fixed point iteration and plotting iteration

I have a map such that ,

where is closed unit disc in and define

where ,

taking , ,

i want a matlab code whch calculates iteration for in a table format and plots these iteration in . i will be thankful if someone could help me with it.

[David Goodmanson]

Hi Akansha, the statement x_n+1 = (1-lambda)x_n + lambda x_n appears to be incorrect since the outcome is the uninteresting x_n+1 = x_n. Also, how does T enter into it?

[Akansha]

sorry my mistake, iteration is given by x_n+1 = (1-lambda)x_n + lambda T(x_n)

Answer 1

[Voss]

Assuming the sequence is defined as in your other question, here's something:

T = @(xy)xy([2 1]).*[-1 1];
n_iterations = 20;
lambda = [0.1 0.2 0.3 0.4];

n_lambda = numel(lambda);
result = cell(1,n_lambda);

for jj = 1:n_lambda
    
    xy = zeros(n_iterations,2);
    xy(1,:) = [0.5 1];
    for n = 1:n_iterations
        xy(n+1,:) = (1-lambda(jj))*xy(n,:) + lambda(jj)*T(xy(n,:));
    end
    
    result{jj} = array2table(xy,'VariableNames',{'x','y'},'RowNames',compose('%d',(0:n_iterations).'));
    
end
    
figure
hold on
for jj = 1:n_lambda
    plot(result{jj}.x,result{jj}.y)
end
legend("lambda = "+lambda,'Location','NorthWest')

result{:}

ans = 21×2 table
             x           y    
          ________    ________

    0          0.5           1
    1         0.35        0.95
    2         0.22        0.89
    3        0.109       0.823
    4       0.0158      0.7516
    5     -0.06094     0.67802
    6     -0.12265     0.60412
    7      -0.1708     0.53145
    8     -0.20686     0.46122
    9      -0.2323     0.39441
    10    -0.24851     0.33174
    11    -0.25683     0.27372
    12    -0.25852     0.22066
    13    -0.25473     0.17274
    14    -0.24654        0.13
    15    -0.23488    0.092343

ans = 21×2 table
              x            y    
          _________    _________

    0           0.5            1
    1           0.2          0.9
    2         -0.02         0.76
    3        -0.168        0.604
    4       -0.2552       0.4496
    5      -0.29408      0.30864
    6      -0.29699       0.1881
    7      -0.27521     0.091078
    8      -0.23839      0.01782
    9      -0.19427    -0.033421
    10     -0.14873    -0.065591
    11     -0.10587     -0.08222
    12    -0.068251     -0.08695
    13    -0.037211     -0.08321
    14    -0.013127     -0.07401
    15    0.0043006    -0.061834

ans = 21×2 table
              x            y    
          _________    _________

    0           0.5            1
    1          0.05         0.85
    2         -0.22         0.61
    3        -0.337        0.361
    4       -0.3442       0.1516
    5      -0.28642      0.00286
    6      -0.20135    -0.083924
    7      -0.11577     -0.11915
    8     -0.045293     -0.11814
    9     0.0037363    -0.096284
    10     0.031501    -0.066278
    11     0.041934    -0.036944
    12     0.040437    -0.013281
    13      0.03229    0.0028345
    14     0.021753     0.011671
    15     0.011726     0.014696

ans = 21×2 table
              x             y     
          __________    __________

    0            0.5             1
    1           -0.1           0.8
    2          -0.38          0.44
    3         -0.404         0.112
    4        -0.2872       -0.0944
    5       -0.13456      -0.17152
    6      -0.012128      -0.15674
    7       0.055418     -0.098893
    8       0.072808     -0.037169
    9       0.058552     0.0068219
    10      0.032402      0.027514
    11     0.0084359      0.029469
    12    -0.0067262      0.021056
    13     -0.012458     0.0099431
    14     -0.011452    0.00098262
    15    -0.0072643    -0.0039913

[Akansha]

Thankyou so much for the help

[Voss]

You're welcome!

[Manahel]

How can I save both the graph and the iteration result tables into a PDF file in MATLAB? Also, where can we find an explanation for each step in the code so we can understand it better