matlab exercise

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES363 P. Casal St., Quiapo, Manila

CHEP 530D1

COMPUTER APPLICATIONS IN

CHEMICAL ENGINEERING

Crispulo G. MarananInstructor

Laboratory Exercise No. 4Familiarization with Matlab Environment, Built-in Functions, Matrices and Plotting

Lim, Justine Kei T. Laboratory Exercise No. 4Solving Ordinary Differential Equations December 14, 2013

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1. Objective(s):The activity aims to solve differential equations using matlab

2. Intended Learning Outcomes (ILOs):The students shall be able to:2.1 solve first order ordinary differential equations using matlab2.2 solve second order ordinary differential equations using matlab2.3 solve third order ordinary differential equations using matlab2.4 obtain general and particular solutions of first, second and third order ordinary differential equations2.5 solve systems of ordinary differential equations.

3. Discussion:Ordinary differential equations tend to arise whenever you need to model changing quantities that depend on the amount of other quantities around it. For example, in chemistry, the time rate of change of concentration ( dx/dt ) of a chemical solution often depends on the concetrations of other chemicals that surround it. In biology, differential equations are often used in population dynamics, to model the evolution and/or extinction of a particular species (like people, animals, bacteria, or even viruses like HIV) (eg., Volterra Equations). In finance, the stock market is often modeled via sets of coupled differential equations (e.g., Black-Scholes equation). In physics, dfq's are everywhere { we've seen them in Cosmology (e.g., Friedmann's Equations, non-linear structure growth and perturbation theory), Classical Dynamics (e.g., the orbits of planets, stars, and galaxies as specialized N-body problems, hydrodynamics),and Radioactive Transfer. Most differential equations are too complicated to write down a solution by hand (an "analytical solution"), so one has to revert to numerics to find any kind of solution at all.

Numerical methods are commonly used for solving mathematical problems that are formulated in science and engineering where it is difficult or even impossible to obtain exact solutions. Only a limited number of differential equations can be solved analytically. Numerical methods, on the other hand, can give an approximate solution to (almost) any equation. An ordinary differential equation (ODE) is an equation that contains an independent variable, a dependent variable and derivatives of the dependent variable.

The MATLAB ODE solvers are written to solve problems of the formdx/dt = F(t,x)

The Matlab ODE solvers are accesses by calling a function of the form[X,T] = ode** (@F, TimeSpan,Xo,Options,P1,P2,P3)

@F A handle to a function which returns a vector of rates of change

Timespan A row vector of times at which the solution is needed OR a vector of the form [start,end]

Xo A vector of initial valuesOptions (if omitted or set to [], the default settings are used

A data structure which allows the user to set various options associated with the ode solver

P1,P2,P3.. These are additional arguments which will be passed to @F

F must have the following formFunction [dx_dt] = F(t,x,P1,P2,P3…)dx_dt = …returnThere are several different ode solvers supplied with matlab.

Solver Implicit/Explicit Accuracy


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ode45 Explicit 4th order, medium accuracyode23 Explicit 2nd/3rd order, low accuracyode113 Explicit Very accurate 913th order)ode15s Implicit Anything from 1st-5th orderode23s Implicit Low accuracy (but may be more

stable than ode15s)ode23tb Implicit Low accuracy (but may be more

stable than ode15s)

ODE45 (an explicit Runge-Kutta method) is efficient, but can become unstable with stiff systems. This will manifest itself by the solver taking shorter and shorter time steps to compensate. The solution will either take a long time, or the time step will be reduced to the point where machine precision causes the routine to fail.The problems of solving an ODE are classified into initial-value problems (IVP) and boundary value

problems (BVP), depending on how the conditions at the endpoints of the domain are specified. All the conditions of an initial-value problem are specified at the initial point. On the other hand, the problem becomes a boundary-value problem if the conditions are needed for both initial and final points. The ODE in the time domain are initial-value problems, so all the conditions are specified at the initial time, such as t = 0 or x = 0. For notations, we use t or x as an independent variable.Some literatures use t as time for independent variable4. Resources:Matlab

5. Procedure:1. Though Matlab is primarily a numeric package, it can solve straightforward differential equations

symbolically. Suppose, for example, we want to solve the first order differential equation y’ = xy where y’ = dy/dx =y’(x).

2. We can use Matlab’s built-in dsolve(). The input for solving this problem in Matlab is given below:>>y = dsolve(‘Dy = y*x’,’x’) where y’(x) must be written as Dy. If it is y” (x), same as d 2 y/ x 2 ,it must be written as D2y.If it is y’’’(x), same as d 3 y/ x3 , it must be written as D3y. It is 8y’(x), same as 8dy/dx, it must be written as 8*Dy. All in Java command window. Press enter and record the results.

3. Notice in particular that MATLAB uses capital D to indicate the derivative and requires that the entire equation appear in single quotes. MATLAB takes t to be the independent variable by default, so here x must be explicitly defined as the independent variable. Alternatively, if you are going to use the same equation a number of times, you might choose to define it as a variable, say eqn 1. >>eqn1 = ‘Dy=y*x’;

>>y = dsolve(eqn1,’x’)

Press enter and record the results.

4. To solve an initial value problem, say, y’(x)=xy with y(1)=1 use >>y =dsolve (eqn1,’y(1)=1’,’x’)


5. To plot the solution to get a rough idea of its behavior.


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>>x = linspace(0,1,20);

>>z= eval(vectorize(y));

>>plot(x,z)


6. Suppose we want to solve and plot the solution to the second order equation y”(x) + 8y’(x) + 2y(x) = cos(x) ; y(0) = 0 , y’(0)=1

7. The following MATLAB code suffices: >>eqn2 = ‘D2y + 8*Dy + 2*y = cos(x)’;

>>inits2 = ‘ y(0)=0, Dy(0) = 1’;

>>y = dsolve(eqn2,inits2,’x’)

Press enter and see the results. Record the results.

>>z = eval(vectorize(y));

>>plot(x,z)


8. Suppose we want to solve and plot the solutions to the system of three ordinary differential equationsx’(t) = x(t) + 2y(t) –z(t)

y’(t) = x(t) + z(t)

z’(t) = 4x(t) – 4y(t) + 5z(t)

To find a general solution, each equation is now braced in its own pair of (single) quotation marks: >> [x,y,z] = dsolve(‘Dx = x +2*y-z’,’Dy = x + z’,’Dz = 4*x – 4*y + 5*z’)Press enter and record the results. Notice that since no independent variable is specified, MATLAB used its default, t.With conditions:

>> inits = ‘x(0)=1, y(0)= 2, z(0)=3’; >> [x,y,z] = dsolve(‘Dx = x +2*y-z’,’Dy = x + z’,’Dz = 4*x – 4*y + 5*z’,inits)

9. Plotting this solution can be accomplished as follows: >> t = linspace (0,0.5,25);

>> xx = eval(vectorize(x));

>> yy = eval(vectorize(y));


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>> zz = eval(vectorize (z));

>> plot (t,xx,t,yy,t,zz)


10. To find numerical solutions, MATLAB has a number of tools for numerically solving ordinary differential equations. Built-in functions ode23 and ode45, which implement versions of Runge-Kutta 2nd/3rd order and Runge-Kutta 4th and 5th order, respectively. Numerically approximate the solution of the first order differential equation

dy/dx = xy 2 + y ; y(0) =1 on the interval x ∈ [0,0.5]For any differential equation in the form y’ = f(x,y), we begin by defining the function f(x,y). For single equations, we can define f(x,y) as an inline function

>> f = inline(‘x*y^2 + y’)Press enter and record the results.

11. The basic usage for MATLAB’s solver ode45 is ode45(function, domain, initial condition). That is , we use >>[x,y] = ode45(f,[0,0.5],1)


12. To plot the values >>plot(x,y)


13. Choosing the partition. In approximating this solution, the algorithm ode 45 has selected a certain partition [0,0.5] and MATLAB has returned a value of y at each point in this partition. It is often the case that we would like to specify the partition of values on which MATLAB returns an approximation. For example, we might only want to approximate y(0.1),y(0.2)…… y(0.5).We can specify this by entering the vector values [0,0.1,0.2,0.3,0.4,0.5] as the domain in ode45. That is, we use >>xvalues = 0:.1:.5

Press enter and see the results. Record the results.

>>[x,y]=ode45(f,xvalues,1)


14. Several options are available for MATLAB’s ode45 solver, giving the user limited control over the algorithm.Two important options are relative and absolute tolerance, respectively RelTol and AbsTol in MATLAB. At each step of the ode45 algorithm, an error is approximated for that step. If yk is the approximation of y(xk) at step k, and ek is the approximate error at this step, then MATLAB chooses its partition to insure

ek ≤ max(RelTol *yk , AbsTol)

where the default values are RelTol=.001 and AbsTol=.000001. As an example for when we might want


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to change these values, observe that if yk becomes large, then the error ek will be allowed to grow quite large. In this case, we increase the value of RelTol. For the equation y’ = xy2 + y, with y(0)=1, the values of y get quite large as x near 1. In fact, with the default error tolerances, we find that the command

>> [x,y] = ode45(f,[0,1],1);Leads to an error message,caused by the fact that the values of y are getting too large as x nears 1.In order to fix this problem,we choose a smaller value for RelTol >>options = odeset(‘RelTol’,1e-10); >>[x,y]=ode45(f,[0,1],1,options); >>max(y)Press enter and record the results.

15. Alternatively, we can solve the same ODE by first defining f(x,y) as an M-file firstode.mfunction yprime = firstode(x,y);

% FIRSTODE: Computes yprime =x*y^2 + y

yprime = x*y^2 + y;

In this case, we only require one change in the ode45 command: we must use a pointer @ to indicate the m-file. That is, we use the following commands

>>xspan=[0,.5];>>y0=1;>.[x,y]=ode23(@firstode,xspan,y0);>>x

Press enter and record the results.16. Solving a system of ODE in MATLAB is quite similar to solving a single equation, though since a

system of equations cannot be defined as an inline function we must define it as an M-file. Solve the system of Lorenz equations,

dy/dt = -σx + σydy/dt = ρx – y -xzdy/dt = -βz + xy

where for the purposes of this example, we will take σ = 10, β = 8/3, and ρ=28, as well as x(0)=-8, y(0)=8, and z(0)=27. The MATLAB M-file containing the Lorenz equations appears belowfunction xprime = Lorenz(t,x);%LORENZ: Computes the derivatives involved in solving the Lorenz equationssig = 10;beta = 8/3;rho=28;xprime=[-sig*x(1) + sig*x(2);rho*x(1)-x(2)-x(1)*x(3);-beta*x(3) +x(1)*x(2)];

17. Observe that x is stored as x(1), y is stored as x(2) and z is stored as x(3).Additionally, xprime is a column vector,as is evident from semicolon following appearance of x(2).In the command window,we type

>>x0=[-8 8 27];

>>tspan=[0,20];

>>[t,x]= ode45(@lorenz,tspan,x0)


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Press enter and record the results.18. The matrix has been denoted x in the statement calling ode45, and in general any coordinate of the

matrix can be specified as x(m,n) where m denotes the row and n denotes the column.What we will be most interested in is referring to the columns x, which correspond with values of the components of the system. Along these lines, we can denote all row or all x by a colon : . For example, x(:,1) refers to all rows in the first column of the matrix x; that is, it refers to all values of our original x component. Using this information, we can easily plot the Lorenz strange attractor, which is a plot of z versus x:

>>plot(x(:,1),x(:,3))


19. We can also plot each component of the solution as a function of t>>subplot(3,1,1)

>>plot(t,x(:,1))

>>subplot(3,1,2)

>>plot(t,x(:,2)

>>subplot(3,1,3)

>>plot(t,x(:,3)

20. In analyzing system of differential equations, we often want to experiment with different parameter values. For example, in studying the Lorenz equations we might want to consider the behavior as a function of the values of σ,β and ρ. Of course, one way to change this is to manually re-open the M-file Lorenz.m each time we want to try new values, but not only is a slow way to do it, it’s unwieldy to automate it. What we can do instead is pass parameter values directly to our M-file through the ode45 call statement.Alter Lorenz.m into lorenz1.m, the latter of which accepts a vector of parameters that we denote p.

Function xprime = lorenz1(t,x,p);

%LORENZ ; Computes the derivatives involved in solving the Lorenz equations.

sig=p(1);beta=p(2);rho=p(3);

xprime=[-sig*x(1) + sig*x(2);rho*x(1)-x(2)-x(1)*x(3);-beta*x(3) +x(1)*x(2)];

21. We can now send parameter values with ode45>>p=[10 8/3 28];

>>[t,x]=ode45(@lorenz1,tspan,x0,[],p)

Press enter and record the results.22. The first step in solving a second (or higher) order ordinary differential equation in MATLAB is to write

the equation as a first order system. For the equation


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y”(x) + 8y’(x) + 2y(x) = cos(x) ; y(0) = 0 , y’(0)=1

Taking y1(x) = y(x) and y2(x) = y’(x)y1‘(x) = y2(x)y2 ‘(x) = -8y2(x) -2y1(x) + cos(x)

Proceed as in Procedure 16.23. Another class of ODE’s that often arise in applications are boundary value problems (BVP’s).

Consider ,for example, the differential y” – 3y’ + 2y = 0

y(0) = 0

y(1)=10

where our conditions y(0)=0 and y(1) = 10 are specified on the boundary of the interval of

interest

interest x ∈ [0,1]. The first step in solving this type of equation is to write it as a first order system with y1 = 1 and y2 = y’, for which we have

y1 ‘ = y2

y2 ‘ = -2y1 + 3y2

24. We record this system in the M-file bvpexample.mFunction yprime = bvpexample(t.y)

%BVPEXAMPLE : Differential equation for boundary value problem example

yprime=[y(2); -2*y(1) + 3*y(2)];

25. Next , we write the boundary conditions as the M-file bc.m, which records boundary residuesFunction res = bc(y0,y1)

%BC: Evaluates the residue of the boundary condition

Res=[y0(1);y1(1)-10];

By residue, we mean the left-hand side of the boundary condition once it has been set to 0.In this case, the second boundary condition is y(1)=10, so its residue is y(1)-10, which is recorded in the second component of the vector that bc.m returns The variables y0 and y1 represent the solution at x=0 and at x=1 respectively, while the 1 in the parenthesis indicates the first component of the vector. In the event that the second boundary condition was y’(1) = 10, we would replace y1(1)-10 with y1(2)-10.

26. We are now in a position to begin solving the boundary value problem. In the following code, we first specify a grid of x values for MATLAB to solve on and an initial guess for the vector that would be given for an initial value problem [y(0),y’(0)].We solve the boundary value problem with MATLAB’s built-in solver bvp4c.

>>sol = bvpinit(linspace(0,1,25),[0 1]);

>>sol = bvp4c(@bvpexample,@bc,sol);


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>>sol.x

Press enter and record the results.27. We observe that in this case MATLAB returns the solution as a structure whose first component sol.x

simply contains the x values we specified.The second component of the structure sol is sol.y, which is the matrix containing as its first row values of y(x) at the grid points we specified, and as its second row the corresponding values of y’(x).

28. For the first order differential equation where the highest derivative of the function is one :

From calculus, we all know that the solution to this equation is y(t) = Ce -5t, where C is some arbitrary constant. If we specified an initial condition (say, y(0)= 1.43), then our analytical solution would be y(t) = 1.43 e-5t.

29. In Matlab, we can use numerical integration techniques to solve differential equations like this one.For the differential equation in Procedure No. 28, you would make two .m files (one will be a function file, and the other will be a script that calls the function file).Using Matlab editor, create the file below and save it as ilovecats.m.

Function dy= ilovecats(t,y)

dy = zeros(1,1);

dy = -5 * y;

Now create another file and save it as happyscript.m.[t,y]=ode45(‘ilovecats’,[0,10],1.43);plot(t,y,’-‘)xlabel(‘time’);ylabel(‘y(t)’);title(‘This plot dedicated to kitties everywhere’);

30. Type ‘help ode45’ at the prompt. As a general rule of thumb, ode45 is the best function to apply as a first try for most problems.Ode 45 is an explicit (4,5) Runge-Kutta integrating technique.At Matlab prompt, type happyscript.m. Press enter and record the results.

31. A 2nd order differential equation is one where the highest derivative term is of order 2:

To integrate this in Matlab, we have to rewrite this single equation into a set of 2 first order differential equations. The reason behind this is because all Runge-Kutta solvers, including ode45, are built to only integrate over equations of the type dy/dt = f(t,y).We can easily do this by hand, by setting:

dy1/ dt = y2

dy2/dt = - ω2 sin(y1)where y1(t) represents θ(t), and y2(t) represents dθ/dt.


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32. Create an m file and save it as pendulumcats.mfunction dy = pendulumcats(t,y)

dy = zeros(2,1);

omega = 1;

dy(1) = y(2);

dy(2) = -omega*omega*sin(y(1));

33. Create another m file and save it as pendulumcatscript.m.[t,y] = ode45(‘pendulumcats’,[0,25],[1.0 1.0 ]);

plot(t,y(:,1),’-‘);

xlabel (‘time’);

ylabel(‘y_{1}(t)’);

title(‘\theta (t)’);

figure;plot(t,y(:,2),'-');xlabel('time');ylabel('y_{2}(t)');title('d \theta / dt (t)');

figure;plot(y(:,1),y(:,2),'-');xlabel('\theta (t)');ylabel('d \theta / dt (t)');title('Phase Plane Portrait for undamped pendulum');

34. The change in the function file, pendulumcats.m, is the initialization part in line two – dy = zeros(2,1); This is because we now have two equations we are integrating over (y1(t) and y2(t), so Matlab will store their data into a matrix with two columns.If you just type y at your Matlab prompt, you will get two columns of data that display.The first column is the set of y(t) (or y1(t)), whose data points you can alone access by typing y(:,1) at your prompt.The second column of y are the datapoints for y2(t), which you can access by themselves by typing y(:,2) at your prompt.

35. Run the commands. Record the results.36. Back in the day, scientists didn't know as much, and thought they could accurately predict the weather

once computers became more powerful. This is because weather people used many sets of differential


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equations to model the weather, and it took a long time to integrate those equations (keep in mind that awesome things like Matlab weren't around in the 50s and 60s { people still used slide rulers and tables to calculate many things, and the computers that were available back in the day had very little computing power, so integrating some ODEs, like those in the pendulum example, would take a crazy long time for the computer to chug through!).

Edward Lorenz was a mathematician and weather forecaster for the US Army Air Corps, and later an MIT professor. For many years, he was interested in solving a simple set of 3 coupled differential equations just because he wanted to find out what the weather would be like during the next week." These equations are called the Lorenz Equations, and were derived from simplified equations of convection rolls rising in the atmosphere. They are pretty simple and can be expressed as:

dx/dt = -Px + Pydy/dt = rx – y – xzdz/dt = xy –bz

where P, r and b are all constants ( P represents the Prandtl number, and r is the ratio of Rayleigh number to the critical Rayleigh number), and x, y and z are all functions of time. We can use Matlab to look at trajectories (i.e. plots of x(t) vs time, y(t) vs. time and z(t) vs. time) or phase plane portraits (i.e. x(t) vs y(t), x(t) vs z(t), and/or y(t) vs z(t) for this system.

