legpoly()ΒΆ

Create polynomial in the Legendre basis to use with QUINOPT.

Syntax:

P = legpoly(x,DEG)
Description:

creates a polynomial P in the independent variable x of degree DEG, expressed in Legendre basis. That is, the polynomial \(P(x)\) is expressed as

\[P(x) = C_1\,\mathcal{L}_0[z(x)] + C_2\,\mathcal{L}_1[z(x)] + \cdots + C_{{\rm DEG}+1}\,\mathcal{L}_{\rm DEG}[z(x)]\]

where \(\mathcal{L}_n(z)\) is the Legendre polynomial of degree \(n\). Since Legendre polynomials are defined over the standard domain \([-1,1]\), the original independent variable \(x\) with domain \([a,b]\) is rescaled to

\[z(x) = \frac{2\,x-b-a}{b-a}\]

The input x must be a valid independent variable (class indvar), and DEG should be a non-negative integer. The coefficients of the polynomial are YALMIP variables (class sdpvar) and can be recovered with the command C = coefficients(P). Finally, P can be displayed symbolically in the standard monomial basis using the command sdisplay(P).
Syntax:[P,C] = legpoly(x,DEG)
Description:also returns the Legendre coefficients of the polynomials in the vector C. These are YALMIP variables (class sdpvar). The coefficients in C are listed in order of increasing degree of the corresponding Legendre polynomial (see above).
Syntax:P = legpoly(x,DEG,COEF)
Description:creates a polynomial P expressed in Legendre basis whose coefficients are specified by COEF. COEF can be a numeric/sdpvar vector, or an \(M\times N\) cell array whose entries are numeric/sdpvar vectors. When COEF is a cell array, an \(M\times N\) matrix of Legendre polynomials is created such that the coefficients of the entry P(i,j) are given by COEF{i,j}.
Syntax:[P,C] = legpoly(x,DEG,M,N)
Description:creates an \(M\times N\) matrix of Legendre polynomials of degree DEG. The output C, containing the coefficients of each entry of P, is optional.