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1. CAINPB - Factorization
- www.mupad.de
- Factorization.
- Factorization Challenges .
- Deterministic factorization of polynomials over special finite fields .
- Arithmetic and factorization of polynomials over F2 .
- Factorization of polynomials modulo small prime powers .
2. Cholesky Factorization
- www.netlib.org
- Next: Performance Results Up: Factorization Routines Previous: QR Factorization .
- Cholesky Factorization.
- Cholesky factorization factors an , symmetric, positive-definite matrix into the product of a lower triangular matrix and its transpose, i. ...
- The block-partitioned form of Cholesky factorization may be inferred inductively as follows. ...
- A snapshot of the block Cholesky factorization algorithm in Figure 5 shows how the column panel ( and ) is computed and how the trailing submatrix is updated. The factorization can be done by recursively applying the steps outlined above to the matrix. ...
- Figure 5: A snapshot of block Cholesky factorization.
- DPOTF2: Compute the Cholesky factorization of the diagonal block. ...
- PDPOTF2: The process , which has the diagonal block , performs the Cholesky factorization of. ...
- Next: Performance Results Up: Factorization Routines Previous: QR Factorization .
3. Computing the prime-power factorization of n!
- www.numbertheory.org
- Computing the prime-power factorization of n!.
4. Parallel Cholesky Factorization of a Block Tridiagonal Matrix
- csdl.computer.org
- 327 Parallel Cholesky Factorization of a Block Tridiagonal Matrix .
- In this paper we discuss the parallel implementation of the Cholesky factorization of a positive definite symmetric matrix when that matrix is block tridiagonal. While parallel implementations for this problem, and closely related problems like the factorization of banded matrices, have been previously reported in the literature, those implementations dealt with the special cases where the block size (bandwidth) was either very large (wide) or very small (narrow). ... Indeed, factorization of a matrix with block size b = 1000 and a total dimension of more than 500,000 takes about 3. ...
5. Gauss Transformation and LU-Factorization
- www.cs.ut.ee
- Gauss Transformation and LU-Factorization.
- The Gaussian elimination considered in the main course of linear algebra for solution of systems of linear equations is applicable also to the LU-factorization. ...
- If 1:k,1: for (k=1:n-1),then has an LU factorization. If the LU factorization exists and A is regular, then the LU factorization is unique and .
- The element akk(k-1) is the k-th pivot of A and 1:k,1: Hence, if A(1:k,1:k) is regular, then and A has an LU factorization. ...
- * Find the LU factorization of the matrix .
- * Find the LU factorization of .
- * By using the LU factorization, solve the system , where .
- 4 there has been found the LU factorization for A: .
- * Find the LU factorization of A if .
- * By using the LU factorization solve the system , where .
- then A has an LU factorization. ...
- As it is known from the main course of algebra, the direct application of the Gaussian elimination, therefore also the direct realization of the LU factorization, fails if at least one of the principal minors is singular. ...
- If and then there exists a permutation matrix such that all the principal minors of PA are nonzero, and consequently, there exists the LU factorization .
- Find for a certain permutation matrix the LU factorization of PA. ...
- * Find for a certain permutation matrix P the LU factorization of PA if .
6. Prime Factorization
- www.gomath.com
- Prime Factorization.
- Prime factorization(or prime decomposition) is the expression of a positive integer as a product of prime numbers. ... For instance, the prime factorization of 99 is 3 × 3 × 11. ... Prime factorization is usually applied to find the greatest common factor(GCF) or the least common multiple(LCM) of two or more integers.
- Example: Find the prime factorization of 210.
- Example: Find the prime factorization of 364.
7. Factorization
- www.vex.net
- Factorization.
- We refer to the latter as a prime factorization if the context has not made the intention clear.
- We feel that 2*5 should be the ``only'' prime factorization of 10, and things like 1*2*5 and (-1)*(-1)*2*5 should be merely insignificant, superfluous variations. By excluding units from primes, we free ourselves from endless strings of reciprocals, and highlight the more significant and interesting aspect of factorization. ...
- Having settled the issues of 0 and units, it can be proved that numbers can indeed be factorized into primes, and the prime factorization of a number is fairly unique. ...
- Theorem: In the natural numbers, each non-zero, non-unit number can be factorized into a product of primes; furthermore, the factorization is unique up to ordering.
- Furthermore, the factorization is unique up to ordering and associates. ...
- Or, appeal to results in abstract algebra - the set of integers is a Euclidean domain, and therefore a principle ideal domain, and therefore a unique factorization domain.
8. Lab for Doolittle's Method for the LU Factorization
- www.ecs.fullerton.edu
- Lab for Doolittle's Method for the LU Factorization.
- The modern way to solve a linear system AX = B is to first find the A = LU factorization. ...
- The Doolittle factorization uses 1's on the diagonal of L.
- Derivation of the Doolittle Factorization.
- Derivation and example of the Crout Factorization.
- Derivation and example of the PreCholesky Factorization.
- The Cholesky factorization subroutine:.
- Find the A = LU factorization for the following matrix using the Doolittle method.
- Solve the linear system AX = B by finding the A = LU factorization with the Doolittle method.
- Find the A = LU factorization for the following matrix using the Cholesky subroutine.
9. prime factor
- www.gomath.com
- Prime Factorization To prime factor a number, begin dividing by the smallest possible prime and continue until the quotient is a prime number.
10. Solving the Indefinite Least Squares Problem by Hyperbolic QR Factorization
- epubs.siam.org
- Solving the Indefinite Least Squares Problem by Hyperbolic QR Factorization.
- Solving the Indefinite Least Squares Problem by Hyperbolic QR Factorization: SIAM Journal on Matrix Analysis and Applications Vol. ... We describe and analyze a method for solving the ILS problem based on hyperbolic QR factorization. ...
- indefinite least squares problem, downdating, hyperbolic rotation, hyperbolic QR factorization, rounding error analysis, forward stability, perturbation theory, condition number .
11. PA = LU Factorization with Pivoting
- mathews.ecs.fullerton.edu
- PA = LU Factorization with Pivoting.
- Definition (LU-Factorization). The nonsingular matrix A has an LU-factorization if it can be expressed as the product of a lower-triangular matrix L and an upper triangular matrix U: .
- It turns out that this factorization (when it exists) is not unique. If L has 1's on it's diagonal, then it is called a Doolittle factorization. If U has 1's on its diagonal, then it is called a Crout factorization. ... In this module we will develop an algorithm that produces a Doolittle factorization.
- Theorem (LU Factorization with NO pivoting). If row interchanges are not needed to solve the linear system AX = B, then A has the LU factorization (illustrated with 4×4 matrices). ...
- A sufficient condition for the factorization to exist is that all principal minors of A are nonsingular.
- Proof LU Factorization LU Factorization .
- Computer Programs LU Factorization LU Factorization .
- Theorem (A = LU; Factorization with NO Pivoting). Assume that A has a Doolittle, Crout or Cholesky factorization. ...
- Proof LU Factorization LU Factorization .
- If row interchanges are permitted then a factorization of a "permuted matrix" will be obtained. ...
12. LUMOD: Updating a dense square factorization L*C = U
- www.stanford.edu
- LUMOD: Updating a dense square factorization L*C = U .
- CONTENTS: Fortran software for updating a dense square factorization L*C = U when rows and columns of C are added, deleted or replaced. ...
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