Cholesky Decomposition
Cholesky decomposition is a method used to factorize a positive definite matrix into the product of a lower triangular matrix and its transpose. This decomposition is often used in numerical simulations and optimization problems, as it simplifies the process of solving linear equations and computing determinants. It is particularly efficient for solving systems of linear equations with symmetric and positive definite matrices.
4 Key excerpts on "Cholesky Decomposition"
eBook - ePubDigital Signal Processing 101
Everything You Need to Know to Get Started
- Michael Parker(Author)
- 2017(Publication Date)
- Newnes(Publisher)
...Because the vectors are very close to being dependent, this will be a matrix that will require a high degree of accuracy in determining the inverse. 13.2. Cholesky Decomposition The Cholesky Decomposition is used in the special case when A is a square, conjugate symmetric matrix. This makes the problem a lot simpler. Recall that a conjugate symmetric matrix is one where the element A jk equals the element A kj conjugated. This is shown as A jk = A kj ∗. If A jk is a real value (not complex), then A jk = A kj. Figure 13.5 Problem to solve: find x. Note: A conjugate is then the complex value with the sign on the imaginary component reversed. For example, the conjugate of 5 + j12 = 5 − j12. And by definition, the diagonal elements must be real (not complex), since A jj = A jj ∗, or more simply, only a real number can be equal to its conjugate. The Cholesky Decomposition floating point math operations per [N × N] matrix is generally estimated as: FLOPS = 4 N 3 / 3 However, the actual computational rate and efficiency depend on implementation details and the architecture details of the computing device used (CPU, FPGA, GPU, DSP…). The problem statement is A · x = b, where A is an [N × N] complex symmetric matrix, x is an unknown complex [N × 1] vector, and b is a known complex [N × 1] vector. The solution is x = A − 1 · b, which requires the inversion of matrix A (Fig. 13.5). As directly computing the inverse of a large matrix is difficult, there is an alternate technique using a transform to make this problem easier and require less computations. The Cholesky Decomposition maps matrix A into the product of A = L · L H where L is the lower triangular matrix and L H is the transposed, complex conjugate or Hermitian, and therefore of upper triangular form (Fig. 13.6). This is true because of the special case of A being a square, conjugate symmetric matrix. The solution to find L requires square root and inverse square root operators...
eBook - ePub- Andreas Binder, Michael Aichinger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
...The matrices U and L are stored in place of the matrix A. The permutation matrix P is dynamically maintained as an array p, where p [ i ] = j indicates that the i -th row of P contains a 1 in column j. 8.1.4 Cholesky Decomposition The Cholesky Decomposition is a fast method for the LU decomposition of symmetric and positive definite coefficient matrices A, where the upper triangular matrix is the transpose of the lower triangular matrix, (8.6) The Cholesky algorithm to calculate the decomposition matrix L is given in Algorithm 3. Algorithm 3 Pseudocode of a basic version of the Cholesky algorithm. The matrix L is computed and stored in place of the lower triangle of matrix A. Since the square-root operation is time consuming standard implementations perform an A = L D L T decomposition where diag (L)=1 instead. 8.2 ITERATIVE SOLVERS For large and/or sparse systems of equations of the form (8.2), direct solvers are time-consuming and might, for a very large number of unknowns n, not be applicable at all. Iterative solvers are typically used to overcome this problem: starting from an initial vector x 0, an iterative method generates a sequence of consecutive vectors x 0 → x 1 → x 2 → · that converges towards the solution. The computational effort for a single iteration step x i → x i +1 is comparable to the effort of multiplying the vector x by the matrix A. For sparse matrices, in particular, this is possible at comparatively little cost. The classical iterative methods – Jacobi, Gauss Seidel and Successive Overrelaxation – are simple to derive, implement, and analyze, but convergence is usually slow, only guaranteed for a limited class of matrices and often depends on the choice of additional parameters that are hard to estimate. Krylov subspace methods, on the other hand, have no additional parameters that influence convergence. The iterative process is stopped after a predefined error estimate 6 is smaller than a chosen (for instance =10 -8)...
