Thanks to everyone for the replies. I found that the excellent "Chebyshev and Fourier Spectral Methods" by J P Boyd discusses computation of derivatives in one of the appendices (A), suggesting 3 practical methods: (i) use the variable transformation in terms of the cosine, (ii) use Gegenbauer polynomials, (iii) use the derivative matrix transformation (as suggested by Spencer Lyon below).
My use case is roughly the following: 1. calculate the Chebyshev polynomials and the 1st (and possibly 2nd) derivatives at a few (10-20) Chebyshev nodes, 2. use this to solve the functional equation (minimizing a residual, Newton's method, etc), 3. evaluate the residual on a denser grid. Since I only need to calculate the polynomials and their derivatives once for 1., and once for 3., the cost is trivial. Boyd gives a neat algorithm for method (i) above (Table 16.1, about 10 LOC, up to 4th derivative), so I went with that. IMO this is not worth optimizing further. AFAIK method (iii) with the derivative matrix can be slightly unstable numerically, there are quite a few papers on this. Also, I decided to wrap up the code I wrote in a library, which I will publish once I have experimented a bit with the interface and found it satisfactory. I also coded up the standard routines for infinite and semi-infinite domains to make solving PDEs less error prone. Best, Tamas On Mon, Apr 06 2015, Alan Edelman <mit.edel...@gmail.com> wrote: > there could be so many ways to do this > that it might be worth asking about the "use case" > > the derivatives, of course, satisfy the same three term recurrence > but with different initial conditions > > > so it might be worth knowing if > 0. whether this is worth optimizing a great deal or just a little bit or > you just want something > to do the job > 1. the degree of the original polynomial was relatively small > 2. whether one derivative in particular or a run of many derivatives were > desired > 3. whether one or many points were desired > 4. if the arguments were well inside or on [-1,1] > > > > > > > > > > On Saturday, April 4, 2015 at 11:40:53 PM UTC-4, Spencer Lyon wrote: >> >> Oh I forgot to say how to use it. If c are your coefficients for a degree >> n interpolant on [a, b] then the coefficients of the derivative of that >> interpolant are der_matrix(n, a, b) * c >> >> On Saturday, April 4, 2015 at 11:39:49 PM UTC-4, Spencer Lyon wrote: >> >> I have a function that will compute the derivative matrix operator that >>> transforms coefficients of a Chebyshev interpolant on interval [a, b] of >>> degree n (so n+1 total Chebyshev basis functions T_0, T_1, … T_n) into the >>> coefficients of a degree n-1 chebyshev interpolant on [a, b] that is the >>> exact derivative of the first interpolant. Maybe it could help you: >>> >>> # derivative matrix >>> function der_matrix(deg::Int, a::Real=-1, b::Real=1) >>> N = deg >>> D = zeros(Float64, N, N+1) >>> for i=1:N, j=1:N+1 >>> if i == 1 >>> if iseven(i + j) >>> continue >>> end >>> D[i, j] = 2*(j-1)/(b-a) >>> else >>> if j < i || iseven(i+j) >>> continue >>> end >>> D[i, j] = 4*(j-1)/(b-a) >>> end >>> end >>> >>> D >>> end >>> >>> On Thursday, April 2, 2015 at 10:38:48 AM UTC-4, Tamas Papp wrote: >>> >>> Hi, >>>> >>>> Can someone point me to some Julia code that calculates a matrix for the >>>> derivatives of the Chebyshev polynomials T_j, at given values, ie >>>> >>>> d^k T_i(x_j) / dx^k for i=1,..n, j for some and vector x. >>>> >>>> The Chebyshev polynomials themselves are very easy to calculate using >>>> the recurrence relation, but derivatives are not. Alternatively, maybe >>>> this can be extracted from ApproxFun but I have not found a way. >>>> >>>> Best, >>>> >>>> Tamas >>>> >>> >>> >> >>
