In this worksheet we examine some of the potential 'problems' with fitting curves to data from section 4.2
restart:
with(plots):
with(linalg):
We pick four common trigonometric functions as our set of functions
F:=A*sin(x)+B*cos(x)+C*sin(2*x)+E*cos(2*x);
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We pick four points, only one of which we know an exact value of sine and cosine for
pts:=[[0,Pi],[4,2],[7,9],[-1,3]];
NyY3JCIiIUkjUGlHJSpwcm90ZWN0ZWRHNyQiIiUiIiM3JCIiKCIiKjckISIiIiIk
simplify(subs({x=pts[1][1],y=pts[1][2]},F=y),trig);
LywmSSJCRzYiIiIiSSJFR0YlRiZJI1BpRyUqcHJvdGVjdGVkRw==
Sys:=seq(simplify(subs({x=pts[j][1],y=pts[j][2]},F=y),trig),j=1..4);
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Solving the system is straighforward....
Soln:=evalf(solve({Sys},{A,B,C,E}));
PCYvSSJBRzYiJCErM0N3Oz0hIiovSSJCR0YlJCIrNCJwJTR4RigvSSJDR0YlJCIrWGxZYF1GKC9JIkVHRiUkIStgayh5YyVGKA==
FSoln:=subs(Soln,F);
LCotSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKCQhKzNDdzs9ISIqLUkkY29zR0YlRikkIis0InAlNHhGLS1GJDYjLCRGKiIiIyQiK1hsWWBdRi0tRi9GMyQhK2BrKHljJUYt
plot1:=pointplot(pts,symbol=circle,symbolsize=30):
plot2:=plot(FSoln,x=-2..8):
Plot the data and the curve, clearly it works.
display({plot1,plot2});
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Now we choose points that we normally see more frequently when using trig functions, e.g. multiples of pi.
pts2:=[[0,1],[Pi/2,2],[Pi,-2],[3*Pi/2,1]];
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We plot our four functions and the points in question. Notice that the x values of these points correspond to zeros of many of these functions. In particular, sin(2x) is the blue curve, and note that sin(2x) is zero for all the x-coordinates of the points found in pts2 data set.
plot3:=plot({sin(x),cos(x),sin(2*x),cos(2*x)},x=0..2*Pi):
plot4:=pointplot(pts2,symbol=circle,symbolsize=30):
display({plot3,plot4});
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We expect there may be a problem with this system, let's see what happens. Notice our system is very 'sparse' to begin with.
Sys2:=seq(simplify(subs({x=pts2[j][1],y=pts2[j][2]},F=y),trig),j=1..4);
NiYvLCZJIkJHNiIiIiJJIkVHRiZGJ0YnLywmSSJBR0YmRidGKCEiIiIiIy8sJkYoRidGJUYsISIjLywmRitGLEYoRixGJw==
solve({Sys2},{A,B,C,E});
Sys2Matrix:=matrix(4,5,[0,1,0,1,1,1,0,0,-1,2,0,-1,0,1,-2,-1,0,0,-1,1]);
PTYiNiQ7IiIiIiIlO0YmIiImRVxbbDU2JEYmIiIjRiY2JEYnRiwiIiE2JEYsRixGLjYkRixGJkYmNiQiIiRGLCEiIjYkRidGKUYmNiRGJkYpRiY2JEYnRiZGMzYkRixGKUYsNiRGJ0YyRi42JEYmRiZGLjYkRjJGMkYuNiRGMkYnRiY2JEYmRidGJjYkRjJGKSEiIzYkRiZGMkYuNiRGLEYyRi42JEYyRiZGLjYkRixGJ0YzNiRGJ0YnRjM=
The rref matrix of the augmented matrix corresponding to our system. We have serious problems! This means there is NO linear combination of the four given functions which will pass through all four of the points.
