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Polynomial interpolation is finding a polynomial whose values ( not coefficients ) agree with a given sequence ; the Hilbert polynomial is an abstract case of this in commutative algebra.
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Polynomial and interpolation
* Polynomial interpolation
Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations.
Polynomial interpolation is also essential to perform sub-quadratic multiplication and squaring such as Karatsuba multiplication and Toom – Cook multiplication, where an interpolation through points on a polynomial which defines the product yields the product itself.
Additionally, the interpolating polynomial is unique, as shown by the unisolvence theorem at Polynomial interpolation.
For more details, Polynomial interpolation.
# redirect Polynomial interpolation
Polynomial and is
* Polynomial GCD, by a reduction to linear algebra using Sylvester matrix ( it is open whether integer GCD is in NC );
* Magma V2. 12 is apparently " Overall Best in the World at Polynomial GCD " :-)
Polynomial division allows for a polynomial to be written in a divisor – quotient form which is often advantageous.
Polynomial long division can be used to find the equation of the line that is tangent to a polynomial at a particular point.
Polynomial of degree 4, on the right one finds a local optimum, on the left is the global optimum.
Her doctoral thesis, " Die Frage der endlich vielen Schritte in der Theorie der Polynomideale " ( in English " The Question of Finitely Many Steps in Polynomial Ideal Theory "), published in Mathematische Annalen, is the foundational paper for computer algebra.
In computational complexity theory, the class IP ( which stands for Interactive Polynomial time ) is the class of problems solvable by an interactive proof system.
Polynomial and finding
Polynomial time algorithms are known for many algorithmic problems on matchings, including maximum matching ( finding a matching that uses as many edges as possible ), maximum weight matching, and stable marriage.
Polynomial and polynomial
* Polynomial SOS, the representation of a non-negative polynomial as the sum of squares of polynomials
Polynomial and with
The Newton Polynomial above can be expressed in a simplified form when are arranged consecutively with equal space.
* Polynomial evaluation ( e. g., with Horner's rule )
Polynomial curves fitting points generated with a sine function.
Polynomials and Polynomial Inequalities ( with Tamas Erdelyi, 1998 ), Pi and the AGM ( 1987 ; reprinted in 1998 ) and A Dictionary of Real Numbers.
Polynomial and given
Polynomial expressions are given on the pages of new moon and full moon.
Polynomial and sequence
* Polynomial sequence
* Polynomial sequence
Polynomial and Hilbert
Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator.
Polynomial and .
Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.
File: Polynomialdeg2. svg | Polynomial of degree 2: f ( x )
File: Polynomialdeg3. svg | Polynomial of degree 3: f ( x )
File: Polynomialdeg4. svg | Polynomial of degree 4: f ( x )
File: Polynomialdeg5. svg | Polynomial of degree 5: f ( x ) =
File: Sextic Graph. png | Polynomial of degree 6: f ( x )
File: Septic Graph. gif | Polynomial of degree 7: f ( x ) =
Polynomial functions are also frequently used to interpolate functions.
* Chapter 19: Polynomial space, pp. 455 – 490.
Their seminal article was entitled " Polynomial Codes over Certain Finite Fields.
* Polynomial rings.
Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics.
Polynomial truncations arise in the study of Taylor polynomials, for example.
# Polynomial codes ( e. g., BCH codes )
interpolation and is
Premultiplied alpha has some practical advantages over normal alpha blending because premultiplied alpha blending is associative and linear interpolation gives better results, although premultiplication can cause a loss of precision and, in extreme cases, a noticeable loss of quality.
and is equivalent to linear interpolation.
* The curve begins at P < sub > 0 </ sub > and ends at P < sub > n </ sub >; this is the so-called endpoint interpolation property.
While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method.
This interpolation is not unique: aliasing implies that one could add N to any of the complex-sinusoid frequencies ( e. g. changing to ) without changing the interpolation property, but giving different values in between the points.
First, it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited.
In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to ( instead of roughly to as above ), similar to the inverse DFT formula.
This interpolation does not minimize the slope, and is not generally real-valued for real ; its use is a common mistake.
" Although the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes.
When that interval is 0. 5 / T, the applicable reconstruction formula is the Whittaker – Shannon interpolation formula.
In most cases, N is chosen equal to the length of non-zero portion of s. Increasing N, known as zero-padding or interpolation, results in more closely spaced samples of one cycle of S < sub > 1 / T </ sub >( ƒ ).
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.
A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function.
Of course, when a simple function is used to estimate data points from the original, interpolation errors are usually present ; however, depending on the problem domain and the interpolation method used, the gain in simplicity may be of greater value than the resultant loss in accuracy.
There is also another very different kind of interpolation in mathematics, namely the " interpolation of operators ".
Points through which curve is spline ( mathematics ) | splined are red ; the blue curve connecting them is interpolation.
The general scholarly view is that while the Testimonium Flavianum is most likely not authentic in its entirety, it is broadly agreed upon that it originally consisted of an authentic nucleus with a reference to the execution of Jesus by Pilate which was then subject to Christian interpolation.
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