Macaulay2 » Documentation
Packages » RealRoots :: hurwitzMatrix
next | previous | forward | backward | up | index | toc

hurwitzMatrix -- the principal submatrix of the Hurwitz matrix of a univariate polynomial

Description

This computes the principal $k\times k$ submatrix $H_{k}$ of the Hurwitz matrix $H$ of a univariate polynomial f of degree $n > 0$ with rational coefficients whose leading coefficient is positive. Note that ring f is allowed to be multivariate.

i1 : R = QQ[x]

o1 = R

o1 : PolynomialRing
 
i2 : f = 3*x^4 - 7*x^3 + 5*x - 7

       4     3
o2 = 3x  - 7x  + 5x - 7

o2 : R
 
i3 : hurwitzMatrix(f) 

o3 = | -7 5  0  0  |
     | 3  0  -7 0  |
     | 0  -7 5  0  |
     | 0  3  0  -7 |

              4       4
o3 : Matrix QQ  <-- QQ
 
i4 : hurwitzMatrix(f,4)

o4 = | -7 5  0  0  |
     | 3  0  -7 0  |
     | 0  -7 5  0  |
     | 0  3  0  -7 |

              4       4
o4 : Matrix QQ  <-- QQ
 
i5 : hurwitzMatrix(f,3)	      

o5 = | -7 5  0  |
     | 3  0  -7 |
     | 0  -7 5  |

              3       3
o5 : Matrix QQ  <-- QQ

We can also use multiple variables to represent unknown coefficients. Note that we create another ring S so that x and y are not considered variables in the same ring and so confuse the monomials $x$ or $y$ with $xy$.

i6 : S = R[y]

o6 = S

o6 : PolynomialRing
 
i7 : g = y^3 + 2*y^2 + y - x + 1

      3     2
o7 = y  + 2y  + y - x + 1

o7 : S
 
i8 : hurwitzMatrix(g,3)

o8 = | 2 -x+1 0    |
     | 1 1    0    |
     | 0 2    -x+1 |

             3      3
o8 : Matrix R  <-- R
 
i9 : hurwitzMatrix(g,2)

o9 = | 2 -x+1 |
     | 1 1    |

             2      2
o9 : Matrix R  <-- R
 
i10 : hurwitzMatrix(g,1)

o10 = | 2 |

              1      1
o10 : Matrix R  <-- R

See also

Ways to use hurwitzMatrix:

  • hurwitzMatrix(RingElement)
  • hurwitzMatrix(RingElement,ZZ)

For the programmer

The object hurwitzMatrix is a method function.

The source of this document is in RealRoots.m2:1111:0.