# ConvergenceDemo

## PURPOSE

ConvergenceDemo(SourceSwitch);

## SYNOPSIS

function ConvergenceDemo(SourceSwitch);

## DESCRIPTION

```ConvergenceDemo(SourceSwitch);

ConvergenceDemo(SourceSwitch) compares the polynomial EVD approximations by
the functions SBR2() and SMD() for different types of parahermitian matrices.
The demo provides the reduction in off-diagonal energy in increments of 5
iteration steps.

Theinput parameter SourceSwitch can be selected as:

ComparisonDemo('Random') uses an arbitrary complex valued 5x5 polynomial
parahermitian matrix of order 10.

ComparisonDemo('Simple1') compares decompositions of a 3x3 real valued
polynomial parahermitian matrix R(z) of order 4:
| 1     -.4*z       0     |
R(z) =   | -.4*z   1     .5*z^{-2} |
| 0     .5*z^2      3     |

ComparisonDemo('Simple2') compares the decompositions of a 5x5 complex valued
polynomial parahermitian matrix R(z) of order 4:
|    1        .5*z^2  -.4j*z^{-1}    0   .2j*z^2 |
| .5*z^{-2}      1         0         0       0   |
R(z) =   |   .4j*z        0         3      .3z^{-1}   0   |
|    0           0       .3*z       .5       0   |
|-.2j*z^{-2}     0         0         0      .25  |
This matrix admits a PEVD with a non-polynomial Gamma(z).

The function generates a plot showing the remaining off-diagonal power,
normalised w.r.t. the total power of the matrix; this indicates how well the
two algorithms perform their diagonalisation task. A second figure compares
the off-diagonal power in relation to the order of the paraunitary matrices
required in order to accomplish this decomposition.

Input parameter:
SourceSwitch       selects parahermitian matrx to be decomposed
default: 'Random'
Output parameter: none```

## CROSS-REFERENCE INFORMATION

This function calls:
• SBR2 [H,Gamma] = SBR2(R,maxiter,epsilon,mu,vers);
• SMD [H,Gamma] = SMD(R,maxiter,epsilon,Mu,vers);
• ParaHerm H = ParaHerm(G)
• PolyMatConv H = PolyMatConv(F,G);
• PolyMatNorm N = PolyMatNorm(H,spec);
This function is called by:

## SOURCE CODE

