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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 asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. | https://en.wikipedia.org/wiki/Partition_function_(number_theory) |
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 Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5. | https://en.wikipedia.org/wiki/Partition_function_(number_theory) |
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 ( n ) {\displaystyle \Omega (n)} (big omega) counts the total number of prime factors of n , {\displaystyle n,} honoring their multiplicity (see arithmetic function). That is, if we have a prime factorization of n {\displaystyle n} of the form n = p 1 α 1 p 2 α 2 ⋯ p k α k {\displaystyle n=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}} for distinct primes p i {\displaystyle p_{i}} ( 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} ), then the respective prime omega functions are given by ω ( n ) = k {\displaystyle \omega (n)=k} and Ω ( n ) = α 1 + α 2 + ⋯ + α k {\displaystyle \Omega (n)=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{k}} . These prime factor counting functions have many important number theoretic relations. | https://en.wikipedia.org/wiki/Big_Omega_function_(prime_factor) |
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. | https://en.wikipedia.org/wiki/Radical_of_an_integer |
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 numbers plus one, which for the numbers 1, 2, 3, ... produces the sequence As well as being related by adding one, these two sequences are related in a different way: the second one can be formed from the first one by removing all the zeros, and the first one can be formed from the second one by adding zeros at the start and between every pair of numbers. For either definition of the ruler function, the rising and falling patterns of the values of this function resemble the lengths of marks on rulers with traditional units such as inches. These functions should be distinguished from Thomae's function, a function on real numbers which behaves similarly to the ruler function when restricted to the dyadic rational numbers. | https://en.wikipedia.org/wiki/Ruler_function |
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 numbered in order by their size, the 1-based ruler function gives the number of the disk to move at each step in an optimal solution to the puzzle. A simulation of the puzzle, in conjunction with other methods for generating its optimal sequence of moves, can be used in an algorithm for generating the sequence of values of the ruler function in constant time per value. | https://en.wikipedia.org/wiki/Ruler_function |
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. | https://en.wikipedia.org/wiki/Second_Hardy–Littlewood_conjecture |
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" approximation of a real number α if the absolute value of the difference between a/b and α may not decrease if a/b is replaced by another rational number with a smaller denominator. | https://en.wikipedia.org/wiki/Diophantine_approximations |
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 nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than the lower bound for algebraic numbers, which is itself larger than the lower bound for all real numbers. | https://en.wikipedia.org/wiki/Diophantine_approximations |
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. | https://en.wikipedia.org/wiki/Diophantine_approximations |
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. | https://en.wikipedia.org/wiki/Diophantine_approximations |
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 denoted by rk(n). | https://en.wikipedia.org/wiki/Sum_of_squares_function |
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 summation: ∑ k = 1 n k 3 = ( ∑ k = 1 n k ) 2 . | https://en.wikipedia.org/wiki/Nicomachus's_theorem |
{\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). | https://en.wikipedia.org/wiki/Nicomachus's_theorem |
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 factor pk, where prime p ≡ 3 ( mod 4 ) {\displaystyle p\equiv 3{\pmod {4}}} and k is odd. In writing a number as a sum of two squares, it is allowed for one of the squares to be zero, or for both of them to be equal to each other, so all squares and all doubles of squares are included in the numbers that can be represented in this way. This theorem supplements Fermat's theorem on sums of two squares which says when a prime number can be written as a sum of two squares, in that it also covers the case for composite numbers. A number may have multiple representations as a sum of two squares, counted by the sum of squares function; for instance, every Pythagorean triple a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} gives a second representation for c 2 {\displaystyle c^{2}} beyond the trivial representation c 2 + 0 2 {\displaystyle c^{2}+0^{2}} . | https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem |
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 ≤ n. | https://en.wikipedia.org/wiki/Landau's_totient_constant |
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 for Dirichlet convolution.It may be described as the "indicator function of 1" within the set of positive integers. It is also written as u(n) (not to be confused with μ(n), which generally denotes the Möbius function). | https://en.wikipedia.org/wiki/Unit_function |
