Calculus on the reference triangle
In this example, we sample a bivariate function:
\[f(x,y) = \frac{1}{1+x^2+y^2},\]
on the reference triangle with vertices $(0,0)$, $(0,1)$, and $(1,0)$ and analyze it in a Proriol series. Then, we find Proriol series for each component of its gradient by term-by-term differentiation of our expansion, and we compare them with the true Proriol series by sampling an exact expression for the gradient.
We analyze $f(x,y)$ on an $N\times M$ mapped tensor product grid defined by:
\[\begin{aligned}
x & = (1+u)/2,\quad{\rm and}\quad y = (1-u)(1+v)/4,\quad {\rm where:}\\
u_n & = \cos\left[(n+\tfrac{1}{2})\pi/N\right],\quad{\rm for}\quad 0\le n < N,\quad{\rm and}\\
v_m & = \cos\left[(m+\tfrac{1}{2})\pi/M\right],\quad{\rm for}\quad 0\le m < M;
\end{aligned}\]
we convert the function samples to mapped Chebyshev² coefficients using plan_tri_analysis; and finally, we transform the mapped Chebyshev² coefficients to Proriol coefficients using plan_tri2cheb.
For the storage pattern of the arrays, please consult the documentation.
using FastTransforms, LinearAlgebra, Plots
const GENFIGS = joinpath(pkgdir(FastTransforms), "docs/src/generated")
!isdir(GENFIGS) && mkdir(GENFIGS)
plotlyjs()Plots.PlotlyJSBackend()Our function $f$ and the Cartesian components of its gradient:
f = (x,y) -> 1/(1+x^2+y^2)
fx = (x,y) -> -2x/(1+x^2+y^2)^2
fy = (x,y) -> -2y/(1+x^2+y^2)^2#5 (generic function with 1 method)The polynomial degree:
N = 15
M = N15The parameters of the Proriol series:
α, β, γ = 0, 0, 0(0, 0, 0)The $u$ grid:
u = [sinpi((N-2n-1)/(2N)) for n in 0:N-1]15-element Vector{Float64}:
0.9945218953682733
0.9510565162951536
0.8660254037844386
0.7431448254773942
0.5877852522924731
0.4067366430758002
0.20791169081775934
0.0
-0.20791169081775934
-0.4067366430758002
-0.5877852522924731
-0.7431448254773942
-0.8660254037844386
-0.9510565162951536
-0.9945218953682733And the $v$ grid:
v = [sinpi((M-2m-1)/(2M)) for m in 0:M-1]15-element Vector{Float64}:
0.9945218953682733
0.9510565162951536
0.8660254037844386
0.7431448254773942
0.5877852522924731
0.4067366430758002
0.20791169081775934
0.0
-0.20791169081775934
-0.4067366430758002
-0.5877852522924731
-0.7431448254773942
-0.8660254037844386
-0.9510565162951536
-0.9945218953682733Instead of using the $u\times v$ grid, we use one with more accuracy near the origin. Defining $x$ by:
x = [sinpi((2N-2n-1)/(4N))^2 for n in 0:N-1]15-element Vector{Float64}:
0.9972609476841365
0.9755282581475768
0.9330127018922194
0.8715724127386971
0.7938926261462365
0.7033683215379002
0.6039558454088797
0.4999999999999999
0.3960441545911204
0.2966316784620999
0.2061073738537634
0.12842758726130288
0.06698729810778066
0.024471741852423214
0.0027390523158633317And $w$ by:
w = [sinpi((2M-2m-1)/(4M))^2 for m in 0:M-1]15-element Vector{Float64}:
0.9972609476841365
0.9755282581475768
0.9330127018922194
0.8715724127386971
0.7938926261462365
0.7033683215379002
0.6039558454088797
0.4999999999999999
0.3960441545911204
0.2966316784620999
0.2061073738537634
0.12842758726130288
0.06698729810778066
0.024471741852423214
0.0027390523158633317We see how the two grids are related:
((1 .+ u)./2 ≈ x) * ((1 .- u).*(1 .+ v')/4 ≈ reverse(x).*w')trueOn the mapped tensor product grid, our function samples are:
F = [f(x[n+1], x[N-n]*w[m+1]) for n in 0:N-1, m in 0:M-1]15×15 Matrix{Float64}:
0.50137 0.50137 0.50137 0.50137 0.50137 0.50137 0.501371 0.501371 0.501371 0.501371 0.501371 0.501371 0.501371 0.501371 0.501371
0.512229 0.512236 0.512249 0.512266 0.512286 0.512308 0.512328 0.512346 0.512361 0.512372 0.512379 0.512383 0.512385 0.512385 0.512386
0.53334 0.533395 0.533499 0.53364 0.533806 0.533979 0.534145 0.534292 0.534412 0.5345 0.534558 0.534592 0.534607 0.534612 0.534613
0.56305 0.563274 0.563699 0.564281 0.564961 0.565675 0.566362 0.56697 0.567464 0.56783 0.568072 0.568211 0.568275 0.568295 0.568299
0.597903 0.598554 0.599792 0.601491 0.603486 0.60559 0.607622 0.609427 0.6109 0.611994 0.612719 0.613134 0.613325 0.613387 0.613397
0.632017 0.633527 0.636406 0.640382 0.645085 0.650086 0.654955 0.659315 0.662898 0.665571 0.66735 0.66837 0.668842 0.668995 0.669018
0.657568 0.660489 0.666088 0.673894 0.683237 0.693308 0.703247 0.712263 0.719753 0.725392 0.729168 0.731342 0.73235 0.732678 0.732728
0.667275 0.672082 0.681371 0.694488 0.710446 0.727971 0.745606 0.761905 0.775667 0.786165 0.79326 0.79737 0.799283 0.799904 0.799999
0.65806 0.664903 0.678251 0.69738 0.721111 0.74777 0.775252 0.801255 0.82368 0.841081 0.85299 0.859943 0.863194 0.864252 0.864414
0.632907 0.641519 0.658477 0.683152 0.714388 0.750331 0.788365 0.825306 0.857936 0.883766 0.901708 0.912284 0.917254 0.918876 0.919123
0.599054 0.608911 0.628482 0.657352 0.694582 0.738395 0.78593 0.833301 0.876165 0.910799 0.935231 0.94978 0.956655 0.958904 0.959246
0.564342 0.574908 0.59603 0.627532 0.668777 0.718232 0.773048 0.82891 0.880558 0.923076 0.953504 0.971796 0.980486 0.983334 0.983768
0.534691 0.545579 0.567447 0.600326 0.643856 0.69679 0.756419 0.818253 0.876403 0.924997 0.960184 0.981503 0.991676 0.995016 0.995526
0.513598 0.524591 0.546739 0.58021 0.624846 0.679621 0.74199 0.80742 0.869666 0.922224 0.960591 0.983966 0.995154 0.998833 0.999394
0.502741 0.513754 0.535975 0.569641 0.614694 0.670229 0.733797 0.800871 0.865052 0.919526 0.959458 0.983854 0.99555 0.999397 0.999985We superpose a surface plot of $f$ on top of the grid:
X = [x for x in x, w in w]
Y = [x[N-n]*w[m+1] for n in 0:N-1, m in 0:M-1]
scatter3d(vec(X), vec(Y), vec(0F); markersize=0.75, markercolor=:blue)
