Liftwave

LaurentMat Class {2 X 2 Matrices}

Initialization

LaurentMat<double> matrix;

Associated Functions

void    setMat (Laurent< T > &AA, Laurent< T > &BB, Laurent< T > &CC, Laurent< T > &DD)
void    Det (Laurent< T > &oup)
void    MatAdd (LaurentMat &X, LaurentMat &Y)
void    MatSub (LaurentMat &X, LaurentMat &Y)
void    MatMult (LaurentMat &X, LaurentMat &Y)
void    TZ (Laurent< T > &t)
void    SZ (Laurent< T > &s)
void    dispMat ()
void    scale (T s)
void    MatInv (LaurentMat &Inv)
void    getLpoly (Laurent< T > &P, int N)

setMat sets the matrix. The Laurent Polynomials are entered as (1,1), (1,2),(2,1) and (2,2).

        Laurent<double> lpd,hpd,lpr,hpr;
        lpoly(name,lpd,hpd,lpr,hpr);
        
        LaurentMat<double> matrix;
        
        matrix.setMat(lpd,lpr,hpd,hpr);

Det finds the determinant of the matrix while you can use dispMat() to display the matrix.

        Laurent<double> determinant;
        matrix.Det(determinant);
        
        cout << "The Matrix : " << endl;
        matrix.dispMat();
        cout << endl;
        cout << " The Determinant : ";
        determinant.dispPoly();

The Output is

The Matrix :
 Laurent Matrix
(1,1) : -0.176777*z^(1)+0.353553*z^(0)+1.06066*z^(-1)+0.353553*z^(-2)-0.176777*z
^(-3)
(1,2) : 0.353553*z^(0)+0.707107*z^(-1)+0.353553*z^(-2)
(2,1) : 0.353553*z^(1)-0.707107*z^(0)+0.353553*z^(-1)
(2,2) : 0.176777*z^(2)+0.353553*z^(1)-1.06066*z^(0)+0.353553*z^(-1)+0.176777*z^(
-2)

 The Determinant : -0.03125*z^(3)+0*z^(2)+0.375*z^(1)+1.38778e-017*z^(0)-0.6875*
z^(-1)-1.38778e-017*z^(-2)+0.375*z^(-3)+0*z^(-4)-0.03125*z^(-5)

TZ and SZ functions set the T(Z) and S(Z) matrices (see the Sweldens/Daubechies paper).

        LaurentMat<double> T1,S1;
        T1.TZ(lpd);
        S1.SZ(hpd);
        cout << "T(Z) matrix : " << endl;
        T1.dispMat();
        cout << "S(Z) matrix : " << endl;
        S1.dispMat();

The Output

T(Z) matrix :
 Laurent Matrix
(1,1) : 1*z^(0)
(1,2) : 0*z^(0)
(2,1) : -0.176777*z^(1)+0.353553*z^(0)+1.06066*z^(-1)+0.353553*z^(-2)-0.176777*z
^(-3)
(2,2) : 1*z^(0)
S(Z) matrix :
 Laurent Matrix
(1,1) : 1*z^(0)
(1,2) : 0.353553*z^(1)-0.707107*z^(0)+0.353553*z^(-1)
(2,1) : 0*z^(0)
(2,2) : 1*z^(0)

Scaling and The Inverse Of a Matrix

I have used the most basic and very restrictive algorithm to invert a 2X2 matrix. It works only if the determinant is a mononomial. Otherwise it returns all zeros. Scaling by a scalar is straightforward as the name implies.

