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I am really stuck here. Here is the question that I have.

Find the cubic near minimax approximation for $f(x)=\sin(x)$ on $(0,\pi/2)$.

So I defined $h(x)=ax^3 + bx^2 + cx + d - sin(x)$

The max of this occurs at $dh/dx=0$, so that means

$$\frac{dh}{dx} = 3ax^2 + 2bx + c - \cos(x) = 0.$$

Now I am stuck. I wanted to solve this for $x$, and then find $h(0), \; h(\text{whatever } x \text{ gives me from the equation above})$, and $h(\pi/2)$. I would then use these solutions to solve for my coefficients. However, I cannot solve $dh/dx = 0$ for $x$. Not even wolfram can do it, it is too computationally intensive. Can someone please help me?

Thank you for your time!

Mark Fantini
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kingneptune117
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  • You need to make a system of equations that simultaneously solves for your $a,b,c,d$, your error, and the two points where $h$ and $f$ agree within the interval. – Alexander Gruber Apr 25 '14 at 02:44
  • I think that "near minimax approximation" refers to the interpolation through Chebyshev points - see for example page 46 of this: http://www.math.byu.edu/~jeffh/computing/na.pdf –  Apr 25 '14 at 02:47
  • Could I do that by finding h(0) and h(pi/2)? That would give me two equations that should be equal to each other. Then I have the equation above for dh/dx=0. Thats three equations, not enough. Could you give me a hand with setting up that system? – kingneptune117 Apr 25 '14 at 02:48
  • @NotNotLogical yes I think you are right. I will see what I can do. – kingneptune117 Apr 25 '14 at 02:52

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