In this short tutorial we want to use common Matlab commands within the BoSSS framework.

1 Problem statement

For our matrix analysis we use the following random matrix: $$ A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} $$ and the symmetric matrix: $$ S = \begin{bmatrix} 1 & 2 & 3\\ 2 & 3 & 2\\ 3 & 2 & 1 \end{bmatrix} $$ We are going to evaluate some exemplary properties of the matrices and check if the matrices are symmetric, both in the BoSSS framework and in Matlab.

2 Solution within the BoSSS framework

First, we have to initialize the new project:

#r "BoSSSpad.dll"
using System;
using System.Collections.Generic;
using System.Linq;
using ilPSP;
using ilPSP.Utils;
using BoSSS.Platform;
using BoSSS.Platform.LinAlg;
using BoSSS.Foundation;
using BoSSS.Foundation.XDG;
using BoSSS.Foundation.Grid;
using BoSSS.Foundation.Grid.Classic;
using BoSSS.Foundation.Grid.RefElements;
using BoSSS.Foundation.IO;
using BoSSS.Solution;
using BoSSS.Solution.Control;
using BoSSS.Solution.GridImport;
using BoSSS.Solution.Statistic;
using BoSSS.Solution.Utils;
using BoSSS.Solution.AdvancedSolvers;
using BoSSS.Solution.Gnuplot;
using BoSSS.Application.BoSSSpad;
using BoSSS.Application.XNSE_Solver;
using static BoSSS.Application.BoSSSpad.BoSSSshell;
Init();

The below script needs to be able to find the current output cell; this is an easy method to get it.

using ilPSP.LinSolvers;
using ilPSP.Connectors.Matlab;

We want to implement the two 3x3 matrices in BoSSSpad

int Dim     = 3;
MsrMatrix A = new MsrMatrix(Dim,Dim);
MsrMatrix S = new MsrMatrix(Dim,Dim);
double[] A_firstRow = new double[]{1,2,3};
double[] A_secondRow = new double[]{4,5,6};
double[] A_thirdRow = new double[]{7,8,9};
 
double[] S_firstRow = new double[]{1,2,3};
double[] S_secondRow = new double[]{2,3,2};
double[] S_thirdRow = new double[]{3,2,1};
 
for(int i=0; i<Dim; i++){
    A[0, i] = A_firstRow[i];
    S[0, i] = S_firstRow[i];
}
 
for(int i=0; i<Dim; i++){
    A[1, i] = A_secondRow[i];
    S[1, i] = S_secondRow[i];
}
 
for(int i=0; i<Dim; i++){
    A[2, i] = A_thirdRow[i];
    S[2, i] = S_thirdRow[i];
}

Test for symmetry in BoSSS

To analyze if the matrices are symmetric, we need to compare the original matrix with the transpose:

MsrMatrix AT    = A.Transpose();
MsrMatrix ST    = S.Transpose();
bool SymmTest_A;
bool SymmTest_S;
for(int i = 0; i<Dim; i++){
    for(int j = 0; j<Dim; j++){
        if(A[i,j] == AT[i,j]){
            SymmTest_A = true;
            }
        else{
            SymmTest_A = false;
            break;
            }
        }
    }
for(int i = 0; i<Dim; i++){
    for(int j = 0; j<Dim; j++){
        if(S[i,j] == ST[i,j]){
            SymmTest_S = true;
            }
        else{
            SymmTest_S = false;
            break;
            }
        }
    }
if(SymmTest_A == true){
Console.WriteLine("Matrix A seems to be symmetric.");
}
else{
Console.WriteLine("Matrix A seems NOT to be symmetric.");
}
if(SymmTest_S == true){
Console.WriteLine("Matrix S seems to be symmetric.");
}
else{
Console.WriteLine("Matrix S seems NOT to be symmetric.");
}

Matrix A seems NOT to be symmetric.

Matrix S seems to be symmetric.

The interface to Matlab

The BatchmodeConnector initializes an interface to Matlab:

Troubleshoot

In case that BoSSS cant find the path to matlab you will need to modify (if Matlab is installed) a file called MatlabConnectorConfig.json which you can find in your BoSSS binary under the following path:

cs
...\.BoSSS\etc\MatlabConnectorConfig.json

Inside the file add a path to your matlab.exe (e.g.): ```py { "MatlabExecuteable": "C:\ProgramFiles\MATLAB\R2021a\bin\win64\MATLAB.exe", "Flav": "Matlab" }

Console.WriteLine("Calling MATLAB/Octave...");
BatchmodeConnector bmc = new BatchmodeConnector();

Calling MATLAB/Octave...