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 obtain an integer, i.e., B 2 n + ∑ ( p − 1 ) | 2 n 1 p ∈ Z . {\displaystyle B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}\in \mathbb {Z} .} This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B2n as the product of all primes p such that p − 1 divides 2n; consequently the denominators are square-free and divisible by 6. These denominators are 6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, ... (sequence A002445 in the OEIS).The sequence of integers B 2 n + ∑ ( p − 1 ) | 2 n 1 p {\displaystyle B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}} is 1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, ... (sequence A000146 in the OEIS). | https://en.wikipedia.org/wiki/Von_Staudt–Clausen_theorem |
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 method and the quadratic sieve. Another such algorithm is the class group relations method proposed by Schnorr, Seysen, and Lenstra, which they proved only assuming the unproved Generalized Riemann Hypothesis (GRH). | https://en.wikipedia.org/wiki/Integer_factorisation |
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. | https://en.wikipedia.org/wiki/Coprime_integers |
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 reduced fraction are coprime, by definition. | https://en.wikipedia.org/wiki/Coprime_integers |
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 multiplicatively dependent. As examples, 36 and 216 are multiplicatively dependent since 36 3 = ( 6 2 ) 3 = ( 6 3 ) 2 = 216 2 {\displaystyle 36^{3}=(6^{2})^{3}=(6^{3})^{2}=216^{2}} , whereas 6 and 12 are multiplicatively independent. | https://en.wikipedia.org/wiki/Multiplicative_independence |
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 to Atle Selberg. In his 1970 Ph.D. | https://en.wikipedia.org/wiki/Prime_geodesic |
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 the prime number theorem are improved upon. | https://en.wikipedia.org/wiki/Prime_geodesic |
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 extension. See Covering map and Splitting of prime ideals in Galois extensions for more details. | https://en.wikipedia.org/wiki/Prime_geodesic |
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}}.} | https://en.wikipedia.org/wiki/Mertens_conjecture |
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 classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv. They proved that for the group Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } of integers modulo n, Explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n, but that the same is not true of multisets of size 2n − 2. | https://en.wikipedia.org/wiki/Zero-sum_problem |
(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 theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by Christian Reiher in 2003), and the weighted EGZ theorem (proved by David J. Grynkiewicz in 2005). | https://en.wikipedia.org/wiki/Zero-sum_problem |
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 pairing two square designations together, one for the starting square and one for the ending square. | https://en.wikipedia.org/wiki/ICCF_numeric_notation |
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 as four digits unless resulting in promotion. | https://en.wikipedia.org/wiki/ICCF_numeric_notation |
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 Black) also exists (e.g. 6762); however, this is considered confusing and contradicts the standard. | https://en.wikipedia.org/wiki/ICCF_numeric_notation |
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. | https://en.wikipedia.org/wiki/ICCF_numeric_notation |
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 − x f ( x ) d x ≈ ∑ i = 1 n w i f ( x i ) {\displaystyle \int _{0}^{+\infty }e^{-x}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})} where xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi is given by w i = x i ( n + 1 ) 2 2 . {\displaystyle w_{i}={\frac {x_{i}}{\left(n+1\right)^{2}\left^{2}}}.} The following Python code with the SymPy library will allow for calculation of the values of x i {\displaystyle x_{i}} and w i {\displaystyle w_{i}} to 20 digits of precision: | https://en.wikipedia.org/wiki/Gauss-Laguerre_quadrature |
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 = − ∞ ∞ f ( k h ) sinc ( x h − k ) {\displaystyle C(f,h)(x)=\sum _{k=-\infty }^{\infty }f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)} where the step size h>0 and where the sinc function is defined by sinc ( x ) = sin ( π x ) π x {\displaystyle {\textrm {sinc}}(x)={\frac {\sin(\pi x)}{\pi x}}} Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers. The truncated Sinc expansion of f is defined by the following series: C M , N ( f , h ) ( x ) = ∑ k = − M N f ( k h ) sinc ( x h − k ) {\displaystyle C_{M,N}(f,h)(x)=\displaystyle \sum _{k=-M}^{N}f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)} . | https://en.wikipedia.org/wiki/Sinc_numerical_methods |
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 inter-cell boundary. In its basic form, Godunov's method is first order accurate in both space and time, yet can be used as a base scheme for developing higher-order methods. | https://en.wikipedia.org/wiki/Godunov_method |
