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c_nopjl44ta7vg | In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4. No closed-form expression for the partition function is known, but it has both asymp... | Partition function (number theory) |
c_xrmjvuiweghu | The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as ... | Partition function (number theory) |
c_d6is0qgckk18 | In number theory, the prime omega functions ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} count the number of prime factors of a natural number n . {\displaystyle n.} Thereby ω ( n ) {\displaystyle \omega (n)} (little omega) counts each distinct prime factor, whereas the related function Ω (... | Big Omega function (prime factor) |
c_7nq27qfj4h7k | In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: The radical plays a central role in the statement of the abc conjecture. | Radical of an integer |
c_amaleyspjigu | In number theory, the ruler function of an integer n {\displaystyle n} can be either of two closely related functions. One of these functions counts the number of times n {\displaystyle n} can be evenly divided by two, which for the numbers 1, 2, 3, ... is Alternatively, the ruler function can be defined as the same nu... | Ruler function |
c_5wt6kx9j4r04 | In advanced mathematics, the 0-based ruler function is the 2-adic valuation of the number, and the lexicographically earliest infinite square-free word over the natural numbers. It also gives the position of the bit that changes at each step of the Gray code.In the Tower of Hanoi puzzle, with the disks of the puzzle nu... | Ruler function |
c_yx048y0az8d0 | In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the first Hardy–Littlewood conjecture, the second Hardy–Littlewood conjecture was proposed by G. H. Hardy and John Edensor Littlewood in 1923. | Second Hardy–Littlewood conjecture |
c_p89vi8klhm99 | In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number a/b is a "good" approxi... | Diophantine approximations |
c_vycjl7xf9b71 | This problem was solved during the 18th century by means of continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the natu... | Diophantine approximations |
c_vgxku4y1v3hs | Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number. This knowledge enabled Liouville, in 1844, to produce the first explicit transcendental number. Later, the proofs that π and e are transcendental were obtained by a similar method. | Diophantine approximations |
c_6llmublqd7kd | Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations. The 2022 Fields Medal was awarded to James Maynard for his work on Diophantine approximation. | Diophantine approximations |
c_7qvn1p3mv3ep | In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different, and is denot... | Sum of squares function |
c_xiulx61s7uzd | In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, 1 3 + 2 3 + 3 3 + ⋯ + n 3 = ( 1 + 2 + 3 + ⋯ + n ) 2 . {\displaystyle 1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(1+2+3+\cdots +n\right)^{2}.} The same equation may be written more compactly using the mathematical notation for ... | Nicomachus's theorem |
c_yxacgr6fac6w | {\displaystyle \sum _{k=1}^{n}k^{3}={\bigg (}\sum _{k=1}^{n}k{\bigg )}^{2}.} This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). | Nicomachus's theorem |
c_05lic9gnxoue | In number theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares, such that n = a2 + b2 for some integers a, b. An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no ... | Sum of two squares theorem |
c_nokakt97zemv | In number theory, the totient summatory function Φ ( n ) {\displaystyle \Phi (n)} is a summatory function of Euler's totient function defined by: Φ ( n ) := ∑ k = 1 n φ ( k ) , n ∈ N {\displaystyle \Phi (n):=\sum _{k=1}^{n}\varphi (k),\quad n\in \mathbf {N} } It is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ... | Landau's totient constant |
c_dykraqgkxb35 | In number theory, the unit function is a completely multiplicative function on the positive integers defined as: ε ( n ) = { 1 , if n = 1 0 , if n ≠ 1 {\displaystyle \varepsilon (n)={\begin{cases}1,&{\mbox{if }}n=1\\0,&{\mbox{if }}n\neq 1\end{cases}}} It is called the unit function because it is the identity element fo... | Unit function |
c_p7bfx7kz9vmn | In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt (1840) and Thomas Clausen (1840). Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B2n for every prime p such that p − 1 divides 2n, we... | Von Staudt–Clausen theorem |
c_j5zp701n6w2g | In number theory, there are many integer factoring algorithms that heuristically have expected running time L n = e ( 1 + o ( 1 ) ) ( log n ) ( log log n ) {\displaystyle L_{n}\left=e^{(1+o(1)){\sqrt {(\log n)(\log \log n)}}}} in little-o and L-notation. Some examples of those algorithms are the elliptic curve m... | Integer factorisation |
c_i2gypa4i7qvg | In number theory, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. | Coprime integers |
c_z5z4yzpzvi4s | One says also a is prime to b or a is coprime with b. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduc... | Coprime integers |
