Differentiating the Method of Conjugate Gradients

Differentiating the Method of Conjugate Gradients

The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations Ax = b, where A ∈ R is symmetric positive definite. Let xk denote the k–th iterate of CG. This is a nonlinear differentiable function of b. In this paper we obtain expressions for Jk, the Jacobian matrix of xk with respect to b. We use these expressions to obtain bounds on ∥Jk∥2...

Linearizing the Method of Conjugate Gradients

The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations Ax = b, where A ∈ R is symmetric positive definite. Let xk denote the k–th iterate of CG. In this paper we obtain an expression for Jk, the Jacobian matrix of xk with respect to b. We use this expression to obtain computable bounds on the spectral norm condition number of xk, an...

Parallelizing the Method of Conjugate Gradients for Shared Memory Architectures

Solving Partial Differential Equations (PDEs) is an important problem in many fields of science and engineering. For most real-world problems modeled by PDEs, we can only approximate the solution using numerical methods. Many of these numerical methods result in very large systems of linear equations. A common way of solving these systems is to use an iterative solver such as the method of conj...

The Method of Conjugate Gradients in Finite Element Applications

buckling of viscoelastic composite plates using the finite strip method

در سال های اخیر، تقاضای استفاده از تئوری خطی ویسکوالاستیسیته بیشتر شده است. با افزایش استفاده از کامپوزیت های پیشرفته در صنایع هوایی و همچنین استفاده روزافزون از مواد پلیمری، اهمیت روش های دقیق طراحی و تحلیل چنین ساختارهایی بیشتر شده است. این مواد جدید از خودشان رفتارهای مکانیکی ارائه می دهند که با تئوری های الاستیسیته و ویسکوزیته، نمی توان آن ها را توصیف کرد. این مواد، خواص ویسکوالاستیک دارند....