37. The function file lorenz.m) should look like:function dy = lorenz(t,y)

dy = zeros(3,1);

P=10;

r=28;

b=8/3

dy(1)=P*(y(2)-y(1));

dy(2)=-y(1)*y(3) + r*y(1) – y(2);

dy(3) = y(1)*y(2) – b*y(3);

38. The script file lorenzscript.m should look like:[t,y] = ode45(‘lorenz’, [0 250], [1.0 1.0 1.0];

subplot(221)

plot (y(:,1),y(:,2),’-);

xlable(‘x(t)’);

ylabel(‘y(t)’);

title(‘ Phase Plane Portrait for Lorenz attractor – y(t) vs x(t)’);


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subplot(222)

plot(y(:,1),y(:,3),’-‘);

xlabel(“x(t)”);

ylabel(‘ z(t)’);

title(‘Phase Plane Portrait for Lorenz attractor – z(t) vs x(t)’);

subplot(223)

plot(‘ y(:,2),y(:,3,)’-’);

xlabel(‘y(t)’);

ylabel(‘z(t)’);

title(‘Phase Plane Portrait for Lorenz attractor – z(t) vs y(t)’);

suplot(224)

plot(0,0,’.’);

xlabel(‘Edward Lorenz’);

ylabel(‘Kitties’);

title(‘Kitties vs Lorenz’);

39. Run the script.It should take a little while to run. Record the results.40. To make a 3D plot ,add the following to the bottom of the script.

plot3(y(:,1),y(:,2),y(:,3),’-‘)

xlabel(‘x(t)’);

ylabel(‘y(t)’);

zlabel(‘z(t));

title(‘3D phase portrait of Lorenz Attractor’);


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Run the script and record the results.

41. The Matlab code to solve dy/dx = y(x) with no initial conditions is shown below:ODE1= ‘dy = y’

ODE1solved=dsolve(ODE1, ‘x’)

Record the results.

42. To specify initial conditions for the ODE is as follows :initConds = ‘y(0) = 5’

ODE1solved = dsolve(ODE1, initConds,’x’)

Record the results.

43. Matlab makes plotting functions easy. To plot the function:x = -5:0.01:5;

y_values = eval(vectorize(ODE1solved);

plot(x,y_values)

Record the results.

44. The same ideas apply to higher ODEs. To solve a second-order ODE with initial values at y(0) and y’(0).Then plot the function in the range [-5,5]ODE2 = ‘3*D2y – Dy + 6*y = 6 *sin(t) + 2 *cos(t)’

initConds = ‘y(0)=1, Dy(0)=2’

ODE2solved = simplify)dissolve(ODE2,initConds));

pretty(ODE2solved)

t=-5 :0.01:5;

y_values=eval(vectorize(ODE2solved));

plot(t,y_values)

Record the results.

45. Systems of ODEs can be solved in a similar manner.One simply defines each equation as before. The only thing that changes is the return of the dsolve function, which is now an array containing the explicit solutions of each of the functions in the systemsysODE1 = ‘Dx = 2*x + 3*z’

sysODE2 = ‘Dy = 6*z – y’


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sysODE3 = ‘Dz = 3*y – 12*x’

initConds = ‘x(1) = 5, y(2)=3, z(9) = 0’

[x,y,z] = dsolve(sysODE1,sysODE2,sysODE3,initConds)

Record the resultCourse: Laboratory Exercise No.:Group No.: Section:Group Members: Date Performed:

Date Submitted:Instructor:

6. Data and Results:

Procedure Matlab Result

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7 >> eqn2 = 'D2y + 8*Dy + 2*y = cos(x)';>> inits2 = 'y(0)=0, Dy(0) = 1';>> y = dsolve(eqn2,inits2,'x') y = (14^(1/2)*exp(4*x - 14^(1/2)*x)*exp(x*(14^(1/2) - 4))*(sin(x) - cos(x)*(14^(1/2) - 4)))/(28*((14^(1/2) - 4)^2 + 1)) - (98*14^(1/2) + 378)/(exp(x*(14^(1/2) + 4))*(868*14^(1/2) + 3136)) - (14^(1/2)*exp(4*x + 14^(1/2)*x)*(sin(x) + cos(x)*(14^(1/2) + 4)))/(28*exp(x*(14^(1/2) + 4))*((14^(1/2) + 4)^2 + 1)) - (exp(x*(14^(1/2) - 4))*(98*14^(1/2) - 378))/(868*14^(1/2) - 3136)


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8 >> [x,y,z] = dsolve('Dx = x +2*y-z','Dy = x + z','Dz = 4*x-4*y + 5*z') x = - (C8*exp(t))/2 - (C10*exp(3*t))/4 - (C9*exp(2*t))/2 y = (C8*exp(t))/2 + (C10*exp(3*t))/4 + (C9*exp(2*t))/4 z = C8*exp(t) + C10*exp(3*t) + C9*exp(2*t)

>> inits = 'x(0)=1, y(0)= 2, z(0)=3';>> [x,y,z] = dsolve('Dx = x +2*y-z','Dy = x + z','Dz = 4*x -4*y + 5*z',inits) x = 6*exp(2*t) - (5*exp(3*t))/2 - (5*exp(t))/2 y =


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(5*exp(3*t))/2 - 3*exp(2*t) + (5*exp(t))/2 z = 10*exp(3*t) - 12*exp(2*t) + 5*exp(t)

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>> f = inline('x*y^2 + y')

f =

Inline function: f(x,y) = x*y^2 + y

>>11 >> [x,y] = ode45(f,[0,0.5],1)

x =

0 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750 0.0875 0.1000 0.1125

y =

1.0000 1.0127 1.0256 1.0390 1.0526 1.0667 1.0811 1.0959 1.1111 1.1268


17

0.1250 0.1375 0.1500 0.1625 0.1750 0.1875 0.2000 0.2125 0.2250 0.2375 0.2500 0.2625 0.2750 0.2875 0.3000 0.3125 0.3250 0.3375 0.3500 0.3625 0.3750 0.3875 0.4000 0.4125 0.4250 0.4375 0.4500 0.4625 0.4750 0.4875 0.5000

1.1429 1.1594 1.1765 1.1940 1.2121 1.2308 1.2500 1.2698 1.2903 1.3115 1.3333 1.3559 1.3793 1.4035 1.4286 1.4545 1.4815 1.5094 1.5385 1.5686 1.6000 1.6327 1.6667 1.7021 1.7391 1.7778 1.8182 1.8605 1.9048 1.9512 2.0000


18

12

13

>> xvalues = 0:.1:.5

xvalues =

0 0.1000 0.2000 0.3000 0.4000 0.5000>> [x,y]=ode45(f,xvalues,1)

x =

0 0.1000 0.2000 0.3000 0.4000 0.5000

y =

1.0000 1.1111 1.2500 1.4286 1.6667 2.0000

14 >> [x,y] = ode45(f,[0,1],1);Warning: Failure at t=9.999897e-001. Unable to meet integrationtolerances without reducing the step size below the smallest valueallowed (1.776357e-015) at time t.


19

> In ode45 at 371>> options = odeset('RelTol',1e-10);>> [x,y]=ode45(f,[0,1],1,options);>> max(y)

ans =

2.4251e+007

>>

15

function yprime = firstode(x,y);% FIRSTODE: Computes yprime =x*y^2 + yyprime = x*y^2 + y;

>> xspan=[0,.5];>> y0=1;>> [x,y]=ode23(@firstode,xspan,y0);>> x

x =

0 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000

16 – 17 t =

0 0.0025 0.0050 0.0075 0.0100 0.0226 0.0352 0.0477 0.0603 0.0716 0.0830 0.0943 0.1057 0.1171

13.5506 13.5654 13.5802 13.5950 13.6120 13.6290 13.6459 13.6629 13.6831 13.7034 13.7236 13.7438 13.7708 13.7978 13.8249 13.8519

8.0187 11.5183 21.0426 8.7392 12.3817 21.9094 9.4995 13.1632 23.0832 10.2453 13.7449 24.5259 10.9334 14.0355 26.1822 11.5134 13.9605 27.9543 11.9341 13.4672 29.7094 12.1431 12.5738 31.2824 12.1068 11.3508 32.5282 11.8240 9.9034 33.3473


20

0.1284 0.1398 0.1511 0.1646 0.1781 0.1916 0.2051 0.2214 0.2377 0.2540 0.2703 0.2919 0.3135 0.3351 0.3567 0.3768 0.3968 0.4169 0.4370 0.4534 0.4698 0.4862 0.5026 0.5181 0.5337 0.5492 0.5648 0.5801 0.5954 0.6108 0.6261 0.6430 0.6599 0.6767 0.6936 0.7134 0.7332 0.7530 0.7728 0.7905 0.8082 0.8259 0.8437 0.8658 0.8880 0.9102 0.9323 0.9545 0.9766

13.8676 13.8834 13.8992 13.9149 13.9307 13.9465 13.9622 13.9780 13.9933 14.0086 14.0239 14.0392 14.0531 14.0670 14.0810 14.0949 14.1085 14.1222 14.1358 14.1494 14.1681 14.1867 14.2054 14.2240 14.2393 14.2545 14.2698 14.2851 14.3015 14.3179 14.3343 14.3507 14.3709 14.3911 14.4113 14.4315 14.4571 14.4826 14.5082 14.5337 14.5593 14.5848 14.6104 14.6359 14.6503 14.6646 14.6790 14.6933 14.7076

11.3769 8.5321 33.6777 10.7768 7.2117 33.6326 10.0654 6.0257 33.2527 9.2887 5.0293 32.6066 8.5030 4.2540 31.7851 7.7371 3.6841 30.8395 7.0210 3.3047 29.8218 6.3759 3.0882 28.7750 5.8211 3.0035 27.7420 5.3536 3.0259 26.7271 4.9749 3.1355 25.7440 4.6837 3.3162 24.8019 4.4327 3.6324 23.6726 4.3081 4.0339 22.6313 4.3025 4.5154 21.6891 4.4068 5.0803 20.8593 4.6164 5.7448 20.1513 4.9319 6.5098 19.5955 5.3541 7.3809 19.2196 5.8818 8.3606 19.0593 6.5492 9.5013 19.1684 7.3311 10.7255 19.6243 8.2146 11.9712 20.4847 9.1694 13.1352 21.7888 9.9374 13.9009 23.1285 10.6852 14.4397 24.7218 11.3689 14.6640 26.5077 11.9388 14.5051 28.3854 12.3455 13.9185 30.2221 12.5396 12.9285 31.8568 12.4899 11.6093 33.1505 12.1951 10.0684 34.0083 11.7189 8.5663 34.3740 11.0787 7.1206 34.3370 10.3182 5.8214 33.9428 9.4853 4.7277 33.2662 8.6482 3.8807 32.4144 7.8276 3.2499 31.4358 7.0545 2.8189 30.3838 6.3508 2.5577 29.3026 5.7384 2.4336 28.2383 5.2124 2.4181 27.1915


21

0.9988 1.0209 1.0378 1.0546 1.0715 1.0883 1.1052 1.1221 1.1389 1.1558 1.1710 1.1863 1.2016 1.2168 1.2310 1.2452 1.2595 1.2737 1.2873 1.3008 1.3144 1.3280 1.3401 1.3523 1.3645 1.3766 1.3915 1.4064 1.4213 1.4361 1.4492 1.4622 1.4752 1.4883 1.5034 1.5186 1.5337 1.5489 1.5654 1.5820 1.5985 1.6150 1.6372 1.6593 1.6815 1.7036 1.7318 1.7601 1.7883

14.7220 14.7363 14.7507 14.7642 14.7777 14.7912 14.8047 14.8165 14.8283 14.8401 14.8519 14.8674 14.8829 14.8984 14.9139 14.9272 14.9405 14.9537 14.9670 14.9803 14.9936 15.0068 15.0201 15.0324 15.0448 15.0571 15.0694 15.0842 15.0990 15.1137 15.1285 15.1457 15.1629 15.1801 15.1973 15.2176 15.2379 15.2583 15.2786 15.3033 15.3280 15.3526 15.3773 15.3924 15.4074 15.4225 15.4375 15.4526 15.4676

4.7742 2.4891 26.1747 4.4226 2.6285 25.1958 4.1045 2.8726 24.0607 3.8999 3.1851 22.9955 3.8005 3.5584 22.0059 3.7971 3.9929 21.0983 3.8942 4.5484 20.2042 4.0956 5.1926 19.4313 4.4011 5.9362 18.7980 4.8094 6.7929 18.3299 5.2978 7.7295 18.0667 5.8877 8.7817 18.0233 6.5810 9.9394 18.2453 7.3748 11.1738 18.7842 8.2914 12.4796 19.7289 9.2824 13.7095 21.1247 10.3040 14.7204 22.9842 11.2851 15.3407 25.2476 11.9562 15.4372 27.1809 12.5001 15.1306 29.1608 12.8684 14.3958 31.0534 13.0247 13.2580 32.7204 12.9453 11.7925 34.0284 12.6242 10.1351 34.8888 12.0784 8.4274 35.2674 11.3481 6.7943 35.1838 10.4461 5.2826 34.6809 9.4653 4.0407 33.8500 8.4677 3.0951 32.7883 7.5030 2.4293 31.5993 6.7313 2.0478 30.5409 6.0279 1.8144 29.4701 5.4022 1.7020 28.4088 4.8590 1.6845 27.3707 4.3547 1.7492 26.2595 3.9473 1.8818 25.1925 3.6309 2.0663 24.1732 3.3984 2.2923 23.2035 3.2375 2.5646 22.2525 3.1515 2.8734 21.3573 3.1348 3.2192 20.5199


22

1.8165 1.8347 1.8529 1.8710 1.8892 1.9034 1.9176 1.9318 1.9459 1.9601 1.9743 1.9885 2.0027 2.0165 2.0303 2.0442 2.0580 2.0694 2.0807 2.0920 2.1033 2.1181 2.1329 2.1477 2.1625 2.1796 2.1967 2.2138 2.2310 2.2448 2.2586 2.2724 2.2862 2.3005 2.3148 2.3291 2.3434 2.3600 2.3766 2.3932 2.4098 2.4294 2.4490 2.4686 2.4882 2.5132 2.5382 2.5632 2.5882

15.4826 15.4977 15.5115 15.5254 15.5392 15.5530 15.5656 15.5782 15.5908 15.6033 15.6144 15.6254 15.6365 15.6475 15.6648 15.6821 15.6993 15.7166 15.7288 15.7410 15.7533 15.7655 15.7777 15.7899 15.8021 15.8143 15.8281 15.8418 15.8556 15.8693 15.8852 15.9011 15.9170 15.9329 15.9515 15.9702 15.9889 16.0075 16.0299 16.0523 16.0747 16.0971 16.1150 16.1328 16.1506 16.1684 16.1863 16.2041 16.2219

3.1817 3.6063 19.7432 3.2881 4.0412 19.0315 3.4534 4.5303 18.3898 3.6779 5.0812 17.8258 3.9625 5.7030 17.3495 4.3932 6.5684 16.9072 4.9236 7.5657 16.6461 5.5608 8.7018 16.6081 6.3103 9.9714 16.8462 7.2518 11.4740 17.4859 8.3227 13.0380 18.6222 9.4960 14.5315 20.3315 10.7104 15.7517 22.6344 11.5918 16.3341 24.7155 12.3890 16.4995 27.0118 13.0395 16.1562 29.3892 13.4825 15.2677 31.6744 13.6655 13.8543 33.6675 13.5519 12.0519 35.1916 13.1353 10.0348 36.1371 12.4486 7.9726 36.4706 11.5809 6.1024 36.2582 10.5746 4.4815 35.6140 9.4974 3.1728 34.6442 8.4092 2.1871 33.4734 7.5037 1.5742 32.3884 6.6523 1.1500 31.2680 5.8711 0.8833 30.1435 5.1704 0.7414 29.0359 4.5385 0.6941 27.9282 3.9951 0.7223 26.8578 3.5367 0.8056 25.8281 3.1582 0.9283 24.8401 2.7881 1.1198 23.6687 2.5182 1.3423 22.5594 2.3369 1.5884 21.5103 2.2319 1.8575 20.5201 2.1932 2.1320 19.6504 2.2076 2.4326 18.8312 2.2717 2.7645 18.0636 2.3826 3.1354 17.3494


23

2.6046 2.6211 2.6376 2.6541 2.6705 2.6870 2.7035 2.7199 2.7367 2.7534 2.7702 2.7869 2.8004 2.8139 2.8274 2.8409 2.8534 2.8659 2.8784 2.8908 2.9039 2.9169 2.9299 2.9429 2.9559 2.9689 2.9819 2.9949 3.0069 3.0189 3.0309 3.0429 3.0573 3.0718 3.0863 3.1007 3.1166 3.1325 3.1484 3.1643 3.1852 3.2061 3.2270 3.2480 3.2764 3.3049 3.3334 3.3618 3.3815

16.2398 16.2545 16.2693 16.2840 16.2988 16.3127 16.3266 16.3405 16.3544 16.3666 16.3788 16.3911 16.4033 16.4182 16.4332 16.4482 16.4631 16.4771 16.4910 16.5049 16.5188 16.5298 16.5407 16.5516 16.5625 16.5734 16.5843 16.5953 16.6062 16.6188 16.6314 16.6440 16.6567 16.6716 16.6865 16.7015 16.7164 16.7341 16.7517 16.7694 16.7871 16.8077 16.8284 16.8491 16.8698 16.8936 16.9175 16.9414 16.9652

2.5399 3.5571 16.6877 2.7451 4.0360 16.0878 3.0008 4.5821 15.5565 3.3104 5.2066 15.1032 3.8032 6.1573 14.6507 4.4101 7.2872 14.3914 5.1463 8.6150 14.3815 6.0264 10.1442 14.6977 6.9820 11.7289 15.3650 8.0735 13.4226 16.4955 9.2858 15.1191 18.1813 10.5769 16.6410 20.4828 11.5698 17.5372 22.6535 12.5196 18.0582 25.1368 13.3666 18.0785 27.8297 14.0445 17.5109 30.5728 14.4884 16.3114 33.1613 14.6375 14.5557 35.3664 14.4572 12.3882 37.0078 13.9542 9.9861 37.9820 13.2432 7.7785 38.2709 12.3345 5.7220 38.0418 11.2843 3.9241 37.3826 10.1514 2.4435 36.4104 9.0915 1.3759 35.3496 8.0478 0.5657 34.1912 7.0496 -0.0155 32.9897 6.1176 -0.4068 31.7855 5.3091 -0.6394 30.6672 4.5772 -0.7667 29.5807 3.9237 -0.8149 28.5327 3.3477 -0.8061 27.5259 2.7859 -0.7497 26.4386 2.3105 -0.6624 25.4013 1.9131 -0.5585 24.4111 1.5846 -0.4479 23.4646 1.3589 -0.3562 22.7149 1.1691 -0.2666 21.9912 1.0111 -0.1806 21.2922 0.8805 -0.0988 20.6167 0.7737 -0.0214 19.9635


24

3.4011 3.4208 3.4404 3.4549 3.4694 3.4838 3.4983 3.5127 3.5272 3.5417 3.5561 3.5693 3.5824 3.5955 3.6086 3.6203 3.6320 3.6438 3.6555 3.6668 3.6781 3.6895 3.7008 3.7175 3.7341 3.7508 3.7674 3.7812 3.7951 3.8089 3.8227 3.8365 3.8503 3.8641 3.8780 3.8936 3.9093 3.9249 3.9406 3.9587 3.9769 3.9950 4.0132 4.0354 4.0577 4.0800 4.1022 4.1197 4.1371

16.9810 16.9968 17.0126 17.0283 17.0441 17.0599 17.0756 17.0914 17.1049 17.1185 17.1320 17.1455 17.1581 17.1706 17.1832 17.1957 17.2062 17.2166 17.2271 17.2375 17.2542 17.2709 17.2875 17.3042 17.3178 17.3314 17.3450 17.3586 17.3722 17.3858 17.3993 17.4129 17.4280 17.4430 17.4580 17.4730 17.4903 17.5075 17.5247 17.5420 17.5626 17.5833 17.6039 17.6246 17.6446 17.6647 17.6847 17.7047 17.7248