c_h75ono30d4jq | In number theory, two positive integers a and b are said to be multiplicatively independent if their only common integer power is 1. That is, for integers n and m, a n = b m {\displaystyle a^{n}=b^{m}} implies n = m = 0 {\displaystyle n=m=0} . Two integers which are not multiplicatively independent are said to be multi... | Multiplicative independence |
c_yfdqfqntty84 | In number theory, various "prime geodesic theorems" have been proved which are very similar in spirit to the prime number theorem. To be specific, we let π(x) denote the number of closed geodesics whose norm (a function related to length) is less than or equal to x; then π(x) ∼ x/ln(x). This result is usually credited ... | Prime geodesic |
c_mbmi7nhbfvlv | thesis, Grigory Margulis proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, Peter Sarnak proved an analogue of Chebotarev's density theorem. There are other similarities to number theory — error estimates are improved upon, in much the same way that error estimates of t... | Prime geodesic |
c_uyn4opn5l8t8 | Also, there is a Selberg zeta function which is formally similar to the usual Riemann zeta function and shares many of its properties. Algebraically, prime geodesics can be lifted to higher surfaces in much the same way that prime ideals in the ring of integers of a number field can be split (factored) in a Galois exte... | Prime geodesic |
c_rpctoud5l02q | In number theory, we define the Mertens function as M ( n ) = ∑ 1 ≤ k ≤ n μ ( k ) , {\displaystyle M(n)=\sum _{1\leq k\leq n}\mu (k),} where μ(k) is the Möbius function; the Mertens conjecture is that for all n > 1, | M ( n ) | < n . {\displaystyle |M(n)|<{\sqrt {n}}.} | Mertens conjecture |
c_lo0w4t1lary2 | In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0. The... | Zero-sum problem |
c_8mdngn1kcgi4 | (Indeed, the lower bound is easy to see: the multiset containing n − 1 copies of 0 and n − 1 copies of 1 contains no n-subset summing to a multiple of n.) This result is known as the Erdős–Ginzburg–Ziv theorem after its discoverers. It may also be deduced from the Cauchy–Davenport theorem.More general results than this... | Zero-sum problem |
c_44opxzlw4wlv | In numeric notation, each square is designated with a two-digit number via a coordinate system. The first digit describes the file and the second digit the rank. Files are numbered 1 to 8 from White's left to White's right, and ranks are numbered 1 to 8 from White's near side to White's far side. A move is defined by p... | ICCF numeric notation |
c_wao2gi7mnql4 | For example, the move that would be written 1.e4 in algebraic notation would be written 1. 5254 in numeric notation: the pawn starts from square 52 (file 5, rank 2) and moves to square 54 (file 5, rank 4). Numeric notation does not specifically mark the type of moving piece, captures, or checks; every move is written a... | ICCF numeric notation |
c_zho6ql23t72h | For promotion, a fifth digit is added to the move's notation: 1 for queen, 2 for rook, 3 for bishop, and 4 for knight. For instance, a pawn on f7 moving to f8 and promoting to a rook would be written as 67682. A variant four-digit notation where the ending rank is omitted (because it is always 8 for White and 1 for Bla... | ICCF numeric notation |
c_hht12ik680ih | Castling is written using the king's start position and end position. Castling kingside is written as 5171 for White and 5878 for Black, and castling queenside is written as 5131 for White and 5838 for Black. The rook's start and end positions are implied. | ICCF numeric notation |
c_0azhpylk18uo | In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind: ∫ 0 + ∞ e − x f ( x ) d x . {\displaystyle \int _{0}^{+\infty }e^{-x}f(x)\,dx.} In this case ∫ 0 + ∞ e ... | Gauss-Laguerre quadrature |
c_wsxtz64wyatt | In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by C ( f , h ) ( x ) = ∑ k = ... | Sinc numerical methods |
c_cnhfms2wfrmq | In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei Godunov in 1959, for solving partial differential equations. One can think of this method as a conservative finite volume method which solves exact, or approximate Riemann problems at each in... | Godunov method |
c_75vjvvypudri | In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high-resolution schemes for the numerical solution of partial differential equations. The theorem states that: Linear numeri... | Godunov's theorem |
c_p6nq6j369mxs | In numerical analysis and computational statistics, rejection sampling is a basic technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in R m {\... | Adaptive rejection sampling |
c_kpmh6mngmjp0 | In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location... | Discrete Wavelet Transform |
c_qpotu6bw60q4 | In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information. The accuracy o... | Digital Signal Processing |
c_e4c9qt4e8nvk | In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form... | LDU decomposition |
c_p5xauinrcek2 | Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938. To quote: "It appears that Gauss and Doolittle applied... | LDU decomposition |
c_ioj6yjysc9yy | In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly eq... | Symmetric sparse matrix |
c_iis8wh806vgl | Conceptually, sparsity corresponds to systems with few pairwise interactions. For example, consider a line of balls connected by springs from one to the next: this is a sparse system as only adjacent balls are coupled. | Symmetric sparse matrix |
