# CHEBYSHEV_POLYNOMIAL Chebyshev Polynomials

CHEBYSHEV_POLYNOMIAL is a FORTRAN90 library which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

The Chebyshev polynomial T(n,x), or Chebyshev polynomial of the first kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:

cos ( t ) = x
T(n,x) = cos ( n * t )

For any value of x, T(n,x) may be evaluated by a three term recurrence:
T(0,x) = 1
T(1,x) = x
T(n+1,x) = 2x T(n,x) - T(n-1,x)

The Chebyshev polynomial U(n,x), or Chebyshev polynomial of the second kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:

cos ( t ) = x
U(n,x) = sin ( ( n + 1 ) t ) / sin ( t )

For any value of x, U(n,x) may be evaluated by a three term recurrence:
U(0,x) = 1
U(1,x) = 2x
U(n+1,x) = 2x U(n,x) - U(n-1,x)

The Chebyshev polynomial V(n,x), or Chebyshev polynomial of the third kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:

cos ( t ) = x
V(n,x) = cos ( (2n+1)*t/2) / cos ( t/2)

For any value of x, V(n,x) may be evaluated by a three term recurrence:
V(0,x) = 1
V(1,x) = 2x-1
V(n+1,x) = 2x V(n,x) - V(n-1,x)

The Chebyshev polynomial W(n,x), or Chebyshev polynomial of the fourth kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:

cos ( t ) = x
W(n,x) = sin((2*n+1)*t/2)/sin(t/2)

For any value of x, W(n,x) may be evaluated by a three term recurrence:
W(0,x) = 1
W(1,x) = 2x+1
W(n+1,x) = 2x W(n,x) - W(n-1,x)

### Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

### Languages:

CHEBYSHEV_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a FORTRAN90 library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV, a FORTRAN90 library which computes the Chebyshev interpolant/approximant to a given function over an interval.

CHEBYSHEV_SERIES, a FORTRAN90 library which can evaluate a Chebyshev series approximating a function f(x), while efficiently computing one, two or three derivatives of the series, which approximate f'(x), f''(x), and f'''(x), by Manfred Zimmer.

CHEBYSHEV1_RULE, a FORTRAN90 program which computes and prints a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a FORTRAN90 program which compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLAUSEN, a FORTRAN90 library which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).

GEGENBAUER_POLYNOMIAL, a FORTRAN90 library which evaluates the Gegenbauer polynomial and associated functions.

HERMITE_POLYNOMIAL, a FORTRAN90 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

INT_EXACTNESS_CHEBYSHEV1, a FORTRAN90 program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2, a FORTRAN90 program which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

JACOBI_POLYNOMIAL, a FORTRAN90 library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a FORTRAN90 library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a FORTRAN90 library which evaluates the Legendre polynomial and associated functions.

LOBATTO_POLYNOMIAL, a FORTRAN90 library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

POLPAK, a FORTRAN90 library which evaluates a variety of mathematical functions.

TEST_VALUES, a FORTRAN90 library which supplies test values of various mathematical functions.

### Reference:

1. Theodore Chihara,
An Introduction to Orthogonal Polynomials,
Gordon and Breach, 1978,
ISBN: 0677041500,
LC: QA404.5 C44.
2. Walter Gautschi,
Orthogonal Polynomials: Computation and Approximation,
Oxford, 2004,
ISBN: 0-19-850672-4,
LC: QA404.5 G3555.
3. John Mason, David Handscomb,
Chebyshev Polynomials,
CRC Press, 2002,
ISBN: 0-8493-035509,
LC: QA404.5.M37.
4. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
NIST Handbook of Mathematical Functions,
Cambridge University Press, 2010,
ISBN: 978-0521192255,
LC: QA331.N57.
5. Gabor Szego,
Orthogonal Polynomials,
American Mathematical Society, 1992,
ISBN: 0821810235,
LC: QA3.A5.v23.

### Examples and Tests:

Some of the tests create graphic files:

### List of Routines:

• DAXPY computes constant times a vector plus a vector.
• DDOT forms the dot product of two vectors.
• DNRM2 returns the euclidean norm of a vector.
• DROT applies a plane rotation.
• DROTG constructs a Givens plane rotation.
• DSCAL scales a vector by a constant.
• DSVDC computes the singular value decomposition of a real rectangular matrix.
• DSWAP interchanges two vectors.
• GET_UNIT returns a free FORTRAN unit number.
• I4_UNIFORM_AB returns a scaled pseudorandom I4 between A and B.
• I4VEC_MAX computes the maximum element of an I4VEC.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_FACTORIAL computes the factorial of N.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HYPER_2F1 evaluates the hypergeometric function F(A,B,C,X).
• R8_MOP returns the I-th power of -1 as an R8.
• R8_PSI evaluates the function Psi(X).
• R8_SIGN returns the sign of an R8.
• R8_UNIFORM_AB returns a scaled pseudorandom R8.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8POLY_PRINT prints out a polynomial.
• R8VEC_IN_AB is TRUE if the entries of an R8VEC are in the range [A,B].
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• R8VEC2_PRINT prints an R8VEC2.
• SVD_SOLVE solves a linear system in the least squares sense.
• T_MASS_MATRIX computes the mass matrix for the Chebyshev T polynomial.
• T_MOMENT: integral ( -1 <= x <= +1 ) x^e / sqrt ( 1 - x^2 ) dx.
• T_POLYNOMIAL evaluates Chebyshev polynomials T(n,x).
• T_POLYNOMIAL_01_VALUES: values of shifted Chebyshev polynomials T(n,x).
• T_POLYNOMIAL_AB: evaluates Chebyshev polynomials TAB(n,x) in [A,B].
• T_POLYNOMIAL_AB_VALUE: evaluates Chebyshev polynomials TAB(n,x) in [A,B].
• T_POLYNOMIAL_COEFFICIENTS: coefficients of the Chebyshev polynomial T(n,x).
• T_POLYNOMIAL_PLOT plots Chebyshev polynomials T(n,x).
• T_POLYNOMIAL_VALUE: returns the single value T(n,x).
• T_POLYNOMIAL_VALUES returns values of Chebyshev polynomials T(n,x).
• T_POLYNOMIAL_ZEROS returns zeroes of the Chebyshev polynomial T(n,x).
• T_PROJECT_COEFFICIENTS: function projected onto Chebyshev polynomials T(n,x).
• T_PROJECT_COEFFICIENTS_AB: function projected onto TAB(n,x) over [a,b].
• T_PROJECT_COEFFICIENTS_DATA: project data onto Chebyshev polynomials T(n,x).
• T_PROJECT_VALUE evaluates an expansion in Chebyshev polynomials T(n,x).
• T_PROJECT_VALUE_AB evaluates an expansion in Chebyshev polynomials TAB(n,x).
• T_QUADRATURE_RULE: quadrature rule for T(n,x).
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TT_PRODUCT: evaluate T(i,x)*T(j,x)
• TT_PRODUCT_INTEGRAL: integral (-1<=x<=1) T(i,x)*T(j,x)/sqrt(1-x^2) dx
• TTT_PRODUCT_INTEGRAL: int (-1<=x<=1) T(i,x)*T(j,x)*T(k,x)/sqrt(1-x^2) dx
• TU_PRODUCT: evaluate T(i,x)*U(j,x)
• U_MASS_MATRIX computes the mass matrix for the Chebyshev U polynomial.
• U_MOMENT: integral ( -1 <= x <= +1 ) x^e sqrt ( 1 - x^2 ) dx.
• U_POLYNOMIAL evaluates Chebyshev polynomials U(n,x).
• U_POLYNOMIAL_01_VALUES: values of shifted Chebyshev polynomials U01(n,x).
• U_POLYNOMIAL_AB: evaluates Chebyshev polynomials UAB(n,x) in [A,B].
• U_POLYNOMIAL_AB_VALUE: evaluates Chebyshev polynomials UAB(n,x) in [A,B].
• U_POLYNOMIAL_COEFFICIENTS: coefficients of Chebyshev polynomials U(n,x).
• U_POLYNOMIAL_PLOT plots Chebyshev polynomials U(n,x).
• U_POLYNOMIAL_VALUE: returns the single value U(n,x).
• U_POLYNOMIAL_VALUES returns values of Chebyshev polynomials U(n,x).
• U_POLYNOMIAL_ZEROS returns zeroes of Chebyshev polynomials U(n,x).
• U_QUADRATURE_RULE: quadrature rule for U(n,x).
• UU_PRODUCT: evaluate U(i,x)*U(j,x)
• UU_PRODUCT_INTEGRAL: integral (-1<=x<=1) U(i,x)*U(j,x)*sqrt(1-x^2) dx
• V_MASS_MATRIX computes the mass matrix for the Chebyshev V polynomial.
• V_MOMENT: integral ( -1 <= x <= +1 ) x^e sqrt(1+x) / sqrt(1-x) dx.
• V_POLYNOMIAL evaluates Chebyshev polynomials V(n,x).
• V_POLYNOMIAL_01_VALUES: values of shifted Chebyshev polynomials V01(n,x).
• V_POLYNOMIAL_AB: evaluates Chebyshev polynomials VAB(n,x) in [A,B].
• V_POLYNOMIAL_AB_VALUE: evaluates Chebyshev polynomials VAB(n,x) in [A,B].
• V_POLYNOMIAL_COEFFICIENTS: coefficients of Chebyshev polynomials V(n,x).
• V_POLYNOMIAL_PLOT plots Chebyshev polynomials V(n,x).
• V_POLYNOMIAL_VALUE: returns the single value V(n,x).
• V_POLYNOMIAL_VALUES returns values of Chebyshev polynomials V(n,x).
• V_POLYNOMIAL_ZEROS returns zeroes of Chebyshev polynomials V(n,x).
• V_QUADRATURE_RULE: quadrature rule for V(n,x).
• VV_PRODUCT_INTEGRAL: int (-1
• W_MASS_MATRIX computes the mass matrix for the Chebyshev W polynomial.
• W_MOMENT: integral ( -1 <= x <= +1 ) x^e sqrt(1-x) / sqrt(1+x) dx.
• W_POLYNOMIAL evaluates Chebyshev polynomials W(n,x).
• W_POLYNOMIAL_01_VALUES: values of shifted Chebyshev polynomials W01(n,x).
• W_POLYNOMIAL_AB: evaluates Chebyshev polynomials WAB(n,x) in [A,B].
• W_POLYNOMIAL_AB_VALUE: evaluates Chebyshev polynomials WAB(n,x) in [A,B].
• W_POLYNOMIAL_COEFFICIENTS: coefficients of Chebyshev polynomials W(n,x).
• W_POLYNOMIAL_PLOT plots Chebyshev polynomials W(n,x).
• W_POLYNOMIAL_VALUE: returns the single value W(n,x).
• W_POLYNOMIAL_VALUES returns values of Chebyshev polynomials W(n,x).
• W_POLYNOMIAL_ZEROS returns zeroes of Chebyshev polynomials W(n,x).
• W_QUADRATURE_RULE: quadrature rule for W(n,x).
• WW_PRODUCT_INTEGRAL: int (-1

You can go up one level to the FORTRAN90 source codes.

Last revised on 21 July 2015.