0.6876 0.0519 19.3318 0.6195 0.1214 18.7205 0.5668 0.1878 18.1291 0.5314 0.2446 17.6224 0.5053 0.3004 17.1301 0.4874 0.3555 16.6519 0.4770 0.4109 16.1873 0.4731 0.4821 15.6200 0.4789 0.5560 15.0731 0.4937 0.6341 14.5460 0.5167 0.7180 14.0382 0.5518 0.8201 13.4954 0.5969 0.9337 12.9751 0.6527 1.0614 12.4770 0.7196 1.2059 12.0006 0.8234 1.4217 11.4207 0.9504 1.6784 10.8764 1.1044 1.9853 10.3688 1.2905 2.3537 9.8999 1.6487 3.0548 9.2748 2.1217 3.9848 8.7675 2.7450 5.2178 8.4241 3.5702 6.8358 8.3275 4.4719 8.5686 8.5368 5.6024 10.7039 9.1495 6.9975 13.2446 10.3601 8.6763 16.0843 12.4369 10.0301 18.1534 14.5962 11.4897 20.0698 17.4470 12.9958 21.5772 21.0209 14.4474 22.3638 25.2265 15.4361 22.2729 28.7207 16.2433 21.4571 32.2547 16.7970 19.8761 35.5963 17.0398 17.5811 38.5078 16.9318 14.7151 40.7598 16.4585 11.5359 42.2151 15.6386 8.3060 42.8386 14.5259 5.2501 42.6861


25

4.1545 4.1719 4.1893 4.2067 4.2241 4.2415 4.2574 4.2733 4.2892 4.3051 4.3203 4.3355 4.3507 4.3658 4.3804 4.3950 4.4096 4.4241 4.4391 4.4541 4.4692 4.4842 4.5029 4.5217 4.5405 4.5593 4.5763 4.5933 4.6103 4.6274 4.6461 4.6647 4.6834 4.7021 4.7289 4.7557 4.7826 4.8094 4.8301 4.8509 4.8716 4.8924 4.9080 4.9236 4.9392 4.9547 4.9703 4.9859 5.0015

17.7448 17.7648 17.7849 17.8011 17.8173 17.8335 17.8497 17.8653 17.8809 17.8965 17.9122 17.9264 17.9407 17.9550 17.9692 17.9838 17.9984 18.0129 18.0275 18.0474 18.0674 18.0873 18.1072 18.1237 18.1401 18.1566 18.1730 18.1909 18.2088 18.2267 18.2446 18.2688 18.2930 18.3172 18.3414 18.3627 18.3840 18.4053 18.4266 18.4426 18.4586 18.4746 18.4906 18.5066 18.5226 18.5386 18.5546 18.5690 18.5835

13.5222 3.1448 42.1329 12.4252 1.3066 41.2933 11.2712 -0.2409 40.2475 10.0932 -1.5001 39.0727 8.9204 -2.4923 37.8350 7.7771 -3.2464 36.5805 6.6818 -3.7963 35.3446 5.6479 -4.1786 34.1513 4.3777 -4.4882 32.6467 3.2406 -4.6312 31.2533 2.2368 -4.6655 29.9717 1.3610 -4.6366 28.7945 0.5015 -4.5691 27.5602 -0.2238 -4.4973 26.4315 -0.8351 -4.4456 25.3943 -1.3531 -4.4301 24.4371 -1.8018 -4.4604 23.5445 -2.1936 -4.5428 22.7189 -2.5435 -4.6817 21.9572 -2.8669 -4.8789 21.2573 -3.1652 -5.1239 20.6437 -3.4601 -5.4253 20.0894 -3.7593 -5.7846 19.5978 -4.0711 -6.2026 19.1733 -4.4559 -6.7584 18.7742 -4.8744 -7.3949 18.4845 -5.3336 -8.1119 18.3198 -5.8399 -8.9053 18.2977 -6.5174 -9.9475 18.4880 -7.2718 -11.0646 18.9584 -8.0977 -12.2112 19.7512 -8.9806 -13.3157 20.8983 -9.8209 -14.2138 22.2795 -10.6527 -14.8948 23.9609 -11.4308 -15.2546 25.8890 -12.1008 -15.2035 27.9617 -12.6058 -14.6730 30.0369 -12.8856 -13.6709 31.9278


26

5.0171 5.0327 5.0484 5.0640 5.0796 5.0925 5.1053 5.1182 5.1310 5.1440 5.1569 5.1698 5.1827 5.1975 5.2122 5.2270 5.2417 5.2565 5.2712 5.2860 5.3007 5.3138 5.3270 5.3401 5.3532 5.3645 5.3758 5.3871 5.3984 5.4092 5.4199 5.4306 5.4413 5.4548 5.4683 5.4818 5.4954 5.5115 5.5277 5.5439 5.5601 5.5794 5.5987 5.6180 5.6373 5.6591 5.6808 5.7025 5.7243

18.5980 18.6124 18.6257 18.6390 18.6523 18.6657 18.6779 18.6902 18.7024 18.7147 18.7304 18.7462 18.7620 18.7778 18.7912 18.8046 18.8180 18.8314 18.8448 18.8582 18.8716 18.8850 18.8961 18.9072 18.9182 18.9293 18.9420 18.9546 18.9673 18.9799 18.9951 19.0102 19.0253 19.0404 19.0583 19.0763 19.0942 19.1121 19.1330 19.1539 19.1748 19.1956 19.2192 19.2427 19.2662 19.2897 19.3075 19.3253 19.3431

-12.8982 -12.2667 33.4657 -12.6351 -10.5751 34.5289 -12.1770 -8.9500 35.0211 -11.5368 -7.3624 35.0801 -10.7572 -5.9144 34.7489 -9.8873 -4.6756 34.1031 -8.9986 -3.6984 33.2550 -8.1151 -2.9543 32.2610 -7.2711 -2.4281 31.1806 -6.4916 -2.0886 30.0635 -5.8139 -1.9028 28.9798 -5.2192 -1.8318 27.9107 -4.7101 -1.8510 26.8688 -4.2861 -1.9399 25.8618 -3.8960 -2.1096 24.7427 -3.6074 -2.3358 23.6796 -3.4122 -2.6082 22.6750 -3.3015 -2.9228 21.7311 -3.2677 -3.3594 20.6798 -3.3368 -3.8634 19.7256 -3.5051 -4.4439 18.8775 -3.7674 -5.1175 18.1495 -4.0712 -5.7941 17.6279 -4.4506 -6.5641 17.2285 -4.9098 -7.4359 16.9737 -5.4530 -8.4145 16.8923 -6.3471 -9.9370 17.1265 -7.4166 -11.6254 17.8839 -8.6464 -13.3667 19.2875 -9.9811 -14.9403 21.4228 -10.8514 -15.7209 23.1946 -11.6822 -16.1929 25.2323 -12.4218 -16.2554 27.4499 -13.0136 -15.8383 29.7165 -13.4029 -14.9063 31.8653 -13.5385 -13.5161


27

5.7476 5.7708 5.7941 5.8174 5.8354 5.8534 5.8714 5.8894 5.9029 5.9163 5.9298 5.9432 5.9566 5.9701 5.9835 5.9970 6.0092 6.0214 6.0335 6.0457 6.0563 6.0669 6.0776 6.0882 6.0997 6.1112 6.1227 6.1342 6.1497 6.1653 6.1808 6.1963 6.2113 6.2263 6.2413 6.2563 6.2713 6.2863 6.3013 6.3163 6.3333 6.3503 6.3673 6.3843 6.4045 6.4246 6.4448 6.4649 6.4856

19.3609 19.3744 19.3879 19.4015 19.4150 19.4285 19.4420 19.4555 19.4690 19.4820 19.4950 19.5081 19.5211 19.5316 19.5421 19.5525 19.5630 19.5783 19.5935 19.6087 19.6240 19.6400 19.6560 19.6720 19.6881 19.7022 19.7164 19.7306 19.7447 19.7601 19.7754 19.7907 19.8061 19.8237 19.8413 19.8590 19.8766 19.8980 19.9194 19.9408 19.9622 19.9717 19.9811 19.9906 20.0000

x =

33.7098 -13.3915 -11.7798 35.1003 -12.9677 -9.8384 35.9466 -12.3501 -8.0027 36.2233 -11.5569 -6.2818 36.0450 -10.6394 -4.7707 35.4812 -9.6513 -3.5237 34.6292 -8.7058 -2.6046 33.6577 -7.7813 -1.9174 32.5816 -6.9064 -1.4377 31.4525 -6.1010 -1.1309 30.3097 -5.3852 -0.9611 29.1918 -4.7543 -0.8966 28.0990 -4.2091 -0.9120 27.0401 -3.7478 -0.9864 26.0196 -3.3335 -1.1167 24.9490 -3.0058 -1.2844 23.9262 -2.7568 -1.4797 22.9508 -2.5779 -1.6975 22.0221 -2.4469 -1.9779 20.9994 -2.3911 -2.2874 20.0385 -2.4042 -2.6296 19.1398 -2.4791 -3.0127 18.3052 -2.5967 -3.4041 17.6071 -2.7623 -3.8446 16.9677 -2.9774 -4.3429 16.3925 -3.2436 -4.9092 15.8889 -3.6471 -5.7157 15.3822 -4.1429 -6.6609 15.0255 -4.7417 -7.7612 14.8556 -5.4544 -9.0265 14.9220 -6.3290 -10.5181 15.3090 -7.3465 -12.1621 16.1159 -8.5012 -13.8823 17.4419 -9.7647 -15.5338 19.3729 -10.7621 -16.6137 21.2720 -11.7502 -17.4022 23.5327 -12.6744 -17.7640


28

6.5064 6.5271 6.5478 6.5685 6.5893 6.6100 6.6307 6.6494 6.6680 6.6866 6.7052 6.7236 6.7420 6.7604 6.7788 6.7970 6.8151 6.8333 6.8515 6.8746 6.8977 6.9208 6.9440 6.9673 6.9907 7.0141 7.0375 7.0621 7.0867 7.1113 7.1359 7.1552 7.1745 7.1938 7.2131 7.2324 7.2516 7.2709 7.2902 7.3081 7.3260 7.3439 7.3619 7.3794 7.3969 7.4144 7.4319 7.4491 7.4664

-8.0000 8.0000 27.0000 -7.6036 7.9571 26.6639 -7.2182 7.9093 26.3387 -6.8436 7.8574 26.0242 -6.4796 7.8021 25.7199 -4.8119 7.4981 24.3412 -3.3774 7.1889 23.1694 -2.1473 6.9160 22.1667 -1.0914 6.7046 21.3016 -0.2618 6.5763 20.6165 0.4684 6.5119 20.0097 1.1163 6.5131 19.4715 1.6982 6.5799 18.9945 2.2291 6.7105 18.5741 2.7204 6.9037 18.2088 3.1825 7.1578 17.8982 3.6255 7.4708 17.6432 4.1388 7.9154 17.4156 4.6484 8.4367 17.2764 5.1634 9.0304 17.2341 5.6926 9.6906 17.2988 6.3616 10.5662 17.5365 7.0672 11.5092 17.9737 7.8113 12.4920 18.6358 8.5916 13.4742 19.5445 9.6642 14.6894 21.1488 10.7406 15.6455 23.2205 11.7532 16.1565 25.6930 12.6109 16.0569 28.3967 13.1833 15.2991 30.8977 13.4538 13.9363 33.1074 13.3694 12.0970 34.7935 12.9326 9.9522 35.8158 12.3397 8.1418 36.1205 11.5704 6.4354 35.9751 10.6745 4.9284 35.4447 9.7046 3.6772 34.6231 8.7653 2.7427 33.6673 7.8442 2.0411 32.6010 6.9707 1.5493 31.4770 6.1655 1.2329 30.3361 5.4502 1.0564 29.2197 4.8195 0.9872 28.1270

26.0863 -13.4685 -17.5833 28.8021 -14.0632 -16.7725 31.4955 -14.3849 -15.3500 33.9247 -14.3823 -13.4160 35.8751 -14.0444 -11.1270 37.1982 -13.4802 -8.9854 37.7832 -12.6988 -6.9120 37.8355 -11.7488 -5.0327 37.4194 -10.6865 -3.4298 36.6363 -9.6345 -2.2038 35.6673 -8.5783 -1.2524 34.5534 -7.5542 -0.5546 33.3610 -6.5885 -0.0728 32.1430 -5.7525 0.2220 31.0099 -4.9923 0.3948 29.9008 -4.3116 0.4740 28.8259 -3.7110 0.4843 27.7904 -3.2116 0.4487 26.8431 -2.7781 0.3824 25.9321 -2.4057 0.2958 25.0560 -2.0890 0.1968 24.2133 -1.8362 0.0972 23.4475 -1.6234 -0.0052 22.7083 -1.4465 -0.1083 21.9944 -1.3012 -0.2107 21.3045 -1.1837 -0.3122 20.6376 -1.0909 -0.4125 19.9927 -1.0200 -0.5121 19.3690 -0.9684 -0.6116 18.7658 -0.9304 -0.7256 18.1072 -0.9119 -0.8425 17.4733 -0.9110 -0.9640 16.8632 -0.9261 -1.0923 16.2764 -0.9603 -1.2464 15.6483 -1.0124 -1.4152 15.0483 -1.0823 -1.6021 14.4760 -1.1703 -1.8113 13.9314 -1.2992 -2.0952 13.3213 -1.4575 -2.4251 12.7521


29

7.4836 7.5009 7.5219 7.5429 7.5640 7.5850 7.6081 7.6312 7.6544 7.6775 7.6997 7.7218 7.7440 7.7661 7.7891 7.8120 7.8350 7.8579 7.8756 7.8932 7.9108 7.9285 7.9461 7.9638 7.9814 7.9991 8.0175 8.0359 8.0543 8.0727 8.0885 8.1043 8.1201 8.1359 8.1534 8.1709 8.1883 8.2058 8.2239 8.2420 8.2601 8.2782 8.2963 8.3143 8.3324 8.3505 8.3726 8.3946 8.4167

4.2745 0.9998 27.0673 3.8136 1.0732 26.0456 3.3977 1.2052 24.9665 3.0701 1.3766 23.9360 2.8229 1.5776 22.9538 2.6474 1.8029 22.0195 2.5215 2.0967 20.9841 2.4740 2.4228 20.0137 2.4987 2.7854 19.1090 2.5884 3.1936 18.2727 2.7203 3.6067 17.5857 2.9021 4.0727 16.9609 3.1355 4.6006 16.4046 3.4227 5.2006 15.9255 3.8633 6.0689 15.4528 4.4045 7.0869 15.1502 5.0572 8.2698 15.0616 5.8325 9.6221 15.2452 6.7366 11.1294 15.7701 7.7751 12.7552 16.7318 8.9361 14.4072 18.2201 10.1829 15.9264 20.3004 11.1525 16.8620 22.2972 12.0912 17.4645 24.6136 12.9427 17.6100 27.1618 13.6426 17.2051 29.7979 14.1267 16.1931 32.3315 14.3321 14.6265 34.5383 14.2197 12.6285 36.2332 13.7901 10.3617 37.3003 13.1508 8.2555 37.6909 12.3118 6.2634 37.5723 11.3253 4.4967 37.0202 10.2481 3.0212 36.1417 9.2130 1.9266 35.1313 8.1878 1.0906 34.0053 7.2042 0.4879 32.8217 6.2848 0.0808 31.6251 5.4803 -0.1636 30.4970 4.7543 -0.2961 29.3966

-1.6487 -2.8107 12.2262 -1.8774 -3.2633 11.7475 -2.3018 -4.0878 11.1364 -2.8473 -5.1403 10.6736 -3.5438 -6.4783 10.4146 -4.4313 -8.1566 10.4502 -5.5165 -10.1506 10.8927 -6.8628 -12.5371 11.9344 -8.4876 -15.2247 13.8207 -10.3600 -17.9208 16.8315 -11.4857 -19.2591 19.1067 -12.6212 -20.3023 21.8074 -13.7150 -20.8952 24.8810 -14.7011 -20.8905 28.2075 -15.5064 -20.1614 31.6057 -16.0474 -18.6714 34.8096 -16.2558 -16.4812 37.5581 -16.0971 -13.7396 39.6402 -15.6238 -10.9510 40.8333 -14.8569 -8.1353 41.3221 -13.8405 -5.4856 41.1612 -12.6360 -3.1429 40.4631 -11.4049 -1.3093 39.4625 -10.1249 0.1643 38.2402 -8.8441 1.2903 36.8938 -7.6002 2.1099 35.5054 -6.5981 2.6057 34.3420 -5.6548 2.9512 33.2080 -4.7778 3.1763 32.1166 -3.9713 3.3085 31.0747 -2.8512 3.3853 29.5454


30

8.4387 8.4626 8.4866 8.5105 8.5344 8.5523 8.5702 8.5881 8.6060 8.6239 8.6418 8.6597 8.6776 8.6949 8.7121 8.7294 8.7467 8.7616 8.7765 8.7913 8.8062 8.8215 8.8368 8.8521 8.8673 8.8863 8.9053 8.9243 8.9433 8.9610 8.9787 8.9964 9.0141 9.0319 9.0498 9.0676 9.0854 9.1086 9.1319 9.1551 9.1783 9.1999 9.2215 9.2431 9.2647 9.2812 9.2976 9.3140 9.3305

4.1092 -0.3440 28.3322 3.5442 -0.3303 27.3078 3.1029 -0.2811 26.4237 2.7193 -0.2075 25.5715 2.3891 -0.1176 24.7501 2.1078 -0.0174 23.9581 1.8230 0.1132 23.0279 1.5961 0.2478 22.1371 1.4196 0.3836 21.2834 1.2863 0.5196 20.4648 1.2000 0.6396 19.7750 1.1390 0.7607 19.1099 1.1005 0.8841 18.4689 1.0821 1.0115 17.8513 1.0831 1.1671 17.1614 1.1066 1.3334 16.5017 1.1515 1.5137 15.8717 1.2167 1.7121 15.2711 1.3107 1.9540 14.6499 1.4296 2.2283 14.0643 1.5754 2.5415 13.5155 1.7503 2.9013 13.0057 2.0359 3.4709 12.3899 2.3900 4.1628 11.8606 2.8258 5.0041 11.4368 3.3600 6.0244 11.1474 4.2145 7.6233 11.0422 5.3073 9.6363 11.3569 6.6783 12.0781 12.2972 8.3526 14.8359 14.1539 9.5848 16.6622 15.9874 10.9078 18.3565 18.4223 12.2687 19.7049 21.4822 13.5799 20.4499 25.0897 14.4962 20.4348 28.1698 15.2454 19.7826 31.2977 15.7607 18.4547 34.2687 15.9895 16.4926 36.8708 15.8950 14.0186 38.8986 15.4644 11.2567 40.2253

-1.8986 3.3635 28.1369 -1.0984 3.2963 26.8383 -0.4322 3.2211 25.6368 0.0219 3.1713 24.7303 0.4141 3.1404 23.8732 0.7545 3.1344 23.0612 1.0533 3.1574 22.2909 1.3201 3.2116 21.5597 1.5623 3.2991 20.8661 1.7869 3.4218 20.2092 2.0011 3.5807 19.5885 2.2338 3.8001 18.9433 2.4680 4.0677 18.3439 2.7099 4.3864 17.7924 2.9660 4.7587 17.2919 3.2853 5.2553 16.7859 3.6386 5.8329 16.3610 4.0333 6.4977 16.0292 4.4773 7.2545 15.8053 5.0809 8.2795 15.7054 5.7752 9.4435 15.8275 6.5670 10.7359 16.2231 7.4597 12.1250 16.9508 8.7931 14.0403 18.5277 10.2487 15.7661 20.9534 11.7205 16.9275 24.2305 13.0073 17.0924 28.0852 13.5721 16.5629 30.3969 13.9398 15.5525 32.5658 14.0694 14.1035 34.4365 13.9414 12.3077 35.8815 13.5547 10.3062 36.8097 12.9289 8.2696 37.1990 12.1032 6.3433 37.0812 11.1310 4.6359 36.5350 10.1054 3.2521 35.7012 9.0503 2.1552 34.6600 8.0105 1.3368 33.4916 7.0202 0.7628 32.2658 6.1817 0.4133 31.1412 5.4160 0.1997 30.0289


