Polynomial calculations

`8 Polynomial calculations`

`8.1` `bezout` `Bezout equation for polynomials`

CALLING SEQUENCE :

[thegcd,U]=bezout(p1,p2)

PARAMETERS :

p1, p2: two real polynomials

DESCRIPTION :

[thegcd,U]=bezout(p1,p2)computes GCD thegcdof p1 and p2 and in addition a (2x2) unimodular matrix U such that:

[p1,p2]*U = [thegcd,0]

The lcm of p1 and p2 is given by:

p1*U(1,2) (or -p2*U(2,2))

EXAMPLE :

x=poly(0,'x');
p1=(x+1)*(x-3)^5;p2=(x-2)*(x-3)^3;
[thegcd,U]=bezout(p1,p2) 
det(U)
clean([p1,p2]*U)
thelcm=p1*U(1,2)
lcm([p1,p2])

See Also : poly X, roots X, simp X, clean X, lcm X

8.2 clean cleans matrices (round to zero small entries)

CALLING SEQUENCE :

[B]=clean(A [,epsa [,epsr]])

PARAMETERS :

A : a numerical matrix (scalar, polynomial, sparse...)
epsa,epsr : real numbers (default values resp. 1.d-10 and 1.d-10)

DESCRIPTION :

This function eliminates (i.e. set to zero) all the coefficients with absolute value < epsa and relative value < epsr (relative means relative w.r.t. 1-norm of coefficients) in a polynomial (possibly matrix polynomial or rational matrix).

Default values are epsa=1.d-10 and epsr=1.d-10;

For a constant (non polynomial) matrix clean(A,epsa) sets to zero all entries of A smaller than epsa.

EXAMPLE :

x=poly(0,'x');
w=[x,1,2+x;3+x,2-x,x^2;1,2,3+x]/3;
w*inv(w)
clean(w*inv(w))

8.3 cmndred common denominator form

CALLING SEQUENCE :

[n,d]=cmndred(num,den)

PARAMETERS :

num, den : two polynomial matrices of same dimensions

DESCRIPTION :

[n,d]=cmndred(num,den) computes a polynomial matrix n and a common denominator polynomial d such that:

n/d=num./den

The rational matrix defined by num./den is n/d

See Also : simp X, clean X

8.4 coffg inverse of polynomial matrix

CALLING SEQUENCE :

[Ns,d]=coffg(Fs)

PARAMETERS :

Fs : square polynomial matrix

DESCRIPTION :

coffg computes Fs^-1 where Fs is a polynomial matrix by co-factors method.

Fs inverse = Ns/d

d = common denominator; Ns = numerator (a polynomial matrix)

(For large matrices,be patient...results are generally reliable)

EXAMPLE :

s=poly(0,'s')
a=[ s, s^2+1; s  s^2-1];
[a1,d]=coffg(a);
(a1/d)-inv(a)

See Also : determ X, detr X, invr X, penlaur X, glever X

Author : F. D.

8.5 colcompr column compression of polynomial matrix

CALLING SEQUENCE :

[Y,rk,ac]=colcompr(A);

PARAMETERS :

A : polynomial matrix
Y : square polynomial matrix (right unimodular basis)
rk : normal rank of A
Ac : Ac=A*Y, polynomial matrix

DESCRIPTION :

column compression of polynomial matrix A (compression to the left)

EXAMPLE :

s=poly(0,'s');
p=[s;s*(s+1)^2;2*s^2+s^3];
[Y,rk,ac]=colcompr(p*p');
p*p'*Y

8.6 denom denominator

CALLING SEQUENCE :

den=denom(r)

PARAMETERS :

r : rational or polynomial or constant matrix.
den : polynomial matrix

DESCRIPTION :

den=denom(r) returns the denominator of a rational matrix.

Since rationals are internally represented as r=list(['r','num','den','dt'],num,den,[]), denom(r) is the same as r(3) or r('den') .

See Also : numer X

8.7 derivat rational matrix derivative

CALLING SEQUENCE :

pd=derivat(p)

PARAMETERS :

p : polynomial or rational matrix

DESCRIPTION :

computes the derivative of the polynomial or rational function matrix w.r.t the dummy variable.

EXAMPLE :

s=poly(0,'s');
derivat(1/s)  // -1/s^2;

8.8 determ determinant of polynomial matrix

CALLING SEQUENCE :

res=determ(W [,k])

PARAMETERS :

W : real square polynomial matrix
k : integer (upper bound for the degree of the determinant of W)

DESCRIPTION :

res=determ(W [,k]) returns the determinant of a real polynomial matrix (computation made by FFT).

k is an integer larger than the actual degree of the determinant of W.

