# VB Hermitian Band Matrix Example

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```Imports System

Imports CenterSpace.NMath.Core

Namespace CenterSpace.NMath.Examples.VisualBasic

A .NET example in Visual Basic demonstrating the features of the banded Hermitian matrix classes.
Module HermitianBandMatrixExample

Sub Main()

Set up the parameters that describe the shape of a Hermitian banded matrix.
Dim HalfBandwidth As Integer = 1
Dim Order As Integer = 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.
Dim A As New FloatComplexBandMatrix(Order, Order, HalfBandwidth, HalfBandwidth)
Dim I As Integer
For I = -A.LowerBandwidth To A.UpperBandwidth
A.Diagonal(I).Set(Slice.All, New FloatComplex(I + 2, I - 1))
Next
Dim B As 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.Write("B[2,2] = ")
Console.WriteLine(B(2, 2))
Console.Write("B[5,0] = ")
Console.WriteLine(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.
Dim Z As New FloatComplex(99, -99)
B(2, 1) = Z
Console.Write("B[2,1] = ")
Console.WriteLine(B(2, 1)) (99, -99)
Console.Write("B[1,2] = ")
Console.WriteLine(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 E As NonModifiableElementException
Console.WriteLine()
Console.WriteLine("NonModifiableElementException: " & E.Message)
End Try

Scalar multiplication and matrix addition/subtraction are supported.
Z = New FloatComplex(0.0001, 0.0001)
Dim C As FloatHermitianBandMatrix = Z * B
Dim D As FloatHermitianBandMatrix = C - B
Console.WriteLine()
Console.WriteLine("D =")
Console.WriteLine(D.ToTabDelimited("F5"))

Matrix/vector inner products too.
Dim Rng As New RandGenUniform(-1, 1)
Rng.Reset(&H124)
Dim X As New FloatComplexVector(B.Cols, Rng)
Dim Y As FloatComplexVector = MatrixFunctions.Product(B, X)
Console.Write("Bx = ")
Console.WriteLine(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.
Dim X2 As FloatComplexVector = MatrixFunctions.Solve(B, Y)

x and x2 should be the same. Lets look at the l2 norm of
their difference.
Dim Residual As FloatComplexVector = FloatComplexVector.Subtract(X, X2)
Dim ResidualL2Norm As Double = Math.Sqrt(NMathFunctions.ConjDot(Residual, Residual).Real)
Console.Write("||x - x2|| = ")
Console.WriteLine(ResidualL2Norm)

You can calculate inverses too.
Dim BInv As FloatComplexMatrix = MatrixFunctions.Inverse(B)
Console.WriteLine()
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.

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.Write("PD ||x - x2|| = ")
Console.WriteLine(ResidualL2Norm)

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

Console.WriteLine()
Console.WriteLine("Press Enter Key")