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LinearAlgebra[Generic]

  

HessenbergAlgorithm

  

apply the Hessenberg algorithm to a square Matrix

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

HessenbergAlgorithm[F](A)

Parameters

F

-

the domain of computation, a field

A

-

square Matrix of values in F

Description

• 

Given an n x n Matrix A of elements in F, a field, HessenbergAlgorithm[F](A) returns a Vector V of n+1 values from F encoding the characteristic polynomial of A as V[1] x^n + V[2] x^(n-1) + .... + V[n] x + V[n+1]

• 

The algorithm converts a copy of A into upper Hessenberg form H using O(n^3) operations in F then expands the determinant of x I - H in a further O(n^3) operations in F. The algorithm requires that F be a field and should only be used if F is finite as there is severe expression swell in computing H.

• 

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

  

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

  

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

  

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

  

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

  

F[`*`]: a procedure for multiplying two elements of F (commutative)

  

F[`/`]: a procedure for dividing two elements of F

  

F[`=`]: a boolean procedure for testing if two elements in F are equal

Examples

withLinearAlgebra[Generic]:

Q`0`,Q`1`,Q`+`,Q`-`,Q`*`,Q`/`,Q`=`0,1,`+`,`-`,`*`,`/`,`=`:

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

A:=214321005

(1)

CHessenbergAlgorithm[Q]A

C:=19215

(2)

x3|x2|x|1.C

x39x2+21x5

(3)

See Also

Hessenberg Form

LinearAlgebra[Generic]

LinearAlgebra[Generic][HessenbergForm]

 


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