The default value of k is the smallest power of 2 which is larger than n*maxi(degree(W)).

Method: evaluate the determinant of W for the Fourier frequencies and apply inverse FFT to the coefficients of the determinant.

EXAMPLE :

s=poly(0,'s');
w=s*rand(10,10);
determ(w)
det(coeff(w,1))*s^10

See Also : det X, detr X, coffg X

Author : F.D.

8.9 detr polynomial determinant

CALLING SEQUENCE :

d=detr(h)

PARAMETERS :

h : polynomial or rational square matrix

DESCRIPTION :

d=detr(h) returns the determinant d of the polynomial or rational function matrix h. Based on Leverrier's algorithm.

See Also : det X, determ X

8.10 diophant diophantine (Bezout) equation

CALLING SEQUENCE :

[x,err]=diophant(p1p2,b)

PARAMETERS :

p1p2 : polynomial vector p1p2 = [p1 p2]
b : polynomial
x : polynomial vector [x1;x2]

DESCRIPTION :

diophant solves the bezout equation:

p1*x1+p2*x2=b with p1p2 a polynomial vector. If the equation is not solvable

err=||p1*x1+p2*x2-b||/||b||

else err=0

EXAMPLE :

s=poly(0,'s');p1=(s+3)^2;p2=(1+s);
x1=s;x2=(2+s);
[x,err]=diophant([p1,p2],p1*x1+p2*x2);
p1*x1+p2*x2-p1*x(1)-p2*x(2)

8.11 factors numeric real factorization

CALLING SEQUENCE :

[lnum,g]=factors(pol [,'flag'])
[lnum,lden,g]=factors(rat [,'flag'])
rat=factors(rat,'flag')

PARAMETERS :

pol : real polynomial
rat : real rational polynomial (rat=pol1/pol2)
lnum : list of polynomials (of degrees 1 or 2)
lden : list of polynomials (of degrees 1 or 2)
g : real number
flag : character string 'c' or 'd'

DESCRIPTION :

returns the factors of polynomial pol in the list lnum and the "gain" g.

One has pol= g times product of entries of the list lnum (if flag is not given). If flag='c' is given, then one has |pol(i omega)| = |g*prod(lnum_j(i omega)|. If flag='d' is given, then one has |pol(exp(i omega))| = |g*prod(lnum_i(exp(i omega))|. If argument of factors is a 1x1 rational rat=pol1/pol2, the factors of the numerator pol1 and the denominator pol2 are returned in the lists lnum and lden respectively.

The "gain" is returned as g,i.e. one has: rat= g times (product entries in lnum) / (product entries in lden).

If flag is 'c' (resp. 'd'), the roots of pol are refected wrt the imaginary axis (resp. the unit circle), i.e. the factors in lnum are stable polynomials.

Same thing if factors is invoked with a rational arguments: the entries in lnum and lden are stable polynomials if flag is given. R2=factors(R1,'c') or R2=factors(R1,'d') with R1 a rational function or SISO syslin list then the output R2 is a transfer with stable numerator and denominator and with same magnitude as R1 along the imaginary axis ('c') or unit circle ('d').

EXAMPLE :

n=poly([0.2,2,5],'z');
d=poly([0.1,0.3,7],'z');
R=syslin('d',n,d);
R1=factors(R,'d')
roots(R1('num'))
roots(R1('den'))
w=exp(2*%i*%pi*[0:0.1:1]);
norm(abs(horner(R1,w))-abs(horner(R,w)))

8.12 gcd gcd calculation

CALLING SEQUENCE :

[pgcd,U]=gcd(p)

PARAMETERS :

p : polynomial row vector p=[p1,..,pn]

DESCRIPTION :

[pgcd,u]=gcd(p) computes the gcd of components of p and a unimodular matrix (with polynomial inverse) U, with minimal degree such that

p*U=[0 ... 0 pgcd]

EXAMPLE :

s=poly(0,'s');
p=[s,s*(s+1)^2,2*s^2+s^3];
[pgcd,u]=gcd(p);
p*u

See Also : bezout X, lcm X, hermit X

8.13 hermit Hermite form

CALLING SEQUENCE :

[Ar,U]=hermit(A)

PARAMETERS :

A : polynomial matrix
Ar : triangular polynomial matrix
U : unimodolar polynomial matrix

DESCRIPTION :

Hermite form: U is an unimodular matrix such that A*U is in Hermite triangular form:

The output variable is Ar=A*U.