        T1.scale(2.2);
        cout << "T(Z) matrix Scaled by 2.2 : " << endl;
        T1.dispMat();
        
        LaurentMat<double> IS;
        
        S1.MatInv(IS);
        cout << "Inverse of S(Z) matrix  : " << endl;
        IS.dispMat();

The Output

T(Z) matrix Scaled by 2.2 :
 Laurent Matrix
(1,1) : 2.2*z^(0)
(1,2) : 0*z^(0)
(2,1) : -0.388909*z^(1)+0.777817*z^(0)+2.33345*z^(-1)+0.777817*z^(-2)-0.388909*z
^(-3)
(2,2) : 2.2*z^(0)
Inverse of S(Z) matrix  :
 Laurent Matrix
(1,1) : 1*z^(0)
(1,2) : -0.353553*z^(1)+0.707107*z^(0)-0.353553*z^(-1)
(2,1) : 0*z^(0)
(2,2) : 1*z^(0)

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Matrix Addition, Subtraction and Multiplication

MatAdd, MatSub and MatMult add, subtract and multiply two matrices respectively.

        LaurentMat<double> addm,subm,multm;
        
        addm.MatAdd(T1,S1);
        subm.MatSub(T1,S1);
        multm.MatMult(T1,S1);
        
        cout << " T(Z)+S(Z) = " << endl;
        addm.dispMat();
        cout << endl;
        
        cout << " T(Z)-S(Z) = " << endl;
        subm.dispMat();
        cout <<endl;
        
        cout << " T(Z)*S(Z) = " << endl;
        multm.dispMat();
        cout <<endl;

The Output

 T(Z)+S(Z) =
 Laurent Matrix
(1,1) : 3.2*z^(0)
(1,2) : 0.353553*z^(1)-0.707107*z^(0)+0.353553*z^(-1)
(2,1) : -0.388909*z^(1)+0.777817*z^(0)+2.33345*z^(-1)+0.777817*z^(-2)-0.388909*z
^(-3)
(2,2) : 3.2*z^(0)

 T(Z)-S(Z) =
 Laurent Matrix
(1,1) : 1.2*z^(0)
(1,2) : -0.353553*z^(1)+0.707107*z^(0)-0.353553*z^(-1)
(2,1) : -0.388909*z^(1)+0.777817*z^(0)+2.33345*z^(-1)+0.777817*z^(-2)-0.388909*z
^(-3)
(2,2) : 1.2*z^(0)

 T(Z)*S(Z) =
 Laurent Matrix
(1,1) : 2.2*z^(0)
(1,2) : 0.777817*z^(1)-1.55563*z^(0)+0.777817*z^(-1)
(2,1) : -0.388909*z^(1)+0.777817*z^(0)+2.33345*z^(-1)+0.777817*z^(-2)-0.388909*z
^(-3)
(2,2) : -0.1375*z^(2)+0.55*z^(1)+2.3375*z^(0)-1.1*z^(-1)+0.1375*z^(-2)+0.55*z^(-
3)-0.1375*z^(-4)


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A Simple Example demonstrating the use of LaurentMat class

This code also includes "filter.h" only because I am using a biorthogonal filter to set the matrix. You may not need to include "filter.h" usually if you are setting up matrices on your own.

#include <iostream>
#include <cmath>
#include <vector>
#include <string>
#include <algorithm>
#include "lwave.h"
#include "filter.h"


using namespace std;

int main()
{
        string name="bior2.2";
        Laurent<double> lpd,hpd,lpr,hpr;
        lpoly(name,lpd,hpd,lpr,hpr);
        
        LaurentMat<double> matrix;
        
        matrix.setMat(lpd,lpr,hpd,hpr);
        Laurent<double> determinant;
        matrix.Det(determinant);
        
        cout << "The Matrix : " << endl;
        matrix.dispMat();
        cout << endl;
        cout << " The Determinant : ";
        determinant.dispPoly();
        
        LaurentMat<double> T1,S1;
        T1.TZ(lpd);
        S1.SZ(hpd);
        cout << "T(Z) matrix : " << endl;
        T1.dispMat();
        cout << "S(Z) matrix : " << endl;
        S1.dispMat();
        
        T1.scale(2.2);
        cout << "T(Z) matrix Scaled by 2.2 : " << endl;
        T1.dispMat();
        
        LaurentMat<double> IS;
        
        S1.MatInv(IS);
        cout << "Inverse of S(Z) matrix  : " << endl;
        IS.dispMat();
        
