CL_integrate

Description

• Integrates a function of one variable defined by a data series.

  The nth integral is evaluated by using derivatives (ydotn) and initial conditions (yini).

  - For a simple integration (n=1), ydotn is the first derivative of y at abscissae x, and yini gives the value of y at first abscissa. The function returns the values of y at abscissae x.

  - For a double integration (n=2), ydotn is the second derivative of y at abscissae x, and yini gives the value of y and ydot at first abscissa. The function returns the values of y and its first derivative ydot at abscissae x.

  The methods used to integrate are:

  - step_l (l as lower bound): The result is the integral of a step function (constant between x(k) and x(k+1)). The constant value between x(k) and x(k+1) is ydotn(k).

  

  - step_m (m as middle): The result is the integral of step function (constant between x(k) and x(k+1)). The constant value between x(k) and x(k+1) is (ydotn(k)+ydotn(k+1)/2.

  

  - lin (linear): The result is the integral of a linear function on each interval (polynomial (P) of degree 1). P(xk)=ydotn(k) and P(xk+1)=ydotn(k+1).

  

  - sl4 : Simpson (1 intermediate point) + Lagrange interpolation using 4 points. %nan is returned if there are not enough input data.

  - sspl : Simpson (1 intermediate point) + interpolation using splines.

  - bl6 : Boole (3 intermediate points) + Lagrange interpolation using 6 points. %nan is returned if there are not enough input data.

Parameters

x:

Abscissae (in strictly increasing order) (1xN or PxN)

ydotn:

Values of nth derivative at abscissae x (1xN or PxN)

yini:

(optional) Initial conditions for y (y0 or [y0, ydot0]). Default is 0. (1xn or Pxn)

meth:

(string, optional) Integration method: "step_l", "step_m", "lin", "sl4", "sspl", "bl6". Default is "lin". (1x1)

n:

(integer, optional) Integration order (= number of successive integrations): 1 or 2. Default is 1. (1x1)

y:

Integrated values (after n successive integrations) at abscissae x (PxN)

ydot:

First derivatives of y at abscissae x ([] if n == 1) (PxM)

Authors

• CNES - DCT/SB

Examples

// Example 1: Simple integration x = linspace(0,2*%pi,100); ydot = cos(x); yini = 0; y1 = CL_integrate(x, ydot, yini, meth="lin"); disp(max(abs(y1-sin(x)))); y2 = CL_integrate(x, ydot, yini, meth="sl4"); disp(max(abs(y2-sin(x)))); y3 = CL_integrate(x, ydot, yini, meth="sspl"); disp(max(abs(y3-sin(x)))); y4 = CL_integrate(x, ydot, yini, meth="bl6"); disp(max(abs(y4-sin(x)))); // Example 2: Double integration (step method) x = linspace(0,2*%pi,100); ydot2 = cos(x); yini = [1, 2]; [y, ydot] = CL_integrate(x, ydot2, yini, meth="step_m", n=2); ydotref = sin(x) + 2; disp(max(abs(ydot-ydotref))); yref = -cos(x) + 2*x + 2; disp(max(abs(y-yref)));