31

9.3469 9.3633 9.3797 9.3962 9.4111 9.4261 9.4411 9.4561 9.4698 9.4835 9.4972 9.5109 9.5238 9.5368 9.5498 9.5627 9.5773 9.5920 9.6066 9.6212 9.6333 9.6453 9.6574 9.6695 9.6815 9.6936 9.7057 9.7177 9.7284 9.7390 9.7497 9.7603 9.7738 9.7872 9.8007 9.8141 9.8291 9.8440 9.8590 9.8739 9.8931 9.9123 9.9315 9.9507 9.9797 10.0088 10.0378 10.0668 10.0914

14.7154 8.4372 40.8130 13.6984 5.7586 40.7052 12.5137 3.4390 40.0492 11.2084 1.5050 38.9936 9.8507 -0.0124 37.6742 8.4982 -1.1379 36.2247 7.4276 -1.8093 35.0149 6.4105 -2.2895 33.8139 5.4595 -2.6133 32.6446 4.5828 -2.8146 31.5211 3.5565 -2.9430 30.1337 2.6610 -2.9683 28.8362 1.8892 -2.9329 27.6248 1.2308 -2.8692 26.4922 0.5912 -2.7890 25.2666 0.0640 -2.7240 24.1238 -0.3711 -2.6903 23.0542 -0.7352 -2.6972 22.0503 -0.9907 -2.7355 21.2853 -1.2201 -2.8056 20.5576 -1.4308 -2.9091 19.8661 -1.6298 -3.0473 19.2100 -1.8311 -3.2282 18.5667 -2.0323 -3.4498 17.9618 -2.2387 -3.7143 17.3965 -2.4560 -4.0244 16.8725 -2.7283 -4.4445 16.3201 -3.0288 -4.9362 15.8325 -3.3640 -5.5063 15.4179 -3.7416 -6.1615 15.0868 -4.2521 -7.0518 14.8226 -4.8429 -8.0811 14.7243 -5.5240 -9.2543 14.8322 -6.3050 -10.5648 15.1961 -7.4523 -12.4001 16.1177 -8.7671 -14.3192 17.7074 -10.2102 -16.1109 20.0810 -11.6827 -17.4384 23.2498 -12.5968 -17.8564 25.7160 -13.3960 -17.7700 28.3514 -14.0139 -17.0997 30.9917

4.7292 0.0925 28.9431 4.1235 0.0656 27.8921 3.6079 0.0959 26.9003 3.1645 0.1665 25.9456 2.7885 0.2652 25.0277 2.4745 0.3828 24.1455 2.1334 0.5637 22.9939 1.8825 0.7588 21.9025 1.7098 0.9630 20.8681 1.6024 1.1770 19.8880 1.5553 1.3621 19.1250 1.5418 1.5579 18.3962 1.5593 1.7676 17.7013 1.6053 1.9950 17.0399 1.6848 2.2650 16.3663 1.7955 2.5666 15.7327 1.9383 2.9063 15.1407 2.1143 3.2919 14.5930 2.3802 3.8415 13.9854 2.7071 4.4902 13.4601 3.1037 5.2567 13.0321 3.5806 6.1611 12.7236 4.3184 7.5291 12.5527 5.2337 9.1904 12.7108 6.3512 11.1531 13.3272 7.6877 13.3610 14.5747 9.2281 15.6852 16.6277 10.9303 17.7905 19.7212 12.6694 19.1883 23.8782 14.1942 19.2899 28.7579 14.8350 18.5630 31.5632 15.2514 17.2515 34.1817 15.3957 15.4110 36.4271 15.2451 13.1565 38.1485 14.7970 10.6635 39.2403 14.0725 8.1424 39.6847 13.1153 5.7704 39.5291 11.9856 3.6770 38.8756 10.8262 2.0313 37.9313 9.6274 0.7178 36.7740 8.4357 -0.2759 35.4942


32

10.1160 10.1406 10.1652 10.1826 10.2001 10.2175 10.2349 10.2482 10.2615 10.2748 10.2881 10.3014 10.3147 10.3279 10.3412 10.3514 10.3616 10.3718 10.3819 10.3921 10.4023 10.4125 10.4226 10.4362 10.4498 10.4634 10.4770 10.4926 10.5081 10.5237 10.5392 10.5549 10.5706 10.5862 10.6019 10.6170 10.6320 10.6471 10.6621 10.6795 10.6968 10.7142 10.7316 10.7524 10.7733 10.7941 10.8150 10.8342 10.8534

-14.3906 -15.8337 33.4427 -14.4778 -14.0292 35.4928 -14.2469 -11.8604 36.9769 -13.7014 -9.5295 37.8089 -12.8826 -7.2224 37.9835 -11.8348 -5.0833 37.5727 -10.6462 -3.2834 36.6998 -9.3963 -1.8774 35.5007 -8.1523 -0.8548 34.1270 -7.1917 -0.2716 32.9771 -6.2907 0.1301 31.8169 -5.4624 0.3837 30.6716 -4.7144 0.5223 29.5569 -4.0958 0.5740 28.5586 -3.5471 0.5709 27.5958 -3.0656 0.5285 26.6691 -2.6474 0.4593 25.7779 -2.2740 0.3690 24.8864 -1.9579 0.2678 24.0297 -1.6932 0.1614 23.2058 -1.4739 0.0536 22.4129 -1.2943 -0.0536 21.6491 -1.1495 -0.1587 20.9130 -1.0352 -0.2613 20.2033 -0.9473 -0.3618 19.5188 -0.8863 -0.4535 18.9082 -0.8425 -0.5447 18.3176 -0.8142 -0.6366 17.7463 -0.7995 -0.7302 17.1937 -0.7981 -0.8472 16.5524 -0.8130 -0.9709 15.9367 -0.8434 -1.1036 15.3462 -0.8884 -1.2483 14.7802 -0.9543 -1.4249 14.1860 -1.0380 -1.6237 13.6210 -1.1408 -1.8493 13.0852 -1.2640 -2.1074 12.5792 -1.4612 -2.5069 11.9606 -1.7043 -2.9889 11.3999 -2.0020 -3.5724 10.9046 -2.3653 -4.2795 10.4871 -2.9910 -5.4800 10.0805

7.2875 -0.9892 34.1682 6.3399 -1.4251 33.0134 5.4576 -1.7167 31.8822 4.6477 -1.8944 30.7879 3.9133 -1.9859 29.7375 3.0631 -2.0144 28.4272 2.3360 -1.9763 27.1940 1.7215 -1.9033 26.0327 1.2071 -1.8183 24.9371 0.8353 -1.7476 24.0461 0.5181 -1.6859 23.1946 0.2475 -1.6378 22.3797 0.0155 -1.6063 21.5986 -0.1853 -1.5930 20.8492 -0.3607 -1.5993 20.1297 -0.5160 -1.6261 19.4388 -0.6566 -1.6738 18.7754 -0.7783 -1.7372 18.1829 -0.8943 -1.8195 17.6127 -1.0074 -1.9216 17.0646 -1.1204 -2.0443 16.5382 -1.2594 -2.2194 15.9364 -1.4055 -2.4279 15.3660 -1.5625 -2.6731 14.8276 -1.7343 -2.9583 14.3224 -1.9589 -3.3469 13.7779 -2.2147 -3.8045 13.2853 -2.5083 -4.3410 12.8510 -2.8474 -4.9672 12.4842 -3.3180 -5.8374 12.1550 -3.8774 -6.8721 11.9654 -4.5406 -8.0915 11.9562 -5.3243 -9.5082 12.1838 -6.4568 -11.4891 12.8791 -7.8098 -13.7274 14.2136 -9.3748 -16.0711 16.3790 -11.0894 -18.1980 19.5349 -12.1575 -19.1903 21.9778 -13.1871 -19.7805 24.7637 -14.1180 -19.8310 27.7894 -14.8820 -19.2402 30.8903


33

10.8726 10.8918 10.9109 10.9301 10.9493 10.9685 10.9857 11.0029 11.0201 11.0373 11.0542 11.0710 11.0878 11.1047 11.1210 11.1373 11.1536 11.1699 11.1888 11.2077 11.2266 11.2455 11.2682 11.2909 11.3137 11.3364 11.3562 11.3760 11.3958 11.4156 11.4430 11.4704 11.4978 11.5252 11.5428 11.5605 11.5781 11.5957 11.6133 11.6310 11.6486 11.6662 11.6826 11.6990 11.7154 11.7318 11.7472 11.7626 11.7780

-3.8044 -7.0369 9.9230 -4.8519 -9.0231 10.1437 -6.1891 -11.4698 10.9508 -7.3055 -13.4133 11.9954 -8.5866 -15.5065 13.5874 -10.0184 -17.6153 15.8534 -11.5539 -19.5094 18.8867 -12.7041 -20.5854 21.6298 -13.8137 -21.2060 24.7526 -14.8169 -21.2145 28.1379 -15.6381 -20.4958 31.5980 -16.2017 -18.9849 34.8872 -16.4338 -16.7616 37.7217 -16.2884 -13.9956 39.8726 -15.7642 -10.9050 41.2028 -14.9858 -8.0533 41.6681 -13.9627 -5.3723 41.4908 -12.7544 -3.0045 40.7802 -11.4266 -1.0306 39.6821 -10.1951 0.3804 38.5009 -8.9617 1.4773 37.2126 -7.7601 2.2915 35.8841 -6.6151 2.8675 34.5652 -5.5770 3.2442 33.3279 -4.6177 3.4800 32.1439 -3.7418 3.6098 31.0213 -2.9504 3.6632 29.9621 -1.9358 3.6544 28.5163 -1.0827 3.5914 27.1894 -0.3716 3.5158 25.9667 0.2195 3.4559 24.8351

-15.4107 -17.9496 33.8515 -15.6371 -16.0231 36.4233 -15.5192 -13.6030 38.4011 -15.0547 -10.8776 39.6574 -14.3440 -8.3102 40.1395 -13.4021 -5.8808 40.0254 -12.2850 -3.7230 39.4088 -11.0549 -1.9155 38.4190 -9.8919 -0.5968 37.3127 -8.7286 0.4256 36.0897 -7.5989 1.1800 34.8151 -6.5276 1.7086 33.5383 -5.6483 2.0239 32.4406 -4.8340 2.2271 31.3771 -4.0884 2.3433 30.3549 -3.4123 2.3947 29.3770 -2.4908 2.3884 27.9360 -1.7215 2.3221 26.5949 -1.0869 2.2344 25.3437 -0.5671 2.1516 24.1725 -0.2573 2.1047 23.3876 0.0122 2.0721 22.6352 0.2479 2.0564 21.9133 0.4556 2.0593 21.2199 0.6412 2.0819 20.5537 0.8093 2.1252 19.9135 0.9640 2.1900 19.2985 1.1097 2.2769 18.7080 1.2678 2.4020 18.0720 1.4235 2.5573 17.4664 1.5810 2.7446 16.8913 1.7444 2.9657 16.3473 1.9462 3.2664 15.7584 2.1664 3.6204 15.2147 2.4102 4.0336 14.7196 2.6837 4.5122 14.2782 3.0513 5.1671 13.8375 3.4777 5.9361 13.4952 3.9728 6.8331 13.2726 4.5482 7.8696 13.1983 5.3602 9.3065 13.3564


34

11.7935 11.8087 11.8239 11.8392 11.8544 11.8711 11.8877 11.9044 11.9211 11.9405 11.9598 11.9792 11.9986 12.0162 12.0338 12.0513 12.0689 12.0905 12.1122 12.1339 12.1555 12.1781 12.2007 12.2233 12.2459 12.2630 12.2801 12.2972 12.3143 12.3314 12.3485 12.3656 12.3827 12.3978 12.4130 12.4281 12.4432 12.4575 12.4718 12.4860 12.5003 12.5137 12.5271 12.5404 12.5538 12.5665 12.5793 12.5920 12.6047

0.6355 3.4302 23.9567 0.9963 3.4335 23.1281 1.3127 3.4704 22.3457 1.5953 3.5437 21.6067 1.8540 3.6550 20.9092 2.0962 3.8062 20.2527 2.3290 3.9991 19.6373 2.5593 4.2348 19.0634 2.8257 4.5555 18.4651 3.1043 4.9365 17.9256 3.4019 5.3815 17.4499 3.7257 5.8938 17.0444 4.1427 6.5754 16.6735 4.6122 7.3583 16.4258 5.1418 8.2461 16.3239 5.7395 9.2370 16.3954 6.5743 10.5815 16.7666 7.5242 12.0361 17.5255 8.5801 13.5258 18.7471 9.7145 14.9220 20.4864 10.6246 15.8365 22.2217 11.5135 16.4734 24.2619 12.3316 16.7208 26.5370 13.0209 16.4885 28.9264 13.5223 15.7145 31.2678 13.7756 14.4255 33.3594 13.7404 12.7129 35.0260 13.4111 10.7125 36.1476 12.8682 8.7895 36.6455 12.1268 6.9339 36.6569 11.2341 5.2584 36.2401 10.2437 3.8357 35.4879 9.2590 2.7429 34.5555 8.2778 1.9039 33.4853 7.3341 1.2985 32.3388 6.4528 0.8914 31.1650 5.6806 0.6493 30.0431 4.9885 0.5242 28.9419 4.3795 0.4889 27.8720 3.8533 0.5200 26.8392 3.3942 0.6020 25.8163

6.3219 10.9624 13.8693 7.4426 12.8043 14.8453 8.7182 14.7383 16.4064 9.8283 16.2415 18.1461 10.9897 17.5645 20.3752 12.1519 18.5350 23.0816 13.2404 18.9649 26.1730 14.1678 18.6676 29.4817 14.8232 17.5480 32.6974 15.1120 15.6342 35.4971 14.9848 13.0747 37.6162 14.5553 10.6675 38.7076 13.8508 8.2242 39.1623 12.9147 5.9209 39.0240 11.8074 3.8871 38.3913 10.6690 2.2872 37.4660 9.4928 1.0133 36.3268 8.3258 0.0537 35.0629 7.2044 -0.6309 33.7507 6.2774 -1.0474 32.6001 5.4176 -1.3218 31.4712 4.6318 -1.4843 30.3771 3.9229 -1.5625 29.3250 3.1589 -1.5775 28.0966 2.5033 -1.5343 26.9331 1.9472 -1.4587 25.8308 1.4802 -1.3693 24.7851 1.1152 -1.2848 23.8578 0.8082 -1.2065 22.9716 0.5510 -1.1391 22.1231 0.3352 -1.0854 21.3098 0.1900 -1.0539 20.6954 0.0626 -1.0322 20.1003 -0.0499 -1.0206 19.5237 -0.1501 -1.0192 18.9647 -0.2404 -1.0280 18.4228 -0.3227 -1.0470 17.8975 -0.3991 -1.0763 17.3882 -0.4710 -1.1160 16.8945 -0.5510 -1.1750 16.3425 -0.6294 -1.2486 15.8105


35

12.6167 12.6287 12.6407 12.6527 12.6647 12.6767 12.6887 12.7007 12.7143 12.7278 12.7414 12.7550 12.7701 12.7853 12.8004 12.8156 12.8335 12.8515 12.8695 12.8875 12.9143 12.9411 12.9679 12.9947 13.0208 13.0469 13.0731 13.0992 13.1138 13.1285 13.1431 13.1577 13.1724 13.1870 13.2017 13.2163 13.2297 13.2432 13.2566 13.2700 13.2825 13.2950 13.3075 13.3200 13.3306 13.3412 13.3518 13.3624 13.3790

3.0132 0.7201 24.8347 2.7038 0.8637 23.8938 2.4589 1.0254 22.9925 2.2320 1.2479 21.9181 2.0837 1.4890 20.9015 2.0052 1.7484 19.9411 1.9868 2.0306 19.0361 2.0157 2.3122 18.2632 2.0864 2.6228 17.5366 2.1980 2.9688 16.8575 2.3497 3.3580 16.2287 2.5631 3.8447 15.6015 2.8294 4.4056 15.0460 3.1535 5.0536 14.5725 3.5415 5.8026 14.1953 4.2257 7.0791 13.8621 5.0843 8.6346 13.8505 6.1413 10.4819 14.2816 7.4139 12.5745 15.3187 8.5444 14.2889 16.6536 9.7832 15.9544 18.5621 11.0851 17.3804 21.0920 12.3670 18.3176 24.2008 13.2499 18.5298 26.8337 13.9952 18.1999 29.5720 14.5389 17.2719 32.2395 14.8274 15.7599 34.6432 14.8203 13.7496 36.5850 14.4990 11.4285 37.9259 13.8744 9.0007 38.6064 12.9914 6.6482 38.6433 11.9105 4.5273 38.1318 10.7053 2.7522 37.2024 9.4479 1.3640 35.9849 8.1988 0.3458 34.6153 7.2132 -0.2531 33.4528 6.2856 -0.6702 32.2856 5.4288 -0.9391 31.1378 4.6504 -1.0928 30.0241 3.9498 -1.1612 28.9462 3.3297 -1.1668 27.9124

-0.7080 -1.3374 15.2979 -0.7888 -1.4423 14.8044 -0.8897 -1.5885 14.2449 -0.9987 -1.7612 13.7117 -1.1184 -1.9633 13.2051 -1.2518 -2.1983 12.7255 -1.4317 -2.5245 12.1947 -1.6402 -2.9113 11.7055 -1.8830 -3.3694 11.2622 -2.1673 -3.9103 10.8711 -2.5641 -4.6666 10.4922 -3.0430 -5.5816 10.2176 -3.6201 -6.6833 10.0794 -4.3152 -7.9979 10.1241 -5.2893 -9.8028 10.4865 -6.4801 -11.9454 11.3207 -7.9064 -14.3770 12.8126 -9.5603 -16.9262 15.1850 -10.7563 -18.5330 17.3423 -11.9968 -19.9044 20.0133 -13.2286 -20.8547 23.1748 -14.3760 -21.1875 26.7196 -15.3509 -20.7138 30.4624 -16.0444 -19.3407 34.0788 -16.3628 -17.1031 37.2333 -16.2538 -14.1615 39.6461 -15.8058 -11.2993 40.9597 -15.0477 -8.3859 41.5344 -14.0235 -5.6305 41.4214 -12.7974 -3.1875 40.7356 -11.5458 -1.2856 39.7297 -10.2404 0.2436 38.4900


36

13.3956 13.4122 13.4288 13.4421 13.4555 13.4689 13.4823 13.4956 13.5090 13.5224 13.5357

