13.7.2 Laplacian
Recall, the Laplacian of a function F of n variables
x1,…,xn is
Also, the n× n discrete Laplacian matrix
(also called the second difference matrix) is the n × n
tridiagonal matrix with 2s on the main diagonal, −1s just above and
below the main diagonal;
| | ⎡
⎢
⎢
⎢
⎢
⎢
⎣ | | 2 | −1 | 0 | ⋯ | 0 |
| −1 | 2 | −1 | ⋯ | 0 |
| ⋮ | ⋱ | ⋱ | ⋱ | ⋮ |
| 0 | ⋯ | −1 | 2 | −1 |
| 0 | ⋯ | 0 | −1 | 2
|
| ⎤
⎥
⎥
⎥
⎥
⎥
⎦ |
|
If L is the n× n discrete Laplacian matrix and
Y is an n × 1 column vector whose kth
coordinate is yi=y(a+kΔ x) for a twice differential
function y, then the kth coordinate of L Y will be
(implicitly assuming that y(a)=y(a+(N+1)Δ x)=0):
| −y(a+(k−1)Δ x)+2 y(a+kΔ x)−y (a+(k−1)Δ x), |
which approximates y′′(a+kΔ x). So L Y is approximately
−Δ x2 Y′′, where Y′′ is the
n × 1 column vector whose kth
coordinate is y′′(a+kδ x).
The laplacian
command can compute the Laplacian operator or the discrete Laplacian matrix.
-
To compute the Laplacian operator, laplacian takes two arguments:
-
expr, an expression involving several variables.
- vars, a list of the variable names.
- laplacian(expr,vars) returns the
Laplacian of the expression.
- To compute the discrete Laplacian matrix, laplacian takes
n, an integer or floating-point integer.
- laplacian(n) returns the n× n discrete
Laplacian matrix.
Examples
Find the Laplacian of F(x,y,z)=2x2y−xz3.
| laplacian(2*x^2*y-x*z^3,[x,y,z]) |