<div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="LibSSH2_jll                        (7) |        started at 2020-12-17T15:29:21.241
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LibSSH2_jll                        (7) |     1.00 |   0.00 |  0.0 |       0.90 |  1581.01
LibUV_jll                          (7) |        started at 2020-12-17T15:29:22.249
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Libdl                              (7) |        started at 2020-12-17T15:29:22.951
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��� Warning: Package LibGit2_jll does not have nghttp2_jll in its dependencies:
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��� Loading nghttp2_jll into LibGit2_jll from project dependency, future warnings for LibGit2_jll are suppressed."><pre class="notranslate"><code class="notranslate">LibSSH2_jll                        (7) |        started at 2020-12-17T15:29:21.241
      From worker 2:	��� Warning: Package LibGit2_jll does not have nghttp2_jll in its dependencies:
      From worker 2:	��� - If you have LibGit2_jll checked out for development and have
      From worker 2:	���   added nghttp2_jll as a dependency but haven't updated your primary
      From worker 2:	���   environment's manifest file, try `Pkg.resolve()`.
      From worker 2:	��� - Otherwise you may need to report an issue with LibGit2_jll
      From worker 2:	��� Loading nghttp2_jll into LibGit2_jll from project dependency, future warnings for LibGit2_jll are suppressed.
LibSSH2_jll                        (7) |     1.00 |   0.00 |  0.0 |       0.90 |  1581.01
LibUV_jll                          (7) |        started at 2020-12-17T15:29:22.249
      From worker 2:	��� Warning: Module MbedTLS_jll with build ID 45447383554825846 is missing from the cache.
      From worker 2:	��� This may mean MbedTLS_jll [c8ffd9c3-330d-5841-b78e-0817d7145fa1] does not support precompilation but is imported by a module that does.
      From worker 2:	��� @ Base loading.jl:1003
LibUV_jll                          (7) |     0.42 |   0.00 |  0.0 |       1.74 |  1581.01
LibUnwind_jll                      (7) |        started at 2020-12-17T15:29:22.670
LibUnwind_jll                      (7) |     0.28 |   0.00 |  0.0 |       0.29 |  1581.01
Libdl                              (7) |        started at 2020-12-17T15:29:22.951
��� Warning: The call to compilecache failed to create a usable precompiled cache file for LibGit2_jll [e37daf67-58a4-590a-8e99-b0245dd2ffc5]
���   exception = Required dependency MbedTLS_jll [c8ffd9c3-330d-5841-b78e-0817d7145fa1] failed to load from a cache file.
��� @ Base loading.jl:1028
��� Warning: Package LibGit2_jll does not have nghttp2_jll in its dependencies:
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���   added nghttp2_jll as a dependency but haven't updated your primary
���   environment's manifest file, try `Pkg.resolve()`.
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</code></pre></div>
<p dir="auto">Seems to be partially a duplicate of <a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="749047464" data-permission-text="Title is private" data-url="https://github.com/JuliaLang/julia/issues/38545" data-hovercard-type="issue" data-hovercard-url="/JuliaLang/julia/issues/38545/hovercard" href="https://github.com/JuliaLang/julia/issues/38545">#38545</a> (caused by <a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="338280425" data-permission-text="Title is private" data-url="https://github.com/JuliaLang/julia/issues/27932" data-hovercard-type="pull_request" data-hovercard-url="/JuliaLang/julia/pull/27932/hovercard" href="https://github.com/JuliaLang/julia/pull/27932">#27932</a>). Possibly means <a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="725936964" data-permission-text="Title is private" data-url="https://github.com/JuliaLang/julia/issues/38116" data-hovercard-type="issue" data-hovercard-url="/JuliaLang/julia/issues/38116/hovercard" href="https://github.com/JuliaLang/julia/issues/38116">#38116</a> needs to be reopened (again)?</p>
<p dir="auto"><a href="https://build.julialang.org/#/builders/4/builds/7130" rel="nofollow">https://build.julialang.org/#/builders/4/builds/7130</a></p>
<p dir="auto">Higham and Deadman have published a <a href="http://eprints.ma.man.ac.uk/2102/" rel="nofollow">list of software</a> implementing algorithms for matrix functions, showing that Julia's library for these functions could be improved greatly relative to the state of the art.</p>
<p dir="auto">The purpose of this issue is to discuss and track the implementation of matrix functions.</p>
<h2 dir="auto">From Higham and Deadman's list:</h2>
<h3 dir="auto">General function evaluations</h3>
<ul class="contains-task-list">
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> General matrix function with derivatives of the underlying scalar function available: Schur–Parlett algorithm (Davies and Higham, 2003)</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> Function of a symmetric or Hermitian matrix by diagonalization</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> Condition number estimate for general functions: <code class="notranslate">cond(f,A)</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> matrix function - vector product evaluation: <code class="notranslate">f(A) * b</code></li>
</ul>
<h3 dir="auto">Specific functions</h3>
<ul class="contains-task-list">
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox" checked=""> <code class="notranslate">expm</code>: scaling and squaring algorithm (Al-Mohy and Higham, 2009)</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox" checked=""> <code class="notranslate">logm</code>: inverse scaling and squaring algorithm (Al-Mohy, Higham, and Relton, 2012,<br>
2013)</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">sqrtm</code>:
<ul class="contains-task-list">
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox" checked=""> Schur algorithm (Björck and Hammarling, 1983)</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> real version for real matrices (Higham, 1987)</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> blocking algorithm for performance improvements (Deadman, Higham, and Ralha, 2013)</li>
</ul>
</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox" checked=""> <code class="notranslate">A^t</code> for real matrix powers: Schur– Padé algorithm (Higham and Lin, 2013) (<a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="217346763" data-permission-text="Title is private" data-url="https://github.com/JuliaLang/julia/issues/21184" data-hovercard-type="pull_request" data-hovercard-url="/JuliaLang/julia/pull/21184/hovercard" href="https://github.com/JuliaLang/julia/pull/21184">#21184</a>)</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> Matrix <a href="http://eprints.ma.man.ac.uk/2094/" rel="nofollow">unwinding function</a></li>
</ul>
<h3 dir="auto"><a href="http://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative" rel="nofollow">Fréchet derivatives</a></h3>
<ul class="contains-task-list">
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> Exponential: scaling and squaring algorithm (Al-Mohy and Higham, 2009)</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> Logarithm: inverse scaling and squaring algorithm (Al-Mohy, Higham, and Relton, 2013)</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> General matrix function: complex step algorithm (Al-Mohy and Higham, 2010) or using block 2 × 2 matrix formula.</li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">A^t</code> for real matrix powers: Schur– Padé algorithm (Higham and Lin, 2013)</li>
</ul>
<h2 dir="auto">Miscellaneous functions</h2>
<ul class="contains-task-list">
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox" checked=""> <code class="notranslate">inv</code> and <code class="notranslate">det</code> for <code class="notranslate">Tridiagonal</code> and <code class="notranslate">SymTridiagonal</code> matrices (Usmani, 1994) (<a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="25438685" data-permission-text="Title is private" data-url="https://github.com/JuliaLang/julia/issues/5358" data-hovercard-type="pull_request" data-hovercard-url="/JuliaLang/julia/pull/5358/hovercard" href="https://github.com/JuliaLang/julia/pull/5358">#5358</a>)</li>
</ul>