Warning: Experimental version

EXAMPLE :

s=poly(0,'s');
p=[s, s*(s+1)^2, 2*s^2+s^3];
[Ar,U]=hermit(p'*p);
clean(p'*p*U), det(U)

8.14 horner polynomial/rational evaluation

CALLING SEQUENCE :

horner(P,x)

PARAMETERS :

P : polynomial or rational matrix
x : real number or polynomial or rational

DESCRIPTION :

evaluates the polynomial or rational matrix P = P(s) when the variable s of the polynomial is replaced by x:

horner(P,x)=P(x)

Example (Bilinear transform): Assume P = P(s) is a rational matrix then the rational matrix P((1+s)/(1-s)) is obtained by horner(P,(1+s)/(1-s)).

To evaluate a rational matrix at given frequencies use preferably the freq primitive.

EXAMPLES :

s=poly(0,'s');M=[s,1/s];
horner(M,1)
horner(M,%i)
horner(M,1/s)

See Also : freq X, repfreq X, evstr X

8.15 hrmt gcd of polynomials

CALLING SEQUENCE :

[pg,U]=hrmt(v)

PARAMETERS :

v : row of polynomials i.e. 1xk polynomial matrix
pg : polynomial
U : unimodular matrix polynomial

DESCRIPTION :

[pg,U]=hrmt(v) returns a unimodular matrix U and pg = gcd of row of polynomials v such that v*U = [pg,0].

EXAMPLE :

x=poly(0,'x');
v=[x*(x+1),x^2*(x+1),(x-2)*(x+1),(3*x^2+2)*(x+1)];
[pg,U]=hrmt(v);U=clean(U)
det(U)

8.16 htrianr triangularization of polynomial matrix

CALLING SEQUENCE :

[Ar,U,rk]=htrianr(A)

PARAMETERS :

A : polynomial matrix
Ar : polynomial matrix
U : unimodular polynomial matrix
rk : integer, normal rank of A

DESCRIPTION :

triangularization of polynomial matrix A.

A is [m,n] , m <= n.

Ar=A*U

Warning: there is an elimination of "small" terms (see function code).

EXAMPLE :

x=poly(0,'x');
M=[x;x^2;2+x^3]*[1,x-2,x^4];
[Mu,U,rk]=htrianr(M)
det(U)
M*U(:,1:2)

8.17 invr inversion of (rational) matrix

CALLING SEQUENCE :

F = invr(H)

PARAMETERS :

H : polynomial or rational matrix
F : polynomial or rational matrix

DESCRIPTION :

If H is a polynomial or rational function matrix, invr computes H^(-1) using Leverrier's algorithm (see function code)

EXAMPLE :

s=poly(0,'s')
H=[s,s*s+2;1-s,1+s]; invr(H)
[Num,den]=coffg(H);Num/den
H=[1/s,(s+1);1/(s+2),(s+3)/s];invr(H)

See Also : glever X, coffg X, inv X

8.18 lcm least common multiple

CALLING SEQUENCE :

[pp,fact]=lcm(p)

PARAMETERS :

p :
fact : polynomial vector
pp : polynomial

DESCRIPTION :

pp=lcm(p) computes the lcm pp of polynomial vector p.

[pp,fact]=lcm(p) computes in addition the vector fact such that:

p.*fact=pp*ones(p)

EXAMPLE :

s=poly(0,'s');
p=[s,s*(s+1)^2,s^2*(s+2)];
[pp,fact]=lcm(p);
p.*fact, pp

8.19 lcmdiag least common multiple diagonal factorization

CALLING SEQUENCE :

[N,D]=lcmdiag(H)
[N,D]=lcmdiag(H,flag)

PARAMETERS :

H : rational matrix
N : polynomial matrix
D : diagonal polynomial matrix
flag : character string: 'row' or 'col' (default)

DESCRIPTION :

[N,D]=lcmdiag(H,'row') computes a factorization D*H=N, i.e. H=D^(-1)*N where D is a diagonal matrix with D(k,k)=lcm of kth row of H('den').

