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Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as $$T_n(x)$$ and $$U_n(x)$$. They can be defined in several equivalent ways, one of which starts with trigonometric functions:

The Chebyshev polynomials of the first kind $$T_n$$ are defined by: $$T_n(\cos \theta) = \cos(n\theta).$$

Similarly, the Chebyshev polynomials of the second kind $$U_n$$ are defined by: $$U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big).$$

That these expressions define polynomials in $$\cos\theta$$ may not be obvious at first sight but follows by rewriting $$\cos(n\theta)$$ and $$\sin\big((n+1)\theta\big)$$ using de Moivre's formula or by using the angle sum formulas for $$\cos$$ and $$\sin$$ repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain $$T_2(\cos\theta) = \cos(2\theta) = 2\cos^2\theta-1$$ and $$U_1(\cos\theta)\sin\theta = \sin(2\theta) = 2\cos\theta\sin\theta$$, which are respectively a polynomial in $$\cos\theta$$ and a polynomial in $$\cos\theta$$ multiplied by $$\sin\theta$$. Hence $$T_2(x) = 2x^2 - 1$$ and $$U_1(x) = 2x$$.

An important and convenient property of the $T_{n}(x)$ is that they are orthogonal with respect to the inner product: $$\langle f, g\rangle = \int_{-1}^1 f(x) \, g(x) \, \frac{\mathrm{d}x}{\sqrt{1 - x^2}},$$ and $U_{n}(x)$ are orthogonal with respect to another, analogous inner product, given below.

The Chebyshev polynomials $T_{n}$ are polynomials with the largest possible leading coefficient whose absolute value on the interval $[−1, 1]$ is bounded by 1. They are also the "extremal" polynomials for many other properties.

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of $T_{n}(x)$, which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature.

These polynomials were named after Pafnuty Chebyshev. The letter $T$ is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

Recurrence definition
The Chebyshev polynomials of the first kind are obtained from the recurrence relation: $$\begin{align} T_0(x) & = 1 \\ T_1(x) & = x \\ T_{n+1}(x) & = 2 x\,T_n(x) - T_{n-1}(x). \end{align}$$ The recurrence also allows to represent them explicitly as the determinant of a tridiagonal matrix of size $$k \times k$$:

$$T_k(x) = \det \begin{bmatrix} x &     1 &      0 & \cdots & 0      \\ 1 &    2x &      1 & \ddots & \vdots \\ 0 &     1 &     2x & \ddots & 0      \\ \vdots & \ddots & \ddots & \ddots & 1     \\ 0 & \cdots &     0 &      1 & 2x \end{bmatrix}$$

The ordinary generating function for $T_{n}$ is: $$\sum_{n=0}^{\infty}T_n(x)\,t^n = \frac{1-tx}{1-2tx+t^2}.$$ There are several other generating functions for the Chebyshev polynomials; the exponential generating function is: $$\sum_{n=0}^{\infty}T_n(x)\frac{t^n}{n!} = \frac{1}{2}\!\left( e^{t\left(x-\sqrt{x^2 - 1} \right)} + e^{t\left(x+\sqrt{x^2 -1}\right)}\right) = e^{tx}\cosh\left(t\sqrt{x^2-1}\right).$$

The generating function relevant for 2-dimensional potential theory and multipole expansion is: $$\sum\limits_{n=1}^{\infty}T_{n}(x)\,\frac{t^n}{n} = \ln\left( \frac{1}{\sqrt{ 1 - 2tx + t^2 }}\right).$$

The Chebyshev polynomials of the second kind are defined by the recurrence relation: $$\begin{align} U_0(x) & = 1 \\ U_1(x) & = 2 x \\ U_{n+1}(x) & = 2 x\,U_n(x) - U_{n-1}(x). \end{align}$$ Notice that the two sets of recurrence relations are identical, except for $$T_1(x) = x$$ vs. $U_1(x) = 2x$. The ordinary generating function for $T_{n}$ is: $$\sum_{n=0}^{\infty}U_n(x)\,t^n = \frac{1}{1 - 2 t x+t^2},$$ and the exponential generating function is: $$\sum_{n=0}^{\infty}U_n(x)\frac{t^n}{n!} = e^{tx}\!\left(\!\cosh\left(t\sqrt{x^2-1}\right) + \frac{x}{\sqrt{x^2-1}} \sinh\left(t\sqrt{x^2-1}\right)\!\right).$$

Trigonometric definition
As described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying: $$T_n(x) = \begin{cases} \cos(n \arccos x)                         & \text{ if }~ |x| \le 1 \\ \cosh(n \operatorname{arcosh} x)          & \text{ if }~ x \ge 1 \\ (-1)^n \cosh(n \operatorname{arcosh}(-x) ) & \text{ if }~ x \le -1 \end{cases}$$ or, in other words, as the unique polynomials satisfying: $$T_n(\cos\theta) = \cos(n\theta)$$ for $n = 0, 1, 2, 3, …$.

The polynomials of the second kind satisfy: $$U_{n-1}(\cos\theta) \sin\theta = \sin(n\theta),$$ or $$U_n(\cos\theta) = \frac{\sin\big((n+1)\,\theta\big)}{\sin\theta},$$ which is structurally quite similar to the Dirichlet kernel $D_{n}(x)$: $$D_n(x) = \frac{\sin\left((2n+1)\dfrac{x}{2}\,\right)}{\sin \dfrac{x}{2}} = U_{2n}\!\!\left(\cos \frac{x}{2}\right).$$ (The Dirichlet kernel, in fact, coincides with what is now known as the Chebyshev polynomial of the fourth kind.)

An equivalent way to state this is via exponentiation of a complex number: given a complex number $z = a + bi$ with absolute value of one: $$z^n = T_n(a) + ib U_{n-1}(a).$$ Chebyshev polynomials can be defined in this form when studying trigonometric polynomials.

That $cos nx$ is an $U_{n}$th-degree polynomial in $cos x$ can be seen by observing that $cos nx$ is the real part of one side of de Moivre's formula: $$\cos n \theta + i \sin n \theta = (\cos \theta + i \sin \theta)^n.$$ The real part of the other side is a polynomial in $cos x$ and $sin x$, in which all powers of $sin x$ are even and thus replaceable through the identity $cos^{2} x + sin^{2} x = 1$. By the same reasoning, $sin nx$ is the imaginary part of the polynomial, in which all powers of $sin x$ are odd and thus, if one factor of $sin x$ is factored out, the remaining factors can be replaced to create a $(n−1)$st-degree polynomial in $cos x$.

Commuting polynomials definition
Chebyshev polynomials can also be characterized by the following theorem:

If $$ F_n(x)$$ is a family of monic polynomials with coefficients in a field of characteristic $$0$$ such that $$ \deg F_n(x) = n$$ and $$ F_m(F_n(x)) = F_n(F_m(x))$$ for all $$m$$ and $$ n$$, then, up to a simple change of variables, either $$ F_n(x) = x^n$$ for all $$ n$$ or $$F_n(x) = 2\cdot T_n(x/2)$$ for all $$ n$$.

Pell equation definition
The Chebyshev polynomials can also be defined as the solutions to the Pell equation: $$T_n(x)^2 - \left(x^2 - 1\right) U_{n-1}(x)^2 = 1$$ in a ring $R[x]$. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution: $$T_n(x) + U_{n-1}(x)\,\sqrt{x^2-1} = \left(x + \sqrt{x^2-1}\right)^n~. $$

Relations between the two kinds of Chebyshev polynomials
The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences $Ṽ_{n}(P, Q)$ and $Ũ_{n}(P, Q)$ with parameters $P = 2x$ and $Q = 1$: $$\begin{align} {\tilde U}_n(2x,1) &= U_{n-1}(x), \\ {\tilde V}_n(2x,1) &= 2\, T_n(x). \end{align}$$ It follows that they also satisfy a pair of mutual recurrence equations: $$\begin{align} T_{n+1}(x) &= x\,T_n(x) - (1 - x^2)\,U_{n-1}(x), \\ U_{n+1}(x) &= x\,U_n(x) + T_{n+1}(x). \end{align}$$

The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give: $$T_n(x) = \frac{1}{2} \big(U_n(x) - U_{n-2}(x)\big).$$

Using this formula iteratively gives the sum formula: $$ U_n(x) = \begin{cases} 2\sum_{\text{ odd }j}^n T_j(x)     & \text{ for odd }n.\\ 2\sum_{\text{ even }j}^n T_j(x) + 1     & \text{ for even }n, \end{cases} $$ while replacing $$U_n(x)$$ and $$U_{n-2}(x)$$ using the derivative formula for $$T_n(x)$$ gives the recurrence relationship for the derivative of $$T_n$$:

$$2\,T_n(x) = \frac{1}{n+1}\, \frac{\mathrm{d}}{\mathrm{d}x}\, T_{n+1}(x) - \frac{1}{n-1}\,\frac{\mathrm{d}}{\mathrm{d}x}\, T_{n-1}(x), \qquad n=2,3,\ldots$$

This relationship is used in the Chebyshev spectral method of solving differential equations.

Turán's inequalities for the Chebyshev polynomials are: $$\begin{align} T_n(x)^2 - T_{n-1}(x)\,T_{n+1}(x)&= 1-x^2 > 0 &&\text{ for } -1 0~. \end{align}$$

The integral relations are $$\begin{align} \int_{-1}^1 \frac{T_n(y)}{y-x} \, \frac{\mathrm{d}y}{\sqrt{1 - y^2}} &= \pi\,U_{n-1}(x)~, \\[1.5ex] \int_{-1}^1\frac{U_{n-1}(y)}{y-x}\, \sqrt{1 - y^2}\mathrm{d}y &= -\pi\,T_n(x) \end{align}$$ where integrals are considered as principal value.

