Would the use of cubic splines increase the number of data points to interpolate from result in smaller error...
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I am implementing the cubic spline method to interpolate the function:
$$f(x)=sin(x); -π≤0≤π$$
I am using this paper by Antony Jameson to construct my code, with a few modifications to correct the various typos on that paper. I am using uniformly spaced anchor points and natural spline boundary conditions, and I am breaking the range of $-π ≤ 0 ≤ π$ into $100$ splines/$101$ points.
When I set the number of data points to interpolate from to be $N=4$ and $N=8$ and compared it to the actual graph of the function, the resulting graph looked like this:
which is fine and dandy. However, when I set the number of data points to be $N=10$ and $N=16$, the result is not as good:
As we can see from the second graph, the interpolation produces zero values on the positive end of the graph when I input $10$ and $16$ data points. When I checked the value of my interpolation, I found that the last three spline-ends yield zeroes like this:
$$smallbegin{bmatrix}
-0.00000 & -0.06274 & -0.12524 & cdots & 0.24669 & 0.18695 & 0.00000 & 0.00000 & 0.00000\
end{bmatrix}$$
I have scoured the net and found something called Runge's phenomenon, and it seems that my implementation of the cubic spline has run into that phenomenon. However, when reading further, I have found that the phenomenon should not occur in cubic splines.
What am I doing wrong? Is there something wrong with my implementation, or is that phenomenon occurs to everyone applying the cubic spline to the sine function?
interpolation spline
asked Dec 8 '18 at 10:36
sagungrpsagungrp
11
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add a comment |
$begingroup$
I am implementing the cubic spline method to interpolate the function:
$$f(x)=sin(x); -π≤0≤π$$
I am using this paper by Antony Jameson to construct my code, with a few modifications to correct the various typos on that paper. I am using uniformly spaced anchor points and natural spline boundary conditions, and I am breaking the range of $-π ≤ 0 ≤ π$ into $100$ splines/$101$ points.
When I set the number of data points to interpolate from to be $N=4$ and $N=8$ and compared it to the actual graph of the function, the resulting graph looked like this:
which is fine and dandy. However, when I set the number of data points to be $N=10$ and $N=16$, the result is not as good:
As we can see from the second graph, the interpolation produces zero values on the positive end of the graph when I input $10$ and $16$ data points. When I checked the value of my interpolation, I found that the last three spline-ends yield zeroes like this:
$$smallbegin{bmatrix}
-0.00000 & -0.06274 & -0.12524 & cdots & 0.24669 & 0.18695 & 0.00000 & 0.00000 & 0.00000\
end{bmatrix}$$
I have scoured the net and found something called Runge's phenomenon, and it seems that my implementation of the cubic spline has run into that phenomenon. However, when reading further, I have found that the phenomenon should not occur in cubic splines.
What am I doing wrong? Is there something wrong with my implementation, or is that phenomenon occurs to everyone applying the cubic spline to the sine function?
interpolation spline
asked Dec 8 '18 at 10:36
sagungrpsagungrp
11
$endgroup$
$begingroup$
Are you using uniformly spaced anchor points? Natural spline boundary conditions? (In the latter case the sine function "cheats" because the real function satisfies these conditions, so also test with $cos x$.)
$endgroup$
– Oscar Lanzi
Dec 8 '18 at 10:45
$begingroup$
Yes, I'm using uniformly spaced anchor points and natural spline boundary conditions. I tried testing with $cos(x)$ a few seconds ago and it produces the same error--the last three spline-ends have zeroes as their value.
$endgroup$
– sagungrp
Dec 8 '18 at 10:51
$begingroup$
Are you using single precision floating number in your codes? If yes, try using double precision floating numbers.
$endgroup$
– fang
Dec 8 '18 at 22:07
add a comment |
$begingroup$
I am implementing the cubic spline method to interpolate the function:
$$f(x)=sin(x); -π≤0≤π$$
I am using this paper by Antony Jameson to construct my code, with a few modifications to correct the various typos on that paper. I am using uniformly spaced anchor points and natural spline boundary conditions, and I am breaking the range of $-π ≤ 0 ≤ π$ into $100$ splines/$101$ points.
When I set the number of data points to interpolate from to be $N=4$ and $N=8$ and compared it to the actual graph of the function, the resulting graph looked like this:
which is fine and dandy. However, when I set the number of data points to be $N=10$ and $N=16$, the result is not as good:
As we can see from the second graph, the interpolation produces zero values on the positive end of the graph when I input $10$ and $16$ data points. When I checked the value of my interpolation, I found that the last three spline-ends yield zeroes like this:
$$smallbegin{bmatrix}
-0.00000 & -0.06274 & -0.12524 & cdots & 0.24669 & 0.18695 & 0.00000 & 0.00000 & 0.00000\
end{bmatrix}$$
I have scoured the net and found something called Runge's phenomenon, and it seems that my implementation of the cubic spline has run into that phenomenon. However, when reading further, I have found that the phenomenon should not occur in cubic splines.
What am I doing wrong? Is there something wrong with my implementation, or is that phenomenon occurs to everyone applying the cubic spline to the sine function?
interpolation spline
asked Dec 8 '18 at 10:36
sagungrpsagungrp
11
$endgroup$
I am implementing the cubic spline method to interpolate the function:
$$f(x)=sin(x); -π≤0≤π$$
I am using this paper by Antony Jameson to construct my code, with a few modifications to correct the various typos on that paper. I am using uniformly spaced anchor points and natural spline boundary conditions, and I am breaking the range of $-π ≤ 0 ≤ π$ into $100$ splines/$101$ points.
