kroghInterpolateFit

kroghInterpolateFit(X, Y, [der=0])

X is a numeric vector representing the x-coordinates of the points to be interpolated. Values in X must be in increasing order. NULL values are not allowed.

Y is a numeric vector of the same length as X, representing the y-coordinates of the points. NULL values are not allowed.

der (optional) is a non-negative integer representing the derivative order to compute. The default value is 0, meaning to compute the polynomial's value.

This function performs polynomial interpolation for a given set of points, ensuring the polynomial passes through all points in the set.

Multiple derivative values at each point of X can be specified by repeating X values and assigning corresponding derivative values as consecutive Y values:

• If an X value appears only once, the corresponding Y is the value of the polynomial f(X).
• If an X value appears multiple times, the first Y is the value of f(X), the second Y is the first derivative f′(X), the third Y is the second derivative f′′(X), and so on. For example, with inputs X=[0,0,1,1] and Y=[1,0,2,3], the interpretation is Y[0]=f(0), Y[1]=f′(0), Y[2]=f(1), Y[3]=f′(1).

Return value: A dictionary with the following keys:

• modelName: A string "kroghInterpolate" indicating the model name.
• X: The numeric vector representing the x-coordinates used for interpolation (i.e., the input X).
• der: The non-negative integer representing the derivative order (i.e., the input der).
• coeffs: A numeric vector containing the polynomial coefficients fitted from the input data points.
• predict: The prediction function of the model. You can call model.predict(X) or predict(model, X) to make predictions with the generated model. It takes the following parameters:
  • model: A dictionary, which is the output of kroghInterpolateFit.
  • X: A numeric vector representing the x-coordinates at which to evaluate the polynomial.

Perform polynomial interpolation for the sin function and compute the polynomial estimates at specific points:

x = 0 1 2 3 4 5
y = sin(x)
model = kroghInterpolateFit(x,y)
model

/*
output:
X->[0,1,2,3,4,5]
der->0
predict->kroghInterpolateFitPredict
modelName->kroghInterpolate
coeffs->[0.0,0.841470984807,-0.386822271395,-0.010393219665,0.032025753923,-0.005411092181,0.0]
*/

Predict with the generated model using a UDF linspace:

def linspace(start, end, num, endpoint=true){
	if(endpoint) return end$DOUBLE\(num-1), start + end$DOUBLE\(num-1)*0..(num-1)
	else return start + end$DOUBLE\(num-1)*0..(num-1)	
}
xx = linspace(0.0, 5.0, 10)[1]
model.predict(xx)

// output: [0,0.515119011157387,0.898231239576709,0.998548648650381,0.793484053410063,0.354287125066207,-0.188319604452395,-0.678504737959061,-0.969692008469677,-0.958924274663139]

Related Functions: predict, kroghInterpolate