[N,D]=lcmdiag(H) or [N,D]=lcmdiag(H,'col) returns H=N*D^(-1) with diagonal D and D(k,k)=lcm of kth col of H('den')

EXAMPLE :

s=poly(0,'s');
H=[1/s,(s+2)/s/(s+1)^2;1/(s^2*(s+2)),2/(s+2)];
[N,D]=lcmdiag(H);
N/D-H

8.20 ldiv polynomial matrix long division

CALLING SEQUENCE :

[x]=ldiv(n,d,k)

PARAMETERS :

n,d : two real polynomial matrices
k : integer

DESCRIPTION :

x=ldiv(n,d,k) gives the k first coefficients of the long division of n by d i.e. the Taylor expansion of the rational matrix [nij(z)/dij(z)] near infinity.

Coefficients of expansion of nij/dij are stored in x((i-1)*n+k,j) k=1:n

EXAMPLE :

wss=ssrand(1,1,3);[a,b,c,d]=abcd(wss);
wtf=ss2tf(wss);
x1=ldiv(numer(wtf),denom(wtf),5)
x2=[c*b;c*a*b;c*a^2*b;c*a^3*b;c*a^4*b]
wssbis=markp2ss(x1',5,1,1);
wtfbis=clean(ss2tf(wssbis))
x3=ldiv(numer(wtfbis),denom(wtfbis),5)

See Also : arl2 X, markp2ss X, pdiv X

8.21 numer numerator

CALLING SEQUENCE :

NUM=numer(R)

PARAMETERS :

R : rational matrix

DESCRIPTION :

Utility fonction. NUM=numer(R) returns the numerator NUM of a rational function matrix R (R may be also a constant or polynomial matrix). numer(R) is equivalent to R(2) or R('num')

See Also : denom X

8.22 pdiv polynomial division

CALLING SEQUENCE :

[R,Q]=pdiv(P1,P2)
[Q]=pdiv(P1,P2)

PARAMETERS :

P1 : polynomial matrix
P2 : polynomial or polynomial matrix
R,Q : two polynomial matrices

DESCRIPTION :

Element-wise euclidan division of the polynomial matrix P1 by the polynomial P2 or by the polynomial matrix P2. Rij is the matrix of remainders, Qij is the matrix of quotients and P1ij = Qij*P2 + Qij or P1ij = Qij*P2ij + Qij.

EXAMPLE :

x=poly(0,'x');
p1=(1+x^2)*(1-x);p2=1-x;
[r,q]=pdiv(p1,p2)
p2*q-p1
p2=1+x;
[r,q]=pdiv(p1,p2)
p2*q+r-p1

8.23 pol2des polynomial matrix to descriptor form

CALLING SEQUENCE :

[N,B,C]=pol2des(Ds)

PARAMETERS :

Ds : polynomial matrix
N, B, C : three real matrices

DESCRIPTION :

Given the polynomial matrix Ds=D_0 +D_1 s +D_2 s^2 +... +D_k s^k, pol2des returns three matrices N, B, C, with N nilpotent such that:

Ds = C (s*N-eye())^-1 B

EXAMPLE :

s=poly(0,'s');
G=[1,s;1+s^2,3*s^3];[N,B,C]=pol2des(G);
G1=clean(C*inv(s*N-eye())*B),G2=numer(G1)

See Also : ss2des X, tf2des X

Author : F.D.

8.24 pol2str polynomial to string conversion

CALLING SEQUENCE :

[str]=pol2str(p)

PARAMETERS :

p : real polynomial
str : character string

DESCRIPTION :

converts polynomial to character string (utility function).

See Also : string X, pol2tex X

8.25 polfact minimal factors

CALLING SEQUENCE :

[f]=polfact(p)

PARAMETERS :

p : polynomial
f : vector [f0 f1 ... fn] such that p=prod(f)

1
f0 : constant
fi : polynomial

0

DESCRIPTION :

f=polfact(p) returns the minimal factors of p i.e. f=[f0 f1 ... fn] such that p=prod(f)

See Also : lcm X, cmndred X, factors X

8.26 residu residue

CALLING SEQUENCE :

[V]=residu(P,Q1,Q2)

PARAMETERS :

P, Q1, Q2 : polynomials or matrix polynomials with real or complex coefficients.

DESCRIPTION :

V=residu(P,Q1,Q2) returns the matrix V such that V(i,j) is the sum of the residues of the rational fraction P(i,j)/(Q1(i,j)*Q2(i,j)) calculated at the zeros of Q1(i,j).

Q1(i,j) and Q2(i,j) must not have any common root.

