Structural conventions of the input and output of multivariate S2FunHarmonic
In this part we deal with multivariate functions of the form
\[ f\colon \bf{S}^2\to \bf{R}^n \].
- the structure of the nodes is always interpreted as a column vector
- the node index is the first dimension
- the dimensions of the
S2FunHarmonic
itself is counted from the second dimension
For example we got four nodes \(v_1, v_2, v_3\) and \(v_4\) and six functions \(f_1, f_2, f_3, f_4, f_5\) and \(f_6\), which we want to store in a 3x2 array, then the following scheme applies to function evaluations:
\[ F(:, :, 1) = \pmatrix{f_1(v_1) & f_2(v_1) & f_3(v_1) \cr f_1(v_2) & f_2(v_2) & f_3(v_2) \cr f_1(v_3) & f_2(v_3) & f_3(v_3) \cr f_1(v_4) & f_2(v_4) & f_3(v_4)} \quad\mathrm{and}\quad F(:, :, 2) = \pmatrix{f_4(v_1) & f_5(v_1) & f_6(v_1) \cr f_4(v_2) & f_5(v_2) & f_6(v_2) \cr f_4(v_3) & f_5(v_3) & f_6(v_3) \cr f_4(v_4) & f_5(v_4) & f_6(v_4)}. \]
For the intern Fourier-coefficient matrix the first dimension is reserved for the Fourier-coefficients of a single function; the dimension of the functions itself begins again with the second dimension.
If \(\bf{\hat f}_1, \bf{\hat f}_2, \bf{\hat f}_3, \bf{\hat f}_4, \bf{\hat f}_5\) and \(\bf{\hat f}_6\) would be the column vectors of the Fourier-coefficients of the functions above, internally they would be stored in \(\hat F\) as follows. \[ \hat F(:, :, 1) = \pmatrix{\bf{\hat f}_1 & \bf{\hat f}_2 & \bf{\hat f}_3} \quad\mathrm{and}\quad \hat F(:, :, 2) = \pmatrix{\bf{\hat f}_4 & \bf{\hat f}_5 & \bf{\hat f}_6}. \]
Defining a multivariate S2FunHarmonic
Definition via function values
At first we need some vertices
Next we define function values for the vertices
Now the actual command to get a 2x1 sF1
of type S2FunHarmonic
Definition via function handle
If we have a function handle for the function we could create a S2FunHarmonic
via quadrature. At first let us define a function handle which takes vector3d
as an argument and returns double:
Now we call the quadrature command to get 4x1 sF2
of type S2FunHarmonic
Definition via Fourier-coefficients
If we already know the Fourier-coefficients, we can simply hand them in the format above to the constructor of S2FunHarmonic
.
- This command stores the nine first spherical harmonics in
sF3
Operations which differ from an univariate S2FunHarmonic
Some default matrix and vector operations
You can concatenate and refer to functions as MATLAB does with vectors and matrices
You can conjugate the Fourier-coefficients and transpose/ctranspose the multivariate S2FunHarmonic
.
Some other operations
sum
and mean
If we do not specify further options to sum
or mean
they give we the integral or the mean value back for each function. You could also calculate the conventional sum or the mean value over a dimension of a multivariate S2FunHarmonic
.
min/max
If the min
or max
command gets a multivariate S2FunHarmonic
the pointwise minimum or maximum can calculated along the dimension specified as third argument.
Remark on the matrix product
At this point the matrix product is implemented per element and not as the usual matrix product.
Visualization of multivariate S2FunHarmonic
The same plot commands as for univariate S2FunHarmonic
work on multivariate as well. The difference is that, now, each component is plotted next to one another.