# chebyshevU

Chebyshev polynomials of the second kind

## Description

example

chebyshevU(n,x) represents the nth degree Chebyshev polynomial of the second kind at the point x.

## Examples

### First Five Chebyshev Polynomials of the Second Kind

Find the first five Chebyshev polynomials of the second kind for the variable x.

syms x
chebyshevU([0, 1, 2, 3, 4], x)
ans =
[ 1, 2*x, 4*x^2 - 1, 8*x^3 - 4*x, 16*x^4 - 12*x^2 + 1]

### Chebyshev Polynomials for Numeric and Symbolic Arguments

Depending on its arguments, chebyshevU returns floating-point or exact symbolic results.

Find the value of the fifth-degree Chebyshev polynomial of the second kind at these points. Because these numbers are not symbolic objects, chebyshevU returns floating-point results.

chebyshevU(5, [1/6, 1/3, 1/2, 2/3, 4/5])
ans =
0.8560    0.9465    0.0000   -1.2675   -1.0982

Find the value of the fifth-degree Chebyshev polynomial of the second kind for the same numbers converted to symbolic objects. For symbolic numbers, chebyshevU returns exact symbolic results.

chebyshevU(5, sym([1/6, 1/4, 1/3, 1/2, 2/3, 4/5]))
ans =
[ 208/243, 33/32, 230/243, 0, -308/243, -3432/3125]

### Evaluate Chebyshev Polynomials with Floating-Point Numbers

Floating-point evaluation of Chebyshev polynomials by direct calls of chebyshevU is numerically stable. However, first computing the polynomial using a symbolic variable, and then substituting variable-precision values into this expression can be numerically unstable.

Find the value of the 500th-degree Chebyshev polynomial of the second kind at 1/3 and vpa(1/3). Floating-point evaluation is numerically stable.

chebyshevU(500, 1/3)
chebyshevU(500, vpa(1/3))
ans =
0.8680

ans =
0.86797529488884242798157148968078

Now, find the symbolic polynomial U500 = chebyshevU(500, x), and substitute x = vpa(1/3) into the result. This approach is numerically unstable.

syms x
U500 = chebyshevU(500, x);
subs(U500, x, vpa(1/3))
ans =
63080680195950160912110845952.0

Approximate the polynomial coefficients by using vpa, and then substitute x = sym(1/3) into the result. This approach is also numerically unstable.

subs(vpa(U500), x, sym(1/3))
ans =
-1878009301399851172833781612544.0

### Plot Chebyshev Polynomials of the Second Kind

Plot the first five Chebyshev polynomials of the second kind.

syms x y
fplot(chebyshevU(0:4, x))
axis([-1.5 1.5 -2 2])
grid on

ylabel('U_n(x)')
legend('U_0(x)', 'U_1(x)', 'U_2(x)', 'U_3(x)', 'U_4(x)', 'Location', 'Best')
title('Chebyshev polynomials of the second kind')

## Input Arguments

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Degree of the polynomial, specified as a nonnegative integer, symbolic variable, expression, or function, or as a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.

Evaluation point, specified as a number, symbolic number, variable, expression, or function, or as a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.

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### Chebyshev Polynomials of the Second Kind

• Chebyshev polynomials of the second kind are defined as follows:

$U\left(n,x\right)=\frac{\mathrm{sin}\left(\left(n+1\right)a\mathrm{cos}\left(x\right)\right)}{\mathrm{sin}\left(a\mathrm{cos}\left(x\right)\right)}$

These polynomials satisfy the recursion formula

$U\left(0,x\right)=1,\text{ }U\left(1,x\right)=2\text{ }x,\text{ }U\left(n,x\right)=2\text{ }x\text{ }U\left(n-1,x\right)-U\left(n-2,x\right)$

• Chebyshev polynomials of the second kind are orthogonal on the interval -1 ≤ x ≤ 1 with respect to the weight function $w\left(x\right)=\sqrt{1-{x}^{2}}$.

• Chebyshev polynomials of the second kind are a special case of the Jacobi polynomials

$U\left(n,x\right)=\frac{{2}^{2n}n!\left(n+1\right)!}{\left(2n+1\right)!}P\left(n,\frac{1}{2},\frac{1}{2},x\right)$

and Gegenbauer polynomials

$U\left(n,x\right)=G\left(n,1,x\right)$

## Tips

• chebyshevU returns floating-point results for numeric arguments that are not symbolic objects.

• chebyshevU acts element-wise on nonscalar inputs.

• At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, then chebyshevU expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

## References

[1] Hochstrasser, U. W. “Orthogonal Polynomials.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.