DCTs are also closely related to Chebyshev polynomials, and fast DCT algorithms ( below ) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in Clenshaw – Curtis quadrature.

Other quadrature methods with varying intervals include Clenshaw – Curtis quadrature ( also called Fejér quadrature ) methods, which do nest.

* Clenshaw – Curtis quadrature

This approximation leads directly to the method of Clenshaw – Curtis quadrature.

If it is possible to change the points at which the integrand is evaluated, then other methods such as Gaussian quadrature and Clenshaw – Curtis quadrature are probably more suitable.

However, for large n a Newton – Cotes rule can sometimes suffer from catastrophic Runge's phenomenon where the error grows exponentially for large n. Methods such as Gaussian quadrature and Clenshaw – Curtis quadrature with unequally spaced points ( clustered at the endpoints of the integration interval ) are stable and much more accurate, and are normally preferred to Newton – Cotes.

For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw – Curtis quadrature are generally far more accurate ; Clenshaw – Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.

If it is possible to evaluate the integrand at unequally-spaced points, then other methods such as Gaussian quadrature and Clenshaw – Curtis quadrature are generally more accurate.

The level index arithmetic of Clenshaw, Olver, and Turner is a scheme based on a generalised logarithm representation.

In numerical analysis, the Clenshaw algorithm is a recursive method to evaluate a linear combination of Chebyshev polynomials.

The Horner scheme is a special case of the Clenshaw algorithm

where it is usually referred to as Clenshaw summation.

* Clenshaw algorithm to evaluate polynomials in Chebyshev form

* 1928 – Dan Curtis, American director and producer ( d. 2006 )

* 1980 – Curtis Woodhouse, English footballer and boxer

* 1983 – Curtis Thigpen, American baseball player

Conceived while Atkinson and Curtis were working on Not the Nine O ' Clock News, the series dealt comically with a number of medieval issues in Britain – witchcraft, Royal succession, European relations, the Crusades and the conflict between the Church and the Crown.

Chaplin has also been the subject of a musical, Limelight – The Story of Charlie Chaplin by Christopher Curtis and Thomas Meehan, which was performed at the La Jolla Playhouse in 2010.

* 1986 – John Curtis Gowan, American psychologist ( b. 1912 )

* 1934 – King Curtis, American saxophonist ( d. 1971 )

* 1983 – Tony Curtis, American football player

* Film – Screenplay ( Richard Curtis )

* Best Original Screenplay – Richard Curtis ( lost to Quentin Tarantino for Pulp Fiction )

* Best Original Screenplay – Richard Curtis ( lost to Quentin Tarantino for Pulp Fiction )

* Best Screenplay – Richard Curtis ( lost to Quentin Tarantino for Pulp Fiction )

* Best Actress in a Musical or Comedy – Andie MacDowell ( lost to Jamie Lee Curtis for True Lies )

* 1860 – Curtis Guild, Jr., American politician ( d. 1915 )

* 1955 – Curtis Strange, American golfer

* 1907 – Charles Curtis of Kansas becomes the first Native American U. S. Senator.

* 1860 – Charles Curtis, Vice President of the United States ( d. 1936 )

* 1925 – Tony Curtis, American actor ( d. 2010 )

* 1942 – Curtis Mayfield, American singer-songwriter, musician, and producer ( The Impressions ) ( d. 1999 )

* 1968 – Cliff Curtis, New Zealand actor

* 1987 – Johnny Curtis, American wrestler

* 1956 – Ian Curtis, English singer-songwriter ( Joy Division ) ( d. 1980 )

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

The Euler – Maclaurin formula is also used for detailed error analysis in numerical quadrature.

* Gauss – Kronrod quadrature formula

A Gaussian quadrature rule is typically more accurate than a Newton – Cotes rule which requires the same number of function evaluations, if the integrand is smooth ( i. e., if it is sufficiently differentiable ).

Gaussian quadrature rules do not nest, but the related Gauss – Kronrod quadrature formulas do.

In numerical analysis, the Newton – Cotes formulae, also called the Newton – Cotes quadrature rules or simply Newton – Cotes rules, are a group of formulae for numerical integration ( also called quadrature ) based on evaluating the integrand at equally-spaced points.