JacobiRoots

Get the roots of shifted Legendre and Radau polynomials

>>> shiftedLegendreRoots 3
Just (fromList [0.11270166537925831,0.5,0.8872983346207417])
>>> shiftedRadauRoots 2
Just (fromList [0.1550510257216822,0.6449489742783178])

The roots are pre-computed and only a finite number are provided

>>> (V.length allShiftedLegendreRoots, V.length allShiftedRadauRoots)
(301,301)
>>> shiftedLegendreRoots 1000000000000000
Nothing

There are N roots in the Nth Jacobi polynomial

>>> import Data.Maybe ( fromJust )
>>> V.length (fromJust (shiftedLegendreRoots 5))
5
>>> all (\k -> k == V.length (fromJust (shiftedLegendreRoots k))) [0..(V.length allShiftedLegendreRoots - 1)]
True
>>> all (\k -> k == V.length (fromJust (shiftedRadauRoots k))) [0..(V.length allShiftedRadauRoots - 1)]
True

Documentation

shiftedLegendreRoots :: Int -> Maybe (Vector Double)

get the roots of the Nth shifted Legendre polynomial

 shiftedLegendreRoots == (allShiftedLegendreRoots !?)

>>> mapM_ (print . shiftedLegendreRoots) [0..3]
Just (fromList [])
Just (fromList [0.5])
Just (fromList [0.2113248654051871,0.7886751345948129])
Just (fromList [0.11270166537925831,0.5,0.8872983346207417])

shiftedRadauRoots :: Int -> Maybe (Vector Double)

get the roots of the Nth shifted Radau polynomial

 shiftedRadauRoots == (allShiftedRadauRoots !?)

>>> mapM_ (print . shiftedRadauRoots) [0..3]
Just (fromList [])
Just (fromList [0.3333333333333333])
Just (fromList [0.1550510257216822,0.6449489742783178])
Just (fromList [8.858795951270394e-2,0.4094668644407347,0.787659461760847])

allShiftedLegendreRoots :: Vector (Vector Double)

roots of shifted Jacobi polynomials with alpha=0, beta=0

allShiftedRadauRoots :: Vector (Vector Double)

roots of shifted Jacobi polynomials with alpha=1, beta=0