# uniform knot vector for splines

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SA-W el 12 de Abr. de 2024
Editada: Bruno Luong el 13 de Abr. de 2024
Suppose we have uniform interpolation points, e.g
x = [1 2 3 4 5 6]
x = 1x6
1 2 3 4 5 6
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The functions for knot generation produce knots of the form
t = aptknt(x,5)
t = 1x11
1.0000 1.0000 1.0000 1.0000 1.0000 3.5000 6.0000 6.0000 6.0000 6.0000 6.0000
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where the first and last knot is repeated k times, but the breaks are no longer uniform. Similar for other knot functions like optknt.
I would like to carry over the property of equal distance between the interp. points to the knots. The knot vector
t = 1 1 1 2 3 4 5 6 6 6 6
satisfies this, as well as the Schoenberg-Whitney conditions. I arbitrarily repeated the first knot 3 times and the last knot four times.
Is such a knot vector plausible and why does MATLAB not offer functions for uniform knot sequences?
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Bruno Luong el 12 de Abr. de 2024
Editada: Bruno Luong el 12 de Abr. de 2024
Your knot sequence seems NOT to be suitable for interpolation. The interpolation matrix is singular.
x = linspace(1,6,6);
xi = linspace(min(x),max(x),61);
k = 5;
k1 = floor(k/2);
k2 = k-k1;
tSAW = [repelem(x(1),1,k1) x repelem(x(end),1,k2)]
tSAW = 1x11
1 1 1 2 3 4 5 6 6 6 6
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t = aptknt(x,k);
for p=0:k-1
y = x.^p;
sSAW = spapi(tSAW,x,y); % Not workingn coefs are NaN
s = spapi(t,x,y);
subplot(3,2,p+1);
yiSAW = fnval(sSAW, xi);
yi = fnval(s, xi);
plot(x, y, 'ro', xi, yiSAW, 'b+', xi, yi, 'g-') % Blue curve not plotted since yiSAW are NaN
end
Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN.
Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN.
Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN.
Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN.
Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN.
It seems Schoenberg-Whitney conditions are violated despite what you have claimed.
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Bruno Luong el 13 de Abr. de 2024
Editada: Bruno Luong el 13 de Abr. de 2024
"That said, there is no way to have uniform knot breaks on the interpolation domain?"
What is the purpose of it? What if your x is non unform to start with, an assumption you never spell out.
You could chose
x = 1:6;
k = 5;
n = length(x);
dt =(max(x)-min(x))/(n-k+1);
teq = min(x) + dt*(-k+1:n)
teq = 1x11
-9.0000 -6.5000 -4.0000 -1.5000 1.0000 3.5000 6.0000 8.5000 11.0000 13.5000 16.0000
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OK = SWtest(x, teq)
OK = logical
1
figure
hold on
for i = 1:n
si = spmak(teq, accumarray(i, 1, [n 1])');
fnplt(si);
end
xlim([min(x) max(x)]);
or
dx = mean(diff(x));
dt = dx;
a = (n+k-1)/2;
teqx = mean(x) + dt*(-a:a)
teqx = 1x11
-1.5000 -0.5000 0.5000 1.5000 2.5000 3.5000 4.5000 5.5000 6.5000 7.5000 8.5000
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OK = SWtest(x, teqx)
OK = logical
1
figure
hold on
for i = 1:n
si = spmak(teqx, accumarray(i, 1, [n 1])');
fnplt(si);
end
xlim([min(x) max(x)]);
SA-W el 13 de Abr. de 2024
Editada: Bruno Luong el 13 de Abr. de 2024
What is the purpose of it? What if your x is non unform to start with, an assumption you never spell out.
I know that my interpolant is barely sampled at some regions and I can circumvent this with the positioning of interpolation points x. For instance, when I do csape(x,y), the interpolation segments are determined by x, whereas for spapi and friends, the knot vector determiens the interpolation segments and I only have semi-control.
Anyway, if x is uniform, the knot vector aptknt(x,k) is uniform as well except at the boundary. So i can live with that. I was just wondering if there are alternatives and you provided some.

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