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Tutorial for basic usageΒΆ

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[[-3.14159265 -1.          2.        ]
 [-3.14159265 -0.5         2.        ]
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 [-0.78539816  0.          1.90056311]
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 [-0.78539816  1.          1.72378641]
 [-0.39269908 -1.          1.27648914]
 [-0.39269908 -0.5         1.707008  ]
 [-0.39269908  0.          1.94618514]
 [-0.39269908  0.5         1.99402057]
 [-0.39269908  1.          1.85051428]
 [ 0.         -1.          2.        ]
 [ 0.         -0.5         2.        ]
 [ 0.          0.          2.        ]
 [ 0.          0.5         2.        ]
 [ 0.          1.          2.        ]
 [ 0.39269908 -1.          2.72351086]
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 [ 0.78539816 -0.5         2.54137863]
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 [ 0.78539816  1.          2.27621359]
 [ 1.57079633 -1.          3.890625  ]
 [ 1.57079633 -0.5         2.765625  ]
 [ 1.57079633  0.          2.140625  ]
 [ 1.57079633  0.5         2.015625  ]
 [ 1.57079633  1.          2.390625  ]
 [ 2.35619449 -1.          3.33687376]
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 [ 2.35619449  0.          2.09943689]
 [ 2.35619449  0.5         2.01104854]
 [ 2.35619449  1.          2.27621359]
 [ 2.74889357 -1.          2.72351086]
 [ 2.74889357 -0.5         2.292992  ]
 [ 2.74889357  0.          2.05381486]
 [ 2.74889357  0.5         2.00597943]
 [ 2.74889357  1.          2.14948572]
 [ 3.14159265 -1.          2.        ]
 [ 3.14159265 -0.5         2.        ]
 [ 3.14159265  0.          2.        ]
 [ 3.14159265  0.5         2.        ]
 [ 3.14159265  1.          2.        ]]

Coefficients all same? True
Knots all same? True

Antiderivative of derivative:
 Coefficients differ by constant? True
Knots all same? True

Derivative of antiderivative:
 Coefficients the same? True
Knots all same? True


  8 import ndsplines
  9 import numpy as np
 10 import matplotlib.pyplot as plt
 11
 12 # generate grid of independent variables
 13 x = np.array([-1, -7/8, -3/4, -1/2, -1/4, -1/8, 0, 1/8, 1/4, 1/2, 3/4, 7/8, 1])*np.pi
 14 y = np.array([-1, -1/2, 0, 1/2, 1])
 15 meshx, meshy = np.meshgrid(x, y, indexing='ij')
 16 gridxy = np.stack((meshx, meshy), axis=-1)
 17
 18
 19 # generate denser grid of independent variables to interpolate
 20 sparse_dense = 2**7
 21 xx = np.concatenate([np.linspace(x[i], x[i+1], sparse_dense) for i in range(x.size-1)]) # np.linspace(x[0], x[-1], x.size*sparse_dense)
 22 yy = np.concatenate([np.linspace(y[i], y[i+1], sparse_dense) for i in range(y.size-1)]) # np.linspace(y[0], y[-1], y.size*sparse_dense)
 23 gridxxyy = np.stack(np.meshgrid(xx, yy, indexing='ij'), axis=-1)
 24
 25 def plots(sparse_data, dense_data, ylabel='f(x,y)'):
 26     fig, axes = plt.subplots(1, 2, constrained_layout=True)
 27     for yidx in range(sparse_data.shape[1]):
 28         axes[0].plot(x, sparse_data[:, yidx], 'o', color='C%d'%yidx, label='y=%.2f'%y[yidx])
 29         axes[0].plot(xx, dense_data[:, np.clip(yidx*sparse_dense, 0, yy.size-1)], color='C%d'%yidx)# label='y=%.1f'%y[yidx])
 30
 31     axes[0].legend()
 32     axes[0].set_xlabel('x')
 33     axes[0].set_ylabel(ylabel)
 34     for xidx in range(sparse_data.shape[0]//2):
 35         axes[1].plot(yy, dense_data[(xidx+3)*sparse_dense, :], '--', color='C%d'%xidx,)# label='x=%.1f'%x[xidx+3])
 36         axes[1].plot(y, sparse_data[xidx+3, :], 'o', color='C%d'%xidx, label='x=%.1f'%x[xidx+3],)
 37
 38     axes[1].legend()
 39     axes[1].set_xlabel('y')
 40     plt.show()
 41
 42 # evaluate a function to interpolate over input grid
 43 meshf = np.sin(meshx) * (meshy-3/8)**2 + 2
 44
 45 # create the interpolating splane
 46 interp = ndsplines.make_interp_spline(gridxy, meshf)
 47
 48 # evaluate spline over denser grid
 49 meshff = interp(gridxxyy)
 50
 51
 52 plots(meshf, meshff)
 53
 54
 55 ##
 56
 57 # as subplots
 58 fig, axes = plt.subplots(1,2, constrained_layout=True)
 59
 60 gridxxy = np.stack(np.meshgrid(xx, y, indexing='ij'), axis=-1)
 61 meshff = interp(gridxxy)
 62
 63 for yidx in range(meshf.shape[1]):
 64     axes[0].plot(x, meshf[:, yidx], 'o', color='C%d'%yidx, label='y=%.1f'%y[yidx])
 65     axes[0].plot(xx, meshff[:, yidx], color='C%d'%yidx)
 66 axes[0].legend()
 67 axes[0].set_xlabel('$x$')
 68 axes[0].set_ylabel('$f(x,y)$')
 69
 70 # y-dir plot
 71 gridxyy = np.stack(np.meshgrid(x, yy, indexing='ij'), axis=-1)
 72
 73 meshff = interp(gridxyy)
 74 for xidx in range(meshf.shape[0]//2):
 75     axes[1].plot(yy, meshff[xidx*1+3, :], '--', color='C%d'%xidx, label='x=%.1f'%x[xidx*1+3])
 76     axes[1].plot(y, meshf[xidx*1+3, :], 'o', color='C%d'%xidx)
 77
 78 axes[1].legend()
 79 axes[1].set_xlabel('$y$')
 80 # plt.ylabel(r'$\frac{\partial f(x,y)}{\partial y}$')
 81 plt.show()
 82
 83 ##
 84
 85 # we could also use tidy data format to make the grid
 86
 87 tidy_data = np.dstack((gridxy, meshf)).reshape((-1,3))
 88 print(tidy_data)
 89
 90 tidy_interp = ndsplines.make_interp_spline_from_tidy(tidy_data, [0,1], [2])
 91
 92 print("\nCoefficients all same?", np.all(tidy_interp.coefficients == interp.coefficients))
 93 print("Knots all same?", np.all([np.all(knot0 == knot1) for knot0, knot1 in zip(tidy_interp.knots, interp.knots)]))
 94
 95 # send to example of least squares
 96 ##
 97 # two ways to evaluate derivative - y direction
 98
 99 deriv_interp = interp.derivative(1)
100 deriv1 = deriv_interp(gridxy)
101 deriv2 = interp(gridxxyy, nus=np.array([0,1]))
102
103 plots(deriv1, deriv2, r'$\frac{\partial f(x,y)}{\partial y}$')
104
105 ##
106 # two ways to evaluate derivatives x-direction: create a derivative spline or call with nus:
107 deriv_interp = interp.derivative(0)
108 deriv1 = deriv_interp(gridxy)
109 deriv2 = interp(gridxxyy, nus=np.array([1,0]))
110
111 plots(deriv1, deriv2, r'$\frac{\partial f(x,y)}{\partial x}$')
112 ##
113
114 # Calculus demonstration
115 interp1 = deriv_interp.antiderivative(0)
116 coeff_diff = interp1.coefficients - interp.coefficients
117 print("\nAntiderivative of derivative:\n","Coefficients differ by constant?", np.allclose(interp1.coefficients+2.0, interp.coefficients))
118 print("Knots all same?", np.all([np.all(knot0 == knot1) for knot0, knot1 in zip(interp1.knots, interp.knots)]))
119
120 antideriv_interp = interp.antiderivative(0)
121
122 interp2 = antideriv_interp.derivative(0)
123 print("\nDerivative of antiderivative:\n","Coefficients the same?", np.allclose(interp2.coefficients, interp.coefficients))
124 print("Knots all same?", np.all([np.all(knot0 == knot1) for knot0, knot1 in zip(interp2.knots, interp.knots)]))

Total running time of the script: (0 minutes 2.784 seconds)

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