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 Encyclopedia of Keywords > Information Technology > Computer Science > Algorithms > Numerical Analysis > Interpolation Michael Charnine

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### DefinitionsUpDw('Definitions','-Abz-');

1. Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points.
2. Interpolation is an old subject since, as noticed in the paper by M. Gasca and T. Sauer, the term was coined by John Wallis in 1655. (Web site)
3. Interpolation is the process of finding a function which goes through some given data points. (Web site)
4. Interpolation is a specific case of curve fitting, in which the function must go exactly through the pre-calculated data points.
5. Interpolation is a well studied subject inside the field of numerical analysis.

### Interpolation FormulaUpDw('INTERPOLATION_FORMULA','-Abz-');

1. The Whittaker---Shannon interpolation formula can be used if the number of data points is infinite.
2. Other forms of interpolation can be constructed by picking a different class of interpolants.
3. The Nyquist-Shannon interpolation formula can be used if the number of data points is infinite.

### PiecewiseUpDw('PIECEWISE','-Abz-');

1. Birkhoff and de Boor, Piecewise polynomial interpolation and approximation, in: H. L. Garabedian (ed.), Proc.
2. Piecewise cubic interpolation.

### Spline InterpolationUpDw('SPLINE_INTERPOLATION','-Abz-');

1. Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation and the interpolant is smoother.
2. These disadvantages can be avoided by using spline interpolation.

### Interpolation MethodsUpDw('INTERPOLATION_METHODS','-Abz-');

1. Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method.
2. In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points.
3. The simplest interpolation method is to locate the nearest data value, and assign the same value. (Web site)

### Polynomial InterpolationUpDw('POLYNOMIAL_INTERPOLATION','-Abz-');

1. This can be generalized to polynomial interpolation, which is sometimes more accurate but suffers from Runge's phenomenon. (Web site)
2. Furthermore, polynomial interpolation may not be so exact after all, especially at the end points (see Runge's phenomenon).
3. Polynomial interpolation Main article: Polynomial interpolation right Polynomial interpolation is a generalization of linear interpolation.

### Data PointsUpDw('DATA_POINTS','-Abz-');

1. The interpolation error is proportional to the distance between the data points to the power n.
2. The interpolation will give a polynomial that has a degree less than or equal to the number of both data points and their derivatives, minus 1.
3. Constructing the interpolation polynomial The red dots denote the data points ( x k, y k), while the blue curve shows the interpolation polynomial.

### Linear InterpolationUpDw('LINEAR_INTERPOLATION','-Abz-');

1. A very simple method is to use linear interpolation, which assumes that the unknown function is linear between every pair of successive points. (Web site)
2. So, we see that polynomial interpolation solves all the problems of linear interpolation.
3. One of the simplest methods is linear interpolation (sometimes known as lerp).
4. Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation and the interpolant is smoother. (Web site)

### Interpolation UsesUpDw('INTERPOLATION_USES','-Abz-');

1. Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together.
2. Remember that linear interpolation uses a linear function for each of intervals [ x k, x k+1].

### Very PreciseUpDw('VERY_PRECISE','-Abz-');

1. The following error estimate shows that linear interpolation is not very precise.
2. Linear interpolation is quick and easy, but it is not very precise.

### Bilinear InterpolationUpDw('BILINEAR_INTERPOLATION','-Abz-');

1. The discrete Fourier transform is a special case of trigonometric interpolation.
2. Methods include bilinear interpolation and bicubic interpolation in two dimensions, and trilinear interpolation in three dimensions.
3. However, the interpolant is easier to evaluate than the high-degree polynomials used in polynomial interpolation.
4. The interpolated surface is smooth in all directions compared to bilinear interpolation and nearest neighbor interpolation.
5. In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables on a regular grid.

