Returns the integral over the specified dimension of a matrix.
Integrals over singleton dimensions will return zero (i.e. the area under a single point is zero). See integrate2 for an alternate behavior.
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Syntax |
Description |
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out = integrate(A, n, x1); |
Integrates A over the nth dimension in the matrix. x1 is the corresponding position vector for that dimension. |
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out = integrate(A, d, x1, x2, ...); |
Calculates the integral of A over the specified list of dimension(s) d. d is a vector containing the dimensions over which to integrate. xi are the position vectors corresponding to the dimensions of A over which the integration is occurring. For example
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Example
In the following example, the integrate command is used to integrate y=x^2 from 0 to 3, where the function is sampled at the points x=0,1,2,3. The integrate function will determine dx from the position vector x. For reference, the value of this integral for the continuous function y=x^2 (as opposed to the discrete samples in this example) is 9. Reducing dx will make this discrete integral approach the continuous result.
Advanced note: The actual calculation in this very simple example will be 0.5*0 + 1*1 + 1*4 + 0.5*9 = 9.5, as illustrated in the figure below. It is interesting to note that the first and last points have a factor of 0.5*dx because they are at the edge of the integration range. Without the factor of 0.5 applied to those points, the integral would effectively be calculated from x=-0.5 to x=3.5
?x=0:3; y=x^2; ?integrate(y,1,x); result: 0 1 2 3 result: 9.5
Next, we demonstrate that the integrate function correctly treats non-uniform sampling. The portion of the function from 0 to 2 is evaluated with a dx=1, while a dx of 0.2 is used from 2 to 3. In this case, the integrate function will calculate 0.5*0 + 1*1 + 0.6*4 + 0.2*4.84 + 0.2*5.76 + 0.2*6.76 + 0.2*7.84 + 0.1*9;
?x=[[0:1]; [2:0.2:3]]; y=x^2; ?integrate(y,1,x); result: 0 1 2 2.2 2.4 2.6 2.8 3 result: 9.34
Lastly, this example shows how to calculate the power transmitted through a y-normal monitor by integrating the Poynting vector. To get transmitted power, we want to integrate the real part of the normal component of the poynting vector (Py). The Py data matrix will have size N x x N y x N z x N f , where Nx, Ny, Nz are the number of mesh point in each direction. If the monitor is Y-normal, Ny=1. Nf is the number of frequency points collected by the monitor. After integrating over the X and Z direction, we are basically left with a 1D function of the transmitted power vs frequency.
Py = getdata("Monitor1","Py");
x = getdata("Monitor1","x");
y = getdata("Monitor1","y");
z = getdata("Monitor1","z");
f = getdata("Monitor1","f");
power = 0.5 * integrate( real(Py), [1,3], x,z );
See Also
List of commands , integrate2 , conv2 , max , min , interp , find , pinch , round , getdata , sum , length