Editing Kernel (linear algebra) (section)

==Numerical computation==
The problem of computing the kernel on a computer depends on the nature of the coefficients.

===Exact coefficients===
If the coefficients of the matrix are exactly given numbers, the [[column echelon form]] of the matrix may be computed with [[Bareiss algorithm]] more efficiently than with Gaussian elimination. It is even more efficient to use [[modular arithmetic]] and [[Chinese remainder theorem]], which reduces the problem to several similar ones over [[finite field]]s (this avoids the overhead induced by the non-linearity of the [[computational complexity]] of integer multiplication).{{Citation needed|date=October 2014}}

For coefficients in a finite field, Gaussian elimination works well, but for the large matrices that occur in [[cryptography]] and [[Gröbner basis]] computation, better algorithms are known, which have roughly the same [[Analysis of algorithms|computational complexity]], but are faster and behave better with modern [[computer hardware]].{{Citation needed|date=October 2014}}

===Floating point computation===
For matrices whose entries are [[floating-point number]]s, the problem of computing the kernel makes sense only for matrices such that the number of rows is equal to their rank: because of the [[rounding error]]s, a floating-point matrix has almost always a [[full rank]], even when it is an approximation of a matrix of a much smaller rank. Even for a full-rank matrix, it is possible to compute its kernel only if it is [[well-conditioned problem|well conditioned]], i.e. it has a low [[condition number]].<ref>{{Cite web |url=https://www.math.ohiou.edu/courses/math3600/lecture11.pdf |title=Archived copy |access-date=2015-04-14 |archive-url=https://web.archive.org/web/20170829031912/http://www.math.ohiou.edu/courses/math3600/lecture11.pdf |archive-date=2017-08-29 |url-status=dead }}</ref>{{Citation needed|date=December 2019}}

Even for a well conditioned full rank matrix, Gaussian elimination does not behave correctly: it introduces rounding errors that are too large for getting a significant result. As the computation of the kernel of a matrix is a special instance of solving a homogeneous system of linear equations, the kernel may be computed with any of the various algorithms designed to solve homogeneous systems. A state of the art software for this purpose is the [[Lapack]] library.{{Citation needed|date=October 2014}}