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 numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate.Professor Sergei Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methods used in computational fluid dynamics (CFD) and other computational fields. One of his major contributions was to prove the theorem (Godunov, 1954; Godunov, 1959), that bears his name. | https://en.wikipedia.org/wiki/Godunov's_theorem |
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 {\displaystyle \mathbb {R} ^{m}} with a density. Rejection sampling is based on the observation that to sample a random variable in one dimension, one can perform a uniformly random sampling of the two-dimensional Cartesian graph, and keep the samples in the region under the graph of its density function. Note that this property can be extended to N-dimension functions. | https://en.wikipedia.org/wiki/Adaptive_rejection_sampling |
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 in time). | https://en.wikipedia.org/wiki/Discrete_Wavelet_Transform |
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 of the joint time-frequency resolution is limited by the uncertainty principle of time-frequency. | https://en.wikipedia.org/wiki/Digital_Signal_Processing |
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 of Gaussian elimination. | https://en.wikipedia.org/wiki/LDU_decomposition |
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 the method only to symmetric equations. More recent authors, for example, Aitken, Banachiewicz, Dwyer, and Crout … have emphasized the use of the method, or variations of it, in connection with non-symmetric problems … Banachiewicz … saw the point … that the basic problem is really one of matrix factorization, or “decomposition” as he called it." It's also referred to as LR decomposition (factors into left and right triangular matrices). | https://en.wikipedia.org/wiki/LDU_decomposition |
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 equal to the number of rows or columns. By contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements (e.g., m × n for an m × n matrix) is sometimes referred to as the sparsity of the matrix. | https://en.wikipedia.org/wiki/Symmetric_sparse_matrix |
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. | https://en.wikipedia.org/wiki/Symmetric_sparse_matrix |
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 data or connections. Large sparse matrices often appear in scientific or engineering applications when solving partial differential equations. | https://en.wikipedia.org/wiki/Symmetric_sparse_matrix |
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. Operations using standard dense-matrix structures and algorithms are slow and inefficient when applied to large sparse matrices as processing and memory are wasted on the zeros. Sparse data is by nature more easily compressed and thus requires significantly less storage. Some very large sparse matrices are infeasible to manipulate using standard dense-matrix algorithms. | https://en.wikipedia.org/wiki/Symmetric_sparse_matrix |
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 method based on s points has order 2s.All Gauss–Legendre methods are A-stable.The Gauss–Legendre method of order two is the implicit midpoint rule. Its Butcher tableau is: The Gauss–Legendre method of order four has Butcher tableau: The Gauss–Legendre method of order six has Butcher tableau: The computational cost of higher-order Gauss–Legendre methods is usually excessive, and thus, they are rarely used. | https://en.wikipedia.org/wiki/Gauss–Legendre_method |
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 error of order one in time. | https://en.wikipedia.org/wiki/Euler_backward_method |
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 multistep method. | https://en.wikipedia.org/wiki/Trapezoidal_rule_(differential_equations) |
In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. | https://en.wikipedia.org/wiki/Truncation_error |
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 hydrodynamics. | https://en.wikipedia.org/wiki/Diffuse_element_method |
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, partial differential equations and thus fluid dynamic problems can be solved. For this, they coined the term diffuse element method (DEM). Advantages over finite element methods are that DEM doesn't rely on a grid, and is more precise in the evaluation of the derivatives of the reconstructed functions. | https://en.wikipedia.org/wiki/Diffuse_element_method |
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 for rectification of the circle, i.e. the calculation of π. It is most useful for accelerating the convergence of a sequence that is converging linearly. | https://en.wikipedia.org/wiki/Aitken_extrapolation |
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. See root-finding algorithm for other algorithms. | https://en.wikipedia.org/wiki/Bairstow's_method |
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 secant method or inverse quadratic interpolation if possible, but it falls back to the more robust bisection method if necessary. | https://en.wikipedia.org/wiki/Brent's_method |
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' method, which performs exponential interpolations instead of quadratic providing a simpler closed formula for the iterations; and the ITP method which is a hybrid between regula-falsi and bisection that achieves optimal worst-case and asymptotic guarantees. | https://en.wikipedia.org/wiki/Brent's_method |
