Quadratic Grid Examples¶
Example Code¶
Representative script:
"""IMSL QD family examples with plots.
Covers:
- QDVAL, QDDER
- QD2VL, QD2DR
- QD3VL, QD3DR
"""
from __future__ import annotations
from pathlib import Path
from typing import Any, Dict
import sys
import matplotlib.pyplot as plt
import numpy as np
REPO_ROOT = Path(__file__).resolve().parent.parent
if str(REPO_ROOT) not in sys.path:
sys.path.insert(0, str(REPO_ROOT))
from interpolation import (
quad_grid_derivative_1d,
quad_grid_derivative_2d,
quad_grid_derivative_3d,
quad_grid_evaluate_1d,
quad_grid_evaluate_2d,
quad_grid_evaluate_3d,
)
def run_demo_imsl_qd_family() -> Dict[str, Any]:
"""Run website-style demos for QD interpolation family.
Returns:
Dict[str, Any]: Computed values, error arrays, and SVG artifact paths.
"""
out: Dict[str, Any] = {}
out_dir = Path("test_output")
out_dir.mkdir(parents=True, exist_ok=True)
# QDVAL / QDDER ----------------------------------------------------------
x0_1d = -1.0
dx_1d = 0.25
x_nodes_1d = np.array([x0_1d + i * dx_1d for i in range(17)], dtype=float)
# Quadratic polynomial is reproduced exactly by local quadratic interpolation.
vals_1d = x_nodes_1d**2 + 2.0 * x_nodes_1d - 1.0
xq_1d = np.linspace(-0.8, 2.6, 28)
qdval = np.array([quad_grid_evaluate_1d(vals_1d, x0_1d, dx_1d, float(xq)) for xq in xq_1d], dtype=float)
qdder = np.array([quad_grid_derivative_1d(vals_1d, x0_1d, dx_1d, float(xq)) for xq in xq_1d], dtype=float)
exact_1d = xq_1d**2 + 2.0 * xq_1d - 1.0
exact_d1d = 2.0 * xq_1d + 2.0
out["QDVAL"] = {
"x": xq_1d,
"values": qdval,
"error": exact_1d - qdval,
}
out["QDDER"] = {
"x": xq_1d,
"values": qdder,
"error": exact_d1d - qdder,
}
fig1, (ax11, ax12) = plt.subplots(2, 1, figsize=(9, 7), sharex=True)
ax11.plot(xq_1d, exact_1d, "k--", linewidth=1.2, label="Exact")
ax11.plot(xq_1d, qdval, color="#1f77b4", linewidth=2.0, label="QDVAL")
ax11.set_ylabel("f(x)")
ax11.grid(True, alpha=0.3)
ax11.legend(loc="best")
ax12.plot(xq_1d, exact_d1d, "k--", linewidth=1.2, label="Exact df/dx")
ax12.plot(xq_1d, qdder, color="#ff7f0e", linewidth=2.0, label="QDDER")
ax12.set_xlabel("x")
ax12.set_ylabel("df/dx")
ax12.grid(True, alpha=0.3)
ax12.legend(loc="best")
fig1.suptitle("IMSL QDVAL / QDDER")
fig1.tight_layout()
path1 = out_dir / "demo_imsl_qd1_family.svg"
fig1.savefig(path1, format="svg")
plt.close(fig1)
out["qd1_plot_path"] = str(path1)
# QD2VL / QD2DR ----------------------------------------------------------
x0_2d, y0_2d = -1.0, -0.5
dx_2d, dy_2d = 0.25, 0.2
nx2, ny2 = 16, 14
x_nodes_2d = x0_2d + dx_2d * np.arange(nx2, dtype=float)
y_nodes_2d = y0_2d + dy_2d * np.arange(ny2, dtype=float)
xx2, yy2 = np.meshgrid(x_nodes_2d, y_nodes_2d, indexing="ij")
vals_2d = xx2**2 + yy2**2 + xx2 * yy2 + 2.0 * xx2 - 3.0 * yy2 + 1.0
xq2 = np.linspace(-0.6, 2.4, 19)
yq2 = np.linspace(-0.2, 1.8, 17)
qd2vl = np.array(
[[quad_grid_evaluate_2d(vals_2d, x0_2d, dx_2d, y0_2d, dy_2d, float(x), float(y)) for y in yq2] for x in xq2],
dtype=float,
)
qd2dr = np.array(
[[quad_grid_derivative_2d(vals_2d, x0_2d, dx_2d, y0_2d, dy_2d, float(x), float(y), dx_order=1, dy_order=1) for y in yq2] for x in xq2],
dtype=float,
)
exact_2d = np.array(
[[x**2 + y**2 + x * y + 2.0 * x - 3.0 * y + 1.0 for y in yq2] for x in xq2],
dtype=float,
)
exact_d2d = np.ones((xq2.size, yq2.size), dtype=float)
out["QD2VL"] = {
"grid_values": qd2vl,
"grid_error": exact_2d - qd2vl,
}
out["QD2DR"] = {
"grid_values": qd2dr,
"grid_error": exact_d2d - qd2dr,
}
fig2, (ax21, ax22) = plt.subplots(1, 2, figsize=(11, 4.5))
c21 = ax21.contourf(xq2, yq2, qd2vl.T, levels=14, cmap="viridis")
fig2.colorbar(c21, ax=ax21, label="QD2VL")
ax21.set_xlabel("x")
ax21.set_ylabel("y")
ax21.set_title("QD2VL")
c22 = ax22.contourf(xq2, yq2, qd2dr.T, levels=14, cmap="plasma")
fig2.colorbar(c22, ax=ax22, label="QD2DR (d2f/dxdy)")