Explicit expressions
Different approaches to defining Chebyshev polynomials lead to different explicit expressions. The trigonometric definition gives an explicit formula as follows: $$\begin{align} T_n(x) & = \begin{cases} \cos(n\arccos x) \qquad \quad & \text{ for }~ -1 \le x \le 1 \\ \cosh(n \operatorname{arcosh}x) \qquad \quad & \text{ for }~ 1 \le x \\ (-1)^n \cosh\big(n \operatorname{arcosh}(-x)\big) \qquad \quad & \text{ for }~ x \le -1 \end{cases} \end{align}$$ From this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold: $$T_0(\cos\theta) = \cos(0\theta) = 1$$ and $$T_1(\cos\theta) = \cos\theta ,$$ and that the product-to-sum identity holds: $$2\cos n\theta\cos\theta = \cos \lbrack (n+1)\theta \rbrack +\cos\lbrack (n-1)\theta\rbrack.$$

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression: $$ T_n(x) = \dfrac{1}{2} \bigg( \Big(x-\sqrt{x^2-1} \Big)^n + \Big(x+\sqrt{x^2-1} \Big)^n \bigg) \qquad \text{ for }~ x \in \mathbb {R}$$ $$ T_n(x) = \dfrac{1}{2} \bigg( \Big(x-\sqrt{x^2-1} \Big)^n + \Big(x-\sqrt{x^2-1} \Big)^{-n} \bigg) \qquad \text{ for }~ x \in \mathbb {R}$$ The two are equivalent because $$(x + \sqrt{x^2 - 1})(x - \sqrt{x^2 - 1}) = 1$$.

An explicit form of the Chebyshev polynomial in terms of monomials $x^{k}$ follows from de Moivre's formula: $$T_n(\cos(\theta)) = \operatorname{Re}(\cos n \theta + i \sin n \theta) = \operatorname{Re}((\cos \theta + i \sin \theta)^n),$$ where $Re$ denotes the real part of a complex number. Expanding the formula, one gets: $$(\cos \theta + i \sin \theta)^n = \sum\limits_{j=0}^n \binom{n}{j} i^j \sin^j \theta \cos^{n-j} \theta.$$ The real part of the expression is obtained from summands corresponding to even indices. Noting $$i^{2j} = (-1)^j$$ and $$\sin^{2j} \theta = (1-\cos^2 \theta)^j$$, one gets the explicit formula: $$\cos n \theta = \sum\limits_{j=0}^{\lfloor n / 2 \rfloor} \binom{n}{2j} (\cos^2 \theta - 1)^j \cos^{n-2j} \theta,$$ which in turn means that: $$T_n(x) = \sum\limits_{j=0}^{\lfloor n / 2 \rfloor} \binom{n}{2j} (x^2-1)^j x^{n-2j}.$$ This can be written as a $_{2}F_{1}$ hypergeometric function: $$\begin{align} T_n(x) & = \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor} \binom{n}{2k} \left (x^2-1 \right )^k x^{n-2k} \\ & = x^n \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor} \binom{n}{2k} \left (1 - x^{-2} \right )^k \\ & = \frac{n}{2} \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} \qquad\qquad \text{ for }~ n > 0 \\ \\ & = n \sum_{k=0}^{n}(-2)^{k} \frac{(n+k-1)!} {(n-k)!(2k)!}(1 - x)^k \qquad\qquad ~ \text{ for }~ n > 0 \\ \\ & = {}_2F_1\!\left(-n,n;\tfrac 1 2; \tfrac{1}{2}(1-x)\right) \\ \end{align}$$ with inverse:

$$x^n = 2^{1-n}\mathop{{\sum}'}^n_{j=0\atop j \,\equiv\, n \pmod 2} \!\!\binom{n}{\tfrac{n-j}{2}}\!\;T_j(x),$$ where the prime at the summation symbol indicates that the contribution of $j = 0$ needs to be halved if it appears.

A related expression for $T_{n}$ as a sum of monomials with binomial coefficients and powers of two is $$T_{n}\left(x\right) = \sum\limits_{m=0}^{\left \lfloor \frac{n}{2} \right \rfloor}\left(-1\right)^{m}\left(\binom{n-m}{m} + \binom{n-m-1}{n-2m}\right)\cdot 2^{n-2m-1}\cdot x^{n-2m}.$$

Similarly, $U_{n}$ can be expressed in terms of hypergeometric functions: $$\begin{align} U_n(x) & = \frac{\left (x+\sqrt{x^2-1} \right )^{n+1} - \left (x-\sqrt{x^2-1} \right )^{n+1}}{2\sqrt{x^2-1}} \\ & = \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{n+1}{2k+1} \left (x^2-1 \right )^k x^{n-2k} \\ & = x^n \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{n+1}{2k+1} \left (1 - x^{-2} \right )^k \\ & = \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{2k-(n+1)}{k}~(2x)^{n-2k} & \text{ for }~ n > 0 \\ & = \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} (-1)^k \binom{n-k}{k}~(2x)^{n-2k} & \text{ for }~ n > 0 \\ & = \sum_{k=0}^{n}(-2)^{k} \frac{(n+k+1)!} {(n-k)!(2k+1)!}(1 - x)^k & \text{ for }~ n > 0 \\ & = (n+1) \ {}_2F_1\left(-n,n+2; \tfrac{3}{2}; \tfrac{1}{2}(1-x) \right). \\ \end{align}$$

Symmetry
$$\begin{align} T_n(-x) &= (-1)^n\, T_n(x) = \begin{cases} T_n(x) \quad & ~\text{ for }~n~\text{ even} \\ -T_n(x) \quad & ~\text{ for }~n~\text{ odd} \end{cases} \\ \\ U_n(-x) &= (-1)^n\, U_n(x) = \begin{cases} U_n(x) \quad & ~\text{ for }~n~\text{ even} \\ -U_n(x) \quad & ~\text{ for }~n~\text{ odd} \end{cases} \end{align}$$

That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of $U_{n}$. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of $n$.

Roots and extrema
A Chebyshev polynomial of either kind with degree $x$ has $x$ different simple roots, called Chebyshev roots, in the interval $n$. The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that: $$\cos\left((2k+1)\frac{\pi}{2}\right)=0$$ one can show that the roots of $n$ are: $$ x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1.$$ Similarly, the roots of $[−1, 1]$ are: $$ x_k = \cos\left(\frac{k}{n+1}\pi\right),\quad k=1,\ldots,n.$$ The extrema of $T_{n}$ on the interval $−1 ≤ x ≤ 1$ are located at: $$ x_k = \cos\left(\frac{k}{n}\pi\right),\quad k=0,\ldots,n.$$

One unique property of the Chebyshev polynomials of the first kind is that on the interval $−1 ≤ x ≤ 1$ all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by: $$\begin{align} T_n(1) &= 1 \\ T_n(-1) &= (-1)^n \\ U_n(1) &= n+1 \\ U_n(-1) &= (-1)^n (n+1). \end{align}$$

The extrema of $$T_n(x)$$ on the interval $$-1 \leq x \leq 1$$ where $$n>0$$ are located at $$n+1$$ values of $$x$$. They are $$ \pm 1$$, or $$ \cos\left(\frac{2\pi k}{d}\right)$$ where $$d > 2$$, $$d \;|\; 2n$$, $$0 < k < d/2$$ and $$(k, d) = 1$$, i.e., $$k$$ and $$d$$ are relatively prime numbers.

Specifically, when $$n$$ is even:
 * $$T_n(x) = 1$$ if $$x = \pm 1$$, or $$d > 2$$ and $$2n/d$$ is even. There are $$n/2 + 1$$ such values of $$x$$.
 * $$T_n(x) = -1$$ if $$d > 2$$ and $$2n/d$$ is odd. There are $$n/2$$ such values of $$x$$.

When $$n$$ is odd:
 * $$T_n(x) = 1$$ if $$x = 1$$, or $$d > 2$$ and $$2n/d$$ is even. There are $$(n+1)/2$$ such values of $$x$$.
 * $$T_n(x) = -1$$ if $$x = -1$$, or $$d > 2$$ and $$2n/d$$ is odd. There are $$(n+1)/2$$ such values of $$x$$.

This result has been generalized to solutions of $$U_n(x) \pm 1 = 0$$, and to $$V_n(x) \pm 1 = 0$$ and $$W_n(x) \pm 1 = 0$$ for Chebyshev polynomials of the third and fourth kinds, respectively.

Differentiation and integration
The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that: $$\begin{align} \frac{\mathrm{d}T_n}{\mathrm{d}x} &= n U_{n - 1} \\ \frac{\mathrm{d}U_n}{\mathrm{d}x} &= \frac{(n + 1)T_{n + 1} - x U_n}{x^2 - 1} \\ \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} &= n\, \frac{n T_n - x U_{n - 1}}{x^2 - 1} = n\, \frac{(n + 1)T_n - U_n}{x^2 - 1}. \end{align}$$

The last two formulas can be numerically troublesome due to the division by zero ($U_{n}$ indeterminate form, specifically) at $x = 1$ and $x = −1$. By L'Hôpital's rule: $$\begin{align} \left. \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} \right|_{x = 1} \!\! &= \frac{n^4 - n^2}{3}, \\ \left. \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} \right|_{x = -1} \!\! &= (-1)^n \frac{n^4 - n^2}{3}. \end{align}$$

More generally, $$\left.\frac{d^p T_n}{d x^p} \right|_{x = \pm 1} \!\! = (\pm 1)^{n+p}\prod_{k=0}^{p-1}\frac{n^2-k^2}{2k+1}~,$$ which is of great use in the numerical solution of eigenvalue problems.

Also, we have: $$\frac{\mathrm{d}^p}{\mathrm{d}x^p}\,T_n(x) = 2^p\,n\mathop{{\sum}'}_{0\leq k\leq n-p\atop k \,\equiv\, n-p \pmod 2} \binom{\frac{n+p-k}{2}-1}{\frac{n-p-k}{2}}\frac{\left(\frac{n+p+k}{2}-1\right)!}{\left(\frac{n-p+k}{2}\right)!}\,T_k(x),~\qquad p \ge 1,$$ where the prime at the summation symbols means that the term contributed by $k = 0$ is to be halved, if it appears.