surface!(X, Y, F; legend=false, xlabel="x", ylabel="y", zlabel="f")
savefig(joinpath(GENFIGS, "proriol.html"))"/home/runner/work/FastTransforms.jl/FastTransforms.jl/docs/src/generated/proriol.html"We precompute a Proriol–Chebyshev² plan:
P = plan_tri2cheb(F, α, β, γ)FastTransforms Proriol--Chebyshev² plan for 15×15-element array of Float64And an FFTW Chebyshev² plan on the triangle:
PA = plan_tri_analysis(F)FastTransforms plan for FFTW Chebyshev analysis on the triangle for 15×15-element array of Float64Its Proriol-$(α,β,γ)$ coefficients are:
U = P\(PA*F)15×15 Matrix{Float64}:
1.53694 -0.193325 -0.0251445 0.0109626 -0.000813095 -0.000299751 8.30397e-5 -1.59722e-6 -3.2141e-6 6.1667e-7 3.19775e-8 -3.21495e-8 4.17144e-9 6.97174e-10 -3.06613e-10
-0.111616 0.0391775 -0.00636098 -0.000168172 0.000311296 -5.50434e-5 -2.99195e-6 2.97964e-6 -4.20688e-7 -5.4534e-8 2.85785e-8 -2.74994e-9 -7.77992e-10 2.56775e-10 -1.20763e-11
-0.0620411 0.00717668 0.00153132 -0.000551474 4.1134e-5 1.40597e-5 -4.21186e-6 2.00844e-7 1.36869e-7 -3.34625e-8 3.38022e-10 1.36685e-9 -2.6459e-10 -1.30929e-11 1.45495e-11
0.0111316 -0.0036578 0.00043735 4.52913e-5 -2.72385e-5 3.78477e-6 3.57508e-7 -2.26663e-7 2.88459e-8 4.02905e-9 -1.98431e-9 2.01016e-10 4.97746e-11 -1.48758e-11 -2.03146e-12
0.00194132 -7.70184e-5 -0.000145948 3.68556e-5 -1.34273e-6 -1.26813e-6 3.06047e-7 -7.57685e-9 -1.11605e-8 2.45432e-9 -1.63544e-12 -1.03205e-10 9.8773e-12 5.45517e-12 -6.15774e-13
-0.000642637 0.000243768 -2.28339e-5 -5.91216e-6 2.28045e-6 -2.36988e-7 -4.55496e-8 1.90448e-8 -1.91046e-9 -4.16463e-10 1.54281e-10 1.00975e-11 -8.91873e-12 -7.74998e-13 5.33203e-13
-4.38389e-5 -1.71425e-5 1.23092e-5 -2.29813e-6 -6.48598e-8 1.17977e-7 -2.21226e-8 -4.58497e-10 1.02068e-9 -1.63471e-10 -4.64131e-11 6.91422e-12 3.5224e-12 -3.14298e-13 -2.28438e-13
2.79805e-5 -1.30453e-5 7.26152e-7 5.87656e-7 -1.71884e-7 1.12463e-8 5.13657e-9 -1.54461e-9 7.05239e-11 8.70592e-11 6.67156e-12 -5.87108e-12 -8.8146e-13 3.5065e-13 6.60112e-14
1.15026e-6 1.8828e-6 -8.87898e-7 1.24893e-7 1.74103e-8 -1.00961e-8 1.44036e-9 1.53829e-10 -9.59941e-11 -3.49153e-11 3.35895e-12 3.178e-12 -1.72628e-14 -2.20905e-13 -4.41417e-15
-1.05261e-6 6.07061e-7 9.40225e-9 -4.93564e-8 1.14023e-8 -2.56371e-10 -4.79206e-10 1.50921e-11 6.48843e-11 1.18558e-11 -4.29767e-12 -1.50757e-12 2.19175e-13 1.20461e-13 -1.21213e-14
-1.0706e-7 -1.2895e-7 5.63778e-8 -5.46716e-9 -1.98245e-9 6.62772e-10 1.96009e-10 -4.05866e-11 -3.72009e-11 -3.44512e-12 3.09917e-12 6.84678e-13 -1.9369e-13 -6.2004e-14 1.20543e-14
4.44232e-8 -2.74042e-8 -3.41194e-9 3.7536e-9 -3.47875e-10 -3.34332e-10 -7.92618e-11 2.84843e-11 1.87606e-11 8.64628e-13 -1.73542e-12 -2.9854e-13 1.16624e-13 2.95553e-14 -7.5397e-15
