In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing new data points within the range of a discrete set of known data points. In engineering and science, one has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable, it is required to interpolate, i.e. estimate the value of that function for an intermediate value of the independent variable. A related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function, still close to the original; the resulting gain in simplicity may outweigh the loss from interpolation error. This table gives some values of an unknown function f. Interpolation provides a means of estimating the function at intermediate points, such as x = 2.5.
We describe some methods of interpolation, differing in such properties as: accuracy, number of data points needed, smoothness of the resulting interpolant function. The simplest interpolation method is to locate the nearest data value, assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation is as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity. One of the simplest methods is linear interpolation. Consider the above example of estimating f. Since 2.5 is midway between 2 and 3, it is reasonable to take f midway between f = 0.9093 and f = 0.1411, which yields 0.5252. Linear interpolation takes two data points, say and, the interpolant is given by: y = y a + x − x a x b − x a at the point y − y a y b − y a = x − x a x b − x a y − y a x − x a = y b − y a x b − x a This previous equation states that the slope of the new line between and is the same as the slope of the line between and Linear interpolation is quick and easy, but it is not precise.
Another disadvantage is that the interpolant is not differentiable at the point xk. The following error estimate shows that linear interpolation is not precise. Denote the function which we want to interpolate by g, suppose that x lies between xa and xb and that g is twice continuously differentiable; the linear interpolation error is | f − g | ≤ C 2 where C = 1 8 max r ∈ | g ″ |. In words, the error is proportional to the square of the distance between the data points; the error in some other methods, including polynomial interpolation and spline interpolation, is proportional to higher powers of the distance between the data points. These methods produce smoother interpolants. Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a linear function. We now replace this interpolant with a polynomial of higher degree. Consider again the problem given above; the following sixth degree polynomial goes through all the seven points: f = − 0.0001521 x 6 − 0.003130 x 5 + 0.07321 x 4 − 0.3577 x
Live in Branson, MO, USA is a live album by American country music artist, Connie Smith. The album was produced by Ralph Jungheim, it was Smith's first official album since 1978's New Horizons. Live in Branson, MO, USA was Connie Smith's first live album; the album was recorded at a theater in Branson, Missouri October 13, 1992. The album consisted of nine tracks, which included a ten-minute medley of her hits. Seven of the album's tracks were Smith's hits, including her #1 single, "Once a Day," and four other recordings from Smith's years at RCA Victor Records; the album includes two singles from her years at Columbia Records: "You've Got Me" and "I've Got My Baby on My Mind." In addition, the album featured two Gospel songs, "How Great Thou Art" and a cover of Martha Carson's "Satisfied."The album was reviewed by Allmusic critic, Dan Cooper, who gave the release three out of five stars. A full description of the release was not included. Jimmy Capps – guitar Rod Ham – bass Mark Pearman – fiddle Connie Smith – lead vocals Gary Smith – piano Jack Smith – steel guitar, leader Steve Turner – drums
The Conmen in Vegas is a 1999 Hong Kong action comedy film produced and directed by Wong Jing and is a sequel to the 1998 film, The Conman. The film stars original returning cast members Andy Lau and Nick Cheung with new cast members Natalis Chan, Kelly Lin, Meggie Yu, Alex Man and Jewel Lee in her debut film role; the film was filmed in the Caesars Palace Resort, Las Vegas. The story directly follows The Conman, where Dragon's sister, has left to study in Canada. After King and Dragon defeat Macau Mon, they become swindling partners. Along with Dragons' older cousin Luk Chard, the trio try swindling a large amount of money from Big Eyed Man at his underground casino. While the trio are enjoying themselves, Man sends people to kill them and kidnap Dragon. King and Luk Chard are brought to the Senior Chinese Front, it turns out that they want to hire them to go to Las Vegas to complete a task: to catch Peter Chu, a man, in charge of multiple Tofu-dreg projects in the past few years in China, stole a large amount of property and fled to the United States.
Because Peter Chu is backed by many American politicians and organised gangs, they could not use diplomatic and legal channels to extradite him, they could only send a few gambling experts to catch him in the casino where he visits daily. King and Luk would receive $120,000,000 as a reward. Since King needs to give $30,000,000 to Big Eyed Man to rescue Dragon, he accepts the task and goes to Las Vegas with Luk Chard. After arriving in Las Vegas and Luk Chard encounters two gorgeous ladies Betty and Sammi; the four of them set a trap and with the help of King's friend, adult film director Handsome Wu, they escort Peter Chu back to China and retrieve stolen money and rescue Dragon. After this and Betty become lovers. Andy Lau as King Natalis Chan as Luk Chard Nick Cheung as Dragon Kelly Lin as Betty Meggie Yu as Sammi Jewel Lee as Fei-fei Alex Man as Peter Chu Nam Yin as Big Eyed Man Wong Jing as Handsome Wu Leung Kei-hei as Big Eyed Man's thug So Wai-nam as Big Eyed Man's thug Only You in My Heart Composer: Duck Lau Lyricist: Duck Lau Singer: Andy Lau The film grossed HK$17,761,670 during its theatrical run from 25 June to 21 July 1999 in Hong Kong.