We consider symmetric positive definite systems of linear equations with multiple right-hand sides. The seed conjugate gradient method solves one right-hand side with the conjugate gradient method and simultaneously projects over the Krylov subspace thus developed for the other right-hand sides. Then the next system is solved and used to seed the remaining ones. Rounding error in the conjugate ...

abstract Lanczos's major contributions to the numerical solution of linear equations are contained in two papers: \An Iteration Method for the Solution of the Eigenvalue Problem of Linear Diierential and Integral Operators" and \Solutions of Linear Equations by Minimized Iterations ," the second of which contains the method of conjugate gradients. In this note we retrace Lanczos's journey from ...

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial discretization scheme for computing the ground states (GS) of rotating Bose-Einstein condensates (BEC), modeled by the Gross-Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoi...

Variational inference is computationally challenging in models that contain both conjugate and non-conjugate terms. Methods specifically designed for conjugate models, even though computationally efficient, find it difficult to deal with non-conjugate terms. On the other hand, stochastic-gradient methods can handle the nonconjugate terms but they usually ignore the conjugate structure of the mo...

In this paper we shall establish the counterpart of Szmidt, Urbanowicz and Zagier’s formula in the sense of the Hecker correspondence. The motivation is the derivation of the values of the Riemann zeta-function at positive even integral arguments from the partial fraction expansion for the hyperbolic cotangent function (or the cotangent function). Since the last is equivalent to the functional ...

The generalization of p-frame in Banach spaces is considered in this paper. The concepts of an $tilde{X}$-frame and a system conjugate to $tilde{X}$-frame were introduced. Analogues of the results on the existence of conjugate system were obtained. The stability of $tilde{X}$-frame having a conjugate system is studied.

For activity quantification based on planar scintillation camera measurements, photon attenuation is an important factor that needs to be corrected for in a patientand organ-specific manner. One possibility for obtaining attenuation correction maps is to use X-ray CT scout images. Since the intensity of scout images is in relative numbers, their image values need to be multiplied by a factor to...

In this paper, a new nonlinear conjugate gradient method is proposed for large-scale unconstrained optimization. The sufficient descent property holds without any line searches. We use some steplength technique which ensures the Zoutendijk condition to be held, this method is proved to be globally convergent. Finally, we improve it, and do further analysis.

We study the problem of complex-valued independent subspace analysis (ISA). We introduce complex flag manifolds to tackle this problem, and, based on Riemannian geometry, propose the natural conjugate gradient method on this class of manifolds. Numerical experiments demonstrate that the natural conjugate gradient method yields better convergence compared to the natural gradient geodesic search ...

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial discretization scheme for computing the ground states (GS) of rotating Bose-Einstein condensates (BEC), modeled by the Gross-Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoi...