## Cubic Spline (cspline) Object

## Main.ObjectCspline History

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(:toggle hide apm button show="Example APMonitor Model":)

(:toggle hide apm button show="Example APMonitor (Python) Source":)

Sample the function `3\sin(x)-(x-3)` at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the maximum of that function.

Sample the function `3\sin(x)-(x-3)` at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the minimum of that function.

#### GEKKO Example

#### GEKKO Python Example

Sample the function `3\sin(x)-(x-3)` at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the maximum of that function.

Sample the function `3\sin(x)-(x-3)` at 50 random points between 5 and 15. Use the randomly sampled points to construct a cubic spline and find the maximum of that function.

#### APM Python Example

#### APM Python Example

Use the following x and y data to construct a cubic spline.

x_data , y_data -1.0000000e+00 , 3.8461538e-02 -8.0000000e-01 , 5.8823529e-02 -5.0000000e-01 , 1.3793103e-01 -2.5000000e-01 , 3.9024390e-01 0.0000000e+00 , 1.0000000e+00 1.0000000e-01 , 8.0000000e-01 2.0000000e-01 , 5.0000000e-01 5.0000000e-01 , 1.3793103e-01

Find the maximum of the interpolated function.

#### APMonitor Example

#### APM Python Example

#### GEKKO Example

#### APMonitor Example

(:toggle hide gekko button show="Example GEKKO (Python) Code":) (:div id=gekko:)

from gekko import gekko

""" minimize y s.t. y = f(x)

using cubic spline with random sampling of data """

- Function to generate data for cspline

def f(x):

return 3*np.sin(x) - (x-3)

- Create model

c = gekko()

- Cubic spline

x = c.Var(value=15) y = c.Var() x_data = np.random.rand(50)*10+10 y_data = f(x_data) c.cspline(x,y,x_data,y_data,True) c.Obj(y)

- Options

c.options.IMODE = 3 c.options.CSV_READ = 0 c.options.SOLVER = 3 c.solve()

- Generate continuous trend for plot

z = np.linspace(10,20,100)

- Check if solved successfully

if c.options.SOLVESTATUS == 1:

plt.figure() plt.plot(z,f(z),'r-',label='original') plt.scatter(x_data,y_data,5,'b',label='data') plt.scatter(x.value,y.value,200,'k','x',label='minimum') plt.legend(loc='best')

else:

print ('Failed to converge!') plt.figure() plt.plot(z,f(z),'r-',label='original') plt.scatter(x_data,y_data,5,'b') plt.legend(loc='best')

plt.show() (:sourceend:) (:divend:)

(:toggle hide apm button show="Example APMonitor Model":) (:div id=apm:)

(:source lang=python:) import numpy as np import matplotlib.pyplot as plt

(:sourceend:)

(:toggle hide gekko button show="Example GEKKO (Python) Code":) (:div id=gekko:)

(:source lang=python:) from gekko import gekko import numpy as np import matplotlib.pyplot as plt

""" minimize y s.t. y = f(x)

using cubic spline with random sampling of data """

- Function to generate data for cspline

def f(x):

return 3*np.sin(x) - (x-3)

- Create model

c = gekko()

- Cubic spline

x = c.Var(value=15) y = c.Var() x_data = np.random.rand(50)*10+10 y_data = f(x_data) c.cspline(x,y,x_data,y_data,True) c.Obj(y)

- Options

c.options.IMODE = 3 c.options.CSV_READ = 0 c.options.SOLVER = 3 c.solve()

- Generate continuous trend for plot

z = np.linspace(10,20,100)

- Check if solved successfully

if c.options.SOLVESTATUS == 1:

plt.figure() plt.plot(z,f(z),'r-',label='original') plt.scatter(x_data,y_data,5,'b',label='data') plt.scatter(x.value,y.value,200,'k','x',label='minimum') plt.legend(loc='best')

else:

print ('Failed to converge!') plt.figure() plt.plot(z,f(z),'r-',label='original') plt.scatter(x_data,y_data,5,'b') plt.legend(loc='best')

plt.show()

(:toggle hide gekko button show="Example GEKKO (Python) Code":) (:div id=gekko:) (:source lang=python:) from gekko import gekko import numpy as np import matplotlib.pyplot as plt

""" minimize y s.t. y = f(x)

using cubic spline with random sampling of data """

- Function to generate data for cspline

def f(x):

return 3*np.sin(x) - (x-3)

- Create model

c = gekko()

- Cubic spline

x = c.Var(value=15) y = c.Var() x_data = np.random.rand(50)*10+10 y_data = f(x_data) c.cspline(x,y,x_data,y_data,True) c.Obj(y)