1.13247e-8 6.98157e-9 -3.2324e-9 -1.06114e-10 1.55319e-10 1.35282e-10 2.8768e-11 -1.28905e-11 -7.59282e-12 -1.99176e-13 7.32664e-13 1.12739e-13 -5.05664e-14 -1.1679e-14 3.30654e-15
-2.57523e-9 1.14806e-9 1.12793e-10 2.51422e-11 -4.12541e-11 -3.49672e-11 -7.18293e-12 3.43673e-12 1.96276e-12 3.98594e-14 -1.91744e-13 -2.85196e-14 1.33316e-14 2.99912e-15 -8.74775e-16
-7.03699e-10 -8.33285e-12 -8.53391e-12 -1.88826e-12 3.13558e-12 2.65053e-12 5.42528e-13 -2.61311e-13 -1.4878e-13 -2.93192e-15 1.45531e-14 2.15661e-15 -1.01287e-15 -2.27009e-16 6.63378e-17Similarly, our function's gradient samples are:
Fx = [fx(x[n+1], x[N-n]*w[m+1]) for n in 0:N-1, m in 0:M-1]15×15 Matrix{Float64}:
-0.501366 -0.501366 -0.501366 -0.501367 -0.501367 -0.501368 -0.501368 -0.501369 -0.501369 -0.501369 -0.501369 -0.501369 -0.501369 -0.50137 -0.50137
-0.511916 -0.511929 -0.511955 -0.51199 -0.51203 -0.512073 -0.512114 -0.51215 -0.512179 -0.512201 -0.512215 -0.512223 -0.512227 -0.512228 -0.512228
-0.530794 -0.530903 -0.53111 -0.531392 -0.531721 -0.532067 -0.532398 -0.532691 -0.532929 -0.533105 -0.533222 -0.533288 -0.533319 -0.533329 -0.53333
-0.552621 -0.553061 -0.553896 -0.555039 -0.556378 -0.557787 -0.559142 -0.560343 -0.56132 -0.562044 -0.562524 -0.562798 -0.562924 -0.562965 -0.562972
-0.567613 -0.568852 -0.571206 -0.574447 -0.578264 -0.582304 -0.586217 -0.589705 -0.59256 -0.594683 -0.596093 -0.5969 -0.597274 -0.597395 -0.597413
-0.561915 -0.564604 -0.569746 -0.576887 -0.585391 -0.594503 -0.603442 -0.611504 -0.618167 -0.623163 -0.626498 -0.628414 -0.629303 -0.629591 -0.629635
-0.522296 -0.526946 -0.535918 -0.548553 -0.563869 -0.580614 -0.59738 -0.612796 -0.625752 -0.635596 -0.64223 -0.646065 -0.647848 -0.648427 -0.648515
-0.445256 -0.451694 -0.464267 -0.482314 -0.504734 -0.529941 -0.555929 -0.580499 -0.60166 -0.618055 -0.629262 -0.635798 -0.638853 -0.639847 -0.639998
-0.343008 -0.350179 -0.364379 -0.385223 -0.411887 -0.442905 -0.476058 -0.508529 -0.537391 -0.560337 -0.576318 -0.585751 -0.590188 -0.591636 -0.591857
-0.237644 -0.244156 -0.257234 -0.276874 -0.302772 -0.334005 -0.368724 -0.404089 -0.436673 -0.463364 -0.482369 -0.49375 -0.499145 -0.500911 -0.501181
-0.14793 -0.152838 -0.162821 -0.178123 -0.198871 -0.22475 -0.254619 -0.286238 -0.316443 -0.341954 -0.360547 -0.371851 -0.377255 -0.37903 -0.379301
-0.0818036 -0.0848957 -0.0912482 -0.101149 -0.114882 -0.132501 -0.153497 -0.176483 -0.199161 -0.218858 -0.233525 -0.242571 -0.246928 -0.248365 -0.248585
-0.0383026 -0.0398784 -0.0431393 -0.0482832 -0.0555393 -0.0650468 -0.0766563 -0.0897011 -0.102903 -0.114631 -0.123518 -0.129064 -0.131754 -0.132643 -0.132779
-0.0129105 -0.013469 -0.0146303 -0.0164765 -0.0191091 -0.0226062 -0.0269458 -0.0319076 -0.0370169 -0.0416263 -0.0451619 -0.0473866 -0.0484703 -0.0488293 -0.0488842