EXAMPLE :

s=poly(0,'s');
H=[s/(s+1)^2,1/(s+2)];N=numer(H);D=denom(H);
w=residu(N.*horner(N,-s),D,horner(D,-s));  //N(s) N(-s) / D(s) D(-s)
sqrt(sum(w))  //This is H2 norm
h2norm(tf2ss(H))
//
p=(s-1)*(s+1)*(s+2)*(s+10);a=(s-5)*(s-1)*(s*s)*((s+1/2)**2);
b=(s-3)*(s+2/5)*(s+3);
residu(p,a,b)+531863/4410    //Exact
z=poly(0,'z');a=z^3+0.7*z^2+0.5*z-0.3;b=z^3+0.3*z^2+0.2*z+0.1;
atild=gtild(a,'d');btild=gtild(b,'d');
residu(b*btild,z*a,atild)-2.9488038   //Exact
a=a+0*%i;b=b+0*%i;
real(residu(b*btild,z*a,atild)-2.9488038) //Complex case

See Also : pfss X, bdiag X, roots X, poly X, gtild X

Author : F.D.

8.27 roots roots of polynomials

CALLING SEQUENCE :

[x]=roots(p)

PARAMETERS :

p : polynomial with real or complex coefficients

DESCRIPTION :

x=roots(p) returns in the complex vector x the roots of the polynomial p. Degree of p must be <=100.

EXAMPLE :

p=poly([0,10,1+%i,1-%i],'x');
roots(p)
A=rand(3,3);roots(poly(A,'x'))    // Evals by characteristic polynomial
spec(A)

8.28 routh_t Routh's table

CALLING SEQUENCE :

r=routh_t(h [,k]).

PARAMETERS :

h : square rational matrix

DESCRIPTION :

r=routh_t(h,k) computes Routh's table of denominator of the system described by transfer matrix SISO h with the feedback by the gain k.

If k=poly(0,'k') we will have a polynomial matrix with dummy variable k, formal expression of the Routh table.

8.29 rowcompr row compression of polynomial matrix

CALLING SEQUENCE :

[X,rk,Ac]=rowcompr(A)

PARAMETERS :

A : polynomial matrix
Y : square polynomial matrix (left unimodular basis)
rk : normal rank of A
Ac : Ac=X*A, polynomial matrix

DESCRIPTION :

row compression of polynomial matrix A .

X is a left polynomial unimodular basis which row compressed thee rows of A. rk is the normal rank of A.

Warning: elimination of "small" terms (use with care!).

See Also : colcompr X

8.30 sfact discrete time spectral factorization

CALLING SEQUENCE :

F=sfact(P)

PARAMETERS :

P : real polynomial matrix

DESCRIPTION :

Finds F, a spectral factor of P. P is a polynomial matrix such that each root of P has a mirror image w.r.t the unit circle. Problem is singular if a root is on the unit circle.

sfact(P) returns a polynomial matrix F(z) which is antistable and such that

P = F(z)* F(1/z) *z^n

For scalar polynomials a specific algorithm is implemented. Algorithms are adapted from Kucera's book.

EXAMPLE :

//Simple polynomial example
z=poly(0,'z');
p=(z-1/2)*(2-z)
w=sfact(p);
w*numer(horner(w,1/z))
//matrix example
F1=[z-1/2,z+1/2,z^2+2;1,z,-z;z^3+2*z,z,1/2-z];
P=F1*gtild(F1,'d');  //P is symmetric
F=sfact(P)    
roots(det(P))  
roots(det(gtild(F,'d')))  //The stable roots
roots(det(F))             //The antistable roots
clean(P-F*gtild(F,'d'))
//Example of continuous time use
s=poly(0,'s');
p=-3*(s+(1+%i))*(s+(1-%i))*(s+0.5)*(s-0.5)*(s-(1+%i))*(s-(1-%i));p=real(p);
//p(s) = polynomial in s^2 , looks for stable f such that p=f(s)*f(-s) 
w=horner(p,(1-s)/(1+s));  // bilinear transform w=p((1-s)/(1+s))
wn=numer(w);              //take the numerator
fn=sfact(wn);f=numer(horner(fn,(1-s)/(s+1))); //Factor and back transform
f=f/sqrt(horner(f*gtild(f,'c'),0));f=f*sqrt(horner(p,0));      //normalization
roots(f)    //f is stable
clean(f*gtild(f,'c')-p)    //f(s)*f(-s) is p(s)

8.31 simp rational simplification

CALLING SEQUENCE :

[N1,D1]=simp(N,D)
H1=simp(H)

PARAMETERS :

N,D : real polynomials or real matrix polynomials
H : rational matrix (i.e matrix with entries n/d ,n and d real polynomials)

DESCRIPTION :

[n1,d1]=simp(n,d) calculates two polynomials n1 and d1 such that n1/d1 = n/d.

