Routh table: Routh

The Routh table allows one to determine if all the roots of a polynomial have a negative real part. The number of sign changes in the first column of the table is the number of roots with positive real parts. The Routh table has n+1 rows, where n is the degree of the polynomial.

The syntax of this procedure is:

 
Routh(poly,Vars,Type,Where,ProcName)

where:

• poly is the polynomial
• Vars: a list of variable name whose first element is the polynomial variable
• Type: see below
• Where: if P(x) is the polynomial we will compute the Routh table for P(x+Where). If Where is not 0 the number of sign changes will indicate the number of roots with real part greater than Where.
• ProcName: the name of the simplification procedure

Type allows to control the output of this procedure:

• 0 : the procedure will simply returns the Routh table of the polynomial
• $n >0$ and lower than the degree of the polynomial +1 : the procedure will create a C++ simplification procedure (whose name is the last argument of the procedure) that will compute an interval evaluation of the $n$ first element of the first column of the Routh table, using the derivatives of these elements. This simplification procedure will return -1 if the polynomial has a root with real part larger than Where. The interval input of this procedure will be an interval vector corresponding to Vars. Note that if $n$ is large the procedure may take a long time to be generated.
• $[k,[n,var1,var2..],[m,var3]]$: if $k>0$ the same simplification principle than described in the above section will be generated. Furthermore we will calculate the sign of the first element of the first row of the Routh table. If this element has a constant sign (say positive) we will consider the element at the n-th row of the first column. Assuming that this element S has a linear term in var1 (S= a var1+b) with a positive we will write that S is positive when var1 is greater than -b/a and we will update var1 accordingly. We will proceed similarly with the element at of row m and with variable var3

Consider the polynomial $x1+(x1+x2)*s+x2^2*s^2$ in the variable $s$. Its Routh table at 0 is:

 
Routh(x1+(x1+x2)*s+x2^2*s^2,[s,x1,x2],0,0);
                   [    2             ]
                   [  x2       x1    0]
                   [                  ]
                   [x1 + x2    0     0]
                   [                  ]
                   [  x1       0     0]

The first element of the Routh table is positive and the second element of the first row is $x1+x2$. Hence a necessary condition for the polynomial for not having root with real part positive is that $x1+x2>0$. Hence writing:

 
Routh(x1+(x1+x2)*s+x2^2*s^2,[s,x1,x2],[2,[2,x1,x2]],0,"ROUTH");

will imply that the simplification procedure will use the simplification rules $x1> -x2$ and $x2>-x1$. If x1=[-5,5] and x2=[0,4] $x1> -x2$ will imply $[-5,5]>[-4,0]$ which leads to the new range [-4,5] for $x1$