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# RecurrenceCoefficients.jl

Compute recurrence coefficients for Chebyshev polynomials defined on multiple intervals.

This respository contains the code for: arxiv.org/abs/2302.12930

A basic example usage follows.

``````bands = [-3.0 -2.0; 2.0 3.0] #intervals of support
(a,b) = get_coeffs(bands,5) #computes coefficients a₅,b₅
(avec,bvec) = get_n_coeffs(bands,5) #computes coefficients a₀,…,a₅,b₀,…,b₅
``````

Optional arguments include the choice of Chebyshev variant, a weight function perturbation, and the number of collocation points used. Example usage of these optional arguments follows.

``````bands = [-3.0 -2.0; 2.0 3.0]
h(j) = x->exp(0.01j*x)
nmat = [100*ones(size(bands,1)) 15*ones(size(bands,1))] .|> Int128
(avec,bvec) = get_n_coeffs(bands,5,"V",h;nmat) #computes coefficients a₀,…,a₅,b₀,…,b₅
``````

To use different Chebyshev variants on each interval, different functions must be called. The following example uses Chebyshev- \$T\$, \$U\$, \$V\$, and \$W\$ weights in that order.

``````bands = [-3.2 -2.2; 0.1 1.1; 2.0 3.0; 3.5 4.0]
typemat = [1 2 3 4]
(a,b) = get_coeffs_mixed(bands,5,typemat) #computes coefficients a₅,b₅
(avec,bvec) = get_n_coeffs_mixed(bands,5,typemat) #computes coefficients a₀,…,a₅,b₀,…,b₅
``````

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