2.7868 -1.1293 26.9228 2.3162 -1.0645 25.9760 1.8586 -0.9719 24.9432 1.4771 -0.8715 23.9587 1.1622 -0.7722 23.0187 0.9037 -0.6795 22.1199 0.6922 -0.5961 21.2592 0.5204 -0.5242 20.4341 0.3817 -0.4644 19.6426 0.2695 -0.4166 18.8828 0.1872 -0.3832 18.2328 0.1187 -0.3582 17.6055 0.0613 -0.3412 17.0001 0.0126 -0.3316 16.4158 -0.0240 -0.3287 15.9279 -0.0567 -0.3306 15.4545 -0.0863 -0.3372 14.9954 -0.1137 -0.3484 14.5501 -0.1383 -0.3630 14.1407 -0.1620 -0.3817 13.7430 -0.1855 -0.4044 13.3566 -0.2091 -0.4313 12.9812 -0.2397 -0.4714 12.5231 -0.2720 -0.5189 12.0815 -0.3067 -0.5746 11.6560 -0.3446 -0.6392 11.2460 -0.3954 -0.7297 10.7753 -0.4534 -0.8367 10.3257 -0.5200 -0.9628 9.8968 -0.5971 -1.1113 9.4882 -0.7062 -1.3232 9.0267 -0.8373 -1.5806 8.5945 -0.9952 -1.8934 8.1928 -1.1863 -2.2737 7.8242 -1.4505 -2.8002 7.4543 -1.7786 -3.4566 7.1406 -2.1861 -4.2745 6.8994 -2.6932 -5.2907 6.7569 -3.3733 -6.6429 6.7600 -4.2292 -8.3332 6.9933 -5.2984 -10.4106 7.5750 -6.6191 -12.8895 8.6836 -7.8319 -15.0588 10.0482 -9.2182 -17.3742 12.0134 -10.7617 -19.6800 14.7158 -12.4107 -21.7156

-8.9315 1.4112 37.1193 -7.6588 2.2602 35.7047 -6.6477 2.7667 34.5378 -5.6945 3.1212 33.4010 -4.8070 3.3539 32.3077 -3.9895 3.4921 31.2648 -2.8326 3.5763 29.7094 -1.8490 3.5567 28.2816 -1.0230 3.4897 26.9692 -0.3356 3.4150 25.7587 0.1379 3.3660 24.8377 0.5462 3.3377 23.9693 0.9003 3.3366 23.1488 1.2112 3.3669 22.3724 1.4892 3.4311 21.6375 1.7422 3.5315 20.9426 1.9779 3.6697 20.2868 2.2037 3.8471 19.6698 2.4500 4.0899 19.0335 2.6994 4.3845 18.4468 2.9584 4.7337 17.9125 3.2340 5.1402 17.4344 3.5793 5.6812 16.9613 3.9624 6.3078 16.5797 4.3908 7.0258 16.3038 4.8725 7.8382 16.1513 5.5300 8.9362 16.1612 6.2831 10.1677 16.4289 7.1360 11.5100 17.0117 8.0864 12.9123 17.9698 9.0887 14.2565 19.3103 10.1425 15.4632 21.1000 11.2008 16.3801 23.3317 12.1936 16.8389 25.9248 13.0384 16.6700 28.7308 13.6294 15.7869 31.4693 13.8777 14.2135 33.8487 13.7406 12.0801 35.6329 13.3425 10.1144 36.5146 12.7069 8.1229 36.8598 11.8752 6.2497 36.7032


37

18.2529 -13.6592 -22.8519 21.4545 -14.8599 -23.4493 25.0694 -15.9403 -23.3309 28.9579 -16.8180 -22.3716 32.9012 -17.4110 -20.5066 36.6160 -17.6402 -17.8403 39.7820 -17.4566 -14.5780 42.1467 -16.8600 -10.9752 43.5660 -15.9893 -7.6797 44.0128 -14.8526 -4.6031 43.7433 -13.5142 -1.9026 42.8908 -12.0447 0.3364 41.6268 -10.7161 1.8938 40.3318 -9.3824 3.1095 38.9427 -8.0769 4.0186 37.5292 -6.8249 4.6688 36.1430 -5.5475 5.1414 34.7068 -4.3682 5.4252 33.3608 -3.2940 5.5709 32.1139 -2.3266 5.6211 30.9659 -1.1840 5.5973 29.5611 -0.2133 5.5219 28.3035 0.6059 5.4381 27.1719 1.2960 5.3752 26.1482 1.8624 5.3503 25.2480 2.3495 5.3705 24.4245 2.7743 5.4420 23.6713 3.1535 5.5682 22.9837 3.5026 5.7497 22.3595 3.8323 5.9878 21.7994 4.1524 6.2827 21.3052 4.4730 6.6335 20.8797 4.8476 7.0967 20.4857 5.2416 7.6289 20.1948 5.6616 8.2268 20.0184

10.9027 4.6012 36.1264 9.8899 3.2839 35.2783 8.8544 2.2482 34.2330 7.8399 1.4835 33.0683 6.8794 0.9547 31.8511 6.0685 0.6385 30.7330 5.3324 0.4528 29.6280 4.6765 0.3690 28.5497 4.1022 0.3620 27.5058 3.5983 0.4123 26.4792 3.1725 0.5045 25.4926 2.8191 0.6259 24.5455 2.5316 0.7675 23.6372 2.2314 0.9848 22.4538 2.0268 1.2216 21.3368 1.9057 1.4750 20.2836 1.8545 1.7488 19.2923 1.8571 1.9955 18.5201 1.8985 2.2644 17.7895 1.9770 2.5607 17.1011 2.0911 2.8908 16.4563 2.2555 3.2969 15.8066 2.4644 3.7612 15.2152 2.7210 4.2942 14.6882 3.0297 4.9081 14.2345 3.5399 5.8877 13.7603 4.1736 7.0718 13.4848 4.9504 8.4875 13.4721 5.8908 10.1445 13.8131 6.8654 11.7887 14.4951 7.9841 13.5646 15.6461 9.2350 15.3687 17.3683 10.5792 17.0213 19.7378 11.6112 18.0171 21.9642 12.6110 18.6456 24.5329 13.5193 18.7716 27.3488 14.2679 18.2925 30.2557 14.7886 17.1458 33.0467 15.0139 15.3893 35.4774 14.9014 13.1580 37.3460 14.4499 10.6312


38

6.1141 8.8839 19.9688 6.6984 9.7261 20.0927 7.3349 10.6161 20.4423

38.5268 13.7724 8.2864 38.9661 12.8786 6.0680 38.8513 11.8228 4.0987 38.2672 10.6645 2.4508 37.3325 9.5607 1.2386 36.2731 8.4614 0.3045 35.0973 7.3993 -0.3782 33.8666 6.3984 -0.8499 32.6282 5.5449 -1.1357 31.5079 4.7637 -1.3082 30.4191 4.0578 -1.3942 29.3694 3.4273 -1.4169 28.3623 2.7220 -1.3834 27.1272 2.1279 -1.3072 25.9589 1.6335 -1.2111 24.8526 1.2259 -1.1108 23.8030 0.9377 -1.0299 22.9516 0.6954 -0.9571 22.1348 0.4923 -0.8953 21.3501 0.3216 -0.8458 20.5955 0.1775 -0.8090 19.8694 0.0552 -0.7852 19.1702 -0.0493 -0.7748 18.4968 -0.1402 -0.7772 17.8481 -0.2075 -0.7886 17.3306 -0.2694 -0.8087 16.8287 -0.3273 -0.8373 16.3420 -0.3826 -0.8747 15.8701 -0.4443 -0.9282 15.3480 -0.5057 -0.9936 14.8443 -0.5683 -1.0717 14.3585 -0.6335 -1.1631 13.8903 -0.7168 -1.2911 13.3541 -0.8080 -1.4420 12.8423 -0.9092 -1.6186 12.3547 -1.0229 -1.8242 11.8915 -1.1784 -2.1117 11.3739 -1.3599 -2.4542 10.8931 -1.5730 -2.8620 10.4524 -1.8244 -3.3466 10.0564 -2.1757 -4.0251 9.6617 -2.6026 -4.8528 9.3532 -3.1214 -5.8594 9.1569 -3.7518 -7.0757 9.1120 -4.6224 -8.7299 9.3122 -5.6967 -10.7327 9.8900 -7.0029 -13.0793 11.0097


39

-8.5547 -15.6777 12.8896 -9.8838 -17.6960 14.9631 -11.3169 -19.5674 17.7134 -12.7952 -21.0419 21.1710 -14.2186 -21.8157 25.2458 -15.1829 -21.7353 28.6191 -15.9690 -20.9559 32.0308 -16.5067 -19.4394 35.2571 -16.7404 -17.2350 38.0686 -16.6316 -14.4808 40.2444 -16.1668 -11.4242 41.6523 -15.3647 -8.3177 42.2574 -14.2782 -5.3771 42.1127 -13.0296 -2.8665 41.3952 -11.6534 -0.7719 40.2668 -10.2187 0.8761 38.8722 -8.7840 2.1047 37.3517 -7.6614 2.8324 36.1111 -6.5884 3.3607 34.8864 -5.5781 3.7251 33.7011 -4.6391 3.9596 32.5688 -3.4107 4.1321 31.0307 -2.3414 4.1714 29.6181 -1.4227 4.1365 28.3233 -0.6417 4.0708 27.1339 0.0515 4.0006 25.9802 0.6329 3.9517 24.9147 1.1226 3.9396 23.9270 1.5412 3.9742 23.0089 1.8674 4.0475 22.2511 2.1630 4.1639 21.5413 2.4374 4.3257 20.8788 2.7000 4.5339 20.2632


40

2.9808 4.8128 19.6507 3.2654 5.1497 19.0970 3.5612 5.5471 18.6062 3.8759 6.0068 18.1836 4.2703 6.6146 17.7907 4.7065 7.3111 17.5114 5.1915 8.0981 17.3639 5.7325 8.9727 17.3696 6.4715 10.1404 17.6173 7.3037 11.4007 18.1845 8.2228 12.6996 19.1266 9.2111 13.9471 20.4867 9.6611 14.4499 21.2271 10.1138 14.9053 22.0533 10.5640 15.2991 22.9618 11.0058 15.6163 23.9464

18

-20 -15 -10 -5 0 5 10 15 205

10

15

20

25

30

35

40

45

19


41

0 2 4 6 8 10 12 14 16 18 20-20

0

20

0 2 4 6 8 10 12 14 16 18 20-50

0

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1


42

21 - 22 t =

0 0.0025 0.0050 0.0075 0.0100 0.0226 0.0352 0.0477 0.0603 0.0716 0.0830 0.0943 0.1057 0.1171 0.1284 0.1398 0.1511 0.1646 0.1781 0.1916 0.2051 0.2214 0.2377 0.2540 0.2703 0.2919 0.3135 0.3351 0.3567 0.3768 0.3968 0.4169

13.5506 13.5654 13.5802 13.5950 13.6120 13.6290 13.6459 13.6629 13.6831 13.7034 13.7236 13.7438 13.7708 13.7978 13.8249 13.8519 13.8676 13.8834 13.8992 13.9149 13.9307 13.9465 13.9622 13.9780 13.9933 14.0086 14.0239 14.0392 14.0531 14.0670 14.0810 14.0949 14.1085 14.1222

8.0187 11.5183 21.0426 8.7392 12.3817 21.9094 9.4995 13.1632 23.0832

10.2453 13.7449 24.5259

10.9334 14.0355 26.1822

11.5134 13.9605 27.9543

11.9341 13.4672 29.7094

12.1431 12.5738 31.2824

12.1068 11.3508 32.5282

11.8240 9.9034 33.3473 11.3769 8.5321 33.6777 10.7768 7.2117 33.6326 10.0654 6.0257 33.2527 9.2887 5.0293 32.6066 8.5030 4.2540 31.7851 7.7371 3.6841 30.8395 7.0210 3.3047 29.8218 6.3759 3.0882 28.7750 5.8211 3.0035 27.7420 5.3536 3.0259 26.7271 4.9749 3.1355 25.7440 4.6837 3.3162 24.8019 4.4327 3.6324 23.6726 4.3081 4.0339 22.6313 4.3025 4.5154 21.6891 4.4068 5.0803 20.8593 4.6164 5.7448 20.1513 4.9319 6.5098 19.5955


43

0.4370 0.4534 0.4698 0.4862 0.5026 0.5181 0.5337 0.5492 0.5648 0.5801 0.5954 0.6108 0.6261 0.6430 0.6599 0.6767 0.6936 0.7134 0.7332 0.7530 0.7728 0.7905 0.8082 0.8259 0.8437 0.8658 0.8880 0.9102 0.9323 0.9545 0.9766 0.9988 1.0209 1.0378 1.0546 1.0715 1.0883 1.1052 1.1221 1.1389 1.1558 1.1710 1.1863 1.2016 1.2168 1.2310 1.2452 1.2595 1.2737

14.1358 14.1494 14.1681 14.1867 14.2054 14.2240 14.2393 14.2545 14.2698 14.2851 14.3015 14.3179 14.3343 14.3507 14.3709 14.3911 14.4113 14.4315 14.4571 14.4826 14.5082 14.5337 14.5593 14.5848 14.6104 14.6359 14.6503 14.6646 14.6790 14.6933 14.7076 14.7220 14.7363 14.7507 14.7642 14.7777 14.7912 14.8047 14.8165 14.8283 14.8401 14.8519 14.8674 14.8829 14.8984 14.9139 14.9272 14.9405 14.9537

5.3541 7.3809 19.2196 5.8818 8.3606 19.0593 6.5492 9.5013 19.1684 7.3311 10.7255 19.6243 8.2146 11.9712 20.4847 9.1694 13.1352 21.7888 9.9374 13.9009 23.1285

10.6852 14.4397 24.7218

11.3689 14.6640 26.5077

11.9388 14.5051 28.3854

12.3455 13.9185 30.2221

12.5396 12.9285 31.8568

12.4899 11.6093 33.1505

12.1951 10.0684 34.0083

11.7189 8.5663 34.3740 11.0787 7.1206 34.3370 10.3182 5.8214 33.9428 9.4853 4.7277 33.2662 8.6482 3.8807 32.4144 7.8276 3.2499 31.4358 7.0545 2.8189 30.3838 6.3508 2.5577 29.3026 5.7384 2.4336 28.2383 5.2124 2.4181 27.1915 4.7742 2.4891 26.1747 4.4226 2.6285 25.1958 4.1045 2.8726 24.0607 3.8999 3.1851 22.9955 3.8005 3.5584 22.0059 3.7971 3.9929 21.0983 3.8942 4.5484 20.2042 4.0956 5.1926 19.4313 4.4011 5.9362 18.7980 4.8094 6.7929 18.3299 5.2978 7.7295 18.0667 5.8877 8.7817 18.0233 6.5810 9.9394 18.2453 7.3748 11.1738 18.7842 8.2914 12.4796 19.7289 9.2824 13.7095 21.1247

10.3040 14.7204 22.9842


44

1.2873 1.3008 1.3144 1.3280 1.3401 1.3523 1.3645 1.3766 1.3915 1.4064 1.4213 1.4361 1.4492 1.4622 1.4752 1.4883 1.5034 1.5186 1.5337 1.5489 1.5654 1.5820 1.5985 1.6150 1.6372 1.6593 1.6815 1.7036 1.7318 1.7601 1.7883 1.8165 1.8347 1.8529 1.8710 1.8892 1.9034 1.9176 1.9318 1.9459 1.9601 1.9743 1.9885 2.0027 2.0165 2.0303 2.0442 2.0580 2.0694

14.9670 14.9803 14.9936 15.0068 15.0201 15.0324 15.0448 15.0571 15.0694 15.0842 15.0990 15.1137 15.1285 15.1457 15.1629 15.1801 15.1973 15.2176 15.2379 15.2583 15.2786 15.3033 15.3280 15.3526 15.3773 15.3924 15.4074 15.4225 15.4375 15.4526 15.4676 15.4826 15.4977 15.5115 15.5254 15.5392 15.5530 15.5656 15.5782 15.5908 15.6033 15.6144 15.6254 15.6365 15.6475 15.6648 15.6821 15.6993 15.7166

11.2851 15.3407 25.2476

11.9562 15.4372 27.1809

12.5001 15.1306 29.1608

12.8684 14.3958 31.0534

13.0247 13.2580 32.7204

12.9453 11.7925 34.0284

12.6242 10.1351 34.8888

12.0784 8.4274 35.2674 11.3481 6.7943 35.1838 10.4461 5.2826 34.6809 9.4653 4.0407 33.8500 8.4677 3.0951 32.7883 7.5030 2.4293 31.5993 6.7313 2.0478 30.5409 6.0279 1.8144 29.4701 5.4022 1.7020 28.4088 4.8590 1.6845 27.3707 4.3547 1.7492 26.2595 3.9473 1.8818 25.1925 3.6309 2.0663 24.1732 3.3984 2.2923 23.2035 3.2375 2.5646 22.2525 3.1515 2.8734 21.3573 3.1348 3.2192 20.5199 3.1817 3.6063 19.7432 3.2881 4.0412 19.0315 3.4534 4.5303 18.3898 3.6779 5.0812 17.8258 3.9625 5.7030 17.3495 4.3932 6.5684 16.9072 4.9236 7.5657 16.6461 5.5608 8.7018 16.6081 6.3103 9.9714 16.8462 7.2518 11.4740 17.4859 8.3227 13.0380 18.6222 9.4960 14.5315 20.3315

10.7104 15.7517 22.6344

11.5918 16.3341 24.7155

12.3890 16.4995 27.0118


45

2.0807 2.0920 2.1033 2.1181 2.1329 2.1477 2.1625 2.1796 2.1967 2.2138 2.2310 2.2448 2.2586 2.2724 2.2862 2.3005 2.3148 2.3291 2.3434 2.3600 2.3766 2.3932 2.4098 2.4294 2.4490 2.4686 2.4882 2.5132 2.5382 2.5632 2.5882 2.6046 2.6211 2.6376 2.6541 2.6705 2.6870 2.7035 2.7199 2.7367 2.7534 2.7702 2.7869 2.8004 2.8139 2.8274 2.8409 2.8534 2.8659

15.7288 15.7410 15.7533 15.7655 15.7777 15.7899 15.8021 15.8143 15.8281 15.8418 15.8556 15.8693 15.8852 15.9011 15.9170 15.9329 15.9515 15.9702 15.9889 16.0075 16.0299 16.0523 16.0747 16.0971 16.1150 16.1328 16.1506 16.1684 16.1863 16.2041 16.2219 16.2398 16.2545 16.2693 16.2840 16.2988 16.3127 16.3266 16.3405 16.3544 16.3666 16.3788 16.3911 16.4033 16.4182 16.4332 16.4482 16.4631 16.4771

13.0395 16.1562 29.3892

13.4825 15.2677 31.6744

13.6655 13.8543 33.6675

13.5519 12.0519 35.1916

13.1353 10.0348 36.1371

12.4486 7.9726 36.4706 11.5809 6.1024 36.2582 10.5746 4.4815 35.6140 9.4974 3.1728 34.6442 8.4092 2.1871 33.4734 7.5037 1.5742 32.3884 6.6523 1.1500 31.2680 5.8711 0.8833 30.1435 5.1704 0.7414 29.0359 4.5385 0.6941 27.9282 3.9951 0.7223 26.8578 3.5367 0.8056 25.8281 3.1582 0.9283 24.8401 2.7881 1.1198 23.6687 2.5182 1.3423 22.5594 2.3369 1.5884 21.5103 2.2319 1.8575 20.5201 2.1932 2.1320 19.6504 2.2076 2.4326 18.8312 2.2717 2.7645 18.0636 2.3826 3.1354 17.3494 2.5399 3.5571 16.6877 2.7451 4.0360 16.0878 3.0008 4.5821 15.5565 3.3104 5.2066 15.1032 3.8032 6.1573 14.6507 4.4101 7.2872 14.3914 5.1463 8.6150 14.3815 6.0264 10.1442 14.6977 6.9820 11.7289 15.3650 8.0735 13.4226 16.4955 9.2858 15.1191 18.1813