### Bicubic InterpolationUpDw('BICUBIC_INTERPOLATION','-Abz-');

1. Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines or cubic convolution algorithm. (Web site)
2. Bicubic interpolation on the square consisting of 9 unit squares patched together. (Web site)
3. Bicubic Smoother - a good method for enlarging images based on Bicubic interpolation but designed to produce smoother results. (Web site)
4. Splines are piecewise polynomial functions that are often used in bicubic interpolation algorithms. (Web site)
5. For instance, rational interpolation is interpolation by rational functions, and trigonometric interpolation is interpolation by trigonometric polynomials.

### Trigonometric PolynomialsUpDw('TRIGONOMETRIC_POLYNOMIALS','-Abz-');

1. The trigonometric polynomials are used in trigonometric interpolation to interpolate periodic functions.
2. For instance, rational interpolation is interpolation by rational function s, and trigonometric interpolation is interpolation by trigonometric polynomial s.

### Interpolating PolynomialUpDw('INTERPOLATING_POLYNOMIAL','-Abz-');

1. Choosing the points of intersection as interpolation nodes we obtain the interpolating polynomial coinciding with the best approximation polynomial.
2. One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients.
3. For more information on formulation of trigonometric interpolating polynomials in the complex plane see , p128 Interpolation using Fourier Polynomials. (Web site)
4. Choosing the points of intersection as interpolation nodes we obtain the interpolating polynomial coinciding with the best approximation polynomial.

### Cubic SplineUpDw('CUBIC_SPLINE','-Abz-');

1. Cubic spline interpolation. (Web site)
2. Interpolation Using Spline Functions - Theory and example of using a natural cubic spline to connect given data with a smooth curve. (Web site)

### Interpolation PolynomialUpDw('INTERPOLATION_POLYNOMIAL','-Abz-');

1. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the problem of Runge's phenomenon.
2. The more precise name is interpolation polynomial in the Lagrange form. (Web site)
3. In other words, the interpolation polynomial is at most a factor ( L +1) worse than the best possible approximation.
4. Runge's phenomenon shows that for high values of n, the interpolation polynomial may oscillate wildly between the data points.
5. When using a monomial basis for Π n we have to solve the Vandermonde matrix to construct the coefficients a k for the interpolation polynomial.

### OperatorsUpDw('OPERATORS','-Abz-');

1. The classical results about interpolation of operators are the Riesz-Thorin theorem and the Marcinkiewicz theorem.
2. It should be mentioned that there is another very different kind of interpolation in mathematics, namely the " interpolation of operators ".
3. Linear interpolation is often used to fill the gaps in a table. (Web site)
4. This section is an introduction to both interpolation (where an exact fit to constraints is expected) and regression analysis.
5. Investopedia Says: Interpolation is most often used in situations where a table of values is missing data.

### NodesUpDw('NODES','-Abz-');

1. This can be avoided by interpolating the inverse of f, resulting in the inverse quadratic interpolation method. (Web site)
2. This suggests that we look for a set of interpolation nodes that L small.
3. Again, values for the edge fluxes can be reconstructed by interpolation or extrapolation of the cell averages.
4. The Lebesgue constant can be used to give another error estimate for polynomial interpolation.
5. We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation.

### ComplicatedUpDw('COMPLICATED','-Abz-');

1. A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function.
2. Using a standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix. (Web site)

### Interpolation FunctionsUpDw('INTERPOLATION_FUNCTIONS','-Abz-');

1. This leads to Hermite interpolation problems.
2. There are many different interpolation methods, some of which are described below.
3. Interpolation answers questions like this.
4. Any means of calculating a new point between two existing data points is therefore interpolation. (Web site)
5. This will give the same result as linear interpolation evaluated at the midpoint.

### Lagrange InterpolationUpDw('LAGRANGE_INTERPOLATION','-Abz-');

1. However, polynomial interpolation also has some disadvantages.
2. The simplest form of interpolation is to take the mean average of x and y of two adjacent points to find the mid point.
3. Interpolation provides a means of estimating the function at intermediate points, such as x  = 2.5. (Web site)
4. Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. (Web site)
5. The packages will perform operations such as polynomial interpolation and at the very least a user should know when to use a specific type.