ax22.set_xlabel("x")
ax22.set_ylabel("y")
ax22.set_title("QD2DR")
fig2.suptitle("IMSL QD2VL / QD2DR")
fig2.tight_layout()
path2 = out_dir / "demo_imsl_qd2_family.svg"
fig2.savefig(path2, format="svg")
plt.close(fig2)
out["qd2_plot_path"] = str(path2)
# QD3VL / QD3DR ----------------------------------------------------------
x0_3d, y0_3d, z0_3d = -1.0, -0.5, -0.25
dx_3d, dy_3d, dz_3d = 0.25, 0.2, 0.15
nx3, ny3, nz3 = 15, 13, 12
x_nodes_3d = x0_3d + dx_3d * np.arange(nx3, dtype=float)
y_nodes_3d = y0_3d + dy_3d * np.arange(ny3, dtype=float)
z_nodes_3d = z0_3d + dz_3d * np.arange(nz3, dtype=float)
xx3, yy3, zz3 = np.meshgrid(x_nodes_3d, y_nodes_3d, z_nodes_3d, indexing="ij")
vals_3d = (
xx3**2
+ yy3**2
+ zz3**2
+ xx3 * yy3
+ yy3 * zz3
+ xx3 * zz3
+ xx3
- 2.0 * yy3
+ 3.0 * zz3
)
xq3 = np.linspace(-0.6, 2.0, 10)
yq3 = np.linspace(-0.2, 1.6, 9)
zq3 = np.linspace(-0.05, 1.2, 8)
qd3vl = np.array(
[
[
[
quad_grid_evaluate_3d(
vals_3d,
x0_3d,
dx_3d,
y0_3d,
dy_3d,
z0_3d,
dz_3d,
float(x),
float(y),
float(z),
)
for z in zq3
]
for y in yq3
]
for x in xq3
],
dtype=float,
)
qd3dr = np.array(
[
[
[
quad_grid_derivative_3d(
vals_3d,
x0_3d,
dx_3d,
y0_3d,
dy_3d,
z0_3d,
dz_3d,
float(x),
float(y),
float(z),
dx_order=1,
dy_order=0,
dz_order=0,
)
for z in zq3
]
for y in yq3
]
for x in xq3
],
dtype=float,
)
exact_3d = np.array(
[
[
[
x**2 + y**2 + z**2 + x * y + y * z + x * z + x - 2.0 * y + 3.0 * z
for z in zq3
]
for y in yq3
]
for x in xq3
],
dtype=float,
)
exact_d3d = np.array(
[[[2.0 * x + y + z + 1.0 for z in zq3] for y in yq3] for x in xq3],
dtype=float,
)
out["QD3VL"] = {
"grid_values": qd3vl,
"grid_error": exact_3d - qd3vl,
}
out["QD3DR"] = {
"grid_values": qd3dr,
"grid_error": exact_d3d - qd3dr,
}
mid_k = zq3.size // 2
fig3, (ax31, ax32) = plt.subplots(1, 2, figsize=(11, 4.5))
c31 = ax31.contourf(xq3, yq3, qd3vl[:, :, mid_k].T, levels=12, cmap="magma")
fig3.colorbar(c31, ax=ax31, label=f"QD3VL (z={zq3[mid_k]:.2f})")
ax31.set_xlabel("x")
ax31.set_ylabel("y")
ax31.set_title("QD3VL Slice")
c32 = ax32.contourf(xq3, yq3, qd3dr[:, :, mid_k].T, levels=12, cmap="cividis")
fig3.colorbar(c32, ax=ax32, label=f"QD3DR d/dx (z={zq3[mid_k]:.2f})")
ax32.set_xlabel("x")
ax32.set_ylabel("y")
ax32.set_title("QD3DR Slice")
fig3.suptitle("IMSL QD3VL / QD3DR")
fig3.tight_layout()
path3 = out_dir / "demo_imsl_qd3_family.svg"
fig3.savefig(path3, format="svg")
plt.close(fig3)
out["qd3_plot_path"] = str(path3)
return out
if __name__ == "__main__":
run_demo_imsl_qd_family()
Input (Console)¶
Run the quadratic-grid scripts from the package root:
python examples/example_imsl_qd_family.py
Plot Output¶
Generated SVG plots:
Output Console¶
Summary console output:
script input setup representative numeric outputs
example_imsl_qd_family.py 1D: 17 nodes and 28 queries; 2D: 19x17 queries; 3D: 10x9x8 queries QD1 head = [-1.96, -1.893772290809328, -1.795829903978052]; QD1 max abs error = 1.77635683940025e-15; QD2 shape = 19x17 with max abs error = 5.329070518200751e-15; QD3 shape = 10x9x8 with max abs error = 7.105427357601002e-15
Detailed integration result tables:
QD 1D full parameter/result output
x value error
-0.8 -1.96 0
-0.674074074074074 -1.893772290809328 0
-0.5481481481481482 -1.795829903978052 0
-0.4222222222222222 -1.666172839506173 0
-0.2962962962962963 -1.50480109739369 0
-0.1703703703703703 -1.311714677640603 0
-0.0444444444444444 -1.086913580246914 2.220446049250313e-16
0.08148148148148149 -0.83039780521262 0
0.2074074074074075 -0.5421673525377227 0
0.3333333333333335 -0.2222222222222218 -5.551115123125783e-17
0.4592592592592595 0.1294375857338827 5.551115123125783e-17
0.5851851851851853 0.5128120713305901 0
0.7111111111111112 0.9279012345679017 1.110223024625157e-16
0.8370370370370372 1.374705075445817 2.220446049250313e-16
0.962962962962963 1.853223593964335 0