c_y62mwnzp8nlj | By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. The concept of sparsity is useful in combinatorics and application areas such as network theory and numerical analysis, which typically have a low density of significant da... | Symmetric sparse matrix |
c_r2uszloxf5ct | When storing and manipulating sparse matrices on a computer, it is beneficial and often necessary to use specialized algorithms and data structures that take advantage of the sparse structure of the matrix. Specialized computers have been made for sparse matrices, as they are common in the machine learning field. Opera... | Symmetric sparse matrix |
c_1rl5hr2x4jv9 | In numerical analysis and scientific computing, the Gauss–Legendre methods are a family of numerical methods for ordinary differential equations. Gauss–Legendre methods are implicit Runge–Kutta methods. More specifically, they are collocation methods based on the points of Gauss–Legendre quadrature. The Gauss–Legendre ... | Gauss–Legendre method |
c_0pr43rdg5igb | In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has e... | Euler backward method |
c_rhkhrbbvw15h | In numerical analysis and scientific computing, the trapezoidal rule is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. The trapezoidal rule is an implicit second-order method, which can be considered as both a Runge–Kutta method and a linear multis... | Trapezoidal rule (differential equations) |
c_j50vgfbeetm7 | In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. | Truncation error |
c_xf4gmanmc1wn | In numerical analysis the diffuse element method (DEM) or simply diffuse approximation is a meshfree method. The diffuse element method was developed by B. Nayroles, G. Touzot and Pierre Villon at the Universite de Technologie de Compiegne, in 1992. It is in concept rather similar to the much older smoothed particle hy... | Diffuse element method |
c_6zjtz8ldbia1 | In the paper they describe a "diffuse approximation method", a method for function approximation from a given set of points. In fact the method boils down to the well-known moving least squares for the particular case of a global approximation (using all available data points). Using this function approximation method,... | Diffuse element method |
c_0f41usxu6868 | In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. Its early form was known to Seki Kōwa (end of 17th century) and was found fo... | Aitken extrapolation |
c_bm8ei1iyna3y | In numerical analysis, Bairstow's method is an efficient algorithm for finding the roots of a real polynomial of arbitrary degree. The algorithm first appeared in the appendix of the 1920 book Applied Aerodynamics by Leonard Bairstow. The algorithm finds the roots in complex conjugate pairs using only real arithmetic. ... | Bairstow's method |
c_symmn5cyiadp | In numerical analysis, Brent's method is a hybrid root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable methods. The algorithm tries to use the potentially fast-converging seca... | Brent's method |
c_s3kd94emtr3i | Brent's method is due to Richard Brent and builds on an earlier algorithm by Theodorus Dekker. Consequently, the method is also known as the Brent–Dekker method. Modern improvements on Brent's method include Chandrupatla's method, which is simpler and faster for functions that are flat around their roots; Ridders' meth... | Brent's method |
c_573mdb6z45ty | In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in k variables. It was originally described by C. G. Broyden in 1965.Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian is a difficult and expensive operation. The idea ... | Broyden's method |
c_9kmxqhfefa6v | In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the effect of Runge's phenomenon. | Chebyshev nodes |
c_ecaeyl7ymbsd | In numerical analysis, Estrin's scheme (after Gerald Estrin), also known as Estrin's method, is an algorithm for numerical evaluation of polynomials. Horner's method for evaluation of polynomials is one of the most commonly used algorithms for this purpose, and unlike Estrin's scheme it is optimal in the sense that it ... | Estrin's scheme |
c_e2c54ij9wspn | In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind: ∫ − ∞ + ∞ e − x 2 f ( x ) d x . {\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx.} In this case ∫ − ∞ + ∞ e − x 2 f ( x ) d x ≈ ∑ i = 1 n w i f ( x i ) {\displayst... | Gauss-Hermite quadrature |
c_1dzsjjhcz481 | π n 2 2 . {\displaystyle w_{i}={\frac {2^{n-1}n! {\sqrt {\pi }}}{n^{2}^{2}}}.} | Gauss-Hermite quadrature |
c_73yqs9x9dm8d | In numerical analysis, Gauss–Legendre quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating over the interval , the rule takes the form: ∫ − 1 1 f ( x ) d x ≈ ∑ i = 1 n w i f ( x i ) {\displaystyle \int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})} w... | Gauss–Legendre quadrature |
c_lqz83y9empqu | This algorithm was popular, but significantly more efficient algorithms exist. Algorithms based on the Newton–Raphson method are able to compute quadrature rules for significantly larger problem sizes. In 2014, Ignace Bogaert presented explicit asymptotic formulas for the Gauss–Legendre quadrature weights and nodes, wh... | Gauss–Legendre quadrature |