### Interpolation PointsUpDw('INTERPOLATION_POINTS','-Abz-');

1. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. (Web site)
2. In fact, extrapolation consists of interpolation at a point outside the interval containing the interpolation points. (Web site)
3. This oscillation is lessened by choosing interpolation points at Chebyshev nodes.
4. One might think that better convergence properties may be obtained by choosing different interpolation nodes.

### Interpolation SearchUpDw('INTERPOLATION_SEARCH','-Abz-');

1. The interpolation search, like the binary search, requires that the values be sorted and randomly accessible. (Web site)

### Interpolation AlgorithmUpDw('INTERPOLATION_ALGORITHM','-Abz-');

1. A special Bayer interpolation algorithm is then used to create separate R, G and B values for every pixel location.

### Cubic InterpolationUpDw('CUBIC_INTERPOLATION','-Abz-');

1. In mathematics, bicubic interpolation is an extension of cubic interpolation for interpolating data points on a two dimensional regular grid. (Web site)
2. A cubic spline is a piecewise cubic polynomial such that the function, its derivative and its second derivative are continuous at the interpolation nodes.

### Interpolation AttackUpDw('INTERPOLATION_ATTACK','-Abz-');

1. Five rounds of a modified version of SHARK can be broken using a interpolation attack (Jakobsen and Knudsen, 1997). (Web site)

### Interpolation FunctionUpDw('INTERPOLATION_FUNCTION','-Abz-');

1. Calculating the interpolating polynomial is computationally expensive (see computational complexity) compared to linear interpolation. (Web site)
2. In order to obtain solution uniqueness to this system and completely determine the spline interpolation function, more relations need to be set. (Web site)
3. Bicubic filtering is similar to bilinear filtering but with a different interpolation function. (Web site)

### Digital Zoom Interpolation MeansUpDw('DIGITAL_ZOOM_INTERPOLATION_MEANS','-Abz-');

1. In addition, Digital Zoom Interpolation means that digital zooming (up to 120x) is clearer, with less distortion than previous types of digital zoom.
2. In addition the digital zoom interpolation means that extreme digital zooming is clearer, with less distortion than previous types of digital zooms.

### Secant MethodUpDw('SECANT_METHOD','-Abz-');

1. Using linear interpolation instead of quadratic interpolation gives the secant method.
2. Finally, Brent's method is a combination of the bisection method, the secant method and inverse quadratic interpolation. (Web site)

### SplineUpDw('SPLINE','-Abz-');

1. The cubic spline interpolation uses third degree polynomials to connect the data points which often results in strikingly smooth curve fits. (Web site)
2. These disadvantages can be avoided by using spline interpolation or restricting attention to Chebyshev polynomials. (Web site)

### InterpolatesUpDw('INTERPOLATES','-Abz-');

1. Using polynomial interpolation, the polynomial of degree n which interpolates the data set is uniquely defined by the data points.
2. In mathematics, Birkhoff interpolation is an extension of polynomial interpolation. (Web site)
3. Interpolation of a data set Linear interpolation on a data set (red points) consists of pieces of linear interpolants (blue lines). (Web site)

### ExtrapolationUpDw('EXTRAPOLATION','-Abz-');

1. Curve Fitting, Data Modelling, Interpolation, Extrapolation This vague heading covers a multitude of closely related endeavors. (Web site)
2. We also obtain an interpolation formula for polynomials in two variables where the interpolation points are Chebyshev nodes. (Web site)
3. The Fast Fourier Transform came in 1965 and Salzer's barycentric interpolation formula for Chebyshev points in 1972 [Salzer 1972]. (Web site)

### InterpolateUpDw('INTERPOLATE','-Abz-');

1. A significant degree of spatial dependence is required to justify interpolation between measured values in ordered sets.
2. In Section 3.4 and Figure 5, it was shown how 1-D piecewise linear interpolation could be used to interpolate values in the 2-D plane. (Web site)
3. The performance of the interpolation algorithms can then be evaluated for these denser grids against the surface function values. (Web site)

### Proposed MethodUpDw('PROPOSED_METHOD','-Abz-');

1. The interpolated surface is smoother than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation. (Web site)
2. Among three interpolation methods, the proposed method has the highest interpolation accuracy regardless of the arrangement of sample points.