Concerning integration, the first derivative of the $T_{n}$ implies that: $$\int U_n\, \mathrm{d}x = \frac{T_{n + 1}}{n + 1}$$ and the recurrence relation for the first kind polynomials involving derivatives establishes that for $n ≥ 2$: $$\int T_n\, \mathrm{d}x = \frac{1}{2}\,\left(\frac{T_{n + 1}}{n + 1} - \frac{T_{n - 1}}{n - 1}\right) = \frac{n\,T_{n + 1}}{n^2 - 1} - \frac{x\,T_n}{n - 1}.$$

The last formula can be further manipulated to express the integral of $0⁄0$ as a function of Chebyshev polynomials of the first kind only: $$\begin{align} \int T_n\, \mathrm{d}x &= \frac{n}{n^2 - 1} T_{n + 1} - \frac{1}{n - 1} T_1 T_n \\ &= \frac{n}{n^2 - 1}\,T_{n + 1} - \frac{1}{2(n - 1)}\,(T_{n + 1} + T_{n - 1}) \\ &= \frac{1}{2(n + 1)}\,T_{n + 1} - \frac{1}{2(n - 1)}\,T_{n - 1}. \end{align}$$

Furthermore, we have: $$\int_{-1}^1 T_n(x)\, \mathrm{d}x = \begin{cases} \frac{(-1)^n + 1}{1 - n^2} & \text{ if }~ n \ne 1 \\ 0                         & \text{ if }~ n = 1. \end{cases}$$

Products of Chebyshev polynomials
The Chebyshev polynomials of the first kind satisfy the relation: $$T_m(x)\,T_n(x) = \tfrac{1}{2}\!\left(T_{m+n}(x) + T_{|m-n|}(x)\right)\!,\qquad \forall m,n \ge 0,$$ which is easily proved from the product-to-sum formula for the cosine: $$2 \cos \alpha \, \cos \beta = \cos (\alpha + \beta) + \cos (\alpha - \beta).$$ For $n = 1$ this results in the already known recurrence formula, just arranged differently, and with $n = 2$ it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest $T_{n}$) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion: $$\begin{align} T_{2n}(x) &= 2\,T_n^2(x)          - T_0(x) &&= 2 T_n^2(x) - 1, \\ T_{2n+1}(x) &= 2\,T_{n+1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n+1}(x)\,T_n(x) - x, \\ T_{2n-1}(x) &= 2\,T_{n-1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n-1}(x)\,T_n(x) - x. \end{align}$$

The polynomials of the second kind satisfy the similar relation: $$ T_m(x)\,U_n(x) = \begin{cases} \frac{1}{2}\left(U_{m+n}(x) + U_{n-m}(x)\right), & ~\text{ if }~ n \ge m-1,\\ \\ \frac{1}{2}\left(U_{m+n}(x) - U_{m-n-2}(x)\right), & ~\text{ if }~ n \le m-2. \end{cases} $$ (with the definition $U_{−1} ≡ 0$ by convention ). They also satisfy: $$ U_m(x)\,U_n(x) = \sum_{k=0}^n\,U_{m-n+2k}(x) = \sum_\underset{\text{ step 2 }}{p=m-n}^{m+n} U_p(x)~.$$ for $m ≥ n$. For $n = 2$ this recurrence reduces to: $$ U_{m+2}(x) = U_2(x)\,U_m(x) - U_m(x) - U_{m-2}(x) = U_m(x)\,\big(U_2(x) - 1\big) - U_{m-2}(x)~,$$ which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether $T_{n}$ starts with 2 or 3.

Composition and divisibility properties
The trigonometric definitions of $T_{n}$ and $U_{n}$ imply the composition or nesting properties: $$\begin{align} T_{mn}(x) &= T_m(T_n(x)),\\ U_{mn-1}(x) &= U_{m-1}(T_n(x))U_{n-1}(x). \end{align} $$ For $T_{mn}$ the order of composition may be reversed, making the family of polynomial functions $T_{n}$ a commutative semigroup under composition.

Since $T_{m}(x)$ is divisible by $m$ if $m$ is odd, it follows that $T_{mn}(x)$ is divisible by $T_{n}(x)$ if $x$ is odd. Furthermore, $U_{mn−1}(x)$ is divisible by $U_{n−1}(x)$, and in the case that $m$ is even, divisible by $T_{n}(x)U_{n−1}(x)$.

Orthogonality
Both $m$ and $m$ form a sequence of orthogonal polynomials. The polynomials of the first kind $T_{n}$ are orthogonal with respect to the weight: $$\frac{1}{\sqrt{1 - x^2}},$$ on the interval $U_{n}$, i.e. we have: $$\int_{-1}^1 T_n(x)\,T_m(x)\,\frac{\mathrm{d}x}{\sqrt{1-x^2}} = \begin{cases} 0            & ~\text{ if }~ n \ne m, \\[5mu] \pi          & ~\text{ if }~ n=m=0, \\[5mu] \frac{\pi}{2} & ~\text{ if }~ n=m \ne 0. \end{cases}$$

This can be proven by letting $x = cos θ$ and using the defining identity $T_{n}(cos θ) = cos(nθ)$.

Similarly, the polynomials of the second kind $T_{n}$ are orthogonal with respect to the weight: $$\sqrt{1-x^2}$$ on the interval $[−1, 1]$, i.e. we have: $$\int_{-1}^1 U_n(x)\,U_m(x)\,\sqrt{1-x^2} \,\mathrm{d}x = \begin{cases} 0            & ~\text{ if }~ n \ne m, \\[5mu] \frac{\pi}{2} & ~\text{ if }~ n = m. \end{cases}$$

(The measure $√1 − x^{2} dx$ is, to within a normalizing constant, the Wigner semicircle distribution.)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the Chebyshev differential equations: $$\begin{align} (1 - x^2)T_n'' - xT_n' + n^2 T_n &= 0, \\[1ex] (1 - x^2)U_n'' - 3xU_n' + n(n + 2) U_n &= 0, \end{align}$$which are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to those equations.)

The $U_{n}$ also satisfy a discrete orthogonality condition: $$\sum_{k=0}^{N-1}{T_i(x_k)\,T_j(x_k)} = \begin{cases} 0          & ~\text{ if }~ i \ne j, \\[5mu] N          & ~\text{ if }~ i = j = 0, \\[5mu] \frac{N}{2} & ~\text{ if }~ i = j \ne 0, \end{cases} $$ where $[−1, 1]$ is any integer greater than $max(i, j)$, and the $x_{k}$ are the $T_{n}$ Chebyshev nodes (see above) of $T_{N }(x)$: $$x_k = \cos\left(\pi\,\frac{2k+1}{2N}\right) \quad ~\text{ for }~ k = 0, 1, \dots, N-1.$$

For the polynomials of the second kind and any integer $N > i + j$ with the same Chebyshev nodes $x_{k}$, there are similar sums: $$\sum_{k=0}^{N-1}{U_i(x_k)\,U_j(x_k)\left(1-x_k^2\right)} = \begin{cases} 0          & \text{ if }~ i \ne j, \\[5mu] \frac{N}{2} & \text{ if }~ i = j, \end{cases}$$ and without the weight function: $$\sum_{k=0}^{N-1}{ U_i(x_k) \, U_j(x_k) } = \begin{cases} 0                        & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu] N \cdot (1 + \min\{i,j\}) & ~\text{ if }~ i \equiv j\pmod{2}. \end{cases} $$

For any integer $N > i + j$, based on the $N$ zeros of $U_{N }(x)$: $$y_k = \cos\left(\pi\,\frac{k+1}{N+1}\right) \quad ~\text{ for }~ k=0, 1, \dots, N-1,$$ one can get the sum: $$\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)(1-y_k^2)} = \begin{cases} 0 & ~\text{ if } i \ne j, \\[5mu] \frac{N+1}{2} & ~\text{ if } i = j, \end{cases}$$ and again without the weight function: $$\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)} = \begin{cases} 0 & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu] \bigl(\min\{i,j\} + 1\bigr)\bigl(N-\max\{i,j\}\bigr) & ~\text{ if }~ i \equiv j\pmod{2}. \end{cases}$$

Minimal $∞$-norm
For any given $n ≥ 1$, among the polynomials of degree $N$ with leading coefficient 1 (monic polynomials): $$f(x) = \frac{1}{\,2^{n-1}\,}\,T_n(x)$$ is the one of which the maximal absolute value on the interval $N$ is minimal.

This maximal absolute value is: $$\frac1{2^{n-1}}$$ and $|f(x)|$ reaches this maximum exactly $n + 1$ times at: $$x = \cos \frac{k\pi}{n}\quad\text{for }0 \le k \le n.$$

$n$

Remark
By the equioscillation theorem, among all the polynomials of degree $w_{n}(x)$, the polynomial $[−1, 1]$ minimizes $1 / 2^{n − 1}$ on $$ if and only if there are $f_{n}(x)$ points $f_{n}(x)$ such that $n − 1$.

Of course, the null polynomial on the interval $f$ can be approximated by itself and minimizes the $n − 1$-norm.

Above, however, $≤ n$ reaches its maximum only $\| f \|_{∞}$ times because we are searching for the best polynomial of degree $n + 2$ (therefore the theorem evoked previously cannot be used).

Chebyshev polynomials as special cases of more general polynomial families
The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials $$C_n^{(\lambda)}(x)$$, which themselves are a special case of the Jacobi polynomials $$P_n^{(\alpha,\beta)}(x)$$: $$\begin{align} T_n(x) &= \frac{n}{2} \lim_{q \to 0} \frac{1}{q}\,C_n^{(q)}(x) \qquad ~\text{ if }~ n \ge 1, \\ &= \frac{1}{\binom{n-\frac{1}{2}}{n}} P_n^{\left(-\frac{1}{2}, -\frac{1}{2}\right)}(x) = \frac{2^{2n}}{\binom{2n}{n}} P_n^{\left(-\frac{1}{2}, -\frac{1}{2}\right)}(x)~, \\[2ex] U_n(x) & = C_n^{(1)}(x)\\ &= \frac{n+1}{\binom{n+\frac{1}{2}}{n}} P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x) = \frac{2^{2n+1}}{\binom{2n+2}{n+1}} P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x)~. \end{align}$$

Chebyshev polynomials are also a special case of Dickson polynomials: $$D_n(2x\alpha,\alpha^2)= 2\alpha^{n}T_n(x) \, $$ $$E_n(2x\alpha,\alpha^2)= \alpha^{n}U_n(x). \, $$ In particular, when $$\alpha=\tfrac{1}{2}$$, they are related by $$D_n(x,\tfrac{1}{4}) = 2^{1-n}T_n(x)$$ and $$E_n(x,\tfrac{1}{4}) = 2^{-n}U_n(x)$$.