10.5769 16.6410 20.4828

11.5698 17.5372 22.6535

12.5196 18.0582 25.1368

13.3666 18.0785


46

2.8784 2.8908 2.9039 2.9169 2.9299 2.9429 2.9559 2.9689 2.9819 2.9949 3.0069 3.0189 3.0309 3.0429 3.0573 3.0718 3.0863 3.1007 3.1166 3.1325 3.1484 3.1643 3.1852 3.2061 3.2270 3.2480 3.2764 3.3049 3.3334 3.3618 3.3815 3.4011 3.4208 3.4404 3.4549 3.4694 3.4838 3.4983 3.5127 3.5272 3.5417 3.5561 3.5693 3.5824 3.5955 3.6086 3.6203 3.6320 3.6438

16.4910 16.5049 16.5188 16.5298 16.5407 16.5516 16.5625 16.5734 16.5843 16.5953 16.6062 16.6188 16.6314 16.6440 16.6567 16.6716 16.6865 16.7015 16.7164 16.7341 16.7517 16.7694 16.7871 16.8077 16.8284 16.8491 16.8698 16.8936 16.9175 16.9414 16.9652 16.9810 16.9968 17.0126 17.0283 17.0441 17.0599 17.0756 17.0914 17.1049 17.1185 17.1320 17.1455 17.1581 17.1706 17.1832 17.1957 17.2062 17.2166

27.8297 14.0445 17.5109

30.5728 14.4884 16.3114

33.1613 14.6375 14.5557

35.3664 14.4572 12.3882

37.0078 13.9542 9.9861 37.9820 13.2432 7.7785 38.2709 12.3345 5.7220 38.0418 11.2843 3.9241 37.3826 10.1514 2.4435 36.4104 9.0915 1.3759 35.3496 8.0478 0.5657 34.1912 7.0496 -0.0155 32.9897 6.1176 -0.4068 31.7855 5.3091 -0.6394 30.6672 4.5772 -0.7667 29.5807 3.9237 -0.8149 28.5327 3.3477 -0.8061 27.5259 2.7859 -0.7497 26.4386 2.3105 -0.6624 25.4013 1.9131 -0.5585 24.4111 1.5846 -0.4479 23.4646 1.3589 -0.3562 22.7149 1.1691 -0.2666 21.9912 1.0111 -0.1806 21.2922 0.8805 -0.0988 20.6167 0.7737 -0.0214 19.9635 0.6876 0.0519 19.3318 0.6195 0.1214 18.7205 0.5668 0.1878 18.1291 0.5314 0.2446 17.6224 0.5053 0.3004 17.1301 0.4874 0.3555 16.6519 0.4770 0.4109 16.1873 0.4731 0.4821 15.6200 0.4789 0.5560 15.0731 0.4937 0.6341 14.5460 0.5167 0.7180 14.0382 0.5518 0.8201 13.4954 0.5969 0.9337 12.9751 0.6527 1.0614 12.4770 0.7196 1.2059 12.0006 0.8234 1.4217 11.4207 0.9504 1.6784 10.8764 1.1044 1.9853 10.3688


47

3.6555 3.6668 3.6781 3.6895 3.7008 3.7175 3.7341 3.7508 3.7674 3.7812 3.7951 3.8089 3.8227 3.8365 3.8503 3.8641 3.8780 3.8936 3.9093 3.9249 3.9406 3.9587 3.9769 3.9950 4.0132 4.0354 4.0577 4.0800 4.1022 4.1197 4.1371 4.1545 4.1719 4.1893 4.2067 4.2241 4.2415 4.2574 4.2733 4.2892 4.3051 4.3203 4.3355 4.3507 4.3658 4.3804 4.3950 4.4096 4.4241

17.2271 17.2375 17.2542 17.2709 17.2875 17.3042 17.3178 17.3314 17.3450 17.3586 17.3722 17.3858 17.3993 17.4129 17.4280 17.4430 17.4580 17.4730 17.4903 17.5075 17.5247 17.5420 17.5626 17.5833 17.6039 17.6246 17.6446 17.6647 17.6847 17.7047 17.7248 17.7448 17.7648 17.7849 17.8011 17.8173 17.8335 17.8497 17.8653 17.8809 17.8965 17.9122 17.9264 17.9407 17.9550 17.9692 17.9838 17.9984 18.0129

1.2905 2.3537 9.8999 1.6487 3.0548 9.2748 2.1217 3.9848 8.7675 2.7450 5.2178 8.4241 3.5702 6.8358 8.3275 4.4719 8.5686 8.5368 5.6024 10.7039 9.1495 6.9975 13.2446 10.3601 8.6763 16.0843 12.4369

10.0301 18.1534 14.5962

11.4897 20.0698 17.4470

12.9958 21.5772 21.0209

14.4474 22.3638 25.2265

15.4361 22.2729 28.7207

16.2433 21.4571 32.2547

16.7970 19.8761 35.5963

17.0398 17.5811 38.5078

16.9318 14.7151 40.7598

16.4585 11.5359 42.2151

15.6386 8.3060 42.8386 14.5259 5.2501 42.6861 13.5222 3.1448 42.1329 12.4252 1.3066 41.2933

11.2712 -0.2409 40.2475

10.0932 -1.5001 39.0727

8.9204 -2.4923 37.8350 7.7771 -3.2464 36.5805 6.6818 -3.7963 35.3446 5.6479 -4.1786 34.1513 4.3777 -4.4882 32.6467 3.2406 -4.6312 31.2533 2.2368 -4.6655 29.9717 1.3610 -4.6366 28.7945 0.5015 -4.5691 27.5602 -0.2238 -4.4973 26.4315 -0.8351 -4.4456 25.3943 -1.3531 -4.4301 24.4371


48

4.4391 4.4541 4.4692 4.4842 4.5029 4.5217 4.5405 4.5593 4.5763 4.5933 4.6103 4.6274 4.6461 4.6647 4.6834 4.7021 4.7289 4.7557 4.7826 4.8094 4.8301 4.8509 4.8716 4.8924 4.9080 4.9236 4.9392 4.9547 4.9703 4.9859 5.0015 5.0171 5.0327 5.0484 5.0640 5.0796 5.0925 5.1053 5.1182 5.1310 5.1440 5.1569 5.1698 5.1827 5.1975 5.2122 5.2270 5.2417 5.2565

18.0275 18.0474 18.0674 18.0873 18.1072 18.1237 18.1401 18.1566 18.1730 18.1909 18.2088 18.2267 18.2446 18.2688 18.2930 18.3172 18.3414 18.3627 18.3840 18.4053 18.4266 18.4426 18.4586 18.4746 18.4906 18.5066 18.5226 18.5386 18.5546 18.5690 18.5835 18.5980 18.6124 18.6257 18.6390 18.6523 18.6657 18.6779 18.6902 18.7024 18.7147 18.7304 18.7462 18.7620 18.7778 18.7912 18.8046 18.8180 18.8314

-1.8018 -4.4604 23.5445 -2.1936 -4.5428 22.7189 -2.5435 -4.6817 21.9572 -2.8669 -4.8789 21.2573 -3.1652 -5.1239 20.6437 -3.4601 -5.4253 20.0894 -3.7593 -5.7846 19.5978 -4.0711 -6.2026 19.1733 -4.4559 -6.7584 18.7742 -4.8744 -7.3949 18.4845 -5.3336 -8.1119 18.3198 -5.8399 -8.9053 18.2977 -6.5174 -9.9475 18.4880

-7.2718 -11.0646 18.9584

-8.0977 -12.2112 19.7512

-8.9806 -13.3157 20.8983

-9.8209 -14.2138 22.2795

-10.6527 -14.8948 23.9609

-11.4308 -15.2546 25.8890

-12.1008 -15.2035 27.9617

-12.6058 -14.6730 30.0369

-12.8856 -13.6709 31.9278

-12.8982 -12.2667 33.4657

-12.6351 -10.5751 34.5289

-12.1770 -8.9500 35.0211

-11.5368 -7.3624 35.0801

-10.7572 -5.9144 34.7489

-9.8873 -4.6756 34.1031 -8.9986 -3.6984 33.2550 -8.1151 -2.9543 32.2610 -7.2711 -2.4281 31.1806 -6.4916 -2.0886 30.0635 -5.8139 -1.9028 28.9798 -5.2192 -1.8318 27.9107 -4.7101 -1.8510 26.8688


49

5.2712 5.2860 5.3007 5.3138 5.3270 5.3401 5.3532 5.3645 5.3758 5.3871 5.3984 5.4092 5.4199 5.4306 5.4413 5.4548 5.4683 5.4818 5.4954 5.5115 5.5277 5.5439 5.5601 5.5794 5.5987 5.6180 5.6373 5.6591 5.6808 5.7025 5.7243 5.7476 5.7708 5.7941 5.8174 5.8354 5.8534 5.8714 5.8894 5.9029 5.9163 5.9298 5.9432 5.9566 5.9701 5.9835 5.9970 6.0092 6.0214

18.8448 18.8582 18.8716 18.8850 18.8961 18.9072 18.9182 18.9293 18.9420 18.9546 18.9673 18.9799 18.9951 19.0102 19.0253 19.0404 19.0583 19.0763 19.0942 19.1121 19.1330 19.1539 19.1748 19.1956 19.2192 19.2427 19.2662 19.2897 19.3075 19.3253 19.3431 19.3609 19.3744 19.3879 19.4015 19.4150 19.4285 19.4420 19.4555 19.4690 19.4820 19.4950 19.5081 19.5211 19.5316 19.5421 19.5525 19.5630 19.5783

-4.2861 -1.9399 25.8618 -3.8960 -2.1096 24.7427 -3.6074 -2.3358 23.6796 -3.4122 -2.6082 22.6750 -3.3015 -2.9228 21.7311 -3.2677 -3.3594 20.6798 -3.3368 -3.8634 19.7256 -3.5051 -4.4439 18.8775 -3.7674 -5.1175 18.1495 -4.0712 -5.7941 17.6279 -4.4506 -6.5641 17.2285 -4.9098 -7.4359 16.9737 -5.4530 -8.4145 16.8923 -6.3471 -9.9370 17.1265

-7.4166 -11.6254 17.8839

-8.6464 -13.3667 19.2875

-9.9811 -14.9403 21.4228

-10.8514 -15.7209 23.1946

-11.6822 -16.1929 25.2323

-12.4218 -16.2554 27.4499

-13.0136 -15.8383 29.7165

-13.4029 -14.9063 31.8653

-13.5385 -13.5161 33.7098

-13.3915 -11.7798 35.1003

-12.9677 -9.8384 35.9466

-12.3501 -8.0027 36.2233

-11.5569 -6.2818 36.0450

-10.6394 -4.7707 35.4812

-9.6513 -3.5237 34.6292 -8.7058 -2.6046 33.6577 -7.7813 -1.9174 32.5816 -6.9064 -1.4377 31.4525 -6.1010 -1.1309 30.3097 -5.3852 -0.9611 29.1918 -4.7543 -0.8966 28.0990


50

6.0335 6.0457 6.0563 6.0669 6.0776 6.0882 6.0997 6.1112 6.1227 6.1342 6.1497 6.1653 6.1808 6.1963 6.2113 6.2263 6.2413 6.2563 6.2713 6.2863 6.3013 6.3163 6.3333 6.3503 6.3673 6.3843 6.4045 6.4246 6.4448 6.4649 6.4856 6.5064 6.5271 6.5478 6.5685 6.5893 6.6100 6.6307 6.6494 6.6680 6.6866 6.7052 6.7236 6.7420 6.7604 6.7788 6.7970 6.8151 6.8333

19.5935 19.6087 19.6240 19.6400 19.6560 19.6720 19.6881 19.7022 19.7164 19.7306 19.7447 19.7601 19.7754 19.7907 19.8061 19.8237 19.8413 19.8590 19.8766 19.8980 19.9194 19.9408 19.9622 19.9717 19.9811 19.9906 20.0000

x =

-8.0000 8.0000 27.0000 -7.6036 7.9571 26.6639 -7.2182 7.9093 26.3387 -6.8436 7.8574 26.0242 -6.4796 7.8021 25.7199 -4.8119 7.4981 24.3412 -3.3774 7.1889 23.1694 -2.1473 6.9160 22.1667 -1.0914 6.7046 21.3016 -0.2618 6.5763 20.6165 0.4684 6.5119 20.0097 1.1163 6.5131 19.4715 1.6982 6.5799 18.9945 2.2291 6.7105 18.5741 2.7204 6.9037 18.2088 3.1825 7.1578 17.8982 3.6255 7.4708 17.6432 4.1388 7.9154 17.4156

-4.2091 -0.9120 27.0401 -3.7478 -0.9864 26.0196 -3.3335 -1.1167 24.9490 -3.0058 -1.2844 23.9262 -2.7568 -1.4797 22.9508 -2.5779 -1.6975 22.0221 -2.4469 -1.9779 20.9994 -2.3911 -2.2874 20.0385 -2.4042 -2.6296 19.1398 -2.4791 -3.0127 18.3052 -2.5967 -3.4041 17.6071 -2.7623 -3.8446 16.9677 -2.9774 -4.3429 16.3925 -3.2436 -4.9092 15.8889 -3.6471 -5.7157 15.3822 -4.1429 -6.6609 15.0255 -4.7417 -7.7612 14.8556 -5.4544 -9.0265 14.9220

-6.3290 -10.5181 15.3090

-7.3465 -12.1621 16.1159

-8.5012 -13.8823 17.4419

-9.7647 -15.5338 19.3729

-10.7621 -16.6137 21.2720

-11.7502 -17.4022 23.5327

-12.6744 -17.7640 26.0863

-13.4685 -17.5833 28.8021

-14.0632 -16.7725 31.4955

-14.3849 -15.3500 33.9247

-14.3823 -13.4160 35.8751

-14.0444 -11.1270 37.1982

-13.4802 -8.9854 37.7832

-12.6988 -6.9120 37.8355

-11.7488 -5.0327 37.4194

-10.6865 -3.4298


51

6.8515 6.8746 6.8977 6.9208 6.9440 6.9673 6.9907 7.0141 7.0375 7.0621 7.0867 7.1113 7.1359 7.1552 7.1745 7.1938 7.2131 7.2324 7.2516 7.2709 7.2902 7.3081 7.3260 7.3439 7.3619 7.3794 7.3969 7.4144 7.4319 7.4491 7.4664 7.4836 7.5009 7.5219 7.5429 7.5640 7.5850 7.6081 7.6312 7.6544 7.6775 7.6997 7.7218 7.7440 7.7661 7.7891 7.8120 7.8350 7.8579

4.6484 8.4367 17.2764 5.1634 9.0304 17.2341 5.6926 9.6906 17.2988 6.3616 10.5662 17.5365 7.0672 11.5092 17.9737 7.8113 12.4920 18.6358 8.5916 13.4742 19.5445 9.6642 14.6894 21.1488

10.7406 15.6455 23.2205

11.7532 16.1565 25.6930

12.6109 16.0569 28.3967

13.1833 15.2991 30.8977

13.4538 13.9363 33.1074

13.3694 12.0970 34.7935

12.9326 9.9522 35.8158 12.3397 8.1418 36.1205 11.5704 6.4354 35.9751 10.6745 4.9284 35.4447 9.7046 3.6772 34.6231 8.7653 2.7427 33.6673 7.8442 2.0411 32.6010 6.9707 1.5493 31.4770 6.1655 1.2329 30.3361 5.4502 1.0564 29.2197 4.8195 0.9872 28.1270 4.2745 0.9998 27.0673 3.8136 1.0732 26.0456 3.3977 1.2052 24.9665 3.0701 1.3766 23.9360 2.8229 1.5776 22.9538 2.6474 1.8029 22.0195 2.5215 2.0967 20.9841 2.4740 2.4228 20.0137 2.4987 2.7854 19.1090 2.5884 3.1936 18.2727 2.7203 3.6067 17.5857 2.9021 4.0727 16.9609 3.1355 4.6006 16.4046 3.4227 5.2006 15.9255 3.8633 6.0689 15.4528 4.4045 7.0869 15.1502 5.0572 8.2698 15.0616 5.8325 9.6221 15.2452

36.6363 -9.6345 -2.2038 35.6673 -8.5783 -1.2524 34.5534 -7.5542 -0.5546 33.3610 -6.5885 -0.0728 32.1430 -5.7525 0.2220 31.0099 -4.9923 0.3948 29.9008 -4.3116 0.4740 28.8259 -3.7110 0.4843 27.7904 -3.2116 0.4487 26.8431 -2.7781 0.3824 25.9321 -2.4057 0.2958 25.0560 -2.0890 0.1968 24.2133 -1.8362 0.0972 23.4475 -1.6234 -0.0052 22.7083 -1.4465 -0.1083 21.9944 -1.3012 -0.2107 21.3045 -1.1837 -0.3122 20.6376 -1.0909 -0.4125 19.9927 -1.0200 -0.5121 19.3690 -0.9684 -0.6116 18.7658 -0.9304 -0.7256 18.1072 -0.9119 -0.8425 17.4733 -0.9110 -0.9640 16.8632 -0.9261 -1.0923 16.2764 -0.9603 -1.2464 15.6483 -1.0124 -1.4152 15.0483 -1.0823 -1.6021 14.4760 -1.1703 -1.8113 13.9314 -1.2992 -2.0952 13.3213 -1.4575 -2.4251 12.7521 -1.6487 -2.8107 12.2262 -1.8774 -3.2633 11.7475 -2.3018 -4.0878 11.1364 -2.8473 -5.1403 10.6736 -3.5438 -6.4783 10.4146 -4.4313 -8.1566 10.4502

-5.5165 -10.1506 10.8927

-6.8628 -12.5371 11.9344

-8.4876 -15.2247 13.8207

-10.3600 -17.9208 16.8315

-11.4857 -19.2591 19.1067

-12.6212 -20.3023 21.8074


52

7.8756 7.8932 7.9108 7.9285 7.9461 7.9638 7.9814 7.9991 8.0175 8.0359 8.0543 8.0727 8.0885 8.1043 8.1201 8.1359 8.1534 8.1709 8.1883 8.2058 8.2239 8.2420 8.2601 8.2782 8.2963 8.3143 8.3324 8.3505 8.3726 8.3946 8.4167 8.4387 8.4626 8.4866 8.5105 8.5344 8.5523 8.5702 8.5881 8.6060 8.6239 8.6418 8.6597 8.6776 8.6949 8.7121 8.7294 8.7467 8.7616

6.7366 11.1294 15.7701 7.7751 12.7552 16.7318 8.9361 14.4072 18.2201

10.1829 15.9264 20.3004

11.1525 16.8620 22.2972

12.0912 17.4645 24.6136

12.9427 17.6100 27.1618

13.6426 17.2051 29.7979

14.1267 16.1931 32.3315

14.3321 14.6265 34.5383

14.2197 12.6285 36.2332

13.7901 10.3617 37.3003

13.1508 8.2555 37.6909 12.3118 6.2634 37.5723 11.3253 4.4967 37.0202 10.2481 3.0212 36.1417 9.2130 1.9266 35.1313 8.1878 1.0906 34.0053 7.2042 0.4879 32.8217 6.2848 0.0808 31.6251 5.4803 -0.1636 30.4970 4.7543 -0.2961 29.3966 4.1092 -0.3440 28.3322 3.5442 -0.3303 27.3078 3.1029 -0.2811 26.4237 2.7193 -0.2075 25.5715 2.3891 -0.1176 24.7501 2.1078 -0.0174 23.9581 1.8230 0.1132 23.0279 1.5961 0.2478 22.1371 1.4196 0.3836 21.2834 1.2863 0.5196 20.4648 1.2000 0.6396 19.7750 1.1390 0.7607 19.1099 1.1005 0.8841 18.4689 1.0821 1.0115 17.8513 1.0831 1.1671 17.1614 1.1066 1.3334 16.5017 1.1515 1.5137 15.8717 1.2167 1.7121 15.2711