rref(Sys2Matrix);
PTYiNiQ7IiIiIiIlO0YmIiImRVxbbDU2JEYmIiIjIiIhNiRGJ0YsRi02JEYsRixGJjYkRixGJkYtNiQiIiRGLEYtNiRGJ0YpRiY2JEYmRilGLTYkRidGJkYtNiRGLEYpRi02JEYnRjJGLTYkRiZGJkYmNiRGMkYyRi02JEYyRidGJjYkRiZGJ0YtNiRGMkYpRi02JEYmRjJGLTYkRixGMkYtNiRGMkYmRi02JEYsRidGLTYkRidGJ0Yt
We modify the set of points only slightly. We change the x-coordinate of the third point to pi/6 instead of pi, which will yield 2 more non-zero entries in the augmented matrix.
pts3:=[[0,1],[Pi/2,2],[Pi/6,-2],[3*Pi/2,1]];
NyY3JCIiISIiIjckLCRJI1BpRyUqcHJvdGVjdGVkRyNGJSIiI0YrNyQsJEYoI0YlIiInISIjNyQsJEYoIyIiJEYrRiU=
Sys3:=seq(simplify(subs({x=pts3[j][1],y=pts3[j][2]},F=y),trig),j=1..4);
NiYvLCZJIkJHNiIiIiJJIkVHRiZGJ0YnLywmSSJBR0YmRidGKCEiIiIiIy8sKkYrI0YnRi0qJkkiQ0dGJkYnIiIkRjBGMEYoRjAqJkYlRidGM0YwRjAhIiMvLCZGK0YsRihGLEYn
soln3:=solve({Sys3},{A,B,C,E});
PCYvSSJBRzYiIyIiIiIiIy9JIkJHRiUjIiImRigvSSJDR0YlLCYqJCIiJEYmISIiIyEiJkYoRicvSSJFR0YlIyEiJEYo
FSoln3:=subs(soln3,F);
LCotSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKCMiIiIiIiMtSSRjb3NHRiVGKSMiIiZGLSomLCYqJCIiJEYrISIiIyEiJkYtRixGLC1GJDYjLCRGKkYtRixGLC1GL0Y6IyEiJEYt
plot5:=pointplot(pts3,symbol=circle,symbolsize=30):
plot6:=plot(FSoln3,x=-Pi/2..2*Pi,color=blue):
display({plot5,plot6});
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
We try this again, but we remove only one zero from the matrix (replace pi/6 by pi/4 in point 3's x-coordinate). Will we still get a solution?
pts4:=[[0,1],[Pi/2,2],[Pi/4,-2],[3*Pi/2,1]];
NyY3JCIiISIiIjckLCRJI1BpRyUqcHJvdGVjdGVkRyNGJSIiI0YrNyQsJEYoI0YlIiIlISIjNyQsJEYoIyIiJEYrRiU=
Sys4:=seq(simplify(subs({x=pts4[j][1],y=pts4[j][2]},F=y),trig),j=1..4);
NiYvLCZJIkJHNiIiIiJJIkVHRiZGJ0YnLywmSSJBR0YmRidGKCEiIiIiIy8sJComRi0jRidGLSwoRitGJyomSSJDR0YmRidGLUYxRidGJUYnRidGMSEiIy8sJkYrRixGKEYsRic=
soln4:=solve({Sys4},{A,B,C,E});
PCYvSSJBRzYiIyIiIiIiIy9JIkJHRiUjIiImRigvSSJDR0YlLCYqJEYoRiYjISIkRighIiNGJy9JIkVHRiVGMQ==
The answer is yes!!! Will that always be the answer. Definitely not, but in this case it works.
FSoln4:=subs(soln4,F);
LCotSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKCMiIiIiIiMtSSRjb3NHRiVGKSMiIiZGLSomLCYqJEYtRisjISIkRi0hIiNGLEYsLUYkNiMsJEYqRi1GLEYsLUYvRjlGNQ==
plot7:=pointplot(pts4,symbol=circle,symbolsize=30):
plot8:=plot(FSoln4,x=-Pi/2..2*Pi,color=green):
display({plot7,plot8});
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