Other properties
The curves given by $−1 ≤ x_{0} < x_{1} < ⋯ < x_{n + 1} ≤ 1$, or equivalently, by the parametric equations $| f(x_{i})| = \| f \|_{∞}$, $∞$, are a special case of Lissajous curves with frequency ratio equal to $[−1, 1]$.

Similar to the formula: $$T_n(\cos\theta) = \cos(n\theta),$$ we have the analogous formula: $$T_{2n+1}(\sin\theta) = (-1)^n \sin\left(\left(2n+1\right)\theta\right).$$

For $| f |$: $$T_n\!\left(\frac{x + x^{-1}}{2}\right) = \frac{x^n+x^{-n}}{2}$$ and: $$x^n = T_n\! \left(\frac{x+x^{-1}}{2}\right) + \frac{x-x^{-1}}{2}\ U_{n-1}\!\left(\frac{x+x^{-1}}{2}\right),$$ which follows from the fact that this holds by definition for $n + 1$.

First kind

The first few Chebyshev polynomials of the first kind are $$ \begin{align} T_0(x) &= 1 \\ T_1(x) &= x \\ T_2(x) &= 2x^2 - 1 \\ T_3(x) &= 4x^3 - 3x \\ T_4(x) &= 8x^4 - 8x^2 + 1 \\ T_5(x) &= 16x^5 - 20x^3 + 5x \\ T_6(x) &= 32x^6 - 48x^4 + 18x^2 - 1 \\ T_7(x) &= 64x^7 - 112x^5 + 56x^3 - 7x \\ T_8(x) &= 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \\ T_9(x) &= 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x \\ T_{10}(x) &= 512x^{10} - 1280x^8 + 1120x^6 - 400x^4 + 50x^2-1 \end{align}$$

Second kind

The first few Chebyshev polynomials of the second kind are $$\begin{align} U_0(x) &= 1 \\ U_1(x) &= 2x \\ U_2(x) &= 4x^2 - 1 \\ U_3(x) &= 8x^3 - 4x \\ U_4(x) &= 16x^4 - 12x^2 + 1 \\ U_5(x) &= 32x^5 - 32x^3 + 6x \\ U_6(x) &= 64x^6 - 80x^4 + 24x^2 - 1 \\ U_7(x) &= 128x^7 - 192x^5 + 80x^3 - 8x \\ U_8(x) &= 256x^8 - 448 x^6 + 240 x^4 - 40 x^2 + 1 \\ U_9(x) &= 512x^9 - 1024 x^7 + 672 x^5 - 160 x^3 + 10 x \\ U_{10}(x) &= 1024x^{10} - 2304 x^8 + 1792 x^6 - 560 x^4 + 60 x^2-1 \end{align}$$

As a basis set

In the appropriate Sobolev space, the set of Chebyshev polynomials form an orthonormal basis, so that a function in the same space can, on $n ≥ 1$, be expressed via the expansion: $$f(x) = \sum_{n = 0}^\infty a_n T_n(x).$$

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients $y = T_{n}(x)$ can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion.

Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart. These attributes include:

 * The Chebyshev polynomials form a complete orthogonal system.
 * The Chebyshev series converges to $y = T_{n}(cos θ) = cos nθ$ if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in $x = cos θ$ and its derivatives.
 * At a discontinuity, the series will converge to the average of the right and left limits.

The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method, often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

Example 1
Consider the Chebyshev expansion of $x ≠ 0$. One can express: $$ \log(1+x) = \sum_{n = 0}^\infty a_n T_n(x)~. $$

One can find the coefficients $x = e^{iθ}$ either through the application of an inner product or by the discrete orthogonality condition. For the inner product: $$\int_{-1}^{+1}\,\frac{T_m(x)\,\log(1 + x)}{\sqrt{1-x^2}}\,\mathrm{d}x = \sum_{n=0}^{\infty}a_n\int_{-1}^{+1}\frac{T_m(x)\,T_n(x)}{\sqrt{1-x^2}}\,\mathrm{d}x,$$ which gives: $$a_n = \begin{cases} -\log 2           & \text{ for }~ n = 0, \\ \frac{-2(-1)^n}{n} & \text{ for }~ n > 0. \end{cases}$$

Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for approximate coefficients: $$a_n \approx \frac{\,2-\delta_{0n}\,}{N}\,\sum_{k=0}^{N-1}T_n(x_k)\,\log(1+x_k),$$ where $[−1, 1]$ is the Kronecker delta function and the $n$ are the $H$ Gauss–Chebyshev zeros of $−1 < x < 1$: $$ x_k = \cos\left(\frac{\pi\left(k+\tfrac{1}{2}\right)}{N}\right) .$$ For any $δ_{ij}$, these approximate coefficients provide an exact approximation to the function at $x_{k}$ with a controlled error between those points. The exact coefficients are obtained with $T_{0}$, thus representing the function exactly at all points in $N$. The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients $N$ very efficiently through the discrete cosine transform: $$a_n \approx \frac{2-\delta_{0n}}{N}\sum_{k=0}^{N-1}\cos\left(\frac{n\pi\left(\,k+\tfrac{1}{2}\right)}{N}\right)\log(1+x_k).$$

Example 2
To provide another example: $$\begin{align} \left(1-x^2\right)^\alpha &= -\frac{1}{\sqrt{\pi}} \, \frac{\Gamma\left(\tfrac{1}{2} + \alpha\right)}{\Gamma(\alpha+1)} + 2^{1-2\alpha}\,\sum_{n=0} \left(-1\right)^n \, {2 \alpha \choose \alpha-n}\,T_{2n}(x) \\[1ex] &= 2^{-2\alpha}\,\sum_{n=0} \left(-1\right)^n \, {2\alpha+1 \choose \alpha-n}\,U_{2n}(x). \end{align}$$

Partial sums
The partial sums of: $$f(x) = \sum_{n = 0}^\infty a_n T_n(x)$$ are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients $x_{k}$ are through the use of the inner product as in Galerkin's method and through the use of collocation which is related to interpolation.

As an interpolant, the $[−1,1]$ coefficients of the $T_{1}$st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by: $$x_k = -\cos\left(\frac{k \pi}{N - 1}\right); \qquad k = 0, 1, \dots, N - 1.$$

Polynomial in Chebyshev form
An arbitrary polynomial of degree $a_{n}$ can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial $T_{2}$ is of the form: $$p(x) = \sum_{n=0}^N a_n T_n(x).$$

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.

Families of polynomials related to Chebyshev polynomials
Polynomials denoted $$C_n(x)$$ and $$S_n(x)$$ closely related to Chebyshev polynomials are sometimes used. They are defined by: $$C_n(x) = 2T_n\left(\frac{x}{2}\right),\qquad S_n(x) = U_n\left(\frac{x}{2}\right)$$ and satisfy: $$C_n(x) = S_n(x) - S_{n-2}(x).$$ A. F. Horadam called the polynomials $$C_n(x)$$ Vieta–Lucas polynomials and denoted them $$v_n(x)$$. He called the polynomials $$S_n(x)$$ Vieta–Fibonacci polynomials and denoted them $V_n(x)$. Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII. The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of $$i$$ and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials $T_{3}$ and $T_{4}$ of imaginary argument.

Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by: $$T_n^*(x) = T_n(2x-1),\qquad U_n^*(x) = U_n(2x-1).$$

When the argument of the Chebyshev polynomial satisfies $T_{5}$ the argument of the shifted Chebyshev polynomial satisfies $−1 < x < 1$. Similarly, one can define shifted polynomials for generic intervals $a_{n}$.

Around 1990 the terms "third-kind" and "fourth-kind" came into use in connection with Chebyshev polynomials, although the polynomials denoted by these terms had an earlier development under the name airfoil polynomials. According to J. C. Mason and G. H. Elliott, the terminology "third-kind" and "fourth-kind" is due to Walter Gautschi, "in consultation with colleagues in the field of orthogonal polynomials." The Chebyshev polynomials of the third kind are defined as: $$V_n(x)=\frac{\cos\left(\left(n+\frac{1}{2}\right)\theta\right)}{\cos\left(\frac{\theta}{2}\right)}=\sqrt\frac{2}{1+x}T_{2n+1}\left(\sqrt\frac{x+1}{2}\right)$$ and the Chebyshev polynomials of the fourth kind are defined as: $$W_n(x)=\frac{\sin\left(\left(n+\frac{1}{2}\right)\theta\right)}{\sin\left(\frac{\theta}{2}\right)}=U_{2n}\left(\sqrt\frac{x+1}{2}\right),$$ where $$\theta=\arccos x$$. In the airfoil literature $$V_n(x)$$ and $$W_n(x)$$ are denoted $$t_n(x)$$ and $$u_n(x)$$. The polynomial families $$T_n(x)$$, $$U_n(x)$$, $$V_n(x)$$, and $$W_n(x)$$ are orthogonal with respect to the weights: $$\left(1-x^2\right)^{-1/2},\quad\left(1-x^2\right)^{1/2},\quad(1-x)^{-1/2}(1+x)^{1/2},\quad(1+x)^{-1/2}(1-x)^{1/2}$$ and are proportional to Jacobi polynomials $$P_n^{(\alpha,\beta)}(x)$$ with: $$(\alpha,\beta)=\left(-\frac{1}{2},-\frac{1}{2}\right),\quad(\alpha,\beta)=\left(\frac{1}{2},\frac{1}{2}\right),\quad(\alpha,\beta)=\left(-\frac{1}{2},\frac{1}{2}\right),\quad(\alpha,\beta)=\left(\frac{1}{2},-\frac{1}{2}\right).$$

All four families satisfy the recurrence $$p_n(x)=2xp_{n-1}(x)-p_{n-2}(x)$$ with $$p_0(x) = 1$$, where $$p_n = T_n$$, $$U_n$$, $$V_n$$, or $$W_n$$, but they differ according to whether $$p_1(x)$$ equals $$x$$, $$2x$$, $$2x-1$$, or $2x+1$.