1.088888888888889 2.363456790123457 4.440892098500626e-16
1.214814814814815 2.905404663923183 4.440892098500626e-16
1.340740740740741 3.479067215363512 -4.440892098500626e-16
1.466666666666667 4.084444444444446 0
1.592592592592593 4.721536351165982 0
1.718518518518519 5.390342935528123 0
1.844444444444445 6.090864197530866 0
1.970370370370371 6.823100137174212 0
2.096296296296297 7.587050754458163 0
2.222222222222222 8.382716049382715 1.77635683940025e-15
2.348148148148148 9.210096021947875 -1.77635683940025e-15
2.474074074074075 10.06919067215364 1.77635683940025e-15
2.6 10.96 0
QD 2D full parameter/result output
i j value error
0 0 0.9199999999999999 2.220446049250313e-16
0 1 0.435625 1.110223024625157e-16
0 2 -0.01749999999999989 4.163336342344337e-17
0 3 -0.4393749999999998 0
0 4 -0.8299999999999998 0
0 5 -1.189375 -2.220446049250313e-16
0 6 -1.5175 0
0 7 -1.814375 0
0 8 -2.08 0
0 9 -2.314374999999999 -8.881784197001252e-16
0 10 -2.5175 -8.881784197001252e-16
0 11 -2.689375 -4.440892098500626e-16
0 12 -2.830000000000001 8.881784197001252e-16
0 13 -2.939375 0
0 14 -3.0175 0
0 15 -3.064375 0
0 16 -3.080000000000001 8.881784197001252e-16
1 0 1.047777777777778 2.220446049250313e-16
1 1 0.5842361111111112 0
1 2 0.1519444444444444 0
1 3 -0.2490972222222223 0
1 4 -0.6188888888888887 -1.110223024625157e-16
1 5 -0.9574305555555553 -2.220446049250313e-16
1 6 -1.264722222222222 0
1 7 -1.540763888888889 -4.440892098500626e-16
1 8 -1.785555555555555 -4.440892098500626e-16
1 9 -1.999097222222222 -6.661338147750939e-16
1 10 -2.181388888888888 -8.881784197001252e-16
1 11 -2.332430555555555 -4.440892098500626e-16
1 12 -2.452222222222222 0
1 13 -2.540763888888889 0
1 14 -2.598055555555556 0
1 15 -2.624097222222222 0
1 16 -2.61888888888889 4.440892098500626e-16
2 0 1.231111111111111 0
2 1 0.7884027777777779 -1.110223024625157e-16
2 2 0.3769444444444444 1.665334536937735e-16
2 3 -0.003263888888888893 1.595945597898663e-16
2 4 -0.3522222222222223 5.551115123125783e-17
2 5 -0.6699305555555556 1.110223024625157e-16
2 6 -0.956388888888889 0
2 7 -1.211597222222222 -4.440892098500626e-16
2 8 -1.435555555555555 -4.440892098500626e-16
2 9 -1.628263888888889 -4.440892098500626e-16
2 10 -1.789722222222222 -2.220446049250313e-16
2 11 -1.919930555555555 -4.440892098500626e-16
2 12 -2.018888888888889 0
2 13 -2.086597222222222 0
2 14 -2.123055555555555 0
2 15 -2.128263888888888 -8.881784197001252e-16
2 16 -2.102222222222222 0
3 0 1.47 4.440892098500626e-16
3 1 1.048125 0
3 2 0.6575 1.110223024625157e-16
3 3 0.2981250000000001 1.110223024625157e-16
3 4 -0.02999999999999993 1.249000902703301e-16
3 5 -0.326875 2.220446049250313e-16
3 6 -0.5925 0
3 7 -0.8268749999999999 -3.33066907387547e-16
3 8 -1.03 -2.220446049250313e-16
3 9 -1.201875 -2.220446049250313e-16
3 10 -1.3425 -4.440892098500626e-16
3 11 -1.451875 -6.661338147750939e-16
3 12 -1.53 -2.220446049250313e-16
3 13 -1.576875 -2.220446049250313e-16
3 14 -1.5925 -4.440892098500626e-16
3 15 -1.576875 -4.440892098500626e-16
3 16 -1.53 -4.440892098500626e-16
4 0 1.764444444444444 4.440892098500626e-16
4 1 1.363402777777778 0
4 2 0.9936111111111111 0
4 3 0.6550694444444445 0
4 4 0.3477777777777779 0
4 5 0.07173611111111111 0
4 6 -0.1730555555555557 -8.326672684688674e-17
4 7 -0.3865972222222223 -1.665334536937735e-16
4 8 -0.568888888888889 0
4 9 -0.7199305555555554 -5.551115123125783e-16
4 10 -0.8397222222222221 -3.33066907387547e-16
4 11 -0.9282638888888886 -4.440892098500626e-16
4 12 -0.9855555555555556 -1.110223024625157e-16
4 13 -1.011597222222222 0
4 14 -1.006388888888889 2.220446049250313e-16
4 15 -0.9699305555555556 -3.33066907387547e-16
4 16 -0.9022222222222226 1.110223024625157e-16
5 0 2.114444444444444 4.440892098500626e-16
5 1 1.734236111111111 -2.220446049250313e-16
5 2 1.385277777777777 2.220446049250313e-16
5 3 1.067569444444445 -2.220446049250313e-16
5 4 0.781111111111111 1.110223024625157e-16
5 5 0.5259027777777776 0
5 6 0.3019444444444442 0
5 7 0.1092361111111109 -2.220446049250313e-16
5 8 -0.05222222222222234 -2.706168622523819e-16
5 9 -0.1824305555555554 -6.661338147750939e-16
5 10 -0.2813888888888889 -4.996003610813204e-16
5 11 -0.3490972222222223 -1.110223024625157e-16
5 12 -0.3855555555555559 4.996003610813204e-16
5 13 -0.3907638888888892 -1.110223024625157e-16
5 14 -0.3647222222222224 5.551115123125783e-17
5 15 -0.3074305555555559 -4.440892098500626e-16
5 16 -0.2188888888888895 1.110223024625157e-16
6 0 2.52 4.440892098500626e-16
6 1 2.160625 0
6 2 1.8325 0
6 3 1.535625 0
6 4 1.27 0
6 5 1.035625 0
6 6 0.8325 0
6 7 0.660625 -2.220446049250313e-16
6 8 0.5200000000000001 -3.33066907387547e-16
6 9 0.4106250000000002 -6.661338147750939e-16
6 10 0.3325000000000002 -6.661338147750939e-16
6 11 0.285625 4.440892098500626e-16
6 12 0.2699999999999999 -3.33066907387547e-16
6 13 0.2856249999999999 -3.33066907387547e-16
6 14 0.3325000000000001 3.885780586188048e-16
6 15 0.4106249999999998 -2.220446049250313e-16
6 16 0.5199999999999997 -1.110223024625157e-16
7 0 2.98111111111111 8.881784197001252e-16
7 1 2.642569444444444 0
7 2 2.335277777777777 0
7 3 2.059236111111111 0
7 4 1.814444444444444 4.440892098500626e-16
7 5 1.600902777777777 0
7 6 1.41861111111111 2.220446049250313e-16
7 7 1.267569444444444 0
7 8 1.147777777777777 -2.220446049250313e-16
7 9 1.059236111111111 -8.881784197001252e-16
7 10 1.001944444444444 0
7 11 0.9759027777777769 0
7 12 0.9811111111111106 -6.661338147750939e-16
7 13 1.017569444444444 0
7 14 1.085277777777777 -2.220446049250313e-16
7 15 1.18423611111111 -2.220446049250313e-16
7 16 1.314444444444443 4.440892098500626e-16
8 0 3.497777777777777 4.440892098500626e-16
8 1 3.180069444444444 4.440892098500626e-16
8 2 2.893611111111111 0
8 3 2.638402777777777 4.440892098500626e-16
8 4 2.414444444444444 4.440892098500626e-16
8 5 2.221736111111111 0
8 6 2.060277777777777 0
8 7 1.930069444444444 -4.440892098500626e-16
8 8 1.831111111111111 -4.440892098500626e-16
8 9 1.763402777777777 0
8 10 1.726944444444444 2.220446049250313e-16
8 11 1.72173611111111 2.220446049250313e-16
8 12 1.747777777777778 -4.440892098500626e-16
8 13 1.805069444444445 -2.220446049250313e-16
8 14 1.893611111111111 4.440892098500626e-16
8 15 2.013402777777778 -4.440892098500626e-16
8 16 2.164444444444444 -8.881784197001252e-16
9 0 4.069999999999999 8.881784197001252e-16
9 1 3.773124999999999 8.881784197001252e-16
9 2 3.507499999999999 8.881784197001252e-16
9 3 3.273125 0
9 4 3.07 0
9 5 2.898125 0
9 6 2.7575 0
9 7 2.648125 -8.881784197001252e-16
9 8 2.57 0
9 9 2.523125 -8.881784197001252e-16
9 10 2.5075 -8.881784197001252e-16
9 11 2.523125 0
9 12 2.57 -8.881784197001252e-16
9 13 2.648125000000001 -1.332267629550188e-15
9 14 2.7575 -8.881784197001252e-16
9 15 2.898125 -8.881784197001252e-16
9 16 3.07 0
10 0 4.697777777777776 8.881784197001252e-16
10 1 4.421736111111111 0
10 2 4.176944444444444 -8.881784197001252e-16
10 3 3.963402777777777 0
10 4 3.78111111111111 4.440892098500626e-16
10 5 3.630069444444443 4.440892098500626e-16
10 6 3.510277777777777 0
10 7 3.42173611111111 0
10 8 3.364444444444444 -8.881784197001252e-16
10 9 3.338402777777778 -8.881784197001252e-16
10 10 3.343611111111111 -8.881784197001252e-16
10 11 3.380069444444444 -8.881784197001252e-16
10 12 3.447777777777776 4.440892098500626e-16
10 13 3.54673611111111 0
10 14 3.676944444444444 0
10 15 3.838402777777776 -4.440892098500626e-16
10 16 4.03111111111111 8.881784197001252e-16
11 0 5.381111111111111 8.881784197001252e-16
11 1 5.125902777777776 1.77635683940025e-15
11 2 4.901944444444444 0
11 3 4.70923611111111 1.77635683940025e-15
11 4 4.547777777777776 1.77635683940025e-15
11 5 4.417569444444444 0
11 6 4.318611111111112 -8.881784197001252e-16
11 7 4.250902777777778 0
11 8 4.214444444444445 -8.881784197001252e-16
11 9 4.209236111111112 -8.881784197001252e-16
11 10 4.235277777777778 0
11 11 4.292569444444444 0
11 12 4.381111111111111 0
11 13 4.500902777777779 -8.881784197001252e-16
11 14 4.651944444444446 -1.77635683940025e-15
11 15 4.834236111111111 8.881784197001252e-16
11 16 5.047777777777778 0
12 0 6.119999999999999 0
12 1 5.885624999999999 0
12 2 5.682499999999998 8.881784197001252e-16
12 3 5.510624999999999 0
12 4 5.369999999999998 8.881784197001252e-16
12 5 5.260624999999999 0
12 6 5.182499999999999 0
12 7 5.135624999999999 0
12 8 5.119999999999999 0
12 9 5.135624999999999 0
12 10 5.182499999999999 0
12 11 5.260624999999997 2.664535259100376e-15
12 12 5.369999999999999 0
12 13 5.510625000000001 -1.77635683940025e-15
12 14 5.6825 -8.881784197001252e-16
12 15 5.885624999999999 0
12 16 6.119999999999999 0
13 0 6.914444444444442 1.77635683940025e-15
13 1 6.700902777777776 0
13 2 6.518611111111108 1.77635683940025e-15
13 3 6.367569444444444 0
13 4 6.247777777777776 0
13 5 6.15923611111111 -8.881784197001252e-16
13 6 6.101944444444443 0
13 7 6.075902777777777 -8.881784197001252e-16
13 8 6.08111111111111 -8.881784197001252e-16
13 9 6.117569444444442 8.881784197001252e-16
13 10 6.185277777777776 0
13 11 6.284236111111108 1.77635683940025e-15
13 12 6.414444444444443 8.881784197001252e-16
13 13 6.575902777777777 -1.77635683940025e-15
13 14 6.768611111111111 -1.77635683940025e-15
13 15 6.992569444444444 0
13 16 7.247777777777776 0
14 0 7.764444444444441 0
14 1 7.571736111111109 0
14 2 7.410277777777775 0
14 3 7.280069444444442 -8.881784197001252e-16
14 4 7.181111111111108 8.881784197001252e-16
14 5 7.113402777777775 0
14 6 7.076944444444443 -8.881784197001252e-16
14 7 7.071736111111108 8.881784197001252e-16
14 8 7.097777777777775 0
14 9 7.155069444444445 -3.552713678800501e-15
14 10 7.243611111111108 -8.881784197001252e-16
14 11 7.363402777777774 1.77635683940025e-15
14 12 7.514444444444441 0
14 13 7.696736111111108 8.881784197001252e-16
14 14 7.910277777777774 -8.881784197001252e-16
14 15 8.155069444444443 -1.77635683940025e-15
14 16 8.431111111111111 -1.77635683940025e-15
15 0 8.67 0
15 1 8.498125 0
15 2 8.357499999999998 0
15 3 8.248124999999998 1.77635683940025e-15
15 4 8.169999999999998 1.77635683940025e-15
15 5 8.123124999999998 1.77635683940025e-15
15 6 8.1075 0
15 7 8.123125 0
15 8 8.17 0
15 9 8.248125000000002 -1.77635683940025e-15
15 10 8.357499999999998 1.77635683940025e-15
15 11 8.498124999999998 1.77635683940025e-15
15 12 8.67 0
15 13 8.873124999999998 1.77635683940025e-15
15 14 9.107499999999998 1.77635683940025e-15
15 15 9.373125 0
15 16 9.670000000000002 -1.77635683940025e-15
16 0 9.63111111111111 -1.77635683940025e-15
16 1 9.480069444444442 0
16 2 9.360277777777776 0
16 3 9.271736111111109 0
16 4 9.214444444444441 1.77635683940025e-15
16 5 9.188402777777776 0
16 6 9.193611111111109 0
16 7 9.230069444444442 0
16 8 9.297777777777776 0
16 9 9.39673611111111 -1.77635683940025e-15
16 10 9.526944444444442 0
16 11 9.688402777777776 0
16 12 9.88111111111111 -1.77635683940025e-15
16 13 10.10506944444444 0
16 14 10.36027777777778 0
16 15 10.64673611111111 -1.77635683940025e-15
16 16 10.96444444444444 0
17 0 10.64777777777777 0
17 1 10.51756944444444 1.77635683940025e-15
17 2 10.41861111111111 0
17 3 10.35090277777778 -1.77635683940025e-15
17 4 10.31444444444444 1.77635683940025e-15
17 5 10.30923611111111 -1.77635683940025e-15
17 6 10.33527777777777 0
17 7 10.39256944444444 -1.77635683940025e-15
17 8 10.48111111111111 1.77635683940025e-15
17 9 10.60090277777778 -3.552713678800501e-15
17 10 10.75194444444444 -1.77635683940025e-15
17 11 10.93423611111111 -1.77635683940025e-15
17 12 11.14777777777777 0
17 13 11.39256944444444 1.77635683940025e-15
17 14 11.66861111111111 -1.77635683940025e-15
17 15 11.97590277777777 0
17 16 12.31444444444444 3.552713678800501e-15
18 0 11.72 1.77635683940025e-15
18 1 11.610625 1.77635683940025e-15
18 2 11.5325 1.77635683940025e-15
18 3 11.485625 0
18 4 11.47 0
18 5 11.485625 0
18 6 11.5325 1.77635683940025e-15
18 7 11.610625 1.77635683940025e-15
18 8 11.72 1.77635683940025e-15
18 9 11.860625 1.77635683940025e-15
18 10 12.0325 1.77635683940025e-15
18 11 12.235625 0
18 12 12.47 1.77635683940025e-15
18 13 12.735625 1.77635683940025e-15
18 14 13.0325 0
18 15 13.360625 5.329070518200751e-15
18 16 13.72 1.77635683940025e-15
QD 3D full parameter/result output
i j k value error
0 0 0 0.2124999999999999 8.326672684688674e-17
0 0 1 0.6193877551020407 1.110223024625157e-16
0 0 2 1.090051020408163 4.440892098500626e-16
0 0 3 1.624489795918367 2.220446049250313e-16
0 0 4 2.222704081632653 0
0 0 5 2.884693877551021 0
0 0 6 3.610459183673469 8.881784197001252e-16
0 0 7 4.4 -8.881784197001252e-16
0 1 0 -0.4231250000000001 5.551115123125783e-17
0 1 1 0.02394132653061225 -4.85722573273506e-17
0 1 2 0.5347831632653062 0
0 1 3 1.109400510204082 0
0 1 4 1.747793367346939 -2.220446049250313e-16
0 1 5 2.449961734693878 0
0 1 6 3.215905612244897 8.881784197001252e-16
0 1 7 4.045624999999999 0
0 2 0 -0.9574999999999999 -1.110223024625157e-16
0 2 1 -0.4702551020408163 0
0 2 2 0.080765306122449 2.775557561562891e-17
0 2 3 0.6955612244897957 2.220446049250313e-16
0 2 4 1.374132653061225 0
0 2 5 2.116479591836735 -4.440892098500626e-16
0 2 6 2.922602040816326 4.440892098500626e-16
0 2 7 3.792499999999999 8.881784197001252e-16
0 3 0 -1.390625 0
0 3 1 -0.8632015306122449 0
0 3 2 -0.2720025510204083 2.220446049250313e-16
0 3 3 0.3829719387755102 0
0 3 4 1.10172193877551 0
0 3 5 1.884247448979592 -8.881784197001252e-16
0 3 6 2.730548469387755 -4.440892098500626e-16
0 3 7 3.640625 0
0 4 0 -1.7225 0
0 4 1 -1.154897959183674 2.220446049250313e-16
0 4 2 -0.5235204081632654 2.220446049250313e-16
0 4 3 0.1716326530612245 -2.498001805406602e-16
0 4 4 0.9305612244897958 -2.220446049250313e-16
0 4 5 1.753265306122449 -6.661338147750939e-16
0 4 6 2.639744897959183 4.440892098500626e-16
0 4 7 3.59 0
0 5 0 -1.953125 4.440892098500626e-16
0 5 1 -1.345344387755102 -4.440892098500626e-16
0 5 2 -0.6737882653061223 1.110223024625157e-16
0 5 3 0.06154336734693884 -6.938893903907228e-17
0 5 4 0.8606505102040818 0
0 5 5 1.723533163265306 -4.440892098500626e-16
0 5 6 2.650191326530612 0
0 5 7 3.640625 0
0 6 0 -2.0825 -4.440892098500626e-16
0 6 1 -1.434540816326531 -2.220446049250313e-16
0 6 2 -0.7228061224489792 -6.661338147750939e-16
0 6 3 0.05270408163265292 -2.012279232133096e-16
0 6 4 0.8919897959183674 0
0 6 5 1.795051020408164 -4.440892098500626e-16
0 6 6 2.76188775510204 0
0 6 7 3.7925 -8.881784197001252e-16
0 7 0 -2.110624999999999 -8.881784197001252e-16
0 7 1 -1.422487244897959 -4.440892098500626e-16
0 7 2 -0.6705739795918364 -4.440892098500626e-16
0 7 3 0.1451147959183675 -5.551115123125783e-17
0 7 4 1.024579081632653 4.440892098500626e-16
0 7 5 1.967818877551021 -8.881784197001252e-16
0 7 6 2.974834183673469 -4.440892098500626e-16
0 7 7 4.045624999999999 0
0 8 0 -2.0375 4.440892098500626e-16
0 8 1 -1.309183673469388 2.220446049250313e-16
0 8 2 -0.5170918367346939 4.440892098500626e-16
0 8 3 0.3387755102040816 2.775557561562891e-16
0 8 4 1.258418367346939 -2.220446049250313e-16
0 8 5 2.241836734693878 -8.881784197001252e-16
0 8 6 3.289030612244898 -4.440892098500626e-16
0 8 7 4.4 0
1 0 0 0.1659567901234568 -2.775557561562891e-17
1 0 1 0.6244318468127992 0
1 0 2 1.146682413706223 0
1 0 3 1.732708490803729 -2.220446049250313e-16
1 0 4 2.382510078105316 0
1 0 5 3.096087175610985 4.440892098500626e-16