-0.00138458 -0.00144591 -0.00157369 -0.0017776 -0.0020699 -0.0024608 -0.00294973 -0.00351363 -0.00409934 -0.00463189 -0.00504292 -0.00530263 -0.00542945 -0.0054715 -0.00547794and:
Fy = [fy(x[n+1], x[N-n]*w[m+1]) for n in 0:N-1, m in 0:M-1]15×15 Matrix{Float64}:
-0.00137327 -0.00134334 -0.0012848 -0.00120019 -0.00109322 -0.000968569 -0.000831675 -0.000688523 -0.000545372 -0.000408476 -0.00028382 -0.000176851 -9.22448e-5 -3.36988e-5 -3.77181e-6
-0.0128066 -0.0125278 -0.0119824 -0.0111941 -0.0101972 -0.00903524 -0.00775884 -0.0064238 -0.00508851 -0.00381138 -0.00264832 -0.00165022 -0.000860756 -0.000314451 -3.51956e-5
-0.0380049 -0.0371844 -0.0355776 -0.0332524 -0.0303075 -0.0268691 -0.0230859 -0.0191228 -0.0151537 -0.0113536 -0.00789053 -0.00491728 -0.00256498 -0.000937054 -0.000104882
-0.0812065 -0.0795001 -0.0761501 -0.0712824 -0.0650859 -0.0578104 -0.0497602 -0.0412837 -0.0327574 -0.0245665 -0.017084 -0.0106504 -0.00555645 -0.00203003 -0.000227217
-0.146958 -0.144069 -0.138361 -0.129983 -0.119184 -0.106332 -0.0919169 -0.0765485 -0.0609267 -0.0457968 -0.0318962 -0.0199018 -0.0103872 -0.00379541 -0.000424822
-0.236327 -0.232283 -0.224184 -0.212046 -0.195994 -0.176349 -0.153701 -0.128945 -0.103249 -0.0779568 -0.0544563 -0.0340361 -0.0177782 -0.00649768 -0.000727317
-0.341557 -0.337089 -0.327887 -0.313516 -0.293548 -0.267799 -0.236589 -0.200921 -0.162512 -0.123634 -0.0868006 -0.0544092 -0.028458 -0.0104055 -0.00116482
-0.444036 -0.44064 -0.433167 -0.420371 -0.400704 -0.372744 -0.335756 -0.290249 -0.238284 -0.183335 -0.129696 -0.0816541 -0.042795 -0.0156582 -0.00175299
-0.521645 -0.520944 -0.518445 -0.512009 -0.498657 -0.475067 -0.438456 -0.387746 -0.32456 -0.253471 -0.181141 -0.114718 -0.0602899 -0.0220791 -0.00247217
-0.561955 -0.56477 -0.56909 -0.572203 -0.569958 -0.557059 -0.528047 -0.479085 -0.410077 -0.325915 -0.235743 -0.15036 -0.0792838 -0.0290664 -0.00325507
-0.568241 -0.574301 -0.585148 -0.597987 -0.608136 -0.608908 -0.592332 -0.551271 -0.482733 -0.39071 -0.286235 -0.183948 -0.0973409 -0.0357278 -0.00400177
-0.553639 -0.562044 -0.577773 -0.598286 -0.618953 -0.632479 -0.629146 -0.598851 -0.535295 -0.440581 -0.326642 -0.211418 -0.112256 -0.0412478 -0.00462083
-0.532026 -0.541842 -0.560603 -0.586131 -0.614126 -0.637241 -0.644834 -0.624688 -0.567634 -0.473605 -0.354585 -0.230866 -0.122928 -0.0452109 -0.00506551
-0.513246 -0.523783 -0.544148 -0.572459 -0.604752 -0.633849 -0.648741 -0.635973 -0.584411 -0.492221 -0.371057 -0.242599 -0.129432 -0.0476343 -0.00533757
-0.502731 -0.513558 -0.534584 -0.564086 -0.5983 -0.630184 -0.648628 -0.639638 -0.591107 -0.500247 -0.378429 -0.247947 -0.132421 -0.0487506 -0.00546294For the partial derivative with respect to $x$, Olver et al. derive simple expressions for the representation of this component using a Proriol-$(α+1,β,γ+1)$ series.
