2023-04-12 Integration

Last time

• Reflect on differentiation

• Second order (Newton type) optimization

• Project discussion

Today

• Midpoint and trapezoid rules

• Extrapolation

• Polynomial interpolation for integration

using LinearAlgebra
using Plots
default(linewidth=4, legendfontsize=12)

function vander_legendre(x, k=nothing)
    if isnothing(k)
        k = length(x) # Square by default
    end
    m = length(x)
    Q = ones(m, k)
    Q[:, 2] = x
    for n in 1:k-2
        Q[:, n+2] = ((2*n + 1) * x .* Q[:, n+1] - n * Q[:, n]) / (n + 1)
    end
    Q
end

CosRange(a, b, n) = (a + b)/2 .+ (b - a)/2 * cos.(LinRange(-pi, 0, n))

CosRange (generic function with 1 method)

Integration

We’re interested in computing definite integrals

\[ \int_a^b f(x) dx \]

and will usually consider finite domains \(-\infty < a <b < \infty\).

• Cost: (usually) how many times we need to evaluate the function \(f(x)\)

• Accuracy

  • compare to a reference value

  • compare to the same method using more evaluations

• Consideration: how smooth is \(f\)?

We need some test functions

F_expx(x) = exp(2x) / (1 + x^2)
f_expx(x) = 2*exp(2x) / (1 + x^2) - 2x*exp(2x)/(1 + x^2)^2

F_dtanh(x) = tanh(x)
f_dtanh(x) = cosh(x)^-2

integrands = [f_expx, f_dtanh]
antiderivatives = [F_expx, F_dtanh]
tests = zip(integrands, antiderivatives)

zip(Function[f_expx, f_dtanh], Function[F_expx, F_dtanh])

plot(integrands, xlims=(-1, 1))

Fundamental Theorem of Calculus

Let \(f(x)\) be a continuous function and define \(F(x)\) by

\[ F(x) = \int_a^x f(s) ds . \]

Then \(F(x)\) is uniformly continuous, differentiable, and

\[ F'(x) = f(x) . \]

We say that \(F\) is an antiderivative of \(f\). This implies that

\[ \int_a^b f(x) dx = F(b) - F(a) . \]

We will test the accuracy of our integration schemes using an antiderivative provided in our tests.

Method of Manufactured Solutions

• Analytically integrating an arbitrary function is hard

  • tends to require trickery

  • not always possible to express in closed form (e.g., elliptic integrals)

  • sometimes needs special functions \(\operatorname{erf} x = \frac{2}{\sqrt\pi} \int_0^x e^{-t^2} dt\)

  • don’t know when to give up

• Analytic differentation

  • involves straightforward application of the product rule and chain rule.

So if we just choose an arbitrary function \(F\) (the antiderivative), we can

1. compute \(f = F'\)

2. numerically integrate \(\int_a^b f\) and compare to \(F(b) - F(a)\)

Newton-Cotes methods

Approximate \(f(x)\) using piecewise polynomials (an interpolation problem) and integrate the polynomials.

Midpoint method

function fint_midpoint(f, a, b; n=20)
    dx = (b - a) / n
    x = LinRange(a + dx/2, b - dx/2, n)
    sum(f.(x)) * dx
end

for (f, F) in tests
    a, b = -2, 2
    I_num = fint_midpoint(f, a, b, n=20)
    I_analytic = F(b) - F(a)
    println("$f: $I_num error=$(I_num - I_analytic)")
end

f_expx: 10.885522849146847 error=-0.03044402970425253
f_dtanh: 1.9285075531458646 error=0.00045239299423083246

How does the accuracy change as we use more points?

function plot_accuracy(fint, tests, ns; ref=[1,2])
    a, b = -2, 2
    p = plot(xscale=:log10, yscale=:log10, xlabel="n", ylabel="error")
    for (f, F) in tests
        Is = [fint(f, a, b, n=n) for n in ns]
        Errors = abs.(Is .- (F(b) - F(a)))
        scatter!(ns, Errors, label=f)
    end
    for k in ref
        plot!(ns, ns.^(-1. * k), label="\$n^{-$k}\$")
    end
    p
end
plot_accuracy(fint_midpoint, tests, 2 .^ (0:10))

Trapezoid Rule

The trapezoid rule uses piecewise linear functions on each interval.

\[\begin{split}\begin{split} \int_a^b \left[ f(a) + \frac{f(b) - f(a)}{b - a} (x - a) \right] &= f(a) (x-a) + \frac{f(b) - f(a)}{2(b - a)} (x - a)^2 \Big|_{x=a}^b \\ &= f(a) (b-a) + \frac{f(b) - f(a)}{2(b - a)} (b-a)^2 \\ &= \frac{b-a}{2} \big( f(a) + f(b) \big) . \end{split} \end{split}\]

• Can you get to the same result using a geometric argument?

• What happens when we sum over a bunch of adjacent intervals?