```0001 function ConvergenceDemo(SourceSwitch);
0002 %ConvergenceDemo(SourceSwitch);
0003 %
0004 %  ConvergenceDemo(SourceSwitch) compares the polynomial EVD approximations by
0005 %  the functions SBR2() and SMD() for different types of parahermitian matrices.
0006 %  The demo provides the reduction in off-diagonal energy in increments of 5
0007 %  iteration steps.
0008 %
0009 %  Theinput parameter SourceSwitch can be selected as:
0010 %
0011 %  ComparisonDemo('Random') uses an arbitrary complex valued 5x5 polynomial
0012 %  parahermitian matrix of order 10.
0013 %
0014 %  ComparisonDemo('Simple1') compares decompositions of a 3x3 real valued
0015 %  polynomial parahermitian matrix R(z) of order 4:
0016 %                | 1     -.4*z       0     |
0017 %       R(z) =   | -.4*z   1     .5*z^{-2} |
0018 %                | 0     .5*z^2      3     |
0019 %
0020 %  ComparisonDemo('Simple2') compares the decompositions of a 5x5 complex valued
0021 %  polynomial parahermitian matrix R(z) of order 4:
0022 %                |    1        .5*z^2  -.4j*z^{-1}    0   .2j*z^2 |
0023 %                | .5*z^{-2}      1         0         0       0   |
0024 %       R(z) =   |   .4j*z        0         3      .3z^{-1}   0   |
0025 %                |    0           0       .3*z       .5       0   |
0026 %                |-.2j*z^{-2}     0         0         0      .25  |
0027 %  This matrix admits a PEVD with a non-polynomial Gamma(z).
0028 %
0029 %  The function generates a plot showing the remaining off-diagonal power,
0030 %  normalised w.r.t. the total power of the matrix; this indicates how well the
0031 %  two algorithms perform their diagonalisation task. A second figure compares
0032 %  the off-diagonal power in relation to the order of the paraunitary matrices
0033 %  required in order to accomplish this decomposition.
0034 %
0035 %  Input parameter:
0036 %       SourceSwitch       selects parahermitian matrx to be decomposed
0037 %                          default: 'Random'
0038 %  Output parameter: none
0039
0040 % S. Weiss, University of Southampton, 8/10/2014
0041
0042 if nargin==0,
0043    SourceSwitch='Random';
0044 end;
0045
0046 disp('SBR2 and SMD Convergence Comparison Demo');
0047 disp('--------------------------------------------');
0048
0049 %---------------------------------------
0050 %  Define Scenario
0051 %---------------------------------------
0052 % create parahermitian matrix to be decomposed
0053 if strcmp(SourceSwitch,'Random')==1,
0054    A = randn(5,5,6) + sqrt(-1)*randn(5,5,6);;
0055    R = PolyMatConv(A,ParaHerm(A));
0056 elseif strcmp(SourceSwitch,'Simple1')==1,
0057    R        = zeros(3,3,5);
0058    R(:,:,3) = diag([1 1 3]);
0059    R(1,2,1) = .5;
0060    R(2,1,5) = .5;
0061    R(1,2,2) = -.4;
0062    R(2,1,4) = -.4;
0063 elseif strcmp(SourceSwitch,'Simple2')==1,
0064    R        = zeros(5,5,5);
0065    R(:,:,3) = diag([1 1 3 .5 .25]);
0066    R(1,2,1) = .5;
0067    R(2,1,5) = .5;
0068    R(1,3,4) = -.4*sqrt(-1);
0069    R(3,1,2) = .4*sqrt(-1);
0070    R(1,5,1) = .2*sqrt(-1);
0071    R(5,1,5) = -.2*sqrt(-1);
0072    R(4,3,2) = .3;
0073    R(3,4,4) = .3;
0074 else
0075    error('option for input parameter SourceSwitch not implemented');
0076 end;
0077
0078 %---------------------------------------
0079 %  Set incremental parameters
0080 %---------------------------------------
0081 % parameters
0082 maxiter = 5;                  % max number of iterations per update
0083 blocks  = 20;
0084 epsilon = 10^(-10);           % alternative stopping criterion
0085                               %    ensure that this is not invoked
0086 mu = 0;                       % truncate true zeroes
0087 GammaSMD = R;                 % initialisations
0088 GammaSBR2 = R;
0089 N1 = PolyMatNorm(R);
0090 N2 = PolyMatNorm(R,'OffDiag');
0091 OffDiagNormSMD(1) = N2/N1;    % normalised remaining off-diag. energy
0092 OffDiagNormSBR2(1) = N2/N1;
0093 OrderSMD(1)=0;
0094 OrderSBR2(1)=0;
0095
0096 %---------------------------------------
0097 %  Recursively decompose in blocks of 5 iterations
0098 %---------------------------------------
0099 for n = 2:blocks,
0100    % SMD iteration
0101    [H,GammaSMD] = SMD(GammaSMD,maxiter,epsilon,mu,'SMD');
0102    OffDiagNormSMD(n) = PolyMatNorm(GammaSMD,'OffDiag')/N1;
0103    OrderSMD(n) = OrderSMD(n-1)+size(H,3)-1;
0104    % SBR2 iteration
0105    [H,GammaSBR2] = SBR2(GammaSBR2,maxiter,epsilon,mu,'SBR2');
0106    OffDiagNormSBR2(n) = PolyMatNorm(GammaSBR2,'OffDiag')/N1;
0107    OrderSBR2(n) = OrderSBR2(n-1)+size(H,3)-1;
0108 end;
0109
0110 %---------------------------------------
0111 %  Display resulting matrices
0112 %---------------------------------------
0113 % display normalised remaining off-diagonal power vs iterations
0114 figure(1); clf;
0115 plot(1:maxiter:maxiter*blocks,5*log10(OffDiagNormSBR2),'bo');
0116 hold on;
0117 plot(1:maxiter:maxiter*blocks,5*log10(OffDiagNormSMD),'r*');
0118 legend('SBR2','SMD');
0119 xlabel('iterations');
0120 ylabel('norm. remaining off-diagonal power / [dB]');
0121 disp('Figure 1: norm. remaining off-diagonal power vs iterations');
0122
0123 % display normalised remaining off-diagonal power vs paraunitary order
0124 figure(2); clf;
0125 plot(OrderSBR2,5*log10(OffDiagNormSBR2),'bo');
0126 hold on;
0127 plot(OrderSMD,5*log10(OffDiagNormSMD),'r*');
0128 legend('SBR2','SMD');
0129 xlabel('paraunitary order (without truncation)');
0130 ylabel('norm. remaining off-diagonal power / [dB]');
0131 disp('Figure 1: norm. remaining off-diagonal power vs paraunitary order');```

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