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 behind Broyden's method is to compute the whole Jacobian only at the first iteration and to do rank-one updates at other iterations. In 1979 Gay proved that when Broyden's method is applied to a linear system of size n × n, it terminates in 2 n steps, although like all quasi-Newton methods, it may not converge for nonlinear systems. | https://en.wikipedia.org/wiki/Broyden's_method |
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. | https://en.wikipedia.org/wiki/Chebyshev_nodes |
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 minimizes the number of multiplications and additions required to evaluate an arbitrary polynomial. On a modern processor, instructions that do not depend on each other's results may run in parallel. Horner's method contains a series of multiplications and additions that each depend on the previous instruction and so cannot execute in parallel. Estrin's scheme is one method that attempts to overcome this serialization while still being reasonably close to optimal. | https://en.wikipedia.org/wiki/Estrin's_scheme |
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 ) {\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})} where n is the number of sample points used. The xi are the roots of the physicists' version of the Hermite polynomial Hn(x) (i = 1,2,...,n), and the associated weights wi are given by w i = 2 n − 1 n ! | https://en.wikipedia.org/wiki/Gauss-Hermite_quadrature |
π n 2 2 . {\displaystyle w_{i}={\frac {2^{n-1}n! {\sqrt {\pi }}}{n^{2}^{2}}}.} | https://en.wikipedia.org/wiki/Gauss-Hermite_quadrature |
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})} where n is the number of sample points used, wi are quadrature weights, and xi are the roots of the nth Legendre polynomial.This choice of quadrature weights wi and quadrature nodes xi is the unique choice that allows the quadrature rule to integrate degree 2n − 1 polynomials exactly. Many algorithms have been developed for computing Gauss–Legendre quadrature rules. The Golub–Welsch algorithm presented in 1969 reduces the computation of the nodes and weights to an eigenvalue problem which is solved by the QR algorithm. | https://en.wikipedia.org/wiki/Gauss–Legendre_quadrature |
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, which are accurate to within double-precision machine epsilon for any choice of n ≥ 21. This allows for computation of nodes and weights for values of n exceeding one billion. | https://en.wikipedia.org/wiki/Gauss–Legendre_quadrature |
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 sequence of approximations to the root; their rate of convergence to the root is cubic. Multidimensional versions of this method exist. Halley's method exactly finds the roots of a linear-over-linear Padé approximation to the function, in contrast to Newton's method or the Secant method which approximate the function linearly, or Muller's method which approximates the function quadratically. | https://en.wikipedia.org/wiki/Bailey's_method_(root_finding) |
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 interpolation computes a polynomial of degree less than mn such that the polynomial and its m − 1 first derivatives have the same values at n given points as a given function and its m − 1 first derivatives. Hermite's method of interpolation is closely related to the Newton's interpolation method, in that both are derived from the calculation of divided differences. | https://en.wikipedia.org/wiki/Hermite_interpolation |
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 theorem § Hermite interpolation. | https://en.wikipedia.org/wiki/Hermite_interpolation |
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 to being a "sure-fire" method, meaning that it is almost guaranteed to always converge to some root of the polynomial, no matter what initial guess is chosen. However, for computer computation, more efficient methods are known, with which it is guaranteed to find all roots (see Root-finding algorithm § Roots of polynomials) or all real roots (see Real-root isolation). This method is named in honour of Edmond Laguerre, a French mathematician. | https://en.wikipedia.org/wiki/Laguerre's_method |
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 corresponding set of integration weights are determined by enforcing the exact integration of polynomials (or equivalently, spherical harmonics) up to a given order, leading to a sequence of increasingly dense grids analogous to the one-dimensional Gauss-Legendre scheme. The Lebedev grid is often employed in the numerical evaluation of volume integrals in the spherical coordinate system, where it is combined with a one-dimensional integration scheme for the radial coordinate. Applications of the grid are found in fields such as computational chemistry and neutron transport. | https://en.wikipedia.org/wiki/Lebedev_grid |
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 defined for a real variable x, the function's derivative f′, and an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then x 1 = x 0 − f ( x 0 ) f ′ ( x 0 ) {\displaystyle x_{1}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}} is a better approximation of the root than x0. Geometrically, (x1, 0) is the intersection of the x-axis and the tangent of the graph of f at (x0, f(x0)): that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as x n + 1 = x n − f ( x n ) f ′ ( x n ) {\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}} until a sufficiently precise value is reached. | https://en.wikipedia.org/wiki/Newton_method |