Even order modified Chebyshev polynomials
Some applications rely on Chebyshev polynomials but may be unable to accommodate the lack of a root at zero, which rules out the use of standard Chebyshev polynomials for these kinds of applications. Even order Chebyshev filter designs using equally terminated passive networks are an example of this. However, even order Chebyshev polynomials may be modified to move the lowest roots down to zero while still maintaining the desirable Chebyshev equi-ripple effect. Such modified polynomials contain two roots at zero, and may be referred to as even order modified Chebyshev polynomials. Even order modified Chebyshev polynomials may be created from the Chebyshev nodes in the same manner as standard Chebyshev polynomials. $$P_N = \prod_{i=1}^N(x-C_i) $$

where

 * $$P_N$$ is an N-th order Chebyshev polynomial
 * $$C_i$$ is the i-th Chebyshev node

In the case of even order modified Chebyshev polynomials, the even order modified Chebyshev nodes are used to construct the even order modified Chebyshev polynomials. $$Pe_N = \prod_{i=1}^N(x-Ce_i) $$

where

 * $$P e_N$$ is an N-th order even order modified Chebyshev polynomial
 * $$Ce_i$$ is the i-th even order modified Chebyshev node

For example, the 4th order Chebyshev polynomial from the example above is $$X^4-X^2+.125 $$, which by inspection contains no roots of zero. Creating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of $$X^4-.828427X^2 $$, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.

title: Chebyshev polynomials


text:
GABA

GABA (γ-Aminobutyric acid) is the chief inhibitory neurotransmitter in the developmentally mature mammalian central nervous system. Its principal role is reducing neuronal excitability throughout the nervous system.

GABA is sold as a dietary supplement in many countries. It has been traditionally thought that exogenous GABA (i.e., taken as a supplement) does not cross the blood–brain barrier, but data obtained from more recent research in rats describes the notion as being unclear.

The carboxylate form of GABA is γ-aminobutyrate.

Neurotransmitter
Two general classes of GABA receptor are known: Neurons that produce GABA as their output are called GABAergic neurons, and have chiefly inhibitory action at receptors in the adult vertebrate. Medium spiny cells are a typical example of inhibitory central nervous system GABAergic cells. In contrast, GABA exhibits both excitatory and inhibitory actions in insects, mediating muscle activation at synapses between nerves and muscle cells, and also the stimulation of certain glands. In mammals, some GABAergic neurons, such as chandelier cells, are also able to excite their glutamatergic counterparts. In addition to fast-acting phasic inhibition, small amounts of extracellular GABA can induce slow timescale tonic inhibition on neurons.
 * GABAA in which the receptor is part of a ligand-gated ion channel complex
 * GABAB metabotropic receptors, which are G protein-coupled receptors that open or close ion channels via intermediaries (G proteins)

GABAA receptors are ligand-activated chloride channels: when activated by GABA, they allow the flow of chloride ions across the membrane of the cell. Whether this chloride flow is depolarizing (makes the voltage across the cell's membrane less negative), shunting (has no effect on the cell's membrane potential), or inhibitory/hyperpolarizing (makes the cell's membrane more negative) depends on the direction of the flow of chloride. When net chloride flows out of the cell, GABA is depolarising; when chloride flows into the cell, GABA is inhibitory or hyperpolarizing. When the net flow of chloride is close to zero, the action of GABA is shunting. Shunting inhibition has no direct effect on the membrane potential of the cell; however, it reduces the effect of any coincident synaptic input by reducing the electrical resistance of the cell's membrane. Shunting inhibition can "override" the excitatory effect of depolarising GABA, resulting in overall inhibition even if the membrane potential becomes less negative. It was thought that a developmental switch in the molecular machinery controlling the concentration of chloride inside the cell changes the functional role of GABA between neonatal and adult stages. As the brain develops into adulthood, GABA's role changes from excitatory to inhibitory.

Brain development
GABA is an inhibitory transmitter in the mature brain; its actions were thought to be primarily excitatory in the developing brain. The gradient of chloride was reported to be reversed in immature neurons, with its reversal potential higher than the resting membrane potential of the cell; activation of a GABA-A receptor thus leads to efflux of Cl− ions from the cell (that is, a depolarizing current). The differential gradient of chloride in immature neurons was shown to be primarily due to the higher concentration of NKCC1 co-transporters relative to KCC2 co-transporters in immature cells. GABAergic interneurons mature faster in the hippocampus and the GABA machinery appears earlier than glutamatergic transmission. Thus, GABA is considered the major excitatory neurotransmitter in many regions of the brain before the maturation of glutamatergic synapses.

In the developmental stages preceding the formation of synaptic contacts, GABA is synthesized by neurons and acts both as an autocrine (acting on the same cell) and paracrine (acting on nearby cells) signalling mediator. The ganglionic eminences also contribute greatly to building up the GABAergic cortical cell population.

GABA regulates the proliferation of neural progenitor cells, the migration and differentiation  the elongation of neurites and the formation of synapses.

GABA also regulates the growth of embryonic and neural stem cells. GABA can influence the development of neural progenitor cells via brain-derived neurotrophic factor (BDNF) expression. GABA activates the GABAA receptor, causing cell cycle arrest in the S-phase, limiting growth.

Beyond the nervous system
Besides the nervous system, GABA is also produced at relatively high levels in the insulin-producing beta cells (β-cells) of the pancreas. The β-cells secrete GABA along with insulin and the GABA binds to GABA receptors on the neighboring islet alpha cells (α-cells) and inhibits them from secreting glucagon (which would counteract insulin's effects).

GABA can promote the replication and survival of β-cells  and also promote the conversion of α-cells to β-cells, which may lead to new treatments for diabetes.

Alongside GABAergic mechanisms, GABA has also been detected in other peripheral tissues including intestines, stomach, fallopian tubes, uterus, ovaries, testicles, kidneys, urinary bladder, the lungs and liver, albeit at much lower levels than in neurons or β-cells.

Experiments on mice have shown that hypothyroidism induced by fluoride poisoning can be halted by administering GABA. The test also found that the thyroid recovered naturally without further assistance after the fluoride had been expelled by the GABA.

Immune cells express receptors for GABA and administration of GABA can suppress inflammatory immune responses and promote "regulatory" immune responses, such that GABA administration has been shown to inhibit autoimmune diseases in several animal models.

In 2018, GABA has shown to regulate secretion of a greater number of cytokines. In plasma of T1D patients, levels of 26 cytokines are increased and of those, 16 are inhibited by GABA in the cell assays.

In 2007, an excitatory GABAergic system was described in the airway epithelium. The system is activated by exposure to allergens and may participate in the mechanisms of asthma. GABAergic systems have also been found in the testis and in the eye lens.

Structure and conformation
GABA is found mostly as a zwitterion (i.e., with the carboxyl group deprotonated and the amino group protonated). Its conformation depends on its environment. In the gas phase, a highly folded conformation is strongly favored due to the electrostatic attraction between the two functional groups. The stabilization is about 50 kcal/mol, according to quantum chemistry calculations. In the solid state, an extended conformation is found, with a trans conformation at the amino end and a gauche conformation at the carboxyl end. This is due to the packing interactions with the neighboring molecules. In solution, five different conformations, some folded and some extended, are found as a result of solvation effects. The conformational flexibility of GABA is important for its biological function, as it has been found to bind to different receptors with different conformations. Many GABA analogues with pharmaceutical applications have more rigid structures in order to control the binding better.

History
In 1883, GABA was first synthesized, and it was first known only as a plant and microbe metabolic product.

In 1950, GABA was discovered as an integral part of the mammalian central nervous system.

In 1959, it was shown that at an inhibitory synapse on crayfish muscle fibers GABA acts like stimulation of the inhibitory nerve. Both inhibition by nerve stimulation and by applied GABA are blocked by picrotoxin.

Biosynthesis

GABA is primarily synthesized from glutamate via the enzyme glutamate decarboxylase (GAD) with pyridoxal phosphate (the active form of vitamin B6) as a cofactor. This process converts glutamate (the principal excitatory neurotransmitter) into GABA (the principal inhibitory neurotransmitter).

GABA can also be synthesized from putrescine by diamine oxidase and aldehyde dehydrogenase.

Historically it was thought that exogenous GABA did not penetrate the blood–brain barrier, but more current research describes the notion as being unclear pending further research.

Metabolism
GABA transaminase enzymes catalyze the conversion of 4-aminobutanoic acid (GABA) and 2-oxoglutarate (α-ketoglutarate) into succinic semialdehyde and glutamate. Succinic semialdehyde is then oxidized into succinic acid by succinic semialdehyde dehydrogenase and as such enters the citric acid cycle as a usable source of energy.

Pharmacology
Drugs that act as allosteric modulators of GABA receptors (known as GABA analogues or GABAergic drugs), or increase the available amount of GABA, typically have relaxing, anti-anxiety, and anti-convulsive effects (with equivalent efficacy to lamotrigine based on studies of mice). Many of the substances below are known to cause anterograde amnesia and retrograde amnesia.

In general, GABA does not cross the blood–brain barrier, although certain areas of the brain that have no effective blood–brain barrier, such as the periventricular nucleus, can be reached by drugs such as systemically injected GABA. At least one study suggests that orally administered GABA increases the amount of human growth hormone (HGH). GABA directly injected to the brain has been reported to have both stimulatory and inhibitory effects on the production of growth hormone, depending on the physiology of the individual. Consequently, considering the potential biphasic effects of GABA on growth hormone production, as well as other safety concerns, its usage is not recommended during pregnancy and lactation.

GABA enhances the catabolism of serotonin into N-acetylserotonin (the precursor of melatonin) in rats. It is thus suspected that GABA is involved in the synthesis of melatonin and thus might exert regulatory effects on sleep and reproductive functions.

Research has indicated that oral supplementation of GABA does not yield any favorable outcomes in terms of stress reduction and enhancement of sleep quality in human subjects.

Chemistry
Although in chemical terms, GABA is an amino acid (as it has both a primary amine and a carboxylic acid functional group), it is rarely referred to as such in the professional, scientific, or medical community. By convention the term "amino acid", when used without a qualifier, refers specifically to an alpha amino acid. GABA is not an alpha amino acid, meaning the amino group is not attached to the alpha carbon. Nor is it incorporated into proteins as are many alpha-amino acids.