eBook - ePubImage Analysis, Classification and Change Detection in Remote Sensing
With Algorithms for Python, Fourth Edition
- Morton John Canty(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
...A Mathematical Tools A.1 Cholesky Decomposition Cholesky Decomposition is used in some of the routines in this book to solve generalized eigenvalue problems associated with the maximum autocorrelation factor (MAF) and maximum noise fraction (MNF) transformations as well as with canonical correlation analysis. We sketch its justification in the following. THEOREM A.1 If the p × p matrix A is symmetric positive definite and if the p × q matrix B, where q ≤ p, has rank q, then B ⊤ AB is positive definite and symmetric. Proof. Choose any q -dimensional vector y ≠ 0 and let x = By. We can write this as x = y 1 b 1 + … + y q b q, where b i is the ith column of B. Since B has rank q, we conclude that x ≠ 0 as well, for otherwise the column vectors would be linearly dependent. But Y ⊤ (B ⊤ A B) y = (B y) ⊤ A (B y) = x ⊤ A x > 0, since A is positive definite. So B ⊤ AB is positive definite (and clearly symmetric). □ A square matrix A is diagonal if a ij = 0 for i ≠ j.It is lower triangular if a ij = 0 for i < j and upper triangular if a ij = 0 for i > j. The product of two lower(upper) triangular matrices is lower(upper) triangular. The inverse of a lower(upper) triangular matrix is lower(upper) triangular. If A is diagonal and positive definite, then all of its diagonal elements are positive. Otherwise if, say, a ij ≤ 0 then, for x = (0…, 1,… 0) ⊤ with the 1 at the ith position, x ⊤ A x ≤ 0 contradicting the fact that A is positive definite. Now we state without proof Theorem A.2. THEOREM A.2 If A is nonsingular, then there exists a nonsingular lower triangular matrix F such that FA is nonsingular upper triangular. The proof is straightforward, but somewhat lengthy; see e.g., Anderson (2003), Appendix A. It follows directly that, if A is symmetric and positive definite, there exists a lower triangular matrix F such that FAF ⊤ is diagonal and positive definite. That is, from Theorem A.2, FA is upper triangular and nonsingular...
eBook - ePubAdvanced Kalman Filtering, Least-Squares and Modeling
A Practical Handbook
- Bruce P. Gibbs(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...That is, (5.2-7) Similar factorizations can also be defined using UU T where U is upper triangular, or as UDU T where U is unit upper triangular (1’s on the diagonal) and D is a diagonal matrix of positive values. The UDU T factorization has the advantage that square roots are not required in the computation. However, we work with U T U because that is the usual definition of Cholesky factors. The factorization can be understood using a three-state example. Example 5.3: Cholesky Factorization Consider the Cholesky factorization of a 3 × 3 positive definite symmetric matrix A = U T U : (5.2-8) It is evident that the U elements can be computed starting from the upper left element: (5.2-9) The general algorithm for a n × m matrix is (5.2-10) Note that the elements of U and the upper triangular portion of A can occupy the same storage locations because A ij is never referenced after setting U ij. The factorization can also be implemented in column order or as vector outer products rather than the row order indicated here (Björck 1996, Section 2.2.2). The stability of the Cholesky factorization is due to the fact that the elements of U must be less than or equal to the diagonals of A. Note that Hence numerical errors do not grow and the pivoting used with other matrix inversion methods is not necessary. However, Björck (1996, Section 2.7.4) shows that pivoting can be used to solve unobservable (singular) least-squares problems when H T R −1 H is only positive semi-definite. In coding Cholesky and other algorithms that work with symmetric or upper triangular matrices, it is customary to store these arrays as upper-triangular-by-column vectors such that Use of vectors for array storage cuts the required memory approximately in half, which is beneficial when the dimension is large. For this nomenclature, the Cholesky algorithm becomes (5.2-11) where floor(x) is the largest integer less than or equal to x...