        LaurentMat<double> addm,subm,multm;
        
        addm.MatAdd(T1,S1);
        subm.MatSub(T1,S1);
        multm.MatMult(T1,S1);
        
        cout << " T(Z)+S(Z) = " << endl;
        addm.dispMat();
        cout << endl;
        
        cout << " T(Z)-S(Z) = " << endl;
        subm.dispMat();
        cout <<endl;
        
        cout << " T(Z)*S(Z) = " << endl;
        multm.dispMat();
        cout <<endl;
        
        
        return 0;
}

The Output Follows

The Matrix :
 Laurent Matrix
(1,1) : -0.176777*z^(1)+0.353553*z^(0)+1.06066*z^(-1)+0.353553*z^(-2)-0.176777*z
^(-3)
(1,2) : 0.353553*z^(0)+0.707107*z^(-1)+0.353553*z^(-2)
(2,1) : 0.353553*z^(1)-0.707107*z^(0)+0.353553*z^(-1)
(2,2) : 0.176777*z^(2)+0.353553*z^(1)-1.06066*z^(0)+0.353553*z^(-1)+0.176777*z^(
-2)

 The Determinant : -0.03125*z^(3)+0*z^(2)+0.375*z^(1)+1.38778e-017*z^(0)-0.6875*
z^(-1)-1.38778e-017*z^(-2)+0.375*z^(-3)+0*z^(-4)-0.03125*z^(-5)
T(Z) matrix :
 Laurent Matrix
(1,1) : 1*z^(0)
(1,2) : 0*z^(0)
(2,1) : -0.176777*z^(1)+0.353553*z^(0)+1.06066*z^(-1)+0.353553*z^(-2)-0.176777*z
^(-3)
(2,2) : 1*z^(0)
S(Z) matrix :
 Laurent Matrix
(1,1) : 1*z^(0)
(1,2) : 0.353553*z^(1)-0.707107*z^(0)+0.353553*z^(-1)
(2,1) : 0*z^(0)
(2,2) : 1*z^(0)
T(Z) matrix Scaled by 2.2 :
 Laurent Matrix
(1,1) : 2.2*z^(0)
(1,2) : 0*z^(0)
(2,1) : -0.388909*z^(1)+0.777817*z^(0)+2.33345*z^(-1)+0.777817*z^(-2)-0.388909*z
^(-3)
(2,2) : 2.2*z^(0)
Inverse of S(Z) matrix  :
 Laurent Matrix
(1,1) : 1*z^(0)
(1,2) : -0.353553*z^(1)+0.707107*z^(0)-0.353553*z^(-1)
(2,1) : 0*z^(0)
(2,2) : 1*z^(0)
 T(Z)+S(Z) =
 Laurent Matrix
(1,1) : 3.2*z^(0)
(1,2) : 0.353553*z^(1)-0.707107*z^(0)+0.353553*z^(-1)
(2,1) : -0.388909*z^(1)+0.777817*z^(0)+2.33345*z^(-1)+0.777817*z^(-2)-0.388909*z
^(-3)
(2,2) : 3.2*z^(0)

 T(Z)-S(Z) =
 Laurent Matrix
(1,1) : 1.2*z^(0)
(1,2) : -0.353553*z^(1)+0.707107*z^(0)-0.353553*z^(-1)
(2,1) : -0.388909*z^(1)+0.777817*z^(0)+2.33345*z^(-1)+0.777817*z^(-2)-0.388909*z
^(-3)
(2,2) : 1.2*z^(0)

 T(Z)*S(Z) =
 Laurent Matrix
(1,1) : 2.2*z^(0)
(1,2) : 0.777817*z^(1)-1.55563*z^(0)+0.777817*z^(-1)
(2,1) : -0.388909*z^(1)+0.777817*z^(0)+2.33345*z^(-1)+0.777817*z^(-2)-0.388909*z
^(-3)
(2,2) : -0.1375*z^(2)+0.55*z^(1)+2.3375*z^(0)-1.1*z^(-1)+0.1375*z^(-2)+0.55*z^(-
3)-0.1375*z^(-4)


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