1 0 6 3.873439783320735 4.440892098500626e-16
1 0 7 4.714567901234568 0
1 1 0 -0.4046682098765432 5.551115123125783e-17
1 1 1 0.09398541824137065 -1.387778780781446e-17
1 1 2 0.6564145565633661 0
1 1 3 1.282619205089443 -2.220446049250313e-16
1 1 4 1.972599363819602 -4.440892098500626e-16
1 1 5 2.726355032753843 -4.440892098500626e-16
1 1 6 3.543886211892165 -8.881784197001252e-16
1 1 7 4.425192901234568 -8.881784197001252e-16
1 2 0 -0.8740432098765432 0
1 2 1 -0.335211010330058 1.110223024625157e-16
1 2 2 0.267396699420509 5.551115123125783e-17
1 2 3 0.9337799193751575 0
1 2 4 1.663938649533888 0
1 2 5 2.457872889896699 0
1 2 6 3.315582640463593 -4.440892098500626e-16
1 2 7 4.237067901234568 0
1 3 0 -1.242168209876543 2.220446049250313e-16
1 3 1 -0.6631574389014867 2.220446049250313e-16
1 3 2 -0.02037115772234821 1.110223024625157e-16
1 3 3 0.6861906336608716 2.220446049250313e-16
1 3 4 1.456527935248173 -2.220446049250313e-16
1 3 5 2.290640747039557 0
1 3 6 3.18852906903502 8.881784197001252e-16
1 3 7 4.150192901234568 -8.881784197001252e-16
1 4 0 -1.509043209876543 4.440892098500626e-16
1 4 1 -0.8898538674729151 0
1 4 2 -0.2068890148652053 1.665334536937735e-16
1 4 3 0.539851347946586 1.110223024625157e-16
1 4 4 1.350367220962459 2.220446049250313e-16
1 4 5 2.224658604182414 -4.440892098500626e-16
1 4 6 3.162725497606449 0
1 4 7 4.164567901234568 0
1 5 0 -1.674668209876544 2.220446049250313e-16
1 5 1 -1.015300296044344 0
1 5 2 -0.2921568720080624 1.110223024625157e-16
1 5 3 0.4947620622323002 1.110223024625157e-16
1 5 4 1.345456506676745 -2.220446049250313e-16
1 5 5 2.259926461325271 0
1 5 6 3.238171926177878 4.440892098500626e-16
1 5 7 4.280192901234567 0
1 6 0 -1.739043209876543 -2.220446049250313e-16
1 6 1 -1.039496724615772 0
1 6 2 -0.2761747291509195 -3.33066907387547e-16
1 6 3 0.5509227765180147 -4.440892098500626e-16
1 6 4 1.44179579239103 4.440892098500626e-16
1 6 5 2.396444318468128 0
1 6 6 3.414868354749307 -8.881784197001252e-16
1 6 7 4.497067901234568 -8.881784197001252e-16
1 7 0 -1.702168209876543 2.220446049250313e-16
1 7 1 -0.9624431531872006 -2.220446049250313e-16
1 7 2 -0.1589425862937764 -1.110223024625157e-16
1 7 3 0.7083334908037293 -2.220446049250313e-16
1 7 4 1.639385078105316 6.661338147750939e-16
1 7 5 2.634212175610985 4.440892098500626e-16
1 7 6 3.692814783320735 8.881784197001252e-16
1 7 7 4.815192901234568 0
1 8 0 -1.564043209876544 6.661338147750939e-16
1 8 1 -0.7841395817586295 1.110223024625157e-16
1 8 2 0.05953955656336637 3.608224830031759e-16
1 8 3 0.9669942050894434 1.110223024625157e-16
1 8 4 1.938224363819602 0
1 8 5 2.973230032753843 4.440892098500626e-16
1 8 6 4.072011211892164 8.881784197001252e-16
1 8 7 5.234567901234568 0
2 0 0 0.2863271604938272 0
2 0 1 0.7963895187704713 0
2 0 2 1.370227387251197 0
2 0 3 2.007840765936004 4.440892098500626e-16
2 0 4 2.709229654824893 4.440892098500626e-16
2 0 5 3.474394053917863 0
2 0 6 4.303333963214915 0
2 0 7 5.196049382716049 0
2 1 0 -0.2192978395061728 0
2 1 1 0.3309430901990427 -5.551115123125783e-17
2 1 2 0.9449595301083398 0
2 1 3 1.622751480221718 2.220446049250313e-16
2 1 4 2.364318940539179 -4.440892098500626e-16
2 1 5 3.169661911060721 0
2 1 6 4.038780391786343 8.881784197001252e-16
2 1 7 4.971674382716049 0
2 2 0 -0.6236728395061728 0
2 2 1 -0.03325333837238589 -2.775557561562891e-17
2 2 2 0.6209416729654825 1.110223024625157e-16
2 2 3 1.338912194507432 2.220446049250313e-16
2 2 4 2.120658226253465 -4.440892098500626e-16
2 2 5 2.966179768203578 -4.440892098500626e-16
2 2 6 3.875476820357772 0
2 2 7 4.848549382716049 0
2 3 0 -0.9267978395061729 1.110223024625157e-16
2 3 1 -0.2961997669438146 0
2 3 2 0.3981738158226255 5.551115123125783e-17
2 3 3 1.156322908793147 -2.220446049250313e-16
2 3 4 1.97824751196775 0
2 3 5 2.863947625346436 -8.881784197001252e-16
