Package org.opensourcephysics.cabrillo.tracker

Class SecondDerivative

java.lang.Object
org.opensourcephysics.cabrillo.tracker.SecondDerivative
All Implemented Interfaces:
Derivative
public class SecondDerivative extends Object implements Derivative
This implements an algorithm for finding a second derivative. Derivation of the 5-point algorithm: 1. Find least-square fit of parabola y = a + b*x + c*x^2. 2. Use eqn 3-86 from Parratt's Probability and Experimental Errors in Science (1961) to find best fit parameters. 3. Use n = 5 so the values of x are -2 thru +2 (units of delta_t), y are corresponding positions. 4. Plug in to obtain c = (2*x[i+2] - x[i+1] - 2*x[i] - x[i-1] + 2*x[i-2])/14. 5. Use derivative expressions y' = b + 2cx and y'' = 2c, and scale y'' by dt^2. 6. Final result is accel[i] = (2*x[i+2] - x[i+1] - 2*x[i] - x[i-1] + 2*x[i-2])/(7*dt^2). Other possible 5-point expressions (finite difference equations)--not used since more sensitive to noisy data: 1. accel[i] = (x[i+2] - 2*x[i] + x[i-2]) / (4*dt^2) 2. accel[i] = (-x[i+2] +16*x[i+1] - 30*x[i] +16*x[i-1] - x[i-2]) / (12*dt^2)
Author:
Douglas Brown

  • Constructor Details

    • SecondDerivative

      public SecondDerivative()

  • Method Details

    • evaluate

      public Object[] evaluate(Object[] data)
      Evaluates the derivative. Input data: data[0] = parameters (int[] {spill, start, stepsize, count}) data[1] = xData (double[]) data[2] = yData (double[]) (may be null) data[3] = validData (boolean[]) Returned result: result[0] = null result[1] = null result[2] = xDeriv (double[]) (invalid values are NaN) result[3] = yDeriv (double[]) (invalid values are NaN) (null if input data[2] is null)
      Specified by:
      evaluate in interface Derivative
      Parameters:
      data - the input data
      Returns:
      Object[] the result