-13.7150 -20.8952 24.8810

-14.7011 -20.8905 28.2075

-15.5064 -20.1614 31.6057

-16.0474 -18.6714 34.8096

-16.2558 -16.4812 37.5581

-16.0971 -13.7396 39.6402

-15.6238 -10.9510 40.8333

-14.8569 -8.1353 41.3221

-13.8405 -5.4856 41.1612

-12.6360 -3.1429 40.4631

-11.4049 -1.3093 39.4625

-10.1249 0.1643 38.2402

-8.8441 1.2903 36.8938 -7.6002 2.1099 35.5054 -6.5981 2.6057 34.3420 -5.6548 2.9512 33.2080 -4.7778 3.1763 32.1166 -3.9713 3.3085 31.0747 -2.8512 3.3853 29.5454 -1.8986 3.3635 28.1369 -1.0984 3.2963 26.8383 -0.4322 3.2211 25.6368 0.0219 3.1713 24.7303 0.4141 3.1404 23.8732 0.7545 3.1344 23.0612 1.0533 3.1574 22.2909 1.3201 3.2116 21.5597 1.5623 3.2991 20.8661 1.7869 3.4218 20.2092 2.0011 3.5807 19.5885 2.2338 3.8001 18.9433 2.4680 4.0677 18.3439 2.7099 4.3864 17.7924 2.9660 4.7587 17.2919 3.2853 5.2553 16.7859 3.6386 5.8329 16.3610 4.0333 6.4977 16.0292


53

8.7765 8.7913 8.8062 8.8215 8.8368 8.8521 8.8673 8.8863 8.9053 8.9243 8.9433 8.9610 8.9787 8.9964 9.0141 9.0319 9.0498 9.0676 9.0854 9.1086 9.1319 9.1551 9.1783 9.1999 9.2215 9.2431 9.2647 9.2812 9.2976 9.3140 9.3305 9.3469 9.3633 9.3797 9.3962 9.4111 9.4261 9.4411 9.4561 9.4698 9.4835 9.4972 9.5109 9.5238 9.5368 9.5498 9.5627 9.5773 9.5920

1.3107 1.9540 14.6499 1.4296 2.2283 14.0643 1.5754 2.5415 13.5155 1.7503 2.9013 13.0057 2.0359 3.4709 12.3899 2.3900 4.1628 11.8606 2.8258 5.0041 11.4368 3.3600 6.0244 11.1474 4.2145 7.6233 11.0422 5.3073 9.6363 11.3569 6.6783 12.0781 12.2972 8.3526 14.8359 14.1539 9.5848 16.6622 15.9874

10.9078 18.3565 18.4223

12.2687 19.7049 21.4822

13.5799 20.4499 25.0897

14.4962 20.4348 28.1698

15.2454 19.7826 31.2977

15.7607 18.4547 34.2687

15.9895 16.4926 36.8708

15.8950 14.0186 38.8986

15.4644 11.2567 40.2253

14.7154 8.4372 40.8130 13.6984 5.7586 40.7052 12.5137 3.4390 40.0492 11.2084 1.5050 38.9936 9.8507 -0.0124 37.6742 8.4982 -1.1379 36.2247 7.4276 -1.8093 35.0149 6.4105 -2.2895 33.8139 5.4595 -2.6133 32.6446 4.5828 -2.8146 31.5211 3.5565 -2.9430 30.1337 2.6610 -2.9683 28.8362 1.8892 -2.9329 27.6248 1.2308 -2.8692 26.4922 0.5912 -2.7890 25.2666 0.0640 -2.7240 24.1238 -0.3711 -2.6903 23.0542 -0.7352 -2.6972 22.0503

4.4773 7.2545 15.8053 5.0809 8.2795 15.7054 5.7752 9.4435 15.8275 6.5670 10.7359 16.2231 7.4597 12.1250 16.9508 8.7931 14.0403 18.5277

10.2487 15.7661 20.9534

11.7205 16.9275 24.2305

13.0073 17.0924 28.0852

13.5721 16.5629 30.3969

13.9398 15.5525 32.5658

14.0694 14.1035 34.4365

13.9414 12.3077 35.8815

13.5547 10.3062 36.8097

12.9289 8.2696 37.1990 12.1032 6.3433 37.0812 11.1310 4.6359 36.5350 10.1054 3.2521 35.7012 9.0503 2.1552 34.6600 8.0105 1.3368 33.4916 7.0202 0.7628 32.2658 6.1817 0.4133 31.1412 5.4160 0.1997 30.0289 4.7292 0.0925 28.9431 4.1235 0.0656 27.8921 3.6079 0.0959 26.9003 3.1645 0.1665 25.9456 2.7885 0.2652 25.0277 2.4745 0.3828 24.1455 2.1334 0.5637 22.9939 1.8825 0.7588 21.9025 1.7098 0.9630 20.8681 1.6024 1.1770 19.8880 1.5553 1.3621 19.1250 1.5418 1.5579 18.3962 1.5593 1.7676 17.7013 1.6053 1.9950 17.0399 1.6848 2.2650 16.3663 1.7955 2.5666 15.7327 1.9383 2.9063 15.1407 2.1143 3.2919 14.5930


54

9.6066 9.6212 9.6333 9.6453 9.6574 9.6695 9.6815 9.6936 9.7057 9.7177 9.7284 9.7390 9.7497 9.7603 9.7738 9.7872 9.8007 9.8141 9.8291 9.8440 9.8590 9.8739 9.8931 9.9123 9.9315 9.9507 9.9797 10.0088 10.0378 10.0668 10.0914 10.1160 10.1406 10.1652 10.1826 10.2001 10.2175 10.2349 10.2482 10.2615 10.2748 10.2881 10.3014 10.3147 10.3279 10.3412 10.3514 10.3616 10.3718

-0.9907 -2.7355 21.2853 -1.2201 -2.8056 20.5576 -1.4308 -2.9091 19.8661 -1.6298 -3.0473 19.2100 -1.8311 -3.2282 18.5667 -2.0323 -3.4498 17.9618 -2.2387 -3.7143 17.3965 -2.4560 -4.0244 16.8725 -2.7283 -4.4445 16.3201 -3.0288 -4.9362 15.8325 -3.3640 -5.5063 15.4179 -3.7416 -6.1615 15.0868 -4.2521 -7.0518 14.8226 -4.8429 -8.0811 14.7243 -5.5240 -9.2543 14.8322

-6.3050 -10.5648 15.1961

-7.4523 -12.4001 16.1177

-8.7671 -14.3192 17.7074

-10.2102 -16.1109 20.0810

-11.6827 -17.4384 23.2498

-12.5968 -17.8564 25.7160

-13.3960 -17.7700 28.3514

-14.0139 -17.0997 30.9917

-14.3906 -15.8337 33.4427

-14.4778 -14.0292 35.4928

-14.2469 -11.8604 36.9769

-13.7014 -9.5295 37.8089

-12.8826 -7.2224 37.9835

-11.8348 -5.0833 37.5727

-10.6462 -3.2834 36.6998

-9.3963 -1.8774 35.5007 -8.1523 -0.8548 34.1270 -7.1917 -0.2716 32.9771 -6.2907 0.1301 31.8169

2.3802 3.8415 13.9854 2.7071 4.4902 13.4601 3.1037 5.2567 13.0321 3.5806 6.1611 12.7236 4.3184 7.5291 12.5527 5.2337 9.1904 12.7108 6.3512 11.1531 13.3272 7.6877 13.3610 14.5747 9.2281 15.6852 16.6277

10.9303 17.7905 19.7212

12.6694 19.1883 23.8782

14.1942 19.2899 28.7579

14.8350 18.5630 31.5632

15.2514 17.2515 34.1817

15.3957 15.4110 36.4271

15.2451 13.1565 38.1485

14.7970 10.6635 39.2403

14.0725 8.1424 39.6847 13.1153 5.7704 39.5291 11.9856 3.6770 38.8756 10.8262 2.0313 37.9313 9.6274 0.7178 36.7740 8.4357 -0.2759 35.4942 7.2875 -0.9892 34.1682 6.3399 -1.4251 33.0134 5.4576 -1.7167 31.8822 4.6477 -1.8944 30.7879 3.9133 -1.9859 29.7375 3.0631 -2.0144 28.4272 2.3360 -1.9763 27.1940 1.7215 -1.9033 26.0327 1.2071 -1.8183 24.9371 0.8353 -1.7476 24.0461 0.5181 -1.6859 23.1946 0.2475 -1.6378 22.3797 0.0155 -1.6063 21.5986 -0.1853 -1.5930 20.8492 -0.3607 -1.5993 20.1297 -0.5160 -1.6261 19.4388 -0.6566 -1.6738 18.7754 -0.7783 -1.7372 18.1829


55

10.3819 10.3921 10.4023 10.4125 10.4226 10.4362 10.4498 10.4634 10.4770 10.4926 10.5081 10.5237 10.5392 10.5549 10.5706 10.5862 10.6019 10.6170 10.6320 10.6471 10.6621 10.6795 10.6968 10.7142 10.7316 10.7524 10.7733 10.7941 10.8150 10.8342 10.8534 10.8726 10.8918 10.9109 10.9301 10.9493 10.9685 10.9857 11.0029 11.0201 11.0373 11.0542 11.0710 11.0878 11.1047 11.1210 11.1373 11.1536 11.1699

-5.4624 0.3837 30.6716 -4.7144 0.5223 29.5569 -4.0958 0.5740 28.5586 -3.5471 0.5709 27.5958 -3.0656 0.5285 26.6691 -2.6474 0.4593 25.7779 -2.2740 0.3690 24.8864 -1.9579 0.2678 24.0297 -1.6932 0.1614 23.2058 -1.4739 0.0536 22.4129 -1.2943 -0.0536 21.6491 -1.1495 -0.1587 20.9130 -1.0352 -0.2613 20.2033 -0.9473 -0.3618 19.5188 -0.8863 -0.4535 18.9082 -0.8425 -0.5447 18.3176 -0.8142 -0.6366 17.7463 -0.7995 -0.7302 17.1937 -0.7981 -0.8472 16.5524 -0.8130 -0.9709 15.9367 -0.8434 -1.1036 15.3462 -0.8884 -1.2483 14.7802 -0.9543 -1.4249 14.1860 -1.0380 -1.6237 13.6210 -1.1408 -1.8493 13.0852 -1.2640 -2.1074 12.5792 -1.4612 -2.5069 11.9606 -1.7043 -2.9889 11.3999 -2.0020 -3.5724 10.9046 -2.3653 -4.2795 10.4871 -2.9910 -5.4800 10.0805 -3.8044 -7.0369 9.9230 -4.8519 -9.0231 10.1437

-6.1891 -11.4698 10.9508

-7.3055 -13.4133 11.9954

-8.5866 -15.5065 13.5874

-10.0184 -17.6153 15.8534

-11.5539 -19.5094 18.8867

-12.7041 -20.5854 21.6298

-13.8137 -21.2060 24.7526

-14.8169 -21.2145 28.1379

-0.8943 -1.8195 17.6127 -1.0074 -1.9216 17.0646 -1.1204 -2.0443 16.5382 -1.2594 -2.2194 15.9364 -1.4055 -2.4279 15.3660 -1.5625 -2.6731 14.8276 -1.7343 -2.9583 14.3224 -1.9589 -3.3469 13.7779 -2.2147 -3.8045 13.2853 -2.5083 -4.3410 12.8510 -2.8474 -4.9672 12.4842 -3.3180 -5.8374 12.1550 -3.8774 -6.8721 11.9654 -4.5406 -8.0915 11.9562 -5.3243 -9.5082 12.1838

-6.4568 -11.4891 12.8791

-7.8098 -13.7274 14.2136

-9.3748 -16.0711 16.3790

-11.0894 -18.1980 19.5349

-12.1575 -19.1903 21.9778

-13.1871 -19.7805 24.7637

-14.1180 -19.8310 27.7894

-14.8820 -19.2402 30.8903

-15.4107 -17.9496 33.8515

-15.6371 -16.0231 36.4233

-15.5192 -13.6030 38.4011

-15.0547 -10.8776 39.6574

-14.3440 -8.3102 40.1395

-13.4021 -5.8808 40.0254

-12.2850 -3.7230 39.4088

-11.0549 -1.9155 38.4190

-9.8919 -0.5968 37.3127 -8.7286 0.4256 36.0897


56

11.1888 11.2077 11.2266 11.2455 11.2682 11.2909 11.3137 11.3364 11.3562 11.3760 11.3958 11.4156 11.4430 11.4704 11.4978 11.5252 11.5428 11.5605 11.5781 11.5957 11.6133 11.6310 11.6486 11.6662 11.6826 11.6990 11.7154 11.7318 11.7472 11.7626 11.7780 11.7935 11.8087 11.8239 11.8392 11.8544 11.8711 11.8877 11.9044 11.9211 11.9405 11.9598 11.9792 11.9986 12.0162 12.0338 12.0513 12.0689 12.0905

-15.6381 -20.4958 31.5980

-16.2017 -18.9849 34.8872

-16.4338 -16.7616 37.7217

-16.2884 -13.9956 39.8726

-15.7642 -10.9050 41.2028

-14.9858 -8.0533 41.6681

-13.9627 -5.3723 41.4908

-12.7544 -3.0045 40.7802

-11.4266 -1.0306 39.6821

-10.1951 0.3804 38.5009

-8.9617 1.4773 37.2126 -7.7601 2.2915 35.8841 -6.6151 2.8675 34.5652 -5.5770 3.2442 33.3279 -4.6177 3.4800 32.1439 -3.7418 3.6098 31.0213 -2.9504 3.6632 29.9621 -1.9358 3.6544 28.5163 -1.0827 3.5914 27.1894 -0.3716 3.5158 25.9667 0.2195 3.4559 24.8351 0.6355 3.4302 23.9567 0.9963 3.4335 23.1281 1.3127 3.4704 22.3457 1.5953 3.5437 21.6067 1.8540 3.6550 20.9092 2.0962 3.8062 20.2527 2.3290 3.9991 19.6373 2.5593 4.2348 19.0634 2.8257 4.5555 18.4651 3.1043 4.9365 17.9256 3.4019 5.3815 17.4499 3.7257 5.8938 17.0444 4.1427 6.5754 16.6735 4.6122 7.3583 16.4258 5.1418 8.2461 16.3239 5.7395 9.2370 16.3954 6.5743 10.5815 16.7666 7.5242 12.0361 17.5255

-7.5989 1.1800 34.8151 -6.5276 1.7086 33.5383 -5.6483 2.0239 32.4406 -4.8340 2.2271 31.3771 -4.0884 2.3433 30.3549 -3.4123 2.3947 29.3770 -2.4908 2.3884 27.9360 -1.7215 2.3221 26.5949 -1.0869 2.2344 25.3437 -0.5671 2.1516 24.1725 -0.2573 2.1047 23.3876 0.0122 2.0721 22.6352 0.2479 2.0564 21.9133 0.4556 2.0593 21.2199 0.6412 2.0819 20.5537 0.8093 2.1252 19.9135 0.9640 2.1900 19.2985 1.1097 2.2769 18.7080 1.2678 2.4020 18.0720 1.4235 2.5573 17.4664 1.5810 2.7446 16.8913 1.7444 2.9657 16.3473 1.9462 3.2664 15.7584 2.1664 3.6204 15.2147 2.4102 4.0336 14.7196 2.6837 4.5122 14.2782 3.0513 5.1671 13.8375 3.4777 5.9361 13.4952 3.9728 6.8331 13.2726 4.5482 7.8696 13.1983 5.3602 9.3065 13.3564 6.3219 10.9624 13.8693 7.4426 12.8043 14.8453 8.7182 14.7383 16.4064 9.8283 16.2415 18.1461

10.9897 17.5645 20.3752

12.1519 18.5350 23.0816

13.2404 18.9649 26.1730

14.1678 18.6676 29.4817

14.8232 17.5480 32.6974

15.1120 15.6342 35.4971

14.9848 13.0747 37.6162


57

12.1122 12.1339 12.1555 12.1781 12.2007 12.2233 12.2459 12.2630 12.2801 12.2972 12.3143 12.3314 12.3485 12.3656 12.3827 12.3978 12.4130 12.4281 12.4432 12.4575 12.4718 12.4860 12.5003 12.5137 12.5271 12.5404 12.5538 12.5665 12.5793 12.5920 12.6047 12.6167 12.6287 12.6407 12.6527 12.6647 12.6767 12.6887 12.7007 12.7143 12.7278 12.7414 12.7550 12.7701 12.7853 12.8004 12.8156 12.8335 12.8515

8.5801 13.5258 18.7471 9.7145 14.9220 20.4864

10.6246 15.8365 22.2217

11.5135 16.4734 24.2619

12.3316 16.7208 26.5370

13.0209 16.4885 28.9264

13.5223 15.7145 31.2678

13.7756 14.4255 33.3594

13.7404 12.7129 35.0260

13.4111 10.7125 36.1476

12.8682 8.7895 36.6455 12.1268 6.9339 36.6569 11.2341 5.2584 36.2401 10.2437 3.8357 35.4879 9.2590 2.7429 34.5555 8.2778 1.9039 33.4853 7.3341 1.2985 32.3388 6.4528 0.8914 31.1650 5.6806 0.6493 30.0431 4.9885 0.5242 28.9419 4.3795 0.4889 27.8720 3.8533 0.5200 26.8392 3.3942 0.6020 25.8163 3.0132 0.7201 24.8347 2.7038 0.8637 23.8938 2.4589 1.0254 22.9925 2.2320 1.2479 21.9181 2.0837 1.4890 20.9015 2.0052 1.7484 19.9411 1.9868 2.0306 19.0361 2.0157 2.3122 18.2632 2.0864 2.6228 17.5366 2.1980 2.9688 16.8575 2.3497 3.3580 16.2287 2.5631 3.8447 15.6015 2.8294 4.4056 15.0460 3.1535 5.0536 14.5725 3.5415 5.8026 14.1953 4.2257 7.0791 13.8621 5.0843 8.6346 13.8505 6.1413 10.4819 14.2816

14.5553 10.6675 38.7076

13.8508 8.2242 39.1623 12.9147 5.9209 39.0240 11.8074 3.8871 38.3913 10.6690 2.2872 37.4660 9.4928 1.0133 36.3268 8.3258 0.0537 35.0629 7.2044 -0.6309 33.7507 6.2774 -1.0474 32.6001 5.4176 -1.3218 31.4712 4.6318 -1.4843 30.3771 3.9229 -1.5625 29.3250 3.1589 -1.5775 28.0966 2.5033 -1.5343 26.9331 1.9472 -1.4587 25.8308 1.4802 -1.3693 24.7851 1.1152 -1.2848 23.8578 0.8082 -1.2065 22.9716 0.5510 -1.1391 22.1231 0.3352 -1.0854 21.3098 0.1900 -1.0539 20.6954 0.0626 -1.0322 20.1003 -0.0499 -1.0206 19.5237 -0.1501 -1.0192 18.9647 -0.2404 -1.0280 18.4228 -0.3227 -1.0470 17.8975 -0.3991 -1.0763 17.3882 -0.4710 -1.1160 16.8945 -0.5510 -1.1750 16.3425 -0.6294 -1.2486 15.8105 -0.7080 -1.3374 15.2979 -0.7888 -1.4423 14.8044 -0.8897 -1.5885 14.2449 -0.9987 -1.7612 13.7117 -1.1184 -1.9633 13.2051 -1.2518 -2.1983 12.7255 -1.4317 -2.5245 12.1947 -1.6402 -2.9113 11.7055 -1.8830 -3.3694 11.2622 -2.1673 -3.9103 10.8711 -2.5641 -4.6666 10.4922 -3.0430 -5.5816 10.2176 -3.6201 -6.6833 10.0794 -4.3152 -7.9979 10.1241 -5.2893 -9.8028 10.4865

