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LinearAlgebra[Modular]

  

IntegerCharacteristicPolynomial

  

computation of the characteristic polynomial of an integer matrix using modular methods

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

IntegerCharacteristicPolynomial(A, lambda)

Parameters

A

-

square matrix with integer entries

lambda

-

variable to use for output characteristic polynomial

Description

• 

The IntegerCharacteristicPolynomial function computes the characteristic polynomial for a square Matrix with integer entries. This is a programmer level function, and it does not perform argument checking. Thus, argument checking must be handled external to this function.

  

Note: The IntegerCharacteristicPolynomial routine uses a probabilistic approach that achieves great gains for structured systems. Information on controlling the probabilistic behavior can be found in EnvProbabilistic.

• 

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form IntegerCharacteristicPolynomial(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][IntegerCharacteristicPolynomial](..).

Examples

withLinearAlgebra[Modular]:

MMatrix2,1,3,4,3,1,2,1,3

M:=213431213

(1)

IntegerCharacteristicPolynomialM,x

x32x28x20

(2)

LinearAlgebra:-CharacteristicPolynomialM,x

x32x28x20

(3)

This function is provided as a high-efficiency function for computation of characteristic polynomials for larger matrices. For example:

MLinearAlgebra:-RandomMatrix50,50,density=0.5,generator=99..99,outputoptions=datatype=integer

M:= 50 x 50 MatrixData Type: integerStorage: rectangularOrder: Fortran_order

(4)

tttime:

p1IntegerCharacteristicPolynomialM,x:

t1timett

t1:=0.015

(5)

_EnvDisableModulartrue:

tttime:

p2LinearAlgebra:-CharacteristicPolynomialM,x:

t2timett

t2:=0.449

(6)

expandp1p2

0

(7)

Speed-up factor:

t2t1

29.93333333

(8)

See Also

LinearAlgebra/Details

LinearAlgebra[CharacteristicPolynomial]

LinearAlgebra[Modular]

 


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