# C# Hermitian Band Matrix Example

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```using System;

using CenterSpace.NMath.Core;

namespace CenterSpace.NMath.Examples.CSharp
{
/// <summary>
/// A .NET example in C# demonstrating the features of the banded Hermitian matrix classes.
/// </summary>
class HermitianBandMatrixExample
{

static void Main( string[] args )
{
// Set up the parameters that describe the shape of a Hermitian banded matrix.
int halfBandwidth = 1;
int order = 7;

// Set up an Hermitian banded matrix B by creating a banded complex matrix and setting
// B equal to the product of the conjugate transpose of that matrix with itself.
var A = new FloatComplexBandMatrix( order, order, halfBandwidth, halfBandwidth );
for ( int i = -A.LowerBandwidth; i <= A.UpperBandwidth; ++i )
{
A.Diagonal( i ).Set( Slice.All, new FloatComplex( i + 2, i - 1 ) );
}
var B = new FloatHermitianBandMatrix( MatrixFunctions.Product( A.ConjTranspose(), A ) );

Console.WriteLine();

Console.WriteLine( "B = " );
Console.WriteLine( B.ToTabDelimited( "G3" ) );

// B =
// (10,0)  (10,6)  (3,6)   (0,0)   (0,0)   (0,0)   (0,0)
// (10,-6) (19,0)  (10,6)  (3,6)   (0,0)   (0,0)   (0,0)
// (3,-6)  (10,-6) (19,0)  (10,6)  (3,6)   (0,0)   (0,0)
// (0,0)   (3,-6)  (10,-6) (19,0)  (10,6)  (3,6)   (0,0)
// (0,0)   (0,0)   (3,-6)  (10,-6) (19,0)  (10,6)  (3,6)
// (0,0)   (0,0)   (0,0)   (3,-6)  (10,-6) (19,0)  (10,6)
// (0,0)   (0,0)   (0,0)   (0,0)   (3,-6)  (10,-6) (14,0)

// Indexer accessor works just like it does for general matrices.
Console.WriteLine( "B[2,2] = {0}", B[2, 2] );
Console.WriteLine( "B[5,0] = {0}", B[5, 0] );

// You can set the values of elements in the bandwidth
// of an Hermitian banded matrix using the indexer. Note that setting
// the element in row i and column j to a value implicitly sets the
// element in column j and row i to the complex conjugate of that value.
var z = new FloatComplex( 99, -99 );
B[2, 1] = z;
Console.WriteLine( "B[2,1] = {0}", B[2, 1] ); // (99, -99)
Console.WriteLine( "B[1,2] = {0}", B[1, 2] ); // (99, 99 )

// But setting an element outside the bandwidth of the
// matrix raises a NonModifiableElementException exception.
try
{
B[6, 0] = z;
}
catch ( NonModifiableElementException e )
{
Console.WriteLine();
Console.WriteLine( "NonModifiableElementException: {0}", e.Message );
}

// Scalar multiplication and matrix addition/subtraction are supported.
z = new FloatComplex( .0001, .0001 );
FloatHermitianBandMatrix C = z * B;
FloatHermitianBandMatrix D = C - B;
Console.WriteLine();
Console.WriteLine( "D =" );
Console.WriteLine( D.ToTabDelimited( "F5" ) );

// Matrix/vector inner products.
var rng = new RandGenUniform( -1, 1 );
rng.Reset( 0x124 );
var x = new FloatComplexVector( B.Cols, rng );
FloatComplexVector y = MatrixFunctions.Product( B, x );
Console.WriteLine();
Console.WriteLine( "Bx = {0}", y.ToString() );

// You can transform the non-zero elements of a banded matrix object by using
// the Transform() method on its data vector.
C.DataVector.Transform( NMathFunctions.FloatComplexExpFunc );
Console.WriteLine();
Console.WriteLine( "exp(C) = " );
Console.WriteLine( C.ToTabDelimited( "F5" ) );

// You can also solve linear systems.
FloatComplexVector x2 = MatrixFunctions.Solve( B, y );

// x and x2 should be the same. Lets look at the l2 norm of
// their difference.
FloatComplexVector residual = x - x2;
double residualL2Norm = Math.Sqrt( NMathFunctions.ConjDot( residual, residual ).Real );
Console.WriteLine();
Console.WriteLine( "||x - x2|| = {0}", residualL2Norm );
Console.WriteLine();

// You can calculate inverses too.
FloatComplexMatrix BInv = MatrixFunctions.Inverse( B );
Console.WriteLine( "BInv = " );
Console.WriteLine( BInv.ToTabDelimited( "F5" ) );

// If your Hermitian banded matrix is positive definite, you can invoke
// the Solve function with a third - the isPositiveDefinite
// parameter - set to true.
// First make sure B is positive definite
B = new FloatHermitianBandMatrix( MatrixFunctions.Product( A.ConjTranspose(), A ) );
y = MatrixFunctions.Product( B, x );
x2 = MatrixFunctions.Solve( B, y, true ); // 3rd parameter isPositiveDefinite set to true.

// Lets see how close Bx is to y by computing the l2 norm of their difference.
residual = x - x2;
residualL2Norm = Math.Sqrt( NMathFunctions.ConjDot( residual, residual ).Real );
Console.WriteLine();
Console.WriteLine( "PD ||x - x2|| = {0}", residualL2Norm );

// You can use the Resize() method to change the bandwidths.
D.Resize( D.Order, 2 );

Console.WriteLine();
Console.WriteLine( "Press Enter Key" );
Console.Read();
}
}
}

```
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