tufte-latex: Full-width algorithm2e blocks

Question

I'm using documentclass{tufte-book} and trying to get a full-width algorithm environment (specifically, using algorithm2e).
Minimum working example:
documentclass{tufte-book}

title{Full-width algorithm2e algorithms}

usepackage{lipsum} % for dummy text only

usepackage[lined, ruled]{algorithm2e}

begin{document}

lipsum[2]

begin{algorithm}[H]
    tcp{Main loop}
    While{not converged}{
        $a = a + 1$ ;
    }
    caption{A text-width algorithm}
end{algorithm}

lipsum[2]

marginnote[10pt]{This algorithm should be full-width, but isn't}
begin{fullwidth}
begin{algorithm}[H]
    tcp{Main loop}
    While{not converged}{
        $a = a + 1$ ;
    }
    caption{A full-width algorithm}
end{algorithm}
end{fullwidth}

lipsum[2]

end{document}

Which renders:

In the algorithm2e.sty source-code, I found the dimension algowidth defined about half way through the document.
%[snip]
newdimenalgowidth%
%[snip]

So I've tried adjusting this length using setlength{algowidth}{XXX} in my example, but it doesn't seem to have any effect.

How can I achieve full-width algorithm2e algorithm blocks with tufte-latex?
Bonus marks if algorithm* can be adjusted to be full-width by default.
Extra bonus marks if the algorithm caption can also be moved to hang in the margin, as is the style with tufte-latex full width figures and tables.

Possibly related questions that don't seem to answer this:

Tufte-latex: Full-width table
Adjust width of algorithm float
Set width of algorithm environment (preferably document wide)

leandriis · Accepted Answer

This might not be the most ideal solution since you loose the floating behaviour of the algorithmby enclosing it in a minipage, but you nevertheless already  deactivated the float mechanism by adding [H]...

documentclass{tufte-book}

title{Full-width algorithm2e algorithms}

usepackage{lipsum} % for dummy text only

usepackage[lined, ruled]{algorithm2e}

newenvironment{widealgorithm}{begin{fullwidth}
begin{minipage}{linewidth}
begin{algorithm}[H]}{end{algorithm}
end{minipage}
end{fullwidth}}

begin{document}

lipsum[2]

begin{algorithm}[H]
    tcp{Main loop}
    While{not converged}{
        $a = a + 1$ ;
    }
    caption{A text-width algorithm}
end{algorithm}

lipsum[2]

begin{fullwidth}
begin{minipage}{linewidth}
begin{algorithm}[H]
    tcp{Main loop}
    While{not converged}{
        $a = a + 1$ ;
    }
    caption{A full-width algorithm}
end{algorithm}
end{minipage}
end{fullwidth}

lipsum[2]

begin{widealgorithm}
    tcp{Main loop}
    While{not converged}{
        $a = a + 1$ ;
    }
    caption{A full-width algorithm}
end{widealgorithm}

end{document}

m00am · Answer

To solve this issue also for floating algorithms (I had one that was also a whole page) you can define a new float and put the minipage into that float. This is a variation of this answer. If I interpret this correctly, this should require the float package, which tufte-latex loads for you.
% Reusing the preamble of leandriis' excellent answer
documentclass{tufte-book}
title{Floaty Full-width algorithm2e algorithms}
usepackage{lipsum} % for dummy text only
usepackage[lined, ruled]{algorithm2e}

% Define a new float with name myalgo
% Default placement tbp and a file extension mya (for temp files)
% Cf. https://ctan.mc1.root.project-creative.net/macros/latex/contrib/float/float.pdf
newfloat{myalgo}{tbp}{mya}

% create a new environment called floatyalgorithm with one parameter (our algorithm code)
% Using this creates a new float through the myalgo float we defined above,
% which is filled with a minipage of full page width.
% The width is defined as the sum of a line, the margin and the sapce in between (since the fullwidth environment cannot help us here).
% For the variables names cf. https://www.overleaf.com/learn/latex/page_size_and_margins
% As in leandriis' answer, use [H] for the algorithm to fill the space.
newenvironment{floatyalgorithm}[1][tbp]%
{begin{myalgo}[#1]
begin{minipage}{linewidth+marginparsep+marginparwidth}
begin{algorithm}[H]}%
{end{algorithm}
end{minipage}
end{myalgo}
}

begin{document}

begin{floatyalgorithm}[tbp]
  $m := m'$;
  caption{lipsum[2]}
end{floatyalgorithm}

end{document}

I have not extensively tested this. For my use case it worked nicely, but there might be some unwanted interactions with full pages and the content of the margin.

tufte-latex: Full-width algorithm2e blocks

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