GABAergic drugs
GABAA receptor ligands are shown in the following table

GABAergic pro-drugs include chloral hydrate, which is metabolised to trichloroethanol, which then acts via the GABAA receptor.

The plant kava contains GABAergic compounds, including kavain, dihydrokavain, methysticin, dihydromethysticin and yangonin.

Other GABAergic modulators include:

 * GABAB receptor ligands.
 * Agonists: baclofen, propofol, GHB, phenibut.
 * Antagonists: phaclofen, saclofen.
 * GABA reuptake inhibitors: deramciclane, hyperforin, tiagabine.
 * GABA transaminase inhibitors: gabaculine, phenelzine, valproate, vigabatrin, lemon balm (Melissa officinalis).
 * GABA analogues: pregabalin, gabapentin, picamilon, progabide

In plants
GABA is also found in plants. It is the most abundant amino acid in the apoplast of tomatoes. Evidence also suggests a role in cell signalling in plants.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: GABA


text:
Lists of people by occupation

This is a list of lists of people by occupation. Each is linked to a list of notable people within that profession.

Lists of lists

 * Actors
 * Star Wars actors
 * Actresses
 * Architects
 * Artists
 * Astronauts
 * Engineers
 * Journalists
 * Kickboxers
 * Mathematicians
 * Musicians
 * Office Holders
 * Lists of office-holders of the United States
 * Painters
 * Philosophers
 * List of photographers
 * Playboy models
 * Professional bodybuilders
 * Scientists
 * Lists of screenwriters
 * Sculptors
 * Lists of sportspeople
 * Pro Bowl players
 * Oldest cricketers
 * List of television reporters
 * List of current BBC newsreaders and reporters
 * List of former BBC newsreaders and reporters

List of...

 * Accordionists
 * Africanists
 * Anthropologists
 * Archaeologists
 * Archivists
 * Astronomers and astrophysicists
 * Au pairs
 * Aviators
 * Biochemists
 * Biographers
 * Biologists
 * Botanists
 * Business theorists
 * Cardiologists
 * Caricaturists
 * Cartographers
 * Cartoonists
 * Censors
 * Centenarians
 * Child actors
 * Chefs
 * Chemists
 * Chess grandmasters
 * Chess players
 * Chief executive officers
 * Choreographers
 * Christian theologians
 * Civil rights leaders
 * Climbers
 * Clinical psychologists
 * Clowns
 * Club DJs
 * Coleopterists
 * Comedians
 * Comic creators
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 * Computer scientists
 * Copywriters
 * Cosmologists
 * Crime bosses
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text:
Samuel Hahnemann

Christian Friedrich Samuel Hahnemann (, ; 10 April 1755 – 2 July 1843) was a German physician, best known for creating the pseudoscientific system of alternative medicine called homeopathy.

Early life
Christian Friedrich Samuel Hahnemann was born in Meissen, Saxony, near Dresden. His father Christian Gottfried Hahnemann was a painter and designer of porcelain, for which the town of Meissen is famous.

As a young man, Hahnemann became proficient in a number of languages, including English, French, Italian, Greek and Latin. He eventually made a living as a translator and teacher of languages, gaining further proficiency in "Arabic, Syriac, Chaldaic and Hebrew".

Hahnemann studied medicine for two years at Leipzig. Citing Leipzig's lack of clinical facilities, he moved to Vienna, where he studied for ten months. His medical professors in Leipzig and Vienna included the physician Joseph von Quarin, later credited for turning Vienna General Hospital into a model European medical institution.

After one term of further study, Hahnemann graduated with a medical degree with honors from the University of Erlangen on 10 August 1779. His poverty may have forced him to choose Erlangen, as the school's fees were lower than in Vienna. Hahnemann's thesis was titled Conspectus adfectuum spasmodicorum aetiologicus et therapeuticus [A Dissertation on the Causes and Treatment of Spasmodic Diseases].

Medical practice
In 1781, Hahnemann took a village doctor's position in the copper-mining area of Mansfeld, Saxony. He soon married Johanna Henriette Kuchler and would eventually have eleven children. After abandoning medical practice, and while working as a translator of scientific and medical textbooks, he translated fifteen books from English, six from French and one each from Latin and Italian from 1777 to 1806. Hahnemann travelled around Saxony for many years, staying in many different towns and villages for varying lengths of time, never living far from the River Elbe and settling at different times in Dresden, Torgau, Leipzig and Köthen (Anhalt) before finally moving to Paris in June 1835.

Creation of homeopathy
Hahnemann was dissatisfied with the state of medicine in his time, and particularly objected to practices such as bloodletting. He claimed that the medicine he had been taught to practice sometimes did the patient more harm than good: "My sense of duty would not easily allow me to treat the unknown pathological state of my suffering brethren with these unknown medicines. The thought of becoming in this way a murderer or malefactor towards the life of my fellow human beings was most terrible to me, so terrible and disturbing that I wholly gave up my practice in the first years of my married life and occupied myself solely with chemistry and writing."

After giving up his practice around 1784, Hahnemann made his living chiefly as a writer and translator, while resolving also to investigate the causes of medicine's alleged errors. While translating William Cullen's A Treatise on the Materia Medica, Hahnemann encountered the claim that cinchona, the bark of a Peruvian tree, was effective in treating malaria because of its astringency. Hahnemann believed that other astringent substances are not effective against malaria and began to research cinchona's effect on the human body by self-application. Noting that the drug induced malaria-like symptoms in himself, he concluded that it would do so in any healthy individual. This led him to postulate a healing principle: "that which can produce a set of symptoms in a healthy individual, can treat a sick individual who is manifesting a similar set of symptoms." This principle, like cures like, became the basis for an approach to medicine which he gave the name homeopathy. He first used the term homeopathy in his essay Indications of the Homeopathic Employment of Medicines in Ordinary Practice, published in Hufeland's Journal in 1807.

Development of homeopathy
Following up the work of the Viennese physician Anton von Störck, Hahnemann tested substances for the effects they produced on a healthy individual, presupposing (as von Störck had claimed) that they may heal the same ills that they caused. His researches led him to agree with von Störck that the toxic effects of ingested substances are often broadly parallel to certain disease states, and his exploration of historical cases of poisoning in the medical literature further implied a more generalised medicinal "law of similars". He later devised methods of diluting the drugs he was testing in order to mitigate their toxic effects. He claimed that these dilutions, when prepared according to his technique of "potentization" using dilution and succussion (vigorous shaking), were still effective in alleviating the same symptoms in the sick. His more systematic experiments with dose reduction really commenced around 1800–01 when, on the basis of his "law of similars," he had begun using Ipecacuanha for the treatment of coughs and Belladonna for scarlet fever.

He first published an article about the homeopathic approach in a German-language medical journal in 1796. Following a series of further essays, he published in 1810 "Organon of the Rational Art of Healing", followed over the years by four further editions entitled The Organon of the Healing Art, the first systematic treatise and containing all his detailed instructions on the subject. A 6th Organon edition, unpublished during his lifetime, and dating from February 1842, was only published many years after his death. It consisted of a 5th Organon containing extensive handwritten annotations. The Organon is widely regarded as a remodelled form of an essay he published in 1806 called "The Medicine of Experience", which had been published in Hufeland's Journal. Of the Organon, Robert Ellis Dudgeon states it "was an amplification and extension of his "Medicine of Experience", worked up with greater care, and put into a more methodical and aphoristic form, after the model of the Hippocratic writings."

Coffee theory of disease
Around the start of the nineteenth century Hahnemann developed a theory, propounded in his 1803 essay On the Effects of Coffee from Original Observations, that many diseases are caused by coffee. Hahnemann later abandoned the coffee theory in favour of the theory that disease is caused by Psora, but it has been noted that the list of conditions Hahnemann attributed to coffee was similar to his list of conditions caused by Psora.

Later life

In early 1811 Hahnemann moved his family back to Leipzig with the intention of teaching his new medical system at the University of Leipzig. As required by the university statutes, to become a faculty member he was required to submit and defend a thesis on a medical topic of his choice. On 26 June 1812, Hahnemann presented a Latin thesis, entitled "A Medical Historical Dissertation on the Helleborism of the Ancients." His thesis very thoroughly examined the historical literature and sought to differentiate between the ancient use of Helleborus niger, or black hellebore, and the medicinal uses of the "white hellebore", botanically Veratrum album, both of which are poisonous plants.

Hahnemann continued practicing and researching homeopathy, as well as writing and lecturing for the rest of his life. He died in 1843 in Paris, at 88 years of age, and is entombed in a mausoleum at Paris's Père Lachaise Cemetery.

Descendants
Hahnemann's daughter, Amelie (1789–1881), had a son: Leopold Suss-Hahnemann. Leopold emigrated to England, and he practised homeopathy in London. He retired to the Isle of Wight and died there at the outbreak of World War I in 1914. Dr Leopold Suss-Hahnemann's youngest daughter, Amalia, had two children, Winifred (born in 1898) and Herbert. William Herbert Tankard-Hahnemann (1922–2009) was Winifred's son. He served as a Major in the British Army during World War II, and then had a career in the city of London. He was at one point appointed as a Freeman of the City of London. Mr William Herbert Tankard-Hahnemann, the great-great-great-grandson of Samuel Hahnemann died on 12 January 2009 (his 87th birthday) after 22 years of active patronage of the British Institute of Homeopathy. The William Tankard-Hahnemann line continues with his son, Charles.

