins: x + y 2z = 4 3x + 5y + z = -2-2x + 3y 0z = 7 We knw frm linear algebra the slutin is X = inv(a) * b S with MATLAB we can slve it in 2 ways: X = inv(a) * b; X = A \ b;

28 Suppse we have the fllwing system f linear equatins: x + y 2z = 4 3x + 5y + z = -2-2x + 3y 0z = 7 We knw frm linear algebra the slutin is X = inv(a) * b S with MATLAB we can slve it in 2 ways: X = inv(a) * b; X = A \ b; NOTE : the \ peratr wrks fr square systems. Fr rectangular systems it gives the least squares slutin. NOTE 2 : we have t check if the system is ver r underdetermined

rank() Cmputes the rank f a matrix (the number f linearly independent rws r clumns) R = rank(m); det() Cmputes the determinant f a matrix, which must be square NOTE: if

29 rank() Cmputes the rank f a matrix (the number f linearly independent rws r clumns) R = rank(m); det() Cmputes the determinant f a matrix, which must be square NOTE: if determinant is nnzer, matrix is invertible d = det(m); trace() Cmputes the trace f a matrix (the sum f its diagnal elements) R = trace(m); inv() Cmputes the inverse f a matrix Ainv = inv(m)

30 Algebra/slving/determinant/cfactr_expansin.html

31 Eigenvalues, Eigenvectrs eig() [eigvect eigval] = eig(m); NOTE: M must be square RESULT: 2 matrices f same dimensin as M, a diagnal matrix eigval whse diagnal elements are the eigenvalues f M a matrix eigvec whse clumns are the eigenvectrs f M eigval*m = eigval*eigvec Singular Value Decmpsitin svd() [U, S, V] = svd(m); NOTE: M des nt have t be square RESULT: a diagnal matrix S f the same dimensin as M, with nnnegative diagnal elements in decreasing rder, and unitary matrices U and V s that M = U*S*V.

32 diff() D x = [:2:]; diff(x)/2; f x df x dx gradient() 2D x = :2; M = x *x [dx dy] = gradient(m); f x, y f x, x f x, y y y

33 Using the trapezidal rule x = 0:0.0:pi; inttx = trapz(x,sin(x)); int X 0 sin x dx

34 Using recursive adaptive Simpsn quadrature int X 0 sin x dx intqx = quad(@sin,0,pi) q = quad(@sin,0,pi,tl);

35 Fit an n th degree plynmial t predefined data plyfit() plyval() Example (fit 2nd degree plynmial t nisy data ) x = [-4:0.:4]; y = x.^2; ynisy = y + randn(size(y)); plt(x,ynisy,. ); ply = plyfit(x,ynisy,2); hld n; plt(x,plyval(ply,x), r );

36 Fit an n th degree plynmial t predefined data plyfit() plyval() Example (fit 2nd degree plynmial t nisy data ) x = [-4:0.:4]; y = x.^2; ynisy = y + randn(size(y)); plt(x,ynisy,. ); ply = plyfit(x,ynisy,2); hld n; plt(x,plyval(ply,x), r ); hld ff;

37 Fit an n th degree plynmial t predefined data plyfit() plyval() Example (fit 2nd degree plynmial t nisy data ) x = [-4:0.:4]; y = x.^2; ynisy = y + randn(size(y)); plt(x,ynisy,. ); ply = plyfit(x,ynisy,2); hld n; plt(x,plyval(ply,x), r ); hld ff;

39 bught = dlmread('grceries.txt',',',0,); fid = fpen('grceries.txt'); fd = textscan(fid,'%s'); fclse(fid); fd = strtk(fd{},','); M = eye(length(fd),length(fd)); fr in=:length(fd) fr in2=:length(fd) if(in ~= in2) M(in,in2) =... cbught(fd,bught,fd{in},fd{in2}) /... end end end numbught(fd,bught,fd{in}); imagesc(m); clrmap('ht'); clrbar

40 functin [num] = numbught(names,mat,index) fr in=:length(names) if(strcmp(names{in},index)) break end end num = sum(mat(in,:)); functin [num] = cbught(names,mat,index,index2) fr in=:length(names) if(strcmp(names{in},index)) in = in; end if(strcmp(names{in},index2)) in2 = in; end end num = sum( mat(in,:).* mat(in2,:) );

41 % series n = :0^5; S = cumsum( 2./ (n.^2 + 2.*n) ); figure subplt(,2,) plt(s) % series 2 S2 = cumsum((-).^n.* n./(n+)); subplt(,2,2) plt(s2)

42 Due at beginning f class, n exceptins Put yur cde (.m files) and additinal files in a single flder, name it yuruni_hw_x and zip it Uplad the zipped flder t CurseWrks Bring a printut f yur cde t class Gd luck and have fun, it s the last ne!

CHM112 Lab Graphing with Excel Grading Rubric

Name CHM112 Lab Graphing with Excel Grading Rubric Criteria Pints pssible Pints earned Graphs crrectly pltted and adhere t all guidelines (including descriptive title, prperly frmatted axes, trendline