Wolfram Function Repository
Instant-use add-on functions for the Wolfram Language
Function Repository Resource:
Generate the Schur matrix of a univariate polynomial
ResourceFunction["SchurMatrix"][poly,x] gives the Schur matrix of poly, treated as a polynomial in x. |
Generate the Schur matrix of a polynomial:
In[1]:= | ![]() |
Out[1]= | ![]() |
Generate the Schur matrix from a polynomial with numeric coefficients:
In[2]:= | ![]() |
Out[2]= | ![]() |
Generate the Schur matrix from a polynomial with symbolic coefficients:
In[3]:= | ![]() |
Out[3]= | ![]() |
The Schur matrix for a cyclotomic polynomial is the zero matrix:
In[4]:= | ![]() |
Out[4]= | ![]() |
Use SchurMatrix to check the stability of a polynomial:
In[5]:= | ![]() |
Out[6]= | ![]() |
In[7]:= | ![]() |
Out[7]= | ![]() |
Verify stability by computing the roots of the polynomial:
In[8]:= | ![]() |
Out[8]= | ![]() |
The Schur matrix of a quadratic polynomial with symbolic coefficients:
In[9]:= | ![]() |
Out[9]= | ![]() |
Use the resource function RationalCholeskyDecomposition to transform the Schur matrix into a congruent diagonal matrix:
In[10]:= | ![]() |
Out[10]= | ![]() |
Determine the conditions for the Schur matrix to be positive definite (which is equivalent to the original polynomial being stable):
In[11]:= | ![]() |
Out[11]= | ![]() |
The Schur matrix is Hermitian:
In[12]:= | ![]() |
Out[12]= | ![]() |
In[13]:= | ![]() |
Out[13]= | ![]() |
The inverse of a Schur matrix is a Toeplitz matrix:
In[14]:= | ![]() |
Out[14]= | ![]() |
A polynomial whose roots are all inside the unit disk:
In[15]:= | ![]() |
The corresponding Schur matrix is positive definite:
In[16]:= | ![]() |
Out[16]= | ![]() |
Apply a Möbius transformation to the polynomial:
In[17]:= | ![]() |
Out[17]= | ![]() |
The transformed polynomial has a Hurwitz matrix whose principal minors are all positive. That is, all of the roots of the transformed polynomial have negative real parts:
In[18]:= | ![]() |
Out[19]= | ![]() |
This work is licensed under a Creative Commons Attribution 4.0 International License