Writings
Hahnemann wrote a number of books, essays, and letters on the homeopathic method, chemistry, and general medicine:
 * Heilkunde der Erfahrung. Norderstedt 2010, ISBN 3-8423-1326-8
 * reprinted in
 * Fragmenta de viribus medicamentorum positivis sive in sano corpore humano obeservitis, a collection of 27 drug "provings" published in Latin in 1805.
 * The Organon of the Healing Art (1810), a detailed delineation of what he saw as the rationale underpinning homeopathic medicine, and guidelines for practice. Hahnemann published the 5th edition in 1833; a revised draft of this (1842) was discovered after Hahnemann's death and finally published as the 6th edition in 1921.
 * Materia Medica Pura, a compilation of "homoeopathic proving" reports, published in six volumes between vol. I in 1811 and vol. VI in 1821; second edition of vol. I to Vol. VI from 1822 to 1827 and third revised editions of volumes I and II were published in 1830 and 1833, respectively.
 * Chronic Diseases (1828), an explanation of the root and cure of chronic disease according to the theory of miasms, together with a compilation of "homoeopathic proving" reports, published in five volumes during the 1830s.
 * The Friend of Health, in which Hahnemann "recommended the use of fresh air, bed rest, proper diet, sunshine, public hygiene and numerous other beneficial measures at a time when many other physicians considered them of no value."
 * Appeal to Thinking Philanthropist Respecting the Mode of Propagation of the Asiatic Choler, in which Hahnemann describes cholera physicians and nurses as the "certain and frequent propagators" of cholera and that whilst deriding nurses' "fumigations with chlorine", promoted the use of "drops of camphorated spirit" as a cure for the disease.
 * Hahnemann also campaigned for the humane treatment of the insane in 1792.
 * John Henry Clarke wrote that "In 1787, Hahnemann discovered the best test for arsenic and other poisons in wine, having pointed out the unreliable nature of the 'Wurtemberg test,' which had been in use up to that date."
 * Samuel Hahnemanns Apothekerlexikon. Vol.2. Crusius, Leipzig 1798–1799 Digital edition by the University and State Library Düsseldorf
 * Reine Arzneimittellehre. Arnold, Dresden (several editions) 1822–1827 Digital edition by the University and State Library Düsseldorf
 * Systematische Darstellung der reinen Arzneiwirkungen aller bisher geprüften Mittel. Vieweg, Braunschweig 1831 – Digital edition by the University and State Library Düsseldorf

title: Samuel Hahnemann


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User talk:Bluelion

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Talk:Radiosity (computer graphics)

Comments

 * There's some bad grammar in the first part of this. The second paragraph in particular needs some work; it's not clear what it is saying. Osmaker 18:59, 20 October 2005 (UTC)
 * I've re-arranged some text. Added a bit here and a bit there. Made it easier to read in mosdt of the areas except the maths sections. Snorbaard 3:04, 24 October 2005 (GMT+2)
 * That looks and reads a lot better; thanks. Osmaker 07:02, 24 October 2005 (UTC)

 * There is a mistake in the discrete radiosity equation. The form-factor inside the sum should be $$F_{ij}$$ instead of $$F_{ji}$$ (Francesc Castro, castro@ima.udg.es)
 * Actually, the mistake happened before! The way $$F_{ji}$$ is defined in this article, the very first formula has to be:$$B_i\, d A_i = E_i\, d A_i + R_i \int_j B_j F_{ij}\, d A_j,\,\!$$. The reciprocity also has to be changed to $$A_j F_{ij} = A_i F_{ji}$$. Finally, in the next 2 formulas, $$F_{ij}$$ has to be changed to $$F_{ji}$$. Wernermarius (talk) 17:05, 17 February 2008 (UTC)
 * An alternative to the above correction is to define $$F_{ij}$$ as the view factor for the radiation leaving $$i$$ and hitting patch $$j$$. This is more common in CG textbooks. With this correction, no formulas need to be changed. Wernermarius (talk) 17:05, 17 February 2008 (UTC)
 * I went ahead and integrated this correction. Wernermarius (talk) 09:38, 18 February 2008 (UTC)

 * When the matrix equation/solution is introduced: $$B=(I-\rho F)^{-1}E$$, it introduces a new variable $$I$$ without defining it anywhere.TobyTheOne (talk) 07:18, 24 April 2017 (UTC)

Maxwell..
(afaik) is a truly physically-based renderer, ie it does not employ the kinds of shortcuts that GI renderers use. Is it a good example of a GI renderer? I know of the following: Lucille, POV-Ray, Pane, Parthenon, Perceptuum, Radiance, toxic, Iguana, and those are the free ones...

 * FYI: GI = "global illumination." Global illumination simply means that the shading at each point depends on more than the location and orientation of that point, in other words, the shading depends on other geometry in the scene. Simple shadows are an example of global illumination, as are simple reflections and simple refractions. Maxwell is an example of an unbiased renderer (or so they claim), i.e., it correctly accounts for all types of light paths described by geometric optics.  I can't think of a good example of a commercial radiosity renderer in widespread use, probably because most major renderers tend towards photon mapping and irradiance caching these days.

Global Illumination and Ray Tracing
The article at the very beginning references direct illumination algorithms and refers to ray tracing as an example of one. I however thought ray tracing was also a global illumination algorithm, and if you look at the wikipedia page on global illumination algorithms, it refers to ray tracing as an example of one. --Dterei 09:22, 17 October 2006 (UTC)

 * You are absolutely correct: ray tracing is a global illumination algorithm. I have fixed this error in the page.

I think ray tracing is used to transfere the light around the scene. How else should global illumination get its F[i,j] ? Arnero 17:41, 28 October 2006 (UTC)

 * Well, ray tracing refers to a broad class of algorithms for computing radiance arriving at the image plane "Whitted" ray tracing is what most people think of when they hear ray tracing, though Monte Carlo ray tracing is also ray tracing. Ray casting, i.e., the process of finding the geometric intersection of a ray and the scene geometry, is often used to compute visibility and transport of radiance from one spot to another.  However, this isn't the only way to do it: radiosity doesn't have to use ray casting to compute its form factors (often they can be computed analytically).

 * As I understand it, ray tracing is neither a global nor direct illumination algorithm in itself, but it can be used to implement either.

 * Well, technically what you're referring to is ray casting, i.e., finding the intersection of rays with surfaces. Ray tracing really is a class of global illumination algorithms which use ray casting as the fundamental operation for computing the transport part of the equation.

 * "Global Illumination" refers to the class of rendering algorithms which are capable of illuminating an object from light scattered from the entire scene, not just from light sources and reflections/caustics. Direct illumination refers to algorithms which can only illuminate objects directly from light sources and sometimes from reflections/caustics. Whether you use ray tracing, or something else to implement direct or global illumination is up to you.

 * Wrong: reflections and caustics (not to mention simple hard shadows) are global phenomena. A phenomenon is "global" if you need information about any surface point other than the one you're shading in order to perform the shading computation.  "Global illumination" algorithms are those which have any kind of global effect.  So, ray tracing is a global illumination algorithm.

 * I'm willing to be wrong about this, and what you say could make sense, but you are the first person I have seen who includes shadow casting as part of global illumination. None of the first few pages on a google for "global illumination" seem to include shadow casting. Do you have a reference for this? Many thanks Rocketmagnet 15:38, 8 January 2007 (UTC)

 * Ok, so I was a bit overzealous about the issue of shadows, but certainly it's a consistent definition and one possible interpretation of "global," i.e., the opposite of local. The more common definition is "global" as in the opposite of direct, and is given in the paper "Fast Separation of Direct and Global Components of a Scene using High Frequency Illumination" published in SIGGRAPH 2006: "When a scene is lit by a source of light, the radiance of each point in the scene can be viewed as having two components, namely, direct and global. The direct component is due to the direct illumination of the point by the source. The global component is due to the illumination of the point by other points in the scene."  By this definition, a specular reflection is a global phenomenon because you're seeing an object lit by a bounce off another object, not an object directly lit by a light source.  Honestly I think the English is wrong -- it should be direct vs. indirect or global vs. local, not global vs. direct, but I'm willing to stick with the convention.

replacing the ball picture
I have a replacement image that I think illustrates the difference between direct and global illumimation better than the ball image. The image is from my own web site, and was generated using a radiosity renderer I wrote myself, therefore I own the copyright. Eric Pierce, what do you think? If nobody minds, I'll replace the image, and adjust the text accordingly. Rocketmagnet 18:10, 7 January 2007 (UTC)

Yes, I think this is a very good example.

EXCELLENT! -- M0llusk (talk) 03:36, 30 July 2008 (UTC)

Different Algorithms
Hi Trevor, Thanks for that, and the other changes. From what you wrote, I think it would be good to have a section, or sections, listing or detailing several algorithms (shooting, texture mapping, etc.) If Agreed, I'll move your bit about shooting, and my bit about texturing into their own sections, which can then be expanded upon in a controlled manner by othe people.Rocketmagnet 09:43, 8 January 2007 (UTC)

Trevorgoodchild 07:10, 13 January 2007 (UTC)
 * Sounds good -- by the way, if you're interested in different methods or variations on methods for solving the radiosity equations you might want to check out these papers:
 * R. Troutman, N. Max, "Radiosity Algorithms Using Higher Order Finite Element Methods", SIGGRAPH 1993.
 * P. Hanrahan and D. Salzman and L. Aupperle, "A Rapid Hierarchical Radiosity Algorithm," SIGGRAPH, 197-206, 1991.
 * B.E. Smits and J.R. Arvo and D.H. Salesin, "An Importance Driven Radiosity Algorithm," SIGGRAPH, 273-282, 1992.
 * D. Lischinski, F. Tampieri and D.P. Greenberg, "A Discontinuity Meshing Algorithm for Accurate Radiosity," IEEE Computer Graphics and Applications, 12, 6, 25-39, 1992.

 * You know, I think part of the problem is not enough authors put their papers up on the web, and the damn ACM portal costs money! As a consequence, only people in academia can even take a look at some of these neat solutions, and everyone else ends up figuring out their own way to do it... which is not the point of research! :P Trevorgoodchild 17:40, 13 January 2007 (UTC)

Caption for Radiosity_Progress.jpg
Sorry Trevor, I have reverted the caption for Radiosity_Progress.jpg. From "As additional bounces are calculated in shooting radiosity the scene becomes progressively brighter." back to "As the algorithm iterates, light can be seen to flow into the scene, as multiple bounces are computed."