c_jvyz7y13n9xx | In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. It is named after its inventor Edmond Halley. The algorithm is second in the class of Householder's methods, after Newton's method. Like the latter, it iteratively produces a s... | Bailey's method (root finding) |
c_qkufuoels78r | In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than n that takes the same value at n given points as a given function. Instead, Hermite inter... | Hermite interpolation |
c_a3y8zhlfq83q | However, there are other methods for computing a Hermite interpolating polynomial. One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see Chinese remainder... | Hermite interpolation |
c_z1qjuxaff4f9 | In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to numerically solve the equation p(x) = 0 for a given polynomial p(x). One of the most useful properties of this method is that it is, from extensive empirical study, very close t... | Laguerre's method |
c_825mx8n5x3jq | In numerical analysis, Lebedev quadrature, named after Vyacheslav Ivanovich Lebedev, is an approximation to the surface integral of a function over a three-dimensional sphere. The grid is constructed so to have octahedral rotation and inversion symmetry. The number and location of the grid points together with a corres... | Lebedev grid |
c_zbepdnbs9jvj | In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f ... | Newton method |
c_y6rbov6cz9ya | The number of correct digits roughly doubles with each step. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations. | Newton method |
c_zy4lc9r6r73g | In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A ∗ = lim h → 0 A ( h ) {\displaystyle A^{\ast }=\lim _{h\to 0}A(h)} . In essence, given the value of A ( h ) {\displaystyle A(h)} for several values of h {\... | Richardson extrapolation |
c_1glb0ddqt03w | In numerical analysis, Ridders' method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a continuous function f ( x ) {\displaystyle f(x)} . The method is due to C. Ridders.Ridders' method is simpler than Muller's method or Brent... | Ridders' method |
c_4vi0nhgzw71c | In numerical analysis, Romberg's method is used to estimate the definite integral by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spa... | Romberg integration |
c_mdzda4gmqhwy | In numerical analysis, Steffensen's method is an iterative method for root-finding named after Johan Frederik Steffensen which is similar to Newton's method, but with certain situational advantages. In particular, Steffensen's method achieves similar quadratic convergence, but without using derivatives as Newton's meth... | Steffensen's method |
c_541pp0jrtf5b | In numerical analysis, Stone's method, also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations. The method uses an incomplete LU decomposition, which approximates the exact LU decomposition, to get an iterative solution of the problem. The method is named af... | Stone method |
c_mppk6dgehksc | The biggest disadvantage is that it fails to take advantage of coefficient matrix to be a sparse matrix. The LU decomposition of a sparse matrix is usually not sparse, thus, for a large system of equations, LU decomposition may require a prohibitive amount of memory and number of arithmetical operations. In the precond... | Stone method |
c_btmelkajiz4a | This brings one to idea of using approximate factorization LU of A as the iteration matrix M. A version of incomplete lower-upper decomposition method was proposed by Stone in 1968. This method is designed for equation system arising from discretisation of partial differential equations and was firstly used for a penta... | Stone method |
c_5dxl2vjzvz08 | In numerical analysis, Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of the roots can be very sensitive to perturbations in the coefficients of the polynomial. The polynomial is w ( x ) = ∏ i = ... | Wilkinson's polynomial |
c_nyqh8nvw8tmq | In numerical analysis, a blossom is a functional that can be applied to any polynomial, but is mostly used for Bézier and spline curves and surfaces. The blossom of a polynomial ƒ, often denoted B , {\displaystyle {\mathcal {B}},} is completely characterised by the three properties: It is a symmetric function of its a... | Blossom (functional) |
c_qhzyuzo73p2f | In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for numerically solving the differential equation, y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 . {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0}.} The explicit midpoint method is given by the formula the implicit mi... | Midpoint method |
c_zmqi93ie1tdk | The explicit midpoint method is sometimes also known as the modified Euler method, the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator. Note that the modified Euler method can refer to Heun's method, for further clarity see List of Runge–Kutta methods... | Midpoint method |
c_mei8didehffy | A geometric interpretation may give a better intuitive understanding of the method (see figure at right). In the basic Euler's method, the tangent of the curve at ( t n , y n ) {\displaystyle (t_{n},y_{n})} is computed using f ( t n , y n ) {\displaystyle f(t_{n},y_{n})} . The next value y n + 1 {\displaystyle y_{n+1}}... | Midpoint method |
