The CoinUtils project is a collection of open-source utilities developed and used by a variety of other projects in the COIN-OR repository. The project includes classes for storing and manipulating sparse matrices and vectors, performing matrix Factorization, parsing input files in standard formats, building representations of mathematical programs, comparing floating point numbers with a tolerance, performing simple presolve operations, and warm starting algorithms for mathematical programs, among others.
The "GEE! It s Simple" package illustrates Gaussian elimination with partial pivoting, which produces a Factorization of P*A into the product L*U where P is a permutation matrix, and L and U are lower and upper triangular, respectively.
The functions in this package are accurate, but they are far slower than their MATLAB equivalents (x=A\b, [L,U,p]=lu(A), and so on). They are presented here merely to illustrate and educate. "Real" production code should use backslash and lu, not this package.
The "GEE! It s Simple" package illustrates Gaussian elimination with partial pivoting, which produces a Factorization of P*A into the product L*U where P is a permutation matrix, and L and U are lower and upper triangular, respectively.
The functions in this package are accurate, but they are far slower than their MATLAB equivalents (x=A\b, [L,U,p]=lu(A), and so on). They are presented here merely to illustrate and educate. "Real" production code should use backslash and lu, not this package.
密碼學界牛人Victor Shoup用C++編寫數論類庫。
NTL is a high-performance, portable C++ library providing data structures and algorithms for arbitrary length integers for vectors, matrices, and polynomials over the integers and over finite fields and for arbitrary precision floating point arithmetic.
NTL provides high quality implementations of state-of-the-art algorithms for:
* arbitrary length integer arithmetic and arbitrary precision floating point arithmetic
* polynomial arithmetic over the integers and finite fields including basic arithmetic, polynomial Factorization, irreducibility testing, computation of minimal polynomials, traces, norms, and more
* lattice basis reduction, including very robust and fast implementations of Schnorr-Euchner, block Korkin-Zolotarev reduction, and the new Schnorr-Horner pruning heuristic for block Korkin-Zolotarev
* basic linear algebra over the integers, finite fields, and arbitrary precision floating point numbers.
SuperLU is a general purpose library for the direct solution of large, sparse, nonsymmetric systems of linear equations on high performance machines. The library is written in C and is callable from either C or Fortran. The library routines will perform an LU decomposition with partial pivoting and triangular system solves through forward and back substitution. The LU Factorization routines can handle non-square matrices but the triangular solves are performed only for square matrices. The matrix columns may be preordered (before Factorization) either through library or user supplied routines. This preordering for sparsity is completely separate from the Factorization. Working precision iterative refinement subroutines are provided for improved backward stability. Routines are also provided to equilibrate the system, estimate the condition number, calculate the relative backward error, and estimate error bounds for the refined solutions.
Recovering 3-D structure from motion in noisy 2-D images is a problem addressed by many vision system researchers. By consistently tracking feature points of interest across multiple images using a methodology first described by Lucas-Kanade, a 3-D shape of the scene can be reconstructed using these features points using the Factorization method developed by Tomasi-Kanade.
RSA ( Rivest Shamir Adleman )is crypthograph system that used to give a secret information and digital signature . Its security based on Integer Factorization Problem (IFP). RSA uses an asymetric key. RSA was created by Rivest, Shamir, and Adleman in 1977. Every user have a pair of key, public key and private key. Public key (e) . You may choose any number for e with these requirements, 1< e <Æ (n), where Æ (n)= (p-1) (q-1) ( p and q are first-rate), gcd (e,Æ (n))=1 (gcd= greatest common divisor). Private key (d). d=(1/e) mod(Æ (n)) Encyption (C) . C=Mª mod(n), a = e (public key), n=pq Descryption (D) . D=C° mod(n), o = d (private key
