{"id":3830,"date":"2025-08-05T07:06:19","date_gmt":"2025-08-05T11:06:19","guid":{"rendered":"https:\/\/blogs.mathworks.com\/matlab\/?p=3830"},"modified":"2025-08-05T07:06:19","modified_gmt":"2025-08-05T11:06:19","slug":"the-fastest-way-to-solve-small-linear-systems-in-matlab","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/matlab\/2025\/08\/05\/the-fastest-way-to-solve-small-linear-systems-in-matlab\/","title":{"rendered":"The fastest way to solve small linear systems in MATLAB"},"content":{"rendered":"
While browsing MATLAB Answers today, I stumbled across <\/span>a discussion about the fastest way to solve small linear systems in MATLAB<\/span><\/a>. As a performance-geek, I usually obsess over very large matrices but MathWorks has been taking an interest in small matrices recently so this discussion was very timely. <\/span><\/div>
One of MathWorks' linear algebra specialists, <\/span>Christine Tobler<\/span><\/a>, gave a detailed reply that contained a lot of useful information that I felt deserved to be disseminated more widely. Hence this post.<\/span><\/div>

The conventional wisdom: Use backslash (mldivide) not inv<\/span><\/h2>
As MATLAB community member <\/span>AB<\/span><\/a> correctly points out, <\/span>y = A\\x<\/span> is usually more accurate than y<\/span> = inv(A)*x<\/span>. For large matrices, it's usually faster too and I have lost count of the number of times I have improved the performance of someone's code by demonstrating this. Let's start by constructing a large-ish random problem<\/span> <\/span><\/div>
A = randn(2000);<\/span><\/span><\/div><\/div>
A = A*A'; <\/span>% Construct an Hermitian matrix from A<\/span><\/span><\/div><\/div>
x = randn(2000,1);<\/span><\/span><\/div><\/div><\/div>
Solving this with <\/span>inv<\/span> doesn't seem too slow<\/span><\/div>
tic<\/span><\/span><\/div><\/div>
yInv = inv(A)*x;<\/span><\/span><\/div><\/div>
invTime = toc<\/span><\/span><\/div>
invTime = <\/span>0.6861<\/div><\/div><\/div><\/div><\/div><\/div>
until you see that backslash is much faster<\/span><\/div>
tic<\/span><\/span><\/div><\/div>
yBackslash = A\\x;<\/span><\/span><\/div><\/div>
backSlashTime = toc<\/span><\/span><\/div>
backSlashTime = <\/span>0.1708<\/div><\/div><\/div><\/div><\/div>
fprintf(<\/span>\"For a large matrix, \\\\ is %.2f times faster than inv\\n\"<\/span>,invTime\/backSlashTime)<\/span><\/span><\/div>
For a large matrix, \\ is 4.02 times faster than inv<\/div><\/div><\/div><\/div><\/div>
Furthermore, in the documentation of <\/span>inv<\/span> we tell you <\/span><\/div>