c_mh232nkwh357 | However, if the second derivative is only positive between t n {\displaystyle t_{n}} and t n + 1 {\displaystyle t_{n+1}} , or only negative (as in the diagram), the curve will increasingly veer away from the tangent, leading to larger errors as h {\displaystyle h} increases. The diagram illustrates that the tangent at ... | Midpoint method |
c_l43lc2qllnfw | Instead, this tangent is estimated by using the original Euler's method to estimate the value of y ( t ) {\displaystyle y(t)} at the midpoint, then computing the slope of the tangent with f ( ) {\displaystyle f()} . Finally, the improved tangent is used to calculate the value of y n + 1 {\displaystyle y_{n+1}} from y n... | Midpoint method |
c_k0nyroprz7v6 | Note that the red chord is not exactly parallel to the green segment (the true tangent), due to the error in estimating the value of y ( t ) {\displaystyle y(t)} at the midpoint. The local error at each step of the midpoint method is of order O ( h 3 ) {\displaystyle O\left(h^{3}\right)} , giving a global error of orde... | Midpoint method |
c_3xrqda99kdqj | In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the corresponding domain interval.Cubic Hermite splines are typically used for interpolation ... | Hermite curve |
c_ppnwx9ydmvbn | The resulting spline will be continuous and will have continuous first derivative. Cubic polynomial splines can be specified in other ways, the Bezier cubic being the most common. However, these two methods provide the same set of splines, and data can be easily converted between the Bézier and Hermite forms; so the na... | Hermite curve |
c_40r13qx1ew7i | Cubic polynomial splines are extensively used in computer graphics and geometric modeling to obtain curves or motion trajectories that pass through specified points of the plane or three-dimensional space. In these applications, each coordinate of the plane or space is separately interpolated by a cubic spline function... | Hermite curve |
c_urdw4s2z92ka | Bicubic splines (Bicubic interpolation) are often used to interpolate data on a regular rectangular grid, such as pixel values in a digital image or altitude data on a terrain. Bicubic surface patches, defined by three bicubic splines, are an essential tool in computer graphics. Cubic splines are often called csplines,... | Hermite curve |
c_q5haati74mfd | In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation m... | Multigrid methods |
c_779cgprjdb15 | The main idea of multigrid is to accelerate the convergence of a basic iterative method (known as relaxation, which generally reduces short-wavelength error) by a global correction of the fine grid solution approximation from time to time, accomplished by solving a coarse problem. The coarse problem, while cheaper to s... | Multigrid methods |
c_0nzy7q879ca1 | This recursive process is repeated until a grid is reached where the cost of direct solution there is negligible compared to the cost of one relaxation sweep on the fine grid. This multigrid cycle typically reduces all error components by a fixed amount bounded well below one, independent of the fine grid mesh size. Th... | Multigrid methods |
c_f4gq3e4mzpe2 | For example, the finite element method may be recast as a multigrid method. In these cases, multigrid methods are among the fastest solution techniques known today. | Multigrid methods |
c_tz2ib5vz3zur | In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and boundary conditions. They do not depend on the separability of the equations or other special properties of the equation. They have also been widely used for more-complicated non-symmetric and nonlinear systems of e... | Multigrid methods |
c_6i6avl3oxls1 | In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. | Numerical method |
c_9g5i9i2isrrm | In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl F... | Gaussian integration |
c_vtdtnhro9r85 | This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on . The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint sin... | Gaussian integration |
c_fp28wkq7sxm2 | Instead, if the integrand can be written as where g(x) is well-approximated by a low-degree polynomial, then alternative nodes xi' and weights wi' will usually give more accurate quadrature rules. These are known as Gauss-Jacobi quadrature rules, i.e., Common weights include 1 1 − x 2 {\textstyle {\frac {1}{\sqrt {1-x^... | Gaussian integration |
c_s0t29v3p48bn | In numerical analysis, a superconvergent or supraconvergent method is one which converges faster than generally expected (superconvergence or supraconvergence). For example, in the Finite Element Method approximation to Poisson's equation in two dimensions, using piecewise linear elements, the average error in the grad... | Superconvergence |
c_uead6cl1a3jb | In numerical analysis, adaptive mesh refinement (AMR) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated. When solutions are calculated numerically, they are often limited to pre-determined quan... | Mesh refinement |
c_r5z4jiw91t10 | Adaptive mesh refinement provides such a dynamic programming environment for adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas of multi-dimensional graphs which need precision while leaving the other regions of the multi-dimensional graphs at lower ... | Mesh refinement |
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