Gx = zeros(Float64, N, M)
for m = 0:M-2
for n = 0:N-2
cf1 = m == 0 ? sqrt((n+1)*(n+2m+α+β+γ+3)/(2m+β+γ+2)*(m+γ+1)*8) : sqrt((n+1)*(n+2m+α+β+γ+3)/(2m+β+γ+1)*(m+β+γ+1)/(2m+β+γ+2)*(m+γ+1)*8)
cf2 = sqrt((n+α+1)*(m+1)/(2m+β+γ+2)*(m+β+1)/(2m+β+γ+3)*(n+2m+β+γ+3)*8)
Gx[n+1, m+1] = cf1*U[n+2, m+1] + cf2*U[n+1, m+2]
end
end
Px = plan_tri2cheb(Fx, α+1, β, γ+1)
Ux = Px\(PA*Fx)15×15 Matrix{Float64}:
-0.773299 0.0719368 0.0119032 -0.00401514 0.00019856 0.000113662 -2.54579e-5 -2.37734e-7 1.03154e-6 -1.60364e-7 -1.48715e-8 9.00448e-9 -8.97984e-10 -2.25712e-10 7.08345e-11
-0.223004 0.012725 0.00860848 -0.00187243 -6.32049e-5 8.83801e-5 -1.1938e-5 -1.70295e-6 7.84936e-7 -6.04351e-8 -2.29491e-8 6.45891e-9 -1.23274e-10 -2.55873e-10 5.02863e-11
0.11832 -0.0184961 -0.000231272 0.000708416 -0.000135183 -8.87531e-7 5.70658e-6 -1.08175e-6 -3.24208e-8 5.24395e-8 -8.50439e-9 -6.81366e-10 4.94759e-10 -5.56212e-11 -1.16264e-11
-0.00167066 0.00241796 -0.00105495 0.000134423 2.32113e-5 -1.10992e-5 1.1835e-6 2.55024e-7 -1.00326e-7 8.05316e-9 2.80938e-9 -8.7387e-10 3.23657e-11 3.71003e-11 -4.71068e-12
-0.00812992 0.00143193 9.5529e-5 -8.6497e-5 1.46322e-5 6.06568e-7 -7.52503e-7 1.2809e-7 6.41825e-9 -6.76798e-9 1.05309e-9 1.02809e-10 -6.38887e-11 -4.60392e-12 3.43243e-12
0.00134269 -0.000440563 9.61255e-5 -3.9763e-6 -4.07863e-6 1.16773e-6 -7.04875e-8 -3.76919e-8 1.09538e-8 -5.76688e-10 -3.72602e-10 6.21711e-11 2.34835e-11 -3.2882e-12 -1.02884e-12
0.000287063 -5.03055e-5 -1.79725e-5 8.218e-6 -1.05921e-6 -1.54886e-7 8.35392e-8 -1.11246e-8 -1.37121e-9 7.34787e-10 2.35493e-11 -4.77271e-11 -3.34076e-12 2.78949e-12 2.71211e-14
-0.000114154 3.91247e-5 -6.41411e-6 -4.57437e-7 4.91696e-7 -1.01409e-7 1.49415e-12 4.73486e-9 -8.87703e-10 -2.29361e-10 4.39281e-11 2.01683e-11 -2.11614e-12 -1.32214e-12 2.00992e-13
6.85033e-7 -1.42592e-6 2.06904e-6 -6.34246e-7 4.7282e-8 2.34694e-8 -7.6778e-9 5.5085e-10 4.78984e-10 2.64819e-11 -3.78611e-11 -5.1067e-12 2.54966e-12 4.06715e-13 -1.864e-13
5.33318e-6 -2.34997e-6 2.9001e-7 9.45302e-8 -4.55097e-8 7.14241e-9 5.63984e-10 -6.44351e-10 -2.12017e-10 3.18388e-11 2.34425e-11 -6.60806e-13 -1.88678e-12 -1.3375e-14 1.29399e-13
