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PDEs
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
Since the Crank–Nicolson method is implicit, we need to solve a system of equations. However, the matrix of the system is TST and its solution by sparse Cholesky factorization can be done in O(M) operations. Recall that generally the Cholesky decomposition or Cholesky triangle is a decomposition of a real symmetric, positive-definite matrix A into the unique product of a lower triangular matrix L and its transpose, where L has strictly positive diagonal entries. Thus we want to find A = LLT. The algorithm starts with i = 1 and A(1) ≔ A. At step i, the matrix A(i) has the form. A(i)=(Ii−1000biibiT0biB(i)),
Regularization Techniques for MR Image Reconstruction
Published in Joseph Suresh Paul, Raji Susan Mathew, Regularized Image Reconstruction in Parallel MRI with MATLAB®, 2019
Joseph Suresh Paul, Raji Susan Mathew
A matrix is symmetric if AT=A, and positive definite if xTAx>0. If the matrix is symmetric, the eigenvalues of A are real and eigenvectors associated with distinct eigenvalues are orthogonal. The matrix A is positive definite (or positive semi-definite) if and only if all eigenvalues of A are positive (or non-negative). The operator arising from the quadratic functional Q(x)=12xTAx−bTx is always symmetric. This symmetric matrix may be positive definite (PD), negative definite, singular point (positive-indefinite) and saddle point (indefinite). A symmetric positive or negative definite matrix operator of a quadratic function guarantees a strictly convex function, which in the first place shows that it is an increasing or decreasing function and in turn gives rise to a unique minimum or maximum. The CG algorithm does not converge directly in the case of negative definite matrices. This is the reason that the CG requires a symmetric positive definite (SPD) matrix.
Flexibility Matrix and Stiffness Matrix
Published in A.I. Rusakov, Fundamentals of Structural Mechanics, Dynamics, and Stability, 2020
is referred to as the quadratic form of matrix A with argument x. In equality (17I.2), the quadratic form is represented first in vector notation and next in scalar form. Superscript “T” denotes transposing of the matrix (in the given case of column vector). Matrix A is referred to as positive semidefinite (positive definite) if its quadratic form (17I.2) is nonnegative (positive) for arbitrary vector x. Positive semidefiniteness (definiteness) of A is briefly written by inequality: A ≥ 0 (A > 0).
A dual active-set proximal Newton algorithm for sparse approximation of correlation matrices
Published in Optimization Methods and Software, 2022
Xiao Liu, Chungen Shen, Li Wang
In the following, we take a look at how to solve the EQP subproblem (20) to get . It is straightforward that the KKT system (21) can be rewritten as Obviously, the row vectors of and are linearly independent, and under the assumption that the generalized Hessian is positive definite on the null space of , the KKT matrix in (23) is non-singular. Therefore, there is a unique solution to the KKT system (23) (see [28, Lemma 16.1]). However, it is time-consuming to solve the linear system exactly. Practically, as mentioned in above subsection, we solve the subproblem (20) approximately by applying the projected conjugate gradient (PCG) method [28, Algorithm 16.3] to the equivalent KKT system (23). Due to the special structure of and , the system (23) can be reduced to a smaller linear system with a positive definite coefficient matrix, and then the standard conjugate gradient method [28, Algorithm 5.2] can also be invoked to solve this smaller linear system.
Comprehensive adaptive modelling of 1-D unsteady pipe network hydraulics
Published in Journal of Hydraulic Research, 2021
Johnathan D. Nault, Bryan W. Karney
Equation (18) also features a saddle point structure similar to Eq. (15). Its solution is thus: where (m2 s−1). To compute , an system of equations must be solved. Given that is positive definite, methods such as Cholesky decomposition or the iterative conjugate gradient method can be applied. For the latter, iterations can be terminated prematurely once the residual to Eq. (19) is sufficiently small, yielding an inexact Newton method (Dembo et al., 1982).
Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach
Published in International Journal of Systems Science, 2018
Alireza Nasiri, Sing Kiong Nguang, Akshya Swain, Dhafer Almakhles
Based on Cholesky factorisation, for any given positive definite matrix, P, we can always decompose P into where, is another positive definite matrix and is lower (upper) triangular matrix. It is worth mentioning that expressing P as does not affect the stability results in linear systems.