Method Detail

• createChebyshevPolynomial

  public static PolynomialFunction createChebyshevPolynomial(int degree)

  Create a Chebyshev polynomial of the first kind.

  Chebyshev polynomials of the first kind are orthogonal polynomials. They can be defined by the following recurrence relations:

  \( T_0(x) = 1 \\ T_1(x) = x \\ T_{k+1}(x) = 2x T_k(x) - T_{k-1}(x) \)

  Parameters:

  degree - degree of the polynomial

  Returns:

  Chebyshev polynomial of specified degree

• createHermitePolynomial

  public static PolynomialFunction createHermitePolynomial(int degree)

  Create a Hermite polynomial.

  Hermite polynomials are orthogonal polynomials. They can be defined by the following recurrence relations:

  \( H_0(x) = 1 \\ H_1(x) = 2x \\ H_{k+1}(x) = 2x H_k(X) - 2k H_{k-1}(x) \)

  Parameters:

  degree - degree of the polynomial

  Returns:

  Hermite polynomial of specified degree

• createLaguerrePolynomial

  public static PolynomialFunction createLaguerrePolynomial(int degree)

  Create a Laguerre polynomial.

  Laguerre polynomials are orthogonal polynomials. They can be defined by the following recurrence relations:

  \( L_0(x) = 1 \\ L_1(x) = 1 - x \\ (k+1) L_{k+1}(x) = (2k + 1 - x) L_k(x) - k L_{k-1}(x) \)

  Parameters:

  degree - degree of the polynomial

  Returns:

  Laguerre polynomial of specified degree

• createLegendrePolynomial

  public static PolynomialFunction createLegendrePolynomial(int degree)

  Create a Legendre polynomial.

  Legendre polynomials are orthogonal polynomials. They can be defined by the following recurrence relations:

  \( P_0(x) = 1 \\ P_1(x) = x \\ (k+1) P_{k+1}(x) = (2k+1) x P_k(x) - k P_{k-1}(x) \)

  Parameters:

  degree - degree of the polynomial

  Returns:

  Legendre polynomial of specified degree

• createJacobiPolynomial

  public static PolynomialFunction createJacobiPolynomial(int degree,
                                          int v,
                                          int w)

  Create a Jacobi polynomial.

  Jacobi polynomials are orthogonal polynomials. They can be defined by the following recurrence relations:

  \( P_0^{vw}(x) = 1 \\ P_{-1}^{vw}(x) = 0 \\ 2k(k + v + w)(2k + v + w - 2) P_k^{vw}(x) = \\ (2k + v + w - 1)[(2k + v + w)(2k + v + w - 2) x + v^2 - w^2] P_{k-1}^{vw}(x) \\ - 2(k + v - 1)(k + w - 1)(2k + v + w) P_{k-2}^{vw}(x) \)

  Parameters:

  degree - degree of the polynomial

  v - first exponent

  w - second exponent

  Returns:

  Jacobi polynomial of specified degree

• shift

  public static double[] shift(double[] coefficients,
               double shift)

  Compute the coefficients of the polynomial \(P_s(x)\) whose values at point x will be the same as the those from the original polynomial \(P(x)\) when computed at x + shift.

  More precisely, let \(\Delta = \) shift and let \(P_s(x) = P(x + \Delta)\). The returned array consists of the coefficients of \(P_s\). So if \(a_0, ..., a_{n-1}\) are the coefficients of \(P\), then the returned array \(b_0, ..., b_{n-1}\) satisfies the identity \(\sum_{i=0}^{n-1} b_i x^i = \sum_{i=0}^{n-1} a_i (x + \Delta)^i\) for all \(x\).

  Parameters:

  coefficients - Coefficients of the original polynomial.

  shift - Shift value.

  Returns:

  the coefficients \(b_i\) of the shifted polynomial.