Samuelson–Berkowitz algorithm

In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an $n\times n$ matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVerrier algorithm, it performs no divisions, so may be applied to a wider range of algebraic structures.

Description of the algorithm

The Samuelson–Berkowitz algorithm applied to a matrix $A$ produces a vector whose entries are the coefficient of the characteristic polynomial of $A$. It computes this coefficients vector recursively as the product of a Toeplitz matrix and the coefficients vector an $(n-1)\times (n-1)$ principal submatrix.

Let $A_{0}$ be an $n\times n$ matrix partitioned so that

$A_{0}=\left[{\begin{array}{c|c}a_{1,1}&R\\\hline C&A_{1}\end{array}}\right]$

The first principal submatrix of $A_{0}$ is the $(n-1)\times (n-1)$ matrix $A_{1}$. Associate with $A_{0}$ the $(n+1)\times n$ Toeplitz matrix $T_{0}$ defined by

$T_{0}=\left[{\begin{array}{c}1\\-a_{1,1}\end{array}}\right]$

if $A_{0}$ is $1\times 1$,

$T_{0}=\left[{\begin{array}{c c}1&0\\-a_{1,1}&1\\-RC&-a_{1,1}\end{array}}\right]$

if $A_{0}$ is $2\times 2$, and in general

$T_{0}=\left[{\begin{array}{c c c c c}1&0&0&0&\cdots \\-a_{1,1}&1&0&0&\cdots \\-RC&-a_{1,1}&1&0&\cdots \\-RA_{1}C&-RC&-a_{1,1}&1&\cdots \\-RA_{1}^{2}C&-RA_{1}C&-RC&-a_{1,1}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right]$

That is, all super diagonals of $T_{0}$ consist of zeros, the main diagonal consists of ones, the first subdiagonal consists of $-a_{1,1}$ and the $k$th subdiagonal consists of $-RA_{1}^{k-2}C$.

The algorithm is then applied recursively to $A_{1}$, producing the Toeplitz matrix $T_{1}$ times the characteristic polynomial of $A_{2}$, etc. Finally, the characteristic polynomial of the $1\times 1$ matrix $A_{n-1}$ is simply $T_{n-1}$. The Samuelson–Berkowitz algorithm then states that the vector $v$ defined by

$v=T_{0}T_{1}T_{2}\cdots T_{n-1}$

contains the coefficients of the characteristic polynomial of $A_{0}$.

Because each of the $T_{i}$ may be computed independently, the algorithm is highly parallelizable.

References

• Berkowitz, Stuart J. (30 March 1984). "On computing the determinant in small parallel time using a small number of processors". Information Processing Letters. 18 (3): 147–150. doi:10.1016/0020-0190(84)90018-8.
• Soltys, Michael; Cook, Stephen (December 2004). "The Proof Complexity of Linear Algebra" (PDF). Annals of Pure and Applied Logic. 130 (1–3): 277–323. CiteSeerX 10.1.1.308.6521. doi:10.1016/j.apal.2003.10.018.
• Kerber, Michael (May 2006). Division-Free computation of sub-resultants using Bezout matrices (PS) (Technical report). Saarbrucken: Max-Planck-Institut für Informatik. Tech. Report MPI-I-2006-1-006.