MTH 343 · 4.4 Composite quadrature rules
Chop It Up
A song about Section 4.4: why splitting [a , b ] into small pieces usually beats a single Newton–Cotes formula with large n .
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♪ Intro
Verse 1
Took one Newton–Cotes and ran it a to b ,
Kept on crankin’ up the degree, thought that’d set me free,
But that polynomial went buckin’ like a bronco off the line,
Equal-spaced and high degree, it wiggles every time.
Chorus
So chop it up, chop it up, little pieces down the road,
Low degree on every one, now that’s a lighter load,
Want it closer? Don’t raise the degree, just make that h small,
More pieces, smaller error, that’s the best way of all.
♪ Instrumental
Verse 2
High-degree weights turn negative, and round-off starts to roar,
And the error needs a derivative you’ve never seen before,
But trapezoid and Simpson’s keep every weight positive and true,
Need just f double-prime or the fourth, and round-off won’t grow on you.
Chorus
So chop it up, chop it up, little pieces down the road,
Low degree on every one, now that’s a lighter load,
Want it closer? Don’t raise the degree, just make that h small,
More pieces, smaller error, that’s the best way of all.
♪ Outro
Try it yourself in the Composite Quadrature Rules Lab .