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LinearAlgebra[Generic]

  

BerkowitzAlgorithm

  

apply the Berkowitz algorithm to a square Matrix

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

BerkowitzAlgorithm[R](A)

Parameters

R

-

a table or module, the domain of computation

A

-

square Matrix of values in R

Description

• 

Given an n x n Matrix A of values from a commutative ring R, BerkowitzAlgorithm[R](A) returns a Vector V of dimension n+1 of values in R with the coefficients of the characteristic polynomial of A.

• 

The characteristic polynomial is the polynomial V[1]*x^n + V[2]*x^(n-1) + ... + V[n]*x + V[n+1].

• 

The Berkowitz algorithm does O(n^4) multiplications and additions in R.

• 

The (indexed) parameter R, which specifies the domain of computation, a commutative ring, must be a Maple table/module which has the following values/exports:

  

R[`0`] : a constant for the zero of the ring R

  

R[`1`] : a constant for the (multiplicative) identity of R

  

R[`+`] : a procedure for adding elements of R (nary)

  

R[`-`] : a procedure for negating and subtracting elements of R (unary and binary)

  

R[`*`] : a procedure for multiplying elements of R (binary and commutative)

  

R[`=`] : a boolean procedure for testing if two elements of R are equal

Examples

withLinearAlgebra[Generic]:

Z`0`,Z`1`,Z`+`,Z`-`,Z`*`,Z`=`0,1,`+`,`-`,`*`,`=`:

AMatrix2,1,4,3,2,1,0,0,5

A:=214321005

(1)

CBerkowitzAlgorithm[Z]A

C:=19215

(2)

x3|x2|x|1.C

x39x2+21x5

(3)

See Also

LinearAlgebra[CharacteristicPolynomial]

LinearAlgebra[Generic]

LinearAlgebra[Generic][CharacteristicPolynomial]

LinearAlgebra[Modular][CharacteristicPolynomial]

 


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