-6.4801 -11.9454 11.3207

-7.9064 -14.3770


58

12.8695 12.8875 12.9143 12.9411 12.9679 12.9947 13.0208 13.0469 13.0731 13.0992 13.1138 13.1285 13.1431 13.1577 13.1724 13.1870 13.2017 13.2163 13.2297 13.2432 13.2566 13.2700 13.2825 13.2950 13.3075 13.3200 13.3306 13.3412 13.3518 13.3624 13.3790 13.3956 13.4122 13.4288 13.4421 13.4555 13.4689 13.4823 13.4956 13.5090 13.5224 13.5357

7.4139 12.5745 15.3187 8.5444 14.2889 16.6536 9.7832 15.9544 18.5621

11.0851 17.3804 21.0920

12.3670 18.3176 24.2008

13.2499 18.5298 26.8337

13.9952 18.1999 29.5720

14.5389 17.2719 32.2395

14.8274 15.7599 34.6432

14.8203 13.7496 36.5850

14.4990 11.4285 37.9259

13.8744 9.0007 38.6064 12.9914 6.6482 38.6433 11.9105 4.5273 38.1318 10.7053 2.7522 37.2024 9.4479 1.3640 35.9849 8.1988 0.3458 34.6153 7.2132 -0.2531 33.4528 6.2856 -0.6702 32.2856 5.4288 -0.9391 31.1378 4.6504 -1.0928 30.0241 3.9498 -1.1612 28.9462 3.3297 -1.1668 27.9124 2.7868 -1.1293 26.9228 2.3162 -1.0645 25.9760 1.8586 -0.9719 24.9432 1.4771 -0.8715 23.9587 1.1622 -0.7722 23.0187 0.9037 -0.6795 22.1199 0.6922 -0.5961 21.2592 0.5204 -0.5242 20.4341 0.3817 -0.4644 19.6426 0.2695 -0.4166 18.8828 0.1872 -0.3832 18.2328 0.1187 -0.3582 17.6055 0.0613 -0.3412 17.0001 0.0126 -0.3316 16.4158 -0.0240 -0.3287 15.9279 -0.0567 -0.3306 15.4545 -0.0863 -0.3372 14.9954 -0.1137 -0.3484 14.5501

12.8126 -9.5603 -16.9262

15.1850 -10.7563 -18.5330

17.3423 -11.9968 -19.9044

20.0133 -13.2286 -20.8547

23.1748 -14.3760 -21.1875

26.7196 -15.3509 -20.7138

30.4624 -16.0444 -19.3407

34.0788 -16.3628 -17.1031

37.2333 -16.2538 -14.1615

39.6461 -15.8058 -11.2993

40.9597 -15.0477 -8.3859

41.5344 -14.0235 -5.6305

41.4214 -12.7974 -3.1875

40.7356 -11.5458 -1.2856

39.7297 -10.2404 0.2436

38.4900 -8.9315 1.4112 37.1193 -7.6588 2.2602 35.7047 -6.6477 2.7667 34.5378 -5.6945 3.1212 33.4010 -4.8070 3.3539 32.3077 -3.9895 3.4921 31.2648 -2.8326 3.5763 29.7094 -1.8490 3.5567 28.2816 -1.0230 3.4897 26.9692 -0.3356 3.4150 25.7587 0.1379 3.3660 24.8377 0.5462 3.3377 23.9693 0.9003 3.3366 23.1488 1.2112 3.3669 22.3724 1.4892 3.4311 21.6375 1.7422 3.5315 20.9426 1.9779 3.6697 20.2868 2.2037 3.8471 19.6698


59

-0.1383 -0.3630 14.1407 -0.1620 -0.3817 13.7430 -0.1855 -0.4044 13.3566 -0.2091 -0.4313 12.9812 -0.2397 -0.4714 12.5231 -0.2720 -0.5189 12.0815 -0.3067 -0.5746 11.6560 -0.3446 -0.6392 11.2460 -0.3954 -0.7297 10.7753 -0.4534 -0.8367 10.3257 -0.5200 -0.9628 9.8968 -0.5971 -1.1113 9.4882 -0.7062 -1.3232 9.0267 -0.8373 -1.5806 8.5945 -0.9952 -1.8934 8.1928 -1.1863 -2.2737 7.8242 -1.4505 -2.8002 7.4543 -1.7786 -3.4566 7.1406 -2.1861 -4.2745 6.8994 -2.6932 -5.2907 6.7569 -3.3733 -6.6429 6.7600 -4.2292 -8.3332 6.9933 -5.2984 -10.4106 7.5750 -6.6191 -12.8895 8.6836

-7.8319 -15.0588 10.0482

-9.2182 -17.3742 12.0134

-10.7617 -19.6800 14.7158

-12.4107 -21.7156 18.2529

-13.6592 -22.8519 21.4545

-14.8599 -23.4493 25.0694

-15.9403 -23.3309 28.9579

-16.8180 -22.3716 32.9012

-17.4110 -20.5066 36.6160

-17.6402 -17.8403 39.7820

-17.4566 -14.5780 42.1467

-16.8600 -10.9752 43.5660

-15.9893 -7.6797

2.4500 4.0899 19.0335 2.6994 4.3845 18.4468 2.9584 4.7337 17.9125 3.2340 5.1402 17.4344 3.5793 5.6812 16.9613 3.9624 6.3078 16.5797 4.3908 7.0258 16.3038 4.8725 7.8382 16.1513 5.5300 8.9362 16.1612 6.2831 10.1677 16.4289 7.1360 11.5100 17.0117 8.0864 12.9123 17.9698 9.0887 14.2565 19.3103

10.1425 15.4632 21.1000

11.2008 16.3801 23.3317

12.1936 16.8389 25.9248

13.0384 16.6700 28.7308

13.6294 15.7869 31.4693

13.8777 14.2135 33.8487

13.7406 12.0801 35.6329

13.3425 10.1144 36.5146

12.7069 8.1229 36.8598 11.8752 6.2497 36.7032 10.9027 4.6012 36.1264 9.8899 3.2839 35.2783 8.8544 2.2482 34.2330 7.8399 1.4835 33.0683 6.8794 0.9547 31.8511 6.0685 0.6385 30.7330 5.3324 0.4528 29.6280 4.6765 0.3690 28.5497 4.1022 0.3620 27.5058 3.5983 0.4123 26.4792 3.1725 0.5045 25.4926 2.8191 0.6259 24.5455 2.5316 0.7675 23.6372 2.2314 0.9848 22.4538 2.0268 1.2216 21.3368 1.9057 1.4750 20.2836 1.8545 1.7488 19.2923 1.8571 1.9955 18.5201


60

44.0128 -14.8526 -4.6031

43.7433 -13.5142 -1.9026

42.8908 -12.0447 0.3364

41.6268 -10.7161 1.8938

40.3318 -9.3824 3.1095 38.9427 -8.0769 4.0186 37.5292 -6.8249 4.6688 36.1430 -5.5475 5.1414 34.7068 -4.3682 5.4252 33.3608 -3.2940 5.5709 32.1139 -2.3266 5.6211 30.9659 -1.1840 5.5973 29.5611 -0.2133 5.5219 28.3035 0.6059 5.4381 27.1719 1.2960 5.3752 26.1482 1.8624 5.3503 25.2480 2.3495 5.3705 24.4245 2.7743 5.4420 23.6713 3.1535 5.5682 22.9837 3.5026 5.7497 22.3595 3.8323 5.9878 21.7994 4.1524 6.2827 21.3052 4.4730 6.6335 20.8797 4.8476 7.0967 20.4857 5.2416 7.6289 20.1948 5.6616 8.2268 20.0184 6.1141 8.8839 19.9688 6.6984 9.7261 20.0927 7.3349 10.6161 20.4423

1.8985 2.2644 17.7895 1.9770 2.5607 17.1011 2.0911 2.8908 16.4563 2.2555 3.2969 15.8066 2.4644 3.7612 15.2152 2.7210 4.2942 14.6882 3.0297 4.9081 14.2345 3.5399 5.8877 13.7603 4.1736 7.0718 13.4848 4.9504 8.4875 13.4721 5.8908 10.1445 13.8131 6.8654 11.7887 14.4951 7.9841 13.5646 15.6461 9.2350 15.3687 17.3683

10.5792 17.0213 19.7378

11.6112 18.0171 21.9642

12.6110 18.6456 24.5329

13.5193 18.7716 27.3488

14.2679 18.2925 30.2557

14.7886 17.1458 33.0467

15.0139 15.3893 35.4774

14.9014 13.1580 37.3460

14.4499 10.6312 38.5268

13.7724 8.2864 38.9661 12.8786 6.0680 38.8513 11.8228 4.0987 38.2672 10.6645 2.4508 37.3325 9.5607 1.2386 36.2731 8.4614 0.3045 35.0973 7.3993 -0.3782 33.8666 6.3984 -0.8499 32.6282 5.5449 -1.1357 31.5079 4.7637 -1.3082 30.4191 4.0578 -1.3942 29.3694 3.4273 -1.4169 28.3623 2.7220 -1.3834 27.1272 2.1279 -1.3072 25.9589 1.6335 -1.2111 24.8526 1.2259 -1.1108 23.8030 0.9377 -1.0299 22.9516


61

0.6954 -0.9571 22.1348 0.4923 -0.8953 21.3501 0.3216 -0.8458 20.5955 0.1775 -0.8090 19.8694 0.0552 -0.7852 19.1702 -0.0493 -0.7748 18.4968 -0.1402 -0.7772 17.8481 -0.2075 -0.7886 17.3306 -0.2694 -0.8087 16.8287 -0.3273 -0.8373 16.3420 -0.3826 -0.8747 15.8701 -0.4443 -0.9282 15.3480 -0.5057 -0.9936 14.8443 -0.5683 -1.0717 14.3585 -0.6335 -1.1631 13.8903 -0.7168 -1.2911 13.3541 -0.8080 -1.4420 12.8423 -0.9092 -1.6186 12.3547 -1.0229 -1.8242 11.8915 -1.1784 -2.1117 11.3739 -1.3599 -2.4542 10.8931 -1.5730 -2.8620 10.4524 -1.8244 -3.3466 10.0564 -2.1757 -4.0251 9.6617 -2.6026 -4.8528 9.3532 -3.1214 -5.8594 9.1569 -3.7518 -7.0757 9.1120 -4.6224 -8.7299 9.3122 -5.6967 -10.7327 9.8900

-7.0029 -13.0793 11.0097

-8.5547 -15.6777 12.8896

-9.8838 -17.6960 14.9631

-11.3169 -19.5674 17.7134

-12.7952 -21.0419 21.1710

-14.2186 -21.8157 25.2458

-15.1829 -21.7353 28.6191

-15.9690 -20.9559 32.0308

-16.5067 -19.4394 35.2571

-16.7404 -17.2350 38.0686


62

-16.6316 -14.4808 40.2444

-16.1668 -11.4242 41.6523

-15.3647 -8.3177 42.2574

-14.2782 -5.3771 42.1127

-13.0296 -2.8665 41.3952

-11.6534 -0.7719 40.2668

-10.2187 0.8761 38.8722

-8.7840 2.1047 37.3517 -7.6614 2.8324 36.1111 -6.5884 3.3607 34.8864 -5.5781 3.7251 33.7011 -4.6391 3.9596 32.5688 -3.4107 4.1321 31.0307 -2.3414 4.1714 29.6181 -1.4227 4.1365 28.3233 -0.6417 4.0708 27.1339 0.0515 4.0006 25.9802 0.6329 3.9517 24.9147 1.1226 3.9396 23.9270 1.5412 3.9742 23.0089 1.8674 4.0475 22.2511 2.1630 4.1639 21.5413 2.4374 4.3257 20.8788 2.7000 4.5339 20.2632 2.9808 4.8128 19.6507 3.2654 5.1497 19.0970 3.5612 5.5471 18.6062 3.8759 6.0068 18.1836 4.2703 6.6146 17.7907 4.7065 7.3111 17.5114 5.1915 8.0981 17.3639 5.7325 8.9727 17.3696 6.4715 10.1404 17.6173 7.3037 11.4007 18.1845 8.2228 12.6996 19.1266 9.2111 13.9471 20.4867 9.6611 14.4499 21.2271

10.1138 14.9053 22.0533

10.5640 15.2991 22.9618

11.0058 15.6163


63

23.9464

22 - 26

Function yprime = bvpexample(t.y)%BVPEXAMPLE : Differential equation for boundary value problem exampleyprime=[y(2); -2*y(1) + 3*y(2)];

Function res = bc(y0,y1)%BC: Evaluates the residue of the boundary conditionRes=[y0(1);y1(1)-10];

>> sol = bvpinit(linspace(0,1,25),[0 1]);>> sol = bvp4c(@bvpexample,@bc,sol);

>> sol.x

ans =

Columns 1 through 9

0 0.0417 0.0833 0.1250 0.1667 0.2083 0.2500 0.2917 0.3333

Columns 10 through 18

0.3750 0.4167 0.4583 0.5000 0.5417 0.5833 0.6250 0.6667 0.7083


0.7500 0.7917 0.8333 0.8750 0.9167 0.9583 1.0000

>> sol.y

ans =


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

27 - 30 >> happyscript.m??? Error using ==> fevalUndefined function or method 'ilovecat' for input arguments of type'double'.

Error in ==> odearguments at 110f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.


64

Error in ==> ode45 at 173[neq, tspan, ntspan, next, t0, tfinal, tdir, y0, f0, odeArgs, odeFcn,...

Error in ==> happyscript at 1[t,y]=ode45('ilovecat',[0,10],1.43);

31 - 35


65

36 - 39


66

40

41 >> ODE1 = 'Dy = y'

ODE1 =

Dy = y

>> ODE1solved = dsolve(ODE1, 'x') ODE1solved =


67

C2*exp(x)

42

>> initConds = 'y(0) = 5'

initConds =

y(0) = 5

>> ODE1solved = dsolve(ODE1, initConds, 'x') ODE1solved = 5*exp(x)

43

44 >> ODE2 = '3*D2y - Dy + 6*y = 6 *sin(t) + 2 *cos(t)'

ODE2 =

3*D2y - Dy + 6*y = 6 *sin(t) + 2 *cos(t)

>> initConds = 'y(0)=1, Dy(0)=2'

initConds =

y(0)=1, Dy(0)=2

>> ODE2solved = (dsolve(ODE2,initConds));>> pretty(ODE2solved) / / 1/2 \ / 1/2 \ | | 71 t | | 71 t | / 1/2 \ | 3 cos| t + ------- | 4 sin| t - ------- | | 71 t | | \ 6 / \ 6 /


68

cos| ------- | | -------------------- + -------------------- + \ 6 / \ 5 5 / 1/2 \ / 1/2 \ | 71 t | | 71 t | 4 sin| t + ------- | 3 cos| ------- - t | \ 6 / \ 6 / -------------------- + -------------------- - 5 5 / 1/2 \ / 1/2 \ 1/2 | 71 t | 1/2 | 71 t | 22 71 cos| t + ------- | 21 71 sin| t - ------- | \ 6 / \ 6 / --------------------------- + --------------------------- - 355 355 / 1/2 \ / 1/2 \ \ 1/2 | 71 t | 1/2 | 71 t | | 21 71 sin| t + ------- | 22 71 cos| ------- - t | | \ 6 / \ 6 / | --------------------------- + --------------------------- | - 355 355 / / / 1/2 \ / 1/2 \ | | 71 t | | 71 t | / 1/2 \ | 4 cos| t + ------- | 3 sin| t - ------- | | 71 t | | \ 6 / \ 6 / sin| ------- | | -------------------- + -------------------- - \ 6 / \ 5 5 / 1/2 \ / 1/2 \ | 71 t | | 71 t | 3 sin| t + ------- | 4 cos| ------- - t | \ 6 / \ 6 / -------------------- - -------------------- - 5 5 / 1/2 \ / 1/2 \ 1/2 | 71 t | 1/2 | 71 t | 21 71 cos| t + ------- | 22 71 sin| t - ------- | \ 6 / \ 6 / --------------------------- + --------------------------- + 355 355 / 1/2 \ / 1/2 \ \ 1/2 | 71 t | 1/2 | 71 t | | 22 71 sin| t + ------- | 21 71 cos| ------- - t | | \ 6 / \ 6 / |


69

--------------------------- - --------------------------- | - 355 355 / / 1/2 \ / 1/2 \ / t \ | 71 t | 1/2 / t \ | 71 t | exp| - | cos| ------- | 13 71 exp| - | sin| ------- | \ 6 / \ 6 / \ 6 / \ 6 / ----------------------- + -------------------------------- 5 355>> t=-5 :0.01:5;>> y_values=eval(vectorize(ODE2solved));>> plot(t,y_values)

45 > sysODE1 = 'Dx = 2*x + 3*z'

sysODE1 =

Dx = 2*x + 3*z

>> sysODE2 = 'Dy = 6*z - y'

sysODE2 =

Dy = 6*z - y

>> sysODE3 = 'Dz = 3*y - 12*x'

sysODE3 =

Dz = 3*y - 12*x

>> initConds = 'x(1) = 5, y(2)=3, z(9) = 0'

initConds =


70

x(1) = 5, y(2)=3, z(9) = 0

>> [x,y,z] = dsolve(sysODE1,sysODE2,sysODE3,initConds)

7. Conclusion:

In conclusion, solving ordinary differential equation in Matlab is not as complicated as solving it anually. In using the program, you may be able to play on its values and its constants. Moreover, graphing it is easy. Familiarization and practice is also a key to programming using the Matlab softare.


71

9. Assessment (Rubric for Laboratory Performance):


72


TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES

RUBRIC FOR LABORATORY PERFORMANCE

CRITERIA BEGINNER1

ACCEPTABLE2

PROFICIENT3

SCORE

I. Laboratory Skills

Manipulative Skills

Members do not demonstrate needed skills.

Members occasionally demonstrate needed skills.

Members always demonstrate needed skills.

Experimental Set-up Members are unable to set-up the materials.

Members are able to set-up the materials with supervision.

Members are able to set-up the material with minimum supervision.

Process Skills Member do not demonstrate targeted process skills.

Members occasionally demonstrate targeted process skills.

Members always demonstrate targeted process skills.

Safety Precautions Members do not follow safety precautions.

Members follow safety precautions most of the time.

Members follow safety precautions at all times.

II. Work Habits

Time Management / Conduct of Experiment

Members do not finish on time with incomplete data.

Members finish on time with incomplete data.

Members finish ahead of time with complete data and time to revise data.

Cooperative and Teamwork

Members do not know their tasks and have no defined responsibilities. Group conflicts have to be settled by the teacher.

Members have defined responsibilities most of the time. Group conflicts are cooperatively managed most of the time.

Members are on tasks and have defined responsibilities at all times. Group conflicts are cooperatively managed at all times.

Neatness and Orderliness

Messy workplace during and after the experiment.

Clean and orderly workplace with occasional mess during and after the experiment.

Clean and orderly workplace at all times during and after the experiment.

Ability to do independent work

Members require supervision by the teacher.

Members require occasional supervision by the teacher.

Members do not need to be supervised by the teacher.

Other Comments / Observations: TOTAL SCORE

RATING = ( ) x 100%

Evaluated by:

_______________________________________ Printed Name and Signature of Faculty Member Date: ___________________________

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T I P – V P A A – 0 5 4 D

Revision Status/Date: 0/2009 September 09

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