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. | https://en.wikipedia.org/wiki/Newton_method |
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 {\displaystyle h} , we can estimate A ∗ {\displaystyle A^{\ast }} by extrapolating the estimates to h = 0 {\displaystyle h=0} . It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century, though the idea was already known to Christiaan Huygens in his calculation of π. In the words of Birkhoff and Rota, "its usefulness for practical computations can hardly be overestimated. "Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezoid rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations. | https://en.wikipedia.org/wiki/Richardson_extrapolation |
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's method but with similar performance. The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall order of convergence of the method is 2 {\displaystyle {\sqrt {2}}} . If the function is not well-behaved, the root remains bracketed and the length of the bracketing interval at least halves on each iteration, so convergence is guaranteed. | https://en.wikipedia.org/wiki/Ridders'_method |
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 spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. 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 method is named after Werner Romberg (1909–2003), who published the method in 1955. | https://en.wikipedia.org/wiki/Romberg_integration |
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 method does. | https://en.wikipedia.org/wiki/Steffensen's_method |
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 after Harold S. Stone, who proposed it in 1968. The LU decomposition is an excellent general-purpose linear equation solver. | https://en.wikipedia.org/wiki/Stone_method |
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 preconditioned iterative methods, if the preconditioner matrix M is a good approximation of coefficient matrix A then the convergence is faster. | https://en.wikipedia.org/wiki/Stone_method |
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 pentadiagonal system of equations obtained while solving an elliptic partial differential equation in a two-dimensional space by a finite difference method. The LU approximate decomposition was looked in the same pentadiagonal form as the original matrix (three diagonals for L and three diagonals for U) as the best match of the seven possible equations for the five unknowns for each row of the matrix. | https://en.wikipedia.org/wiki/Stone_method |
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 = 1 20 ( x − i ) = ( x − 1 ) ( x − 2 ) ⋯ ( x − 20 ) . {\displaystyle w(x)=\prod _{i=1}^{20}(x-i)=(x-1)(x-2)\cdots (x-20).} Sometimes, the term Wilkinson's polynomial is also used to refer to some other polynomials appearing in Wilkinson's discussion. | https://en.wikipedia.org/wiki/Wilkinson's_polynomial |
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 arguments: B ( u 1 , … , u d ) = B ( π ( u 1 , … , u d ) ) , {\displaystyle {\mathcal {B}}(u_{1},\dots ,u_{d})={\mathcal {B}}{\big (}\pi (u_{1},\dots ,u_{d}){\big )},\,} (where π is any permutation of its arguments).It is affine in each of its arguments: B ( α u + β v , … ) = α B ( u , … ) + β B ( v , … ) , when α + β = 1. {\displaystyle {\mathcal {B}}(\alpha u+\beta v,\dots )=\alpha {\mathcal {B}}(u,\dots )+\beta {\mathcal {B}}(v,\dots ),{\text{ when }}\alpha +\beta =1.\,} It satisfies the diagonal property: B ( u , … , u ) = f ( u ) . {\displaystyle {\mathcal {B}}(u,\dots ,u)=f(u).\,} | https://en.wikipedia.org/wiki/Blossom_(functional) |
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 midpoint method by for n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\dots } Here, h {\displaystyle h} is the step size — a small positive number, t n = t 0 + n h , {\displaystyle t_{n}=t_{0}+nh,} and y n {\displaystyle y_{n}} is the computed approximate value of y ( t n ) . {\displaystyle y(t_{n}).} | https://en.wikipedia.org/wiki/Midpoint_method |
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. The name of the method comes from the fact that in the formula above, the function f {\displaystyle f} giving the slope of the solution is evaluated at t = t n + h / 2 = t n + t n + 1 2 , {\displaystyle t=t_{n}+h/2={\tfrac {t_{n}+t_{n+1}}{2}},} the midpoint between t n {\displaystyle t_{n}} at which the value of y ( t ) {\displaystyle y(t)} is known and t n + 1 {\displaystyle t_{n+1}} at which the value of y ( t ) {\displaystyle y(t)} needs to be found. | https://en.wikipedia.org/wiki/Midpoint_method |
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}} is found where the tangent intersects the vertical line t = t n + 1 {\displaystyle t=t_{n+1}} . | https://en.wikipedia.org/wiki/Midpoint_method |
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 the midpoint (upper, green line segment) would most likely give a more accurate approximation of the curve in that interval. However, this midpoint tangent could not be accurately calculated because we do not know the curve (that is what is to be calculated). | https://en.wikipedia.org/wiki/Midpoint_method |
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 {\displaystyle y_{n}} . This last step is represented by the red chord in the diagram. | https://en.wikipedia.org/wiki/Midpoint_method |
