Problem 81. Mandelbrot Numbers

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Mandelbrot_set\"><w:r><w:t>Mandelbrot Set</w:t></w:r></w:hyperlink><w:r><w:t> is built around a simple iterative equation.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ z(1) = c\n z(n+1) = z(n)^2 + c]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For any complex c, we can continue this iteration until either abs(z(n+1)) &gt; 2 or n == lim, then return the iteration count n.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>If c = 0 and lim = 3, then z = [0 0 0] and n = 3.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>If c = 1 and lim = 5, then z = [1 2], and n = length(z) or 2.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>If c = 0.5 and lim = 5, then z = [0.5000 0.7500 1.0625 1.6289] and n = 4.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For a matrix of complex numbers C, return a corresponding matrix N such that each element of N is the iteration count n for each complex number c in the matrix C, subject to the iteration count limit of lim.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>If C = [0 0.5; 1 4] and lim = 5, then N = [5 4; 2 1]</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Cleve Moler has a whole chapter on the Mandelbrot set in his book Experiments with MATLAB:</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://www.mathworks.com/moler/exm/chapters/mandelbrot.pdf\"><w:r><w:t>Chapter 10, Mandelbrot Set (PDF)</w:t></w:r></w:hyperlink></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

The <a href="http://en.wikipedia.org/wiki/Mandelbrot_set">Mandelbrot Set</a> is built around a simple iterative equation.<pre> z(1) = c
 z(n+1) = z(n)^2 + c</pre>For any complex c, we can continue this iteration until either
abs(z(n+1)) > 2 or n == lim, then return the iteration count n.<ul><li>If c = 0 and lim = 3, then z = [0 0 0] and n = 3.</li><li>If c = 1 and lim = 5, then z = [1 2], and n = length(z) or 2.</li><li>If c = 0.5 and lim = 5, then z = [0.5000 0.7500 1.0625 1.6289] and n = 4.</li></ul>For a matrix of complex numbers C, return a corresponding matrix N such
that each element of N is the iteration count n for each complex number c
in the matrix C, subject to the iteration count limit of lim.If C = [0 0.5; 1 4] and lim = 5, then N = [5 4; 2 1]Cleve Moler has a whole chapter on the Mandelbrot set in his book Experiments
with MATLAB: <a href="http://www.mathworks.com/moler/exm/chapters/mandelbrot.pdf">Chapter 10, Mandelbrot Set (PDF)</a>

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Last Solution submitted on Sep 25, 2020

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Nicholas Howe on 5 Oct 2012

For c==4 and other numbers where abs(c)>2, I think the function should be defined to return 0 rather than 1.

Sam Perry on 13 Dec 2017

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Ratchet_Hamster on 20 Jul 2018

I am really stuck on this one. I know it is likley me not the question but I cannot figure out why in the final validation, -2i should give N=1, I get it to be N=2? any help is appriciated, this is the only situlation where code fails.

rolf harkes on 6 Jan 2019

For people like me that hoped this challenge would end with a pretty picture:
`[X,Y]=meshgrid(-2:0.0025:2,-2:0.0025:2);C=X+i.*Y;N=mandelbrot(C,50);imagesc(N)`

Asif Newaz on 24 Dec 2019

@Ratchet_Hamster
for complex no, u need to take the absolute value to check if it is greater than 2

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Problem 81. Mandelbrot Numbers

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