### InterpolatedUpDw('INTERPOLATED','-Abz-');

1. The function to be interpolated is known at given points and the interpolation problem consist of yielding values at arbitrary points. (Web site)
2. The Discrete Fourier Transform of the mddp(k) signal, DFT[mddp(k)], is input to the interpolation determiner 22 for analysis. (Web site)
3. Interpolated values can be limited to a given range by entering a minimum and maximum interpolation value. (Web site)

### PixelUpDw('PIXEL','-Abz-');

1. Polynomial interpolation: Discusses limitations of the polynomial interpolation for the data values above; MATLAB codes for Lagrange interpolation. (Web site)
2. In places where gradations have been lost because of whiteout, interpolation techniques are used to calculate pixel values from the surrounding pixels.
3. If the interpolated point is greater than any of the three used for the interpolation, then it is replaced with the point with the smallest function value.

### Complex InterpolationUpDw('COMPLEX_INTERPOLATION','-Abz-');

1. A method for arbitrary Banach spaces is the complex interpolation method. (Web site)
2. Equivalently, complex interpolation yields the same H^s(X) spaces so long as X has an extension operator.
3. InterReg allows you to do complex interpolation and regression calculations just with some point-and-click. (Web site)

### Interpolation MethodUpDw('INTERPOLATION_METHOD','-Abz-');

1. The term Phong shading is used indiscriminately to describe both an illumination model and an interpolation method in 3D computer graphics. (Web site)
2. By dividing the quadrilaterals into triangles and rendering them with Gouraud shading, an approximation of bilinear interpolation can be done. (Web site)
3. Implement Phong interpolation of normals on triangle meshes. (Web site)
4. The Phong reflection model may be used in conjunction with any of these interpolation methods.

### Cubic SplinesUpDw('CUBIC_SPLINES','-Abz-');

1. Interpolation and polynomial approximation (Lagrange polynomials, cubic splines). (Web site)
2. The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. (Web site)
3. This paper gives a recent overview on numerical aspects of multivariate interpolation and approximation by radial basis functions.

### WaveletsUpDw('WAVELETS','-Abz-');

1. Other interpolation methods use localized functions like splines or wavelets.
2. Written by authoritative international mathematicians, this book presents many important results in classical analysis, wavelets, and interpolation theory. (Web site)

### KrigingUpDw('KRIGING','-Abz-');

1. The empirical variogram is used in geostatistics as a first estimate of the (theoretical) variogram needed for spatial interpolation by kriging. (Web site)
2. In this situation, the kriging method may be a better choice for spatial interpolation.

### Curve FittingUpDw('CURVE_FITTING','-Abz-');

1. Linear interpolation is a method of curve fitting using linear polynomials.
2. It is possible to create derived datasets by equations, interpolation, curve fitting, and numerical integration and differentiation.
3. Calculus topics: numerical differentiation and integration, interpolation, solutions of nonlinear equations f(x) = 0. (Web site)
4. FindGraph will help you to quickly create linear and non-linear curve fitting models either for interpolation or for simulation and analyzing graphed data.
5. The function interpft performs one-dimensional interpolation using an FFT-based method. (Web site)

### InterpolationUpDw('INTERPOLATION','-Abz-');

1. In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of interpolating data points as a polynomial function. (Web site)

### CategoriesUpDw('Categories','-Abz-');

1. Information Technology > Computer Science > Algorithms > Numerical Analysis
2. Encyclopedia of Keywords > Information > Science > Mathematics
3. Information > Information Technology > Computer Science > Algorithms
4. Glossaries > Glossary of Numerical Analysis /

### SubcategoriesUpDw('Subcategories','-Abz-');

 Curve Fitting Kriging
1. Books about "Interpolation" in Amazon.com  Short phrases about "Interpolation"   Originally created: March 20, 2008.   Links checked: February 20, 2013.   Please send us comments and questions by this Online Form   Please click on to move good phrases up.
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