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In the following Diophantine equations, w, x, y, and z are the unknowns and the other letters are given constants: a x + b y = c {\displaystyle ax+by=c} This is a linear Diophantine equation or Bézout's identity. w 3 + x 3 = y 3 + z 3 {\displaystyle w^ {3}+x^ {3}=y^ {3}+z^ {3}} The smallest nontrivial solution in positive integers is 123 + 13 ...
Finding one root. The most widely used method for computing a root is Newton's method, which consists of the iterations of the computation of. + = ′ {\displaystyle x_ {n+1}=x_ {n}- {\frac {f (x_ {n})} {f' (x_ {n})}},} by starting from a well-chosen value. If f is a polynomial, the computation is faster when using Horner's method or evaluation ...
Moore–Penrose inverse. In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. [1] It was independently described by E. H. Moore in 1920, [2] Arne Bjerhammar in 1951, [3] and Roger Penrose in 1955. [4]
A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if M is the product of the moduli, there is, in an interval of length M, exactly one integer having any given set of modular values.
Bernstein's theorem (polynomials) In mathematics, Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei Bernstein while he was working on approximation theory.
or the discrete time algebraic Riccati equation (DARE): {\displaystyle P=A^ {T}PA- (A^ {T}PB) (R+B^ {T}PB)^ {-1} (B^ {T}PA)+Q.\,} P is the unknown n by n symmetric matrix and A, B, Q, R are known real coefficient matrices, with Q and R symmetric. Though generally this equation can have many solutions, it is usually specified that we want to ...
The solutions of the system are in one-to-one correspondence with the roots of h and the multiplicity of each root of h equals the multiplicity of the corresponding solution. The solutions of the system are obtained by substituting the roots of h in the other equations. If h does not have any multiple root then g 0 is the derivative of h.
Kunerth's algorithm. Kunerth's algorithm is an algorithm for computing the modular square root of a given number. [1][2] The algorithm does not require the factorization of the modulus, and relies on modular operations that is often easy when the given number is prime.
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