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 order O ( h 2 ) {\displaystyle O\left(h^{2}\right)} . Thus, while more computationally intensive than Euler's method, the midpoint method's error generally decreases faster as h → 0 {\displaystyle h\to 0} . The methods are examples of a class of higher-order methods known as Runge–Kutta methods. | https://en.wikipedia.org/wiki/Midpoint_method |
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 of numeric data specified at given argument values x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} , to obtain a continuous function. The data should consist of the desired function value and derivative at each x k {\displaystyle x_{k}} . (If only the values are provided, the derivatives must be estimated from them.) The Hermite formula is applied to each interval ( x k , x k + 1 ) {\displaystyle (x_{k},x_{k+1})} separately. | https://en.wikipedia.org/wiki/Hermite_curve |
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 names are often used as if they were synonymous. | https://en.wikipedia.org/wiki/Hermite_curve |
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 of a separate parameter t. Cubic polynomial splines are also used extensively in structural analysis applications, such as Euler–Bernoulli beam theory. Cubic polynomial splines have also been applied to mortality analysis and mortality forecasting.Cubic splines can be extended to functions of two or more parameters, in several ways. | https://en.wikipedia.org/wiki/Hermite_curve |
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, especially in computer graphics. Hermite splines are named after Charles Hermite. | https://en.wikipedia.org/wiki/Hermite_curve |
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 methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid. MG methods can be used as solvers as well as preconditioners. | https://en.wikipedia.org/wiki/Multigrid_methods |
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 solve, is similar to the fine grid problem in that it also has short- and long-wavelength errors. It can also be solved by a combination of relaxation and appeal to still coarser grids. | https://en.wikipedia.org/wiki/Multigrid_methods |
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. The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions.Multigrid methods can be applied in combination with any of the common discretization techniques. | https://en.wikipedia.org/wiki/Multigrid_methods |
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. | https://en.wikipedia.org/wiki/Multigrid_methods |
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 equations, like the Lamé equations of elasticity or the Navier-Stokes equations. | https://en.wikipedia.org/wiki/Multigrid_methods |
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. | https://en.wikipedia.org/wiki/Numerical_method |
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 Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights wi for i = 1, ..., n. The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826. The most common domain of integration for such a rule is taken as , so the rule is stated as which is exact for polynomials of degree 2n − 1 or less. | https://en.wikipedia.org/wiki/Gaussian_integration |
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 singularities. | https://en.wikipedia.org/wiki/Gaussian_integration |
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^{2}}}}} (Chebyshev–Gauss) and 1 − x 2 {\textstyle {\sqrt {1-x^{2}}}} . One may also want to integrate over semi-infinite (Gauss-Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature). It can be shown (see Press, et al., or Stoer and Bulirsch) that the quadrature nodes xi are the roots of a polynomial belonging to a class of orthogonal polynomials (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights. | https://en.wikipedia.org/wiki/Gaussian_integration |
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 gradient is first order. However under certain conditions it's possible to recover the gradient at certain locations within each element to second order. | https://en.wikipedia.org/wiki/Superconvergence |
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 quantified grids as in the Cartesian plane which constitute the computational grid, or 'mesh'. Many problems in numerical analysis, however, do not require a uniform precision in the numerical grids used for graph plotting or computational simulation, and would be better suited if specific areas of graphs which needed precision could be refined in quantification only in the regions requiring the added precision. | https://en.wikipedia.org/wiki/Mesh_refinement |
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 levels of precision and resolution. This dynamic technique of adapting computation precision to specific requirements has been accredited to Marsha Berger, Joseph Oliger, and Phillip Colella who developed an algorithm for dynamic gridding called local adaptive mesh refinement. The use of AMR has since then proved of broad use and has been used in studying turbulence problems in hydrodynamics as well as in the study of large scale structures in astrophysics as in the Bolshoi Cosmological Simulation. | https://en.wikipedia.org/wiki/Mesh_refinement |
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