Incorporate Steven's comments #2
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stevengj
reviewed
Dec 24, 2016
| gauss([T,] N) | ||
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| Return a pair `(x, w)` of `N` quadrature points `x[i]` and weights `w[i]` to | ||
| integrate functions on the interval `(-1, 1)`, i.e. `dot(f(x), w)` |
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dot(w, f.(x)), so that you don't conjugate f. Or sum(w .* f.(x)) is probably more natural here.
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`T` is an optional parameter specifying the floating-point type, defaulting to `Float64`.
Arbitrary precision (`BigFloat`) is also supported.
stevengj
reviewed
Dec 24, 2016
| Return a pair `(x, w)` of `N` quadrature points `x[i]` and weights `w[i]` to | ||
| integrate functions on the interval `(-1, 1)`, i.e. `dot(f(x), w)` | ||
| approximates the integral. Uses the method described in Trefethen & | ||
| Bau, Numerical Linear Algebra, to find the `N`-point Gaussian quadrature. |
stevengj
reviewed
Dec 24, 2016
| Since the rule is symmetric, this only returns the `n+1` points with `x <= 0`. | ||
| The function Also computes the embedded `n`-point Gauss quadrature weights `gw` | ||
| (again for `x <= 0`), corresponding to the points `x[2:2:end]`. Returns `(x,w,wg)`. | ||
| """ |
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Given these points and weights, the estimated integral `I` and error `E` can be computed for an
integrand `f(x)` as follows:
x,w,wg = kronrod(n)
fx⁰ = f(x[end]) # f(0)
x⁻ = x[1:end-1] # the x < 0 Kronrod points
fx = f.(x⁻) .+ f.((-).(x⁻)) # f(x < 0) + f(x > 0)
I = sum(fx .* w[1:end-1]) + fx⁰ * w[end]
if isodd(n)
E = abs(sum(fx[2:2:end] .* wg[1:end-1]) + fx⁰*wg[end] - I)
else
E = abs(sum(fx[2:2:end] .* wg[1:end])- I)
end
Also, should have a similar comment to above explaining T and noting that it is O(N^2).
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See also my suggestion on the |
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How are things looking now? |
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Shouldn't |
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Good call, thanks! |
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LGTM |
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Fantastic. Thanks for all your help! |
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I'll put all of the requested changes into this PR so that it can be tagged once this is merged.
Fixes #1