- Options

c.options.IMODE = 3 c.options.CSV_READ = 0 c.options.SOLVER = 3 c.solve()

- Generate continuous trend for plot

z = np.linspace(10,20,100)

- Check if solved successfully

if c.options.SOLVESTATUS == 1:

plt.figure() plt.plot(z,f(z),'r-',label='original') plt.scatter(x_data,y_data,5,'b',label='data') plt.scatter(x.value,y.value,200,'k','x',label='minimum') plt.legend(loc='best')

else:

print ('Failed to converge!') plt.figure() plt.plot(z,f(z),'r-',label='original') plt.scatter(x_data,y_data,5,'b') plt.legend(loc='best')

plt.show() (:sourceend:) (:divend:)

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/s1jSLpDXvzs" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe> (:htmlend:)

The function is evaluated at the points x_data = [-1.0 -0.8 -0.5 -0.25 0.0 0.1 0.2 0.5].

The function is evaluated at the points x_data = [-1.0 -0.8 -0.5 -0.25 0.0 0.1 0.2 0.5]. Evaluating at additional points shows the cubic spline interpolation function. The maximum of the original function is at *x*=0 with a result *y*=1. Because the cubic spline has only 8 points, there is some approximation error and the optimal solution of the cubic spline is slightly to the left of the true solution.

A cubic spline intersects the points to create the function approximations in the range of x between -1.0 and 0.5. There is extrapolation error outside of this range, as expected. Bounds on *x* should be added or additional cubic spline sample points should be added to avoid problems with optimizer performance in the extrapolation region.

The cubic spline intersects the points to create the function approximations in the range of x between -1.0 and 0.5. There is extrapolation error outside of this range, as expected. Bounds on *x* should be added or additional cubic spline sample points should be added to avoid problems with optimizer performance in the extrapolation region.

Find the maximum of a function defined by 8 points that approximate the true function.

$$y(x) = \frac{1}{1+25 x^2}$$

The function is evaluated at the points x_data = [-1.0 -0.8 -0.5 -0.25 0.0 0.1 0.2 0.5].

A cubic spline intersects the points to create the function approximations in the range of x between -1.0 and 0.5. There is extrapolation error outside of this range, as expected. Bounds on *x* should be added or additional cubic spline sample points should be added to avoid problems with optimizer performance in the extrapolation region.

Objects

(:source lang=python:) import numpy as np import matplotlib.pyplot as plt from APMonitor.apm import *

s = 'https://byu.apmonitor.com' a = 'cspline'

model = ''' Objects

End Objects File c.csv

End Objects

File c.csv

End File Connections

End File

Connections

End Connections Variables

End Connections

Parameters End Parameters

Variables

End Variables Equations

End Variables

Equations

End Equations

End Equations '''

- write file

fid = open('model.apm','w') fid.write(model) fid.close()

- clear prior, load new model

apm(s,a,'clear all') apm_load(s,a,'model.apm')

- set steady state optimiation and solve

apm_option(s,a,'apm.imode',3) output = apm(s,a,'solve') print(output)

- retrieve solution

z = apm_sol(s,a)

- print solution

print('x: ' + str(z['x'])) print('y: ' + str(z['y'])) (:sourceend:)

(:title Cubic Spline (cspline) Object:) (:keywords Cubic spline, Object, APMonitor, Option, Configure, Default, Description:) (:description One dimensional cubic spline for nonlinear function approximation with multiple interpolating functions that have continuous first and second derivatives:)

Type: Object Data: Two data vectors that define 1D function points Inputs: Name of first data column (e.g. x) Outputs: Name of second data column (e.g. y) Description: Cubic spline for nonlinear function approximation

A cubic spline is a nonlinear function constructed of multiple third-order polynomials. These polynomials pass through a set of control points and have continuous first and second derivatives everywhere. The second derivative is set to zero at the left and right endpoints, to provide a boundary condition to complete the system of equations. There is poor extrapolation when function retrievals are requested outside of the data points. The input should be constrained or else additional data points added to avoid extrapolation.

**Example Usage**

Objects c = cspline End Objects File c.csv x_data , y_data -1.0000000e+00 , 3.8461538e-02 -8.0000000e-01 , 5.8823529e-02 -5.0000000e-01 , 1.3793103e-01 -2.5000000e-01 , 3.9024390e-01 0.0000000e+00 , 1.0000000e+00 1.0000000e-01 , 8.0000000e-01 2.0000000e-01 , 5.0000000e-01 5.0000000e-01 , 1.3793103e-01 End File Connections x = c.x_data y = c.y_data End Connections Variables x = -0.5 >= -1 <= 0.5 y End Variables Equations maximize y End Equations