This is because I believe the original caption to be the more correct one. This image was not rendered using shooting radiosity. It was rendered using texture mapping radiosity. Secondly, the image is not simply getting quantatively brighter. Rather a qualitative change is happening: light flowing through the scene.Rocketmagnet 09:43, 8 January 2007 (UTC)

 * Fair enough. Though I should point out that whether you remesh or use a texture map doesn't fundamentally change the algorithm: either way you're discretizing space in order to get a piecewise constant solution.  Using a texture vs. using a mesh is just an implementation detail.  Another thing to note (if you're actually writing code using the texture mapping approach) is that evaluating all the elements (texels) naively will lead to slower convergence than doing something like a shooting method where you "shoot" from the element with greatest energy each time.  However, I like the idea of the texture mapping approach because it's pedagogical and easy to understand for someone with a little graphics background.Trevorgoodchild 07:19, 13 January 2007 (UTC)

 * yes, I'm a fan of the texturing approach, not because it's "better", simply because I think that it's easy to understand, especially for someone 'with' graphics background. I've had several people write to me, saying that they thought Radiosity was totally beyond them, until they read my article TGLTLSBFFSP:Radiosity. Plus you get hardware acceleration trivially, and using OpenGL means most of the code is already written. Rocketmagnet 12:36, 13 January 2007 (UTC)

 * Oh yeah, I remember seeing your article a long time ago -- it's great! I think that's the perfect introduction to radiosity. Trevorgoodchild 17:25, 13 January 2007 (UTC)

Iterative Algorithm
Trevor, I'm not sure why you deleted the line about radiosity being an iterative algorithm. The only other place I can see this mentioned is in the paragraph about shooting radiosity. I think that the fact that it's iterative is the highest level thing one can say about the working of the algorithm.Rocketmagnet 09:43, 8 January 2007 (UTC)

 * Actually, you're wrong about this one -- the radiosity method simply defines a system of simultaneous linear equations whose solution is a solution to the radiosity equation. Just like any linear system, there are a multitude of ways to solve this system.  Although you might use an iterative solver, you can solve the system exactly (and non-iteratively) as well.Trevorgoodchild 07:14, 13 January 2007 (UTC)

 * I did not know that. I'll cede, but, in most cases, it's done iteratively isn't it? Rocketmagnet 12:41, 13 January 2007 (UTC)

 * Well, many of the solvers used for the FEM (and in fact, many of the solvers used for large systems of equations in general) are iterative, but not in the way you're thinking. For example, in Troutman & Max's paper they use Gauss-Seidel to solve a system representing the radiosity equation, and Gauss-Seidel is an iterative solver.  But that isn't necessarily the same as these visual iterations where you "look at" the previous solution step to get the next one.Trevorgoodchild 17:37, 13 January 2007 (UTC)

 * Of course. When I say "iterative", I mean that the code loops over the solution many times, bring it closer and closer, and eventually decides to stop when two iterations are similar enough. I don't mean that it had to be visual. Gauss-Seidel is iterative in exactly the way I meant. Can we put the iterative sentence back in? Rocketmagnet 00:56, 14 January 2007 (UTC)

 * Besides, you could render the result of each iteration of the Gauss-Seidel algorithm. Then it would be visual too. Rocketmagnet 01:17, 14 January 2007 (UTC)

Quadratic Increase in Computation Time
"Early methods used a hemicube (an imaginary cube centered upon the first surface to which the second surface was projected, devised by Cohen and Greenberg in 1985) to approximate the form factor, which also solved the intervening patch problem. This is quite computationally expensive, because ideally form factors must be derived for every possible pair of patches, leading to a quadratic increase in computation with added geometry."

I don't think this is correct in all cases. If each patch takes a constant time to render to the hemicube, then it's correct. Using the texture map method, then each 'surface' takes a constant time, and the number of patches makes practically no difference. Therefore it can be linear.

I think that it would be better to have a section detailing various methods. Rocketmagnet 12:38, 8 January 2007 (UTC)

Are shadows part of Global Illumination?
Can anyone find a reference or paper stating that hard shadows are considered to be part of Global Illumination? Rocketmagnet 14:52, 9 January 2007 (UTC)

 * Ok, I'll cede that they're not. See the note above.Trevorgoodchild 07:22, 13 January 2007 (UTC)

 * There's some discussion about this on the Global Illumination Talk page. Rocketmagnet 00:07, 11 May 2007 (UTC)

What is "Real" Radiosity?
I'm not totally sure about this statement:

"Likewise, Radiosity, by Hugo Elias describes a deterministic path tracing algorithm that takes advantage of graphics hardware to accelerate irradiance gathering, by using hemicubes. While this method does indeed compute the radiosity of patches, it does not compute form factors or explicitly construct a system of equations, thus it should not be confused with the classic radiosity method."

Do you have a reference for this? Where do you get the definition of "classic radiosity"? Perhaps "full matrix" radiosity was the method described in the first paper on the technique, but the hemicube method is still considered radiosity. Surely a radiosity method would be any which computes the radiosity of patches. What the algorithm does then is up to it. Indeed, SIGGRAPH 1993 Education Slide Set: Radiosity Overview Page 2, by Stephen Spencer talks about both "full matrix" and "progressive" methods. and the next page talks about "Progressive Radiosity Variants". Therefore pointing to one implementation of radiosity and calling it something else seems odd. Unless a reference can be found, I'll revert this in a couple of days. Rocketmagnet 07:36, 10 May 2007 (UTC)

 * Unfortunately, Rocketmagnet, I think the burden is on you to show that your method is radiometrically valid... that's how science works. ;) —The preceding unsigned comment was added by Trevorgoodchild (talk • contribs) 22:16, 10 May 2007 (UTC).

 * Can people please stop calling it my method? I'd love to take the credit for it, but basically I implemented radiosity from what I learned by reading articles and papers. As far as I can tell, it's one of the ways it's done. I believe it's discussed in the links above. The SIGGRAPH paper calls it radiosity. What more proof can I give that people call it radiosity? Rocketmagnet 23:58, 10 May 2007 (UTC)

 * As for the burden of proof being on me, that the method I used is radiometrically valid, that's way out of my league, and doesn't interest either (should be obvious from the pictures that it works). The fact that other people (who use real equations on slides and everything) seem to say it is, is good enough for me. (Even Mr. 169.229.69.173 seemed to agree that it's radiometrically valid). Rocketmagnet 23:58, 10 May 2007 (UTC)

Radiosity is first and foremost a physics term
User:Ryulong has been reverting my edits on an attempt to move this page to Radiosity (computer graphics). In my view, radiosity (should that be capital R?) does not deserve to hold this page title.

First and foremost this is a term borrowed from physics. The term that describes a fundamental phenomenon in physics with applications in engineering, optics, thermodynamics, and astronomy. In terms of the broad population or wikipedia users, it seems clear to me that this more fundamental meaning should precede the specialised use relating to this computer graphics algorithm. A disambiguation page is the correct approach.

Without the physical concept described by the physics term, there would *be* no computer algorithm. These are not two different meanings. The computer algorithm is an implementation of computation directional specular radiative transfer that borrows the name of one of the phenomena it models. It does not usurp the name.

My reference is the *very* popular (for mechanical engineers) undergraduate text Incropera and Dewitt, Fundamentals of Heat and Mass Transfer. The term 'radiosity is introduced in the first few pages of the radiation topic, as one of the foundation concepts of radiative transfer. Jdpipe 10:31, 11 June 2007 (UTC)

 * I agree. "Radiosity" should be the physics article, and "Radiosity_(Graphics)" should be this one. How do we go about getting it changed? Rocketmagnet 14:14, 11 June 2007 (UTC)

 * Hi Rocketmagnet. I see that you are one of the key authors of this page -- and a good page it is too. Your agreement will be a help on this. There is currently a block on moving the page but after 18 Jun we can progress it, providing there are no objections raised here. Perhaps you will be able to help with filling out that page (Radiosity (heat transfer)). Specifically, mechanical engineer use a matrix-based radiosity calculation to estiamte temperatures of surfaces in a grey-body radiative transfer situation. Thought it might be good to put that on that page, and I imagine there would be some good parallels with your stuff here. Cheers, Jdpipe 11:55, 14 June 2007 (UTC)

Re: Advantages
I think you missed one important thing: calculation results are viewpoint-independend. —Preceding unsigned comment added by 83.25.52.170 (talk) 19:08, 27 February 2008 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: moved to Radiosity (computer graphics). Favonian (talk) 23:12, 12 February 2012 (UTC)

Radiosity (3D computer graphics) → ??? – Disambiguator is overly precise. Radiosity (computer graphics) or Radiosity (algorithm) are acceptable. Pnm (talk) 21:09, 5 February 2012 (UTC)

 * Comment – "Disambiguator is overly precise" seems like an odd complaint. Can you say what's bad about it?  Perhaps not sufficiently concise?  Dicklyon (talk) 22:48, 5 February 2012 (UTC)
 * Over-precision is a valid complaint: the first sentence of WP:PRECISION says, "when additional precision is necessary to distinguish an article title from other uses of the topic name, over-precision should be avoided." (emphasis mine) Not sufficiently concise is valid too, per WP:NAMINGCRITERIA. – Pnm (talk) 22:57, 5 February 2012 (UTC)
 * Move to Radiosity (computer graphics). Disambiguation tags need to be sufficiently precise, but not more specific than necessary. —Ruud 19:32, 6 February 2012 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

title: Talk:Radiosity (computer graphics)


text:
Lists of tennis players

There are several lists of tennis players:

 * Men:
 * List of male singles tennis players
 * List of male doubles tennis players
 * Rankings
 * List of ATP number 1 ranked singles tennis players
 * List of ATP number 1 ranked doubles tennis players
 * World number 1 ranked male tennis players
 * Top ten ranked male tennis players
 * Top ten ranked male tennis players (1912–1972)
 * Women
 * List of female tennis players
 * Rankings
 * List of WTA number 1 ranked singles tennis players
 * List of WTA number 1 ranked doubles tennis players
 * World number 1 ranked female tennis players
 * Top ten ranked female tennis players
 * Top ten ranked female tennis players (1921–1974)

 * Tournament winners:
 * List of Australian Open champions
 * List of French Open champions
 * List of Wimbledon champions
 * List of US Open champions
 * List of Olympic tennis medalists

 * Professional tours
 * Tennis players with most titles in the Open Era

title: Lists of tennis players