The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze
The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze
Degrees and leading coefficients R = 2 + 3s + 6s^2 + 5s^3 In this way, new coefficients may be inserted. Thus yields R{5}=7 R = 2 + 3s + 6s^2 + 5s^3 + 7s^5 For polynomial matrices, it works similarly. Given T = [ 1+s 2 s^2 s 3 4 s^3 0 ]; the coefficient matrix of s 2 may be retrieved as T{2} ans = 0 0 1 0 0 0 0 0 Taking coefficients and taking submatrices may be combined. So the coefficients of the (1,3) entry of T follow as T{:}(1,3) ans = 0 0 1 0 The degree of a polynomial matrix is available by deg(T) ans = 3 The leading coefficient matrix, in our example that of lcoef(T) ans = 0 0 0 0 0 0 1 0 3 s may be obtained as
Constant, zero and empty matrices Values Derivative and integral A special case of polynomial matrix is that of degree 0; in effect, it is just a standard MATLAB matrix. The Polynomial Toolbox treats both these forms of data interchangeably: when a polynomial is expected, the standard matrix is also accepted. For explicit conversions, pol and double commands are available: P = pol([1 2;3 4]); D = double(P); Another special case is the zero polynomial; its degree is –Inf . Still another case: the empty polynomial matrix, similar to empty standard matrix. Its degree is empty. A polynomial may also be considered as a function of real or complex variable. For evaluating the polynomial function, the value command is available: F=2+3*s+4*s^2; X=2; Y=value(F,X) Y = 24 When F or X is a matrix then the values are evaluated entrywise: X=[2 3]; Y=value(F,X) Y = 24 47 A bit different command is mvalue. For scalar polynomial F and square matrix X, it computes the matrix value Y according to the matrix algebra rules: X=[1 2;0 1]; Y=mvalue(F,X) Y = 9 22 0 9 For matrix polynomial function Fs , the derivative and the integral are computed deriv(F) ans =
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- Page 14 and 15: ans = 1 0 0 1 s^2 s^3 1 - s 1 The c
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- Page 22 and 23: 3 + 8s integral(F) ans = 2s + 1.5s^
- Page 24 and 25: isunimod(U) ans = 1 Also the adjoin
- Page 26 and 27: Bases and null spaces ans = 1 confi
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- Page 34 and 35: Greatest common left divisor 0 -3 +
- Page 36 and 37: Dual concepts M = lrm(A,B) M = 0.32
- Page 38 and 39: 0 1 s^2 -s^3 -s 0 Reduced and canon
- Page 40 and 41: Another triangular form Hermite for
- Page 42 and 43: Invariant polynomials The entries a
- Page 44 and 45: Bézout equations Matrix polynomial
- Page 46 and 47: Factorizations returns X = Y = 0.25
- Page 48 and 49: zpplot(1-s^2), grid zpplot(1+2*s^2+
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Degrees and<br />
leading<br />
coefficients<br />
R =<br />
2 + 3s + 6s^2 + 5s^3<br />
In this way, new coefficients may be inserted. Thus<br />
yields<br />
R{5}=7<br />
R =<br />
2 + 3s + 6s^2 + 5s^3 + 7s^5<br />
For polynomial matrices, it works similarly. Given<br />
T = [ 1+s 2 s^2 s<br />
3 4 s^3 0 ];<br />
the coefficient matrix of s 2 may be retrieved as<br />
T{2}<br />
ans =<br />
0 0 1 0<br />
0 0 0 0<br />
Taking coefficients and taking submatrices may be combined. So the coefficients of<br />
the (1,3) entry of T follow as<br />
T{:}(1,3)<br />
ans =<br />
0 0 1 0<br />
<strong>The</strong> degree of a polynomial matrix is available by<br />
deg(T)<br />
ans =<br />
3<br />
<strong>The</strong> leading coefficient matrix, in our example that of<br />
lcoef(T)<br />
ans =<br />
0 0 0 0<br />
0 0 1 0<br />
3<br />
s may be obtained as