Interpolation

Julia code: click here.

# Julia code
# See A3.jl

Log utility

Interpolation Log Fn

Square root utility

Interpolation Square Root Fn

CRRA utility

\(\sigma = 2\)

Interpolation CRRA sigma=2 Fn

\(\sigma = 5\)

Interpolation CRRA sigma=5 Fn

\(\sigma = 10\)

Interpolation CRRA sigma=10 Fn

Summary of interpolation errors

I’m using the Eucledian Norm of the differences between the original function and the interpolation method. Specifically \[ || f(x)-q(x) ||^2 = \left[ \sum_i \left( f(x_i)-q(x_i) \right)^2 \right]^{\frac{1}{2}} \]

Log	n=4	n=6	n=11	n=21
Newton Basis Polynomials	\(6.286\)	\(2.28\)	\(0.4082\)	\(0.03642\)
Natural Cubic Splines	\(8.375\)	\(4.379\)	\(1.606\)	\(0.4928\)
Shape-preserving Schumaker Splines	\(1.176\)	\(0.533\)	\(0.1543\)	\(0.03563\)

Square Root	n=4	n=6	n=11	n=21
Newton Basis Polynomials	\(0.579\)	\(0.166\)	\(0.02202\)	\(0.001462\)
Natural Cubic Splines	\(0.8726\)	\(0.3989\)	\(0.1245\)	\(0.03344\)
Shape-preserving Schumaker Splines	\(0.098\)	\(0.03925\)	\(0.00975\)	\(0.001983\)

CES \(\sigma = 2\)	n=4	n=6	n=11	n=21
Newton Basis Polynomials	\(98.2\)	\(48.55\)	\(13.68\)	\(2.012\)
Natural Cubic Splines	\(117.2\)	\(73.55\)	\(34.44\)	\(13.26\)
Shape-preserving Schumaker Splines	\(30.79\)	\(16.31\)	\(5.628\)	\(1.481\)

CES \(\sigma = 5\)	n=4	n=6	n=11	n=21
Newton Basis Polynomials	\(3.195 \cdot 10^{5}\)	\(2.074 \cdot 10^{5}\)	\(1.024 \cdot 10^{5}\)	\(3.559 \cdot 10^{4}\)
Natural Cubic Splines	\(3.592 \cdot 10^{5}\)	\(2.647 \cdot 10^{5}\)	\(1.655 \cdot 10^{5}\)	\(9.166 \cdot 10^{4}\)
Shape-preserving Schumaker Splines	\(1.419 \cdot 10^{5}\)	\(9.976 \cdot 10^{4}\)	\(5.497 \cdot 10^{4}\)	\(2.357 \cdot 10^{4}\)

CES \(\sigma = 10\)	n=4	n=6	n=11	n=21
Newton Basis Polynomials	\(4.805 \cdot 10^{11}\)	\(3.295 \cdot 10^{11}\)	\(1.908 \cdot 10^{11}\)	\(9.752 \cdot 10^{10}\)
Natural Cubic Splines	\(5.35 \cdot 10^{11}\)	\(4.067 \cdot 10^{11}\)	\(2.748 \cdot 10^{11}\)	\(1.768 \cdot 10^{11}\)
Shape-preserving Schumaker Splines	\(2.214 \cdot 10^{11}\)	\(1.661 \cdot 10^{11}\)	\(1.08 \cdot 10^{11}\)	\(6.367 \cdot 10^{10}\)