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See more related video:Origin VT0010 Interpolation
Contents 
Interpolation is a method of estimating and constructing new data points from a discrete set of known data points. Given an X vector, this function interpolates vector Y base on the input curve (XY Range). Origin provides three options for data interpolation: Linear, Cubic spline and Cubic Bspline methods
Linear interpolation is the simplest and fastest data interpolation method. In linear interpolation, the arithmetic mean of two adjacent data points is calculated. This method is useful in situations where low precision can be tolerated. Linear interpolation is also useful for extremely large data sets, because the calculations are not time or computationpower intensive.
The generalization of linear interpolation is polynomial interpolation. Polynomial interpolation requires much more computation power than linear interpolation, and when the polynomial order is high, the fit of the data oscillates wildly. These disadvantages can be avoided by using loworder polynomial fitting, or spline interpolation.
The Cubic spline method uses 3^{rd} order polynomials, and executes datafitting in a piecewise fashion. spline interpolation incurs less error than linear interpolation, and the interpolant is smoother.
Similar to Cubic spline interpolation, Cubic Bspline interpolation also fits the data in a piecewise fashion, but it uses 3^{rd} order Bezier splines to approximate the data. Cubic BSplines allow the accurate modeling of more general classes of geometry.
Note: To generate uniform linear spaced interpolated values, use the interp1xy XFunction.
To use this function:
In fact, the interp1 XFunction is called to complete the calculation.
X Values to Interpolate 
The vector to interpolate on. 

Input 
The XY range to be interpolated. 
Method 
Specify interpolation methods

Boundary 
Boundary condition is only available in cubic spline method.

Smoothing Factor 
Smoothing is only available in Cubic BSpline method. 
Result of interpolation 
The output vector. 
Coefficients 
Spline coefficients when using spline or Bspline method. 
Given a sequence of distinct pairs of data (, ), where . we are looking for the interpolated at by the following methods:
1. Linear interpolation (interp1q)
For
For
For
2. Cubic spline (spline)
Origin uses the natural cubic spline to do interpolation:
where:
And can be generated from:
For boundary points, we set and equal to zero.
3. Cubic Bspline (bspline)
For or perform linear interpolation.
For
Here, Failed to parse (Missing texvc executable; please see math/README to configure.): N(x)\!
denotes the normalized cubic Bspline defined upon the knots , , ..., , And denotes the coefficient of the corresponding function.
The total number of these knots and their values , ..., are chosen automatically by the function. The knots , ..., are the interior knots; they divide the approximation interval [, ] in to subintervals. The coefficients , , ..., are then determined as the solution of the following constrained minimization problem:
minimize
subject to the constraint
where stands for the discontinuity jump in the third order derivative of at the interior knot , denotes the weighted residual , and S is a nonnegatative number to be specified by the user.
The quantity can be seen as a measure of the (lack of) smoothness of , while closeness of fit is measured through . By means of the parameter , 'the smoothing factor', the user will then control the balance between these two (usually conflicting) properties. If is too large, the spline will be too smooth and signal will be lost (underfit); if is too small, the spline will pick up too much noise (overfit). In the extreme cases the function will return an interpolating spline (=0) is is set to zero, and the weighted leastsquares cubic polynomial (=0) is if set very large. Experimenting with values between these two extremes should result in a good compromise.
1. Michelle Schatzman. Numerical Analysis: A Mathematical Introduction, Chapters 4 and 6. Clarendon Press, Oxford (2002).
2. William H. Press, etc. Numerical Recipes in C++. 2^{nd} Edition. Cambridge University Press (2002).
3. Nag C Library Function Document, nag_1d_spline_fit (e02bec).