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We discuss the direct use of cubic-matrix splines to obtain continuous approximations to the unique solution of matrix models of the type $Y(x) = f(x,Y(x))$. For numerical illustration, an estimation of the approximation error, an algorithm for its implementation, and an example are given.
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