When I set the number of data points to interpolate from to be $N=4$ and $N=8$ and compared it to the actual graph of the function, the resulting graph looked like this:
which is fine and dandy. However, when I set the number of data points to be $N=10$ and $N=16$, the result is not as good:
As we can see from the second graph, the interpolation produces zero values on the positive end of the graph when I input $10$ and $16$ data points. When I checked the value of my interpolation, I found that the last three spline-ends yield zeroes like this:
$$smallbegin{bmatrix}
-0.00000 & -0.06274 & -0.12524 & cdots & 0.24669 & 0.18695 & 0.00000 & 0.00000 & 0.00000\
end{bmatrix}$$
I have scoured the net and found something called Runge's phenomenon, and it seems that my implementation of the cubic spline has run into that phenomenon. However, when reading further, I have found that the phenomenon should not occur in cubic splines.
What am I doing wrong? Is there something wrong with my implementation, or is that phenomenon occurs to everyone applying the cubic spline to the sine function?
interpolation spline
interpolation spline
asked Dec 8 '18 at 10:36
sagungrpsagungrp
11
asked Dec 8 '18 at 10:36
sagungrpsagungrp
11
edited Dec 8 '18 at 10:54
sagungrp
asked Dec 8 '18 at 10:36
sagungrpsagungrp
11
asked Dec 8 '18 at 10:36
sagungrpsagungrp
11
asked Dec 8 '18 at 10:36
sagungrpsagungrp
11
11
$begingroup$
Are you using uniformly spaced anchor points? Natural spline boundary conditions? (In the latter case the sine function "cheats" because the real function satisfies these conditions, so also test with $cos x$.)
$endgroup$
– Oscar Lanzi
Dec 8 '18 at 10:45
$begingroup$
Yes, I'm using uniformly spaced anchor points and natural spline boundary conditions. I tried testing with $cos(x)$ a few seconds ago and it produces the same error--the last three spline-ends have zeroes as their value.
$endgroup$
– sagungrp
Dec 8 '18 at 10:51
$begingroup$
Are you using single precision floating number in your codes? If yes, try using double precision floating numbers.
$endgroup$
– fang
Dec 8 '18 at 22:07
add a comment |
$begingroup$
Are you using uniformly spaced anchor points? Natural spline boundary conditions? (In the latter case the sine function "cheats" because the real function satisfies these conditions, so also test with $cos x$.)
$endgroup$
– Oscar Lanzi
Dec 8 '18 at 10:45
$begingroup$
Yes, I'm using uniformly spaced anchor points and natural spline boundary conditions. I tried testing with $cos(x)$ a few seconds ago and it produces the same error--the last three spline-ends have zeroes as their value.
$endgroup$
– sagungrp
Dec 8 '18 at 10:51
$begingroup$
Are you using single precision floating number in your codes? If yes, try using double precision floating numbers.
$endgroup$
– fang
Dec 8 '18 at 22:07
$begingroup$
Are you using uniformly spaced anchor points? Natural spline boundary conditions? (In the latter case the sine function "cheats" because the real function satisfies these conditions, so also test with $cos x$.)
$endgroup$
– Oscar Lanzi
Dec 8 '18 at 10:45
$begingroup$
Are you using uniformly spaced anchor points? Natural spline boundary conditions? (In the latter case the sine function "cheats" because the real function satisfies these conditions, so also test with $cos x$.)
$endgroup$
– Oscar Lanzi
Dec 8 '18 at 10:45
$begingroup$
Yes, I'm using uniformly spaced anchor points and natural spline boundary conditions. I tried testing with $cos(x)$ a few seconds ago and it produces the same error--the last three spline-ends have zeroes as their value.
$endgroup$
– sagungrp
Dec 8 '18 at 10:51
$begingroup$
Yes, I'm using uniformly spaced anchor points and natural spline boundary conditions. I tried testing with $cos(x)$ a few seconds ago and it produces the same error--the last three spline-ends have zeroes as their value.
$endgroup$
– sagungrp
Dec 8 '18 at 10:51
$begingroup$
Are you using single precision floating number in your codes? If yes, try using double precision floating numbers.
$endgroup$
– fang
Dec 8 '18 at 22:07
$begingroup$
Are you using single precision floating number in your codes? If yes, try using double precision floating numbers.
$endgroup$
– fang
Dec 8 '18 at 22:07
add a comment |
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$begingroup$
Are you using uniformly spaced anchor points? Natural spline boundary conditions? (In the latter case the sine function "cheats" because the real function satisfies these conditions, so also test with $cos x$.)
$endgroup$
– Oscar Lanzi
Dec 8 '18 at 10:45
$begingroup$
Yes, I'm using uniformly spaced anchor points and natural spline boundary conditions. I tried testing with $cos(x)$ a few seconds ago and it produces the same error--the last three spline-ends have zeroes as their value.
$endgroup$
– sagungrp
Dec 8 '18 at 10:51
$begingroup$
Are you using single precision floating number in your codes? If yes, try using double precision floating numbers.
$endgroup$
– fang
Dec 8 '18 at 22:07