5 ms·

>The reason why the Apollo team chose these terms instead of the Taylor ones was because these were actually optimized for minimum error in the interval −π/2<y<

by prvc 5y ago

>The reason why the Apollo team chose these terms instead of the Taylor ones was because these were actually optimized for minimum error in the interval −π/2<y<π/2 How would these coefficients be derived in the first place?

vegesm 5y ago
There is a blogpost linked in the annotations: http://jeanmariechauvet.com/papers/curttrig.html http://jeanmariechauvet.com/papers/curttrig.html
- throwaway744678 5y ago
  Here are the graphs [0] for both functions from the article: + 1/2 * sin(x * pi / 2) + 0.785313 * x - 0.321615 x^3 + 0.0363551 x^5 [0] https://www.desmos.com/calculator/x7tm7i2qtj https://www.desmos.com/calculator/x7tm7i2qtj
  - deleted 5y ago
    [deleted]
  - hexomancer 5y ago
    Also here is the same graph with the classic taylor expansion of sin with 3 terms (x + (x^3)/(3!) + (x^5)/(5!)): https://www.desmos.com/calculator/emfz64hjhv https://www.desmos.com/calculator/emfz64hjhv
    - jacobolus 5y ago
      Showing relative error: https://www.desmos.com/calculator/a3whptxkqu https://www.desmos.com/calculator/a3whptxkqu
deleted 5y ago
[deleted]
jacobolus 5y ago
They used the Remez exchange algorithm to find the polynomial with minimax relative error. https://en.wikipedia.org/wiki/Remez_algorithm https://en.wikipedia.org/wiki/Remez_algorithm If you want to play with this sort of thing and you have access to a copy of Matlab, check out Chebfun, https://www.chebfun.org/examples/approx/ https://www.chebfun.org/examples/approx/
martincmartin 5y ago
We did this in my quantum mechanics class, 8.05x on edX. Consider the functions f_n(x) = x^n to be basis vectors in a vector space. Define the dot product of two vectors (aka functions) to be the integral of their product between -pi/2 and +pi/2. You'll see that your basis vectors aren't orthogonal. So, use Graham Schmidt to make them orthogonal, and normalize them while you're at it. Now you just take the dot product of these new basis vectors with your target function.
- klodolph 5y ago
  Worth noting that this minimizes expected error. IMO, it's much more common to want to minimize maximum (worst-case) error, which requires an algorithm like Remez exchange. You can modify this algorithm to work with relative error (as suggested in the article) instead of absolute error.
- mangamadaiyan 5y ago
  Nitpick: Gram-Schmidt.
  - Syzygies 5y ago
    If one understands linear algebra (and has Haskell output of matrices with nice eigenvalues), then reviewing this spelling is the only preparation needed to teach a semester course.
madcaptenor 5y ago
Also, something doesn't add up here because they were calculating sin(pi*y/2). I suspect it's optimized for -1 < y < 1, so the argument to sin is in [-pi/2, pi/2].
- deleted 5y ago
  [deleted]

by prvc 5y ago

vegesm 5y ago

There is a blogpost linked in the annotations: http://jeanmariechauvet.com/papers/curttrig.html http://jeanmariechauvet.com/papers/curttrig.html

throwaway744678 5y ago
Here are the graphs [0] for both functions from the article: + 1/2 * sin(x * pi / 2) + 0.785313 * x - 0.321615 x^3 + 0.0363551 x^5 [0] https://www.desmos.com/calculator/x7tm7i2qtj https://www.desmos.com/calculator/x7tm7i2qtj
- deleted 5y ago
  [deleted]
- hexomancer 5y ago
  Also here is the same graph with the classic taylor expansion of sin with 3 terms (x + (x^3)/(3!) + (x^5)/(5!)): https://www.desmos.com/calculator/emfz64hjhv https://www.desmos.com/calculator/emfz64hjhv
  - jacobolus 5y ago
    Showing relative error: https://www.desmos.com/calculator/a3whptxkqu https://www.desmos.com/calculator/a3whptxkqu

deleted 5y ago

[deleted]

jacobolus 5y ago

They used the Remez exchange algorithm to find the polynomial with minimax relative error. https://en.wikipedia.org/wiki/Remez_algorithm https://en.wikipedia.org/wiki/Remez_algorithm If you want to play with this sort of thing and you have access to a copy of Matlab, check out Chebfun, https://www.chebfun.org/examples/approx/ https://www.chebfun.org/examples/approx/

martincmartin 5y ago

We did this in my quantum mechanics class, 8.05x on edX. Consider the functions f_n(x) = x^n to be basis vectors in a vector space. Define the dot product of two vectors (aka functions) to be the integral of their product between -pi/2 and +pi/2. You'll see that your basis vectors aren't orthogonal. So, use Graham Schmidt to make them orthogonal, and normalize them while you're at it. Now you just take the dot product of these new basis vectors with your target function.

klodolph 5y ago
Worth noting that this minimizes expected error. IMO, it's much more common to want to minimize maximum (worst-case) error, which requires an algorithm like Remez exchange. You can modify this algorithm to work with relative error (as suggested in the article) instead of absolute error.
mangamadaiyan 5y ago
Nitpick: Gram-Schmidt.
- Syzygies 5y ago
  If one understands linear algebra (and has Haskell output of matrices with nice eigenvalues), then reviewing this spelling is the only preparation needed to teach a semester course.

madcaptenor 5y ago

Also, something doesn't add up here because they were calculating sin(pi*y/2). I suspect it's optimized for -1 < y < 1, so the argument to sin is in [-pi/2, pi/2].

deleted 5y ago
[deleted]