Trapezoid in code

function fint_trapezoid(f, a, b; n=20)
    dx = (b - a) / (n - 1)
    x = LinRange(a, b, n)
    fx = f.(x)
    fx[1] /= 2
    fx[end] /= 2
    sum(fx) * dx
end

plot_accuracy(fint_trapezoid, tests, 2 .^ (1:10))

Extrapolation

Let’s switch our plot around to use \(h = \Delta x\) instead of number of points \(n\).

function plot_accuracy_h(fint, tests, ns; ref=[1,2])
    a, b = -2, 2
    p = plot(xscale=:log10, yscale=:log10, xlabel="h", ylabel="error",
        legend=:bottomright)
    hs = (b - a) ./ ns
    for (f, F) in tests
        Is = [fint(f, a, b, n=n) for n in ns]
        Errors = abs.(Is .- (F(b) - F(a)))
        scatter!(hs, Errors, label=f)
    end
    for k in ref
        plot!(hs, .1 * hs.^k, label="\$h^{$k}\$")
    end
    p
end

plot_accuracy_h (generic function with 1 method)

plot_accuracy_h(fint_midpoint, tests, 2 .^ (2:10))

Extrapolation math

The trapezoid rule with \(n\) points has an interval spacing of \(h = 1/(n-1)\). Let \(I_h\) be the value of the integral approximated using an interval \(h\). We have numerical evidence that the leading error term is \(O(h^2)\), i.e.,

\[ I_h - I_0 = c h^2 + O(h^3) \]

for some as-yet unknown constant \(c\) that will depend on the function being integrated and the domain of integration. If we can determine \(c\) from two approximations, say \(I_h\) and \(I_{2h}\), then we can extrapolate to \(h=0\). For sufficiently small \(h\), we can neglect \(O(h^3)\) and write

\[\begin{split}\begin{split} I_h - I_0 &= c h^2 \\ I_{2h} - I_0 &= c (2h)^2 . \end{split}\end{split}\]

Subtracting these two lines, we have

\[ I_{h} - I_{2h} = c (h^2 - 4 h^2) \]

which can be solved for \(c\) as

\[ c = \frac{I_{h} - I_{2h}}{h^2 - 4 h^2} . \]

Substituting back into the first equation, we solve for \(I_0\) as

\[ I_0 = I_h - c h^2 = I_h + \frac{I_{h} - I_{2h}}{4 - 1} .\]

This is called Richardson extrapolation.

Extrapolation code

function fint_richardson(f, a, b; n=20)
    n = div(n, 2) * 2 + 1
    h = (b - a) / (n - 1)
    x = LinRange(a, b, n)
    fx = f.(x)
    fx[[1, end]] /= 2
    I_h = sum(fx) * h
    I_2h = sum(fx[1:2:end]) * 2h
    I_h + (I_h - I_2h) / 3
end
plot_accuracy_h(fint_richardson, tests, 2 .^ (2:10), ref=1:5)

• we now have a sequence of accurate approximations

• it’s possible to apply extrapolation recursively

• works great if you have a power of 2 number of points

  • and your function is nice enough

F_sin(x) = sin(30x)
f_sin(x) = 30*cos(30x)
plot_accuracy_h(fint_richardson, zip([f_sin], [F_sin]), 2 .^ (2:10), ref=1:5)

Polynomial interpolation for integration

x = LinRange(-1, 1, 100)
P = vander_legendre(x, 10)
plot(x, P, legend=:none)

Idea

• Sample the function \(f(x)\) at some points \(x \in [-1, 1]\)

• Fit a polynomial through those points

• Return the integral of that interpolating polynomial

Question

• What points do we sample on?

• How do we integrate the interpolating polynomial?

Recall that the Legendre polynomials \(P_0(x) = 1\), \(P_1(x) = x\), …, are pairwise orthogonal

\[\int_{-1}^1 P_m(x) P_n(x) = 0, \quad \forall m\ne n.\]

Integration using Legendre polynomials

function plot_accuracy_n(fint, tests, ns; ref=[1,2])
    a, b = -2, 2
    p = plot(xscale=:log10, yscale=:log10, xlabel="n", ylabel="error",
        legend=:bottomright)
    for (f, F) in tests
        Is = [fint(f, a, b, n=n) for n in ns]
        Errors = abs.(Is .- (F(b) - F(a)))
        scatter!(ns, Errors, label=f)
    end
    p
end

plot_accuracy_n (generic function with 1 method)

function fint_legendre(f, a, b; n=20)
    x = CosRange(-1, 1, n)
    P = vander_legendre(x)
    x_ab = (a+b)/2 .+ (b-a)/2*x
    c = P \ f.(x_ab)
    (b - a) * c[1]
end

fint_legendre(x -> 1 + x, -1, 1, n=2)

2.0

p = plot_accuracy_h(fint_legendre, tests, 4:20, ref=1:5)