-7.178e-7 3.37971e-7 -1.76689e-7 3.81304e-8 -1.11295e-9 -2.48346e-9 1.24297e-10 4.4888e-10 8.62743e-11 -3.44107e-11 -1.28198e-11 1.79781e-12 1.16066e-12 -8.54664e-14 -7.76696e-14
-1.15794e-7 9.93748e-8 -5.08059e-9 -9.45944e-9 2.65499e-9 1.0741e-9 -1.63514e-10 -2.32795e-10 -3.2556e-11 2.03989e-11 5.96492e-12 -1.21808e-12 -5.75474e-13 6.48293e-14 3.81294e-14
3.9894e-8 -2.50603e-8 1.18665e-8 -3.69658e-10 -9.43959e-10 -3.54061e-10 7.05516e-11 8.25944e-11 9.98239e-12 -7.57296e-12 -2.02892e-12 4.70119e-13 2.00988e-13 -2.57424e-14 -1.32678e-14
-2.62946e-9 -2.09989e-9 -2.15583e-10 7.49659e-11 1.79606e-10 6.58835e-11 -1.40885e-11 -1.57067e-11 -1.81381e-12 1.45833e-12 3.81132e-13 -9.14684e-14 -3.80521e-14 5.0455e-15 2.50858e-15
-2.54665e-10 1.01738e-11 9.71511e-12 -3.41148e-12 -8.12007e-12 -2.97149e-12 6.40143e-13 7.10069e-13 8.15917e-14 -6.60158e-14 -1.72077e-14 4.14496e-15 1.71962e-15 -2.2876e-16 -1.13402e-16For the partial derivative with respect to y, the analogous formulae result in a Proriol-$(α,β+1,γ+1)$ series.
Gy = zeros(Float64, N, M)
for m = 0:M-2
for n = 0:N-2
Gy[n+1, m+1] = 4*sqrt((m+1)*(m+β+γ+2))*U[n+1, m+2]
end
end
Py = plan_tri2cheb(Fy, α, β+1, γ+1)
Uy = Py\(PA*Fy)15×15 Matrix{Float64}:
-1.09361 -0.246364 0.151902 -0.0145451 -0.00656722 0.00215264 -4.78099e-5 -0.00010909 2.3401e-5 1.34141e-6 -1.47744e-6 2.0787e-7 3.67354e-8 -1.65807e-8 1.30376e-9
0.221621 -0.0623246 -0.00233026 0.00556863 -0.00120594 -7.75602e-5 8.91904e-5 -1.42786e-5 -2.06939e-6 1.19894e-6 -1.26335e-7 -3.89258e-8 1.42118e-8 -5.60413e-10 -5.94789e-10
0.0405975 0.0150038 -0.00764145 0.000735827 0.000308032 -0.000109184 6.01193e-6 4.6455e-6 -1.26982e-6 1.41852e-8 6.28061e-8 -1.31763e-8 -5.9232e-10 7.38193e-10 -7.35368e-11
-0.0206916 0.00428514 0.000627575 -0.000487257 8.29203e-5 9.26771e-6 -6.78478e-6 9.79044e-7 1.52891e-7 -8.32462e-8 9.23683e-9 2.37699e-9 -9.33549e-10 -2.54744e-11 5.13027e-11
-0.000435682 -0.00142999 0.000510686 -2.40196e-5 -2.77834e-5 7.93359e-6 -2.26768e-7 -3.78776e-7 9.31367e-8 -7.2912e-11 -4.72764e-9 9.19194e-10 1.83342e-10 -5.07975e-11 -9.73934e-12