If N and D are polynomial matrices the calculation is performed element-wise.

H1=simp(H) is also valid (each entry of H is simplified in H1).

Caution:

-no threshold is given i.e. simp cannot forces a simplification.

-For linear dynamic systems which include integrator(s) simplification changes the static gain. (H(0) for continuous systems or H(1) for discrete systems)

-for complex data, simp returns its input(s).

-rational simplification is called after nearly each operations on rationals. It is possible to toggle simplification on or off using simp_mode function.

EXAMPLES :

s=poly(0,'s');
[n,d]=simp((s+1)*(s+2),(s+1)*(s-2))

simp_mode(%F);hns=s/s
simp_mode(%T);hns=s/s

See Also : roots X, trfmod X, poly X, clean X, simp_mode X

8.32 simp_mode toggle rational simplification

CALLING SEQUENCE :

mod=simp_mode()
simp_mode(mod)

PARAMETERS :

mod : a boolean

DESCRIPTION :

rational simplification is called after nearly each operations on rationals. It is possible to toggle simplification on or off using simp_mode function.

simp_mod(%t) set rational simplification mode on

simp_mod(%f) set rational simplification mode off

mod=simp_mod() returns in mod the current rational simplification mode

EXAMPLES :

s=poly(0,'s');
mod=simp_mode()
simp_mode(%f);hns=s/s
simp_mode(%t);hns=s/s
simp_mode(mod);

8.33 sylm Sylvester matrix

CALLING SEQUENCE :

[S]=sylm(a,b)

PARAMETERS :

a,b : two polynomials
S : matrix

DESCRIPTION :

sylm(a,b) gives the Sylvester matrix associated to polynomials a and b, i.e. the matrix S such that:

coeff( a*x + b*y )' = S * [coeff(x)';coeff(y)'].

Dimension of S is equal to degree(a)+degree(b).

If a and b are coprime polynomials then

rank(sylm(a,b))=degree(a)+degree(b)) and the instructions

  u = sylm(a,b) \\ eye(na+nb,1)
  x = poly(u(1:nb),'z','coeff')
  y = poly(u(nb+1:na+nb),'z','coeff')

compute Bezout factors x and y of minimal degree such that a*x+b*y = 1

8.34 systmat system matrix

CALLING SEQUENCE :

[Sm]=systmat(Sl);

PARAMETERS :

Sl : linear system (syslin list) or descriptor system
Sm : matrix pencil

DESCRIPTION :

System matrix of the linear system Sl (syslin list) in state-space form (utility function).

 Sm = [-sI + A   B;
      [    C     D]

For a descriptor system (Sl=list('des',A,B,C,D,E)), systmat returns:

 Sm = [-sE + A   B;
      [    C     D]

See Also : ss2des X, sm2des X, sm2ss X

8 Polynomial calculations

8.1 bezout Bezout equation for polynomials

8.2 clean cleans matrices (round to zero small entries)

8.3 cmndred common denominator form

8.4 coffg inverse of polynomial matrix

8.5 colcompr column compression of polynomial matrix

8.6 denom denominator

8.7 derivat rational matrix derivative

8.8 determ determinant of polynomial matrix

8.9 detr polynomial determinant

8.10 diophant diophantine (Bezout) equation

8.11 factors numeric real factorization

8.12 gcd gcd calculation

8.13 hermit Hermite form

8.14 horner polynomial/rational evaluation

8.15 hrmt gcd of polynomials

8.16 htrianr triangularization of polynomial matrix

8.17 invr inversion of (rational) matrix

8.18 lcm least common multiple

8.19 lcmdiag least common multiple diagonal factorization

8.20 ldiv polynomial matrix long division

8.21 numer numerator

8.22 pdiv polynomial division

8.23 pol2des polynomial matrix to descriptor form

8.24 pol2str polynomial to string conversion

8.25 polfact minimal factors

8.26 residu residue

8.27 roots roots of polynomials

8.28 routh_t Routh's table

8.29 rowcompr row compression of polynomial matrix

8.30 sfact discrete time spectral factorization

8.31 simp rational simplification

8.32 simp_mode toggle rational simplification

8.33 sylm Sylvester matrix

8.34 systmat system matrix

`8 Polynomial calculations`

`8.1` `bezout` `Bezout equation for polynomials`