We have to transfer out matrices to Matlab:

bmc.PutSparseMatrix(A, "Matrix_A");
bmc.PutSparseMatrix(S, "Matrix_S");
/// Now we can do calculations in Matlab within the \BoSSSpad{} using the \code{Cmd} command. It commits the Matlab commands as a string. We can calculate e.g. the rank of the matrix or the eigenvalues:
bmc.Cmd("Full_A = full(Matrix_A)");
bmc.Cmd("Full_S = full(Matrix_S)");
bmc.Cmd("Rank_A = rank(Full_A)");
bmc.Cmd("Rank_S = rank(Full_S)");
bmc.Cmd("EV_A = eig(Full_A)");
bmc.Cmd("EV_S = eig(Full_S)");
bmc.Cmd("Det_A = det(Full_A)");
bmc.Cmd("Det_S = det(Full_S)");
bmc.Cmd("Trace_A = trace(Full_A)");
bmc.Cmd("Trace_S = trace(Full_S)");
/// We can transfer matrices or arrays from Matlab to \BoSSSpad{} as well, here we want to have the results:
MultidimensionalArray Results = MultidimensionalArray.Create(2, 3);
bmc.Cmd("Results = [Rank_A, Det_A, Trace_A; Rank_S,  Det_S,  Trace_S]");
bmc.GetMatrix(Results, "Results");
/// and the eigenvalues:
MultidimensionalArray EV_A = MultidimensionalArray.Create(3, 1);
bmc.GetMatrix(EV_A, "EV_A");
MultidimensionalArray EV_S = MultidimensionalArray.Create(3, 1);
bmc.GetMatrix(EV_S, "EV_S");
/// After finishing using Matlab we need to close the interface to Matlab:
bmc.Execute(false);
Console.WriteLine("MATLAB/Octave closed, return to BoSSSPad");

MATLAB/Octave closed, return to BoSSSPad

And here are our results back in the BoSSSpad:

double Rank_A  = Results[0,0];
double Rank_S  = Results[1,0];
double Det_A   = Results[0,1];
double Det_S   = Results[1,1];
double Trace_A = Results[0,2];
double Trace_S = Results[1,2];
Console.WriteLine("The results of matrix A are: rank: " + Rank_A + ", trace: " + Trace_A + ", dterminant: " + Det_A);
Console.WriteLine("The results of matrix S are: rank: " + Rank_S + ", trace: " + Trace_S + ", determinant: " + Det_S);
Console.WriteLine();
Console.WriteLine("The eigenvalues of matrix A are: " + EV_A[0,0] + ", " + EV_A[1,0] + " and " + EV_A[2,0]);
Console.WriteLine("The eigenvalues of matrix S are: " + EV_S[0,0] + ", " + EV_S[1,0] + " and " + EV_S[2,0]);

The results of matrix A are: rank: 2, trace: 15, dterminant: 6.661338147750939E-16

The results of matrix S are: rank: 3, trace: 5, determinant: -7.999999999999998



The eigenvalues of matrix A are: 16.11684396980704, -1.116843969807043 and -1.303677726474702E-15

The eigenvalues of matrix S are: -1.999999999999999, 0.6277186767309848 and 6.372281323269013

Test for symmetry within Matlab using the BatchmodeConnector

We do the same test for symmetry for both matrices. In Matlab we can use the convenient command isequal:

Console.WriteLine("Calling MATLAB/Octave...");
BatchmodeConnector bmc = new BatchmodeConnector();
bmc.PutSparseMatrix(A, "Matrix_A");
bmc.PutSparseMatrix(S, "Matrix_S");
bmc.Cmd("Full_A = full(Matrix_A)");
bmc.Cmd("Full_S = full(Matrix_S)");
bmc.Cmd("A_Transpose = transpose(Full_A)");
bmc.Cmd("S_Transpose = transpose(Full_S)");
bmc.Cmd("SymmTest_A = isequal(Full_A, A_Transpose)");
bmc.Cmd("SymmTest_S = isequal(Full_S, S_Transpose)");
 
MultidimensionalArray SymmTest_A = MultidimensionalArray.Create(1, 1);
bmc.GetMatrix(SymmTest_A, "SymmTest_A");
MultidimensionalArray SymmTest_S = MultidimensionalArray.Create(1, 1);
bmc.GetMatrix(SymmTest_S, "SymmTest_S");
bmc.Execute(false);
Console.WriteLine("MATLAB/Octave closed, return to BoSSSPad");

Calling MATLAB/Octave...

MATLAB/Octave closed, return to BoSSSPad

if(SymmTest_A[0,0] == 1){
Console.WriteLine("Matrix A seems to be symmetric.");
}
else{
Console.WriteLine("Matrix A seems NOT to be symmetric.");
}    
if(SymmTest_S[0,0] == 1){
Console.WriteLine("Matrix S seems to be symmetric.");
}
else{
Console.WriteLine("Matrix S seems NOT to be symmetric.");
}

Matrix A seems NOT to be symmetric.

Matrix S seems to be symmetric.