0.00137896 -0.000223726 -8.19213e-5 4.07942e-5 -5.19191e-6 -1.18071e-6 5.70025e-7 -6.48815e-8 -1.5805e-8 6.74933e-9 -5.22325e-10 -3.74577e-10 2.04346e-11 2.31629e-11 -1.45177e-12
-9.69728e-5 0.000120605 -3.18437e-5 -1.1605e-6 2.58441e-6 -5.73589e-7 -1.36421e-8 3.47148e-8 -6.95597e-9 -3.86259e-10 4.86604e-10 7.26716e-11 -3.26991e-11 -5.56081e-12 2.18954e-12
-7.37954e-5 7.11448e-6 8.14267e-6 -3.07439e-6 2.46945e-7 1.33361e-7 -4.63869e-8 3.49405e-9 1.69907e-9 -3.52262e-10 -2.0628e-10 1.07045e-11 1.73416e-11 -1.8863e-13 -1.24582e-12
1.06509e-5 -8.69913e-6 1.73077e-6 3.10766e-7 -2.22347e-7 3.70118e-8 3.9365e-9 -2.94133e-9 -2.1645e-10 2.94085e-10 6.42746e-11 -2.03559e-11 -6.99253e-12 1.20561e-12 5.53977e-13
3.43376e-6 9.14735e-8 -6.84306e-7 2.05405e-7 -4.3435e-9 -1.35246e-8 2.85641e-9 1.15843e-9 -9.56842e-11 -1.59766e-10 -1.45705e-11 1.36984e-11 2.42348e-12 -9.13606e-13 -2.1722e-13
-7.29088e-7 5.54006e-7 -7.49665e-8 -3.61409e-8 1.78364e-8 -4.16571e-10 -1.36673e-9 -4.10814e-10 8.7904e-11 6.99201e-11 1.76879e-12 -6.65703e-12 -7.49913e-13 4.67293e-13 7.67801e-14
-1.56131e-7 -3.60972e-8 5.21717e-8 -9.4804e-9 -1.42828e-9 2.71687e-10 4.84151e-10 1.24081e-10 -3.76632e-11 -2.38535e-11 1.8587e-13 2.38967e-12 2.02458e-13 -1.7174e-13 -2.30964e-14
4.09267e-8 -3.39594e-8 -3.06743e-9 7.64151e-10 3.1345e-10 -7.59843e-11 -1.14715e-10 -2.73663e-11 9.55255e-12 5.56757e-12 -1.22703e-13 -5.69904e-13 -4.18788e-14 4.1357e-14 5.08049e-15
8.57039e-9 6.52951e-9 -7.39973e-11 -9.29778e-11 -3.74097e-11 9.74901e-12 1.40429e-11 3.27131e-12 -1.19385e-12 -6.78321e-13 1.80885e-14 6.99172e-14 4.88939e-15 -5.08953e-15 -6.06728e-16
-2.86296e-9 1.21911e-13 2.03235e-12 2.54849e-12 1.02299e-12 -2.68873e-13 -3.8519e-13 -8.94733e-14 3.28268e-14 1.85955e-14 -5.06217e-16 -1.9183e-15 -1.33318e-16 1.39692e-16 1.65859e-17The $2$-norm relative error in differentiating the Proriol series for $f(x,y)$ term-by-term and its sampled gradient is:
hypot(norm(Ux-Gx), norm(Uy-Gy))/hypot(norm(Ux), norm(Uy))1.8769900773500217e-8This error can be improved upon by increasing $N$ and $M$.
This page was generated using Literate.jl.