2 3 6 3.813423248929201 0
2 3 7 4.826674382716051 -8.881784197001252e-16
2 4 0 -1.128672839506173 4.440892098500626e-16
2 4 1 -0.457896195515243 0
2 4 2 0.2766559586797686 -2.220446049250313e-16
2 4 3 1.074983623078861 0
2 4 4 1.937086797682036 2.220446049250313e-16
2 4 5 2.862965482489292 0
2 4 6 3.852619677500629 4.440892098500626e-16
2 4 7 4.906049382716048 1.77635683940025e-15
2 5 0 -1.229297839506173 2.220446049250313e-16
2 5 1 -0.5183426240866715 -1.110223024625157e-16
2 5 2 0.2563881015369113 1.110223024625157e-16
2 5 3 1.094894337364576 0
2 5 4 1.997176083396322 0
2 5 5 2.963233339632149 8.881784197001252e-16
2 5 6 3.993066106072059 -8.881784197001252e-16
2 5 7 5.086674382716051 -1.77635683940025e-15
2 6 0 -1.228672839506173 -2.220446049250313e-16
2 6 1 -0.4775390526581 -2.220446049250313e-16
2 6 2 0.3373702443940543 -4.440892098500626e-16
2 6 3 1.21605505165029 0
2 6 4 2.158515369110608 -4.440892098500626e-16
2 6 5 3.164751196775006 0
2 6 6 4.234762534643486 8.881784197001252e-16
2 6 7 5.368549382716049 0
2 7 0 -1.126797839506173 0
2 7 1 -0.3354854812295286 1.110223024625157e-16
2 7 2 0.5196023872511972 -2.220446049250313e-16
2 7 3 1.438465765936004 -2.220446049250313e-16
2 7 4 2.421104654824893 0
2 7 5 3.467519053917864 0
2 7 6 4.577708963214915 0
2 7 7 5.751674382716049 0
2 8 0 -0.923672839506173 8.881784197001252e-16
2 8 1 -0.09218190980095714 2.775557561562891e-16
2 8 2 0.8030845301083404 3.33066907387547e-16
2 8 3 1.762126480221719 4.440892098500626e-16
2 8 4 2.784943940539179 8.881784197001252e-16
2 8 5 3.871536911060721 -4.440892098500626e-16
2 8 6 5.021905391786345 -8.881784197001252e-16
2 8 7 6.23604938271605 0
3 0 0 0.5736111111111112 0
3 0 1 1.135260770975057 2.220446049250313e-16
3 0 2 1.760685941043084 -2.220446049250313e-16
3 0 3 2.449886621315193 0
3 0 4 3.202862811791383 0
3 0 5 4.019614512471655 0
3 0 6 4.900141723356009 0
3 0 7 5.844444444444445 -8.881784197001252e-16
3 1 0 0.1329861111111111 8.326672684688674e-17
3 1 1 0.7348143424036284 -2.220446049250313e-16
3 1 2 1.400418083900227 -4.440892098500626e-16
3 1 3 2.129797335600907 0
3 1 4 2.922952097505668 4.440892098500626e-16
3 1 5 3.779882369614513 -1.332267629550188e-15
3 1 6 4.700588151927438 0
3 1 7 5.685069444444444 0
3 2 0 -0.2063888888888888 0
3 2 1 0.4356179138321996 5.551115123125783e-17
3 2 2 1.14140022675737 -2.220446049250313e-16
3 2 3 1.910958049886621 4.440892098500626e-16
3 2 4 2.744291383219955 0
3 2 5 3.641400226757369 0
3 2 6 4.602284580498866 0
3 2 7 5.626944444444444 0
3 3 0 -0.4445138888888889 0
3 3 1 0.2376714852607711 -1.387778780781446e-16
3 3 2 0.9836323696145127 0
3 3 3 1.793368764172336 0
3 3 4 2.666880668934241 -4.440892098500626e-16
3 3 5 3.604168083900228 -1.332267629550188e-15
3 3 6 4.605231009070295 -8.881784197001252e-16
3 3 7 5.670069444444445 -1.77635683940025e-15
3 4 0 -0.581388888888889 2.220446049250313e-16
3 4 1 0.1409750566893425 -1.110223024625157e-16
3 4 2 0.9271145124716553 2.220446049250313e-16
3 4 3 1.77702947845805 0
3 4 4 2.690719954648526 0
3 4 5 3.668185941043085 -4.440892098500626e-16
3 4 6 4.709427437641724 -8.881784197001252e-16
3 4 7 5.814444444444444 0
3 5 0 -0.617013888888889 1.110223024625157e-16
3 5 1 0.145528628117914 -2.498001805406602e-16
3 5 2 0.9718466553287983 0
3 5 3 1.861940192743764 2.220446049250313e-16
3 5 4 2.815809240362813 -1.332267629550188e-15
3 5 5 3.833453798185941 4.440892098500626e-16
3 5 6 4.914873866213151 0
3 5 7 6.060069444444444 0
3 6 0 -0.5513888888888886 -4.440892098500626e-16
3 6 1 0.2513321995464856 1.110223024625157e-16
3 6 2 1.117828798185942 -8.881784197001252e-16
3 6 3 2.048100907029478 8.881784197001252e-16
3 6 4 3.042148526077097 4.440892098500626e-16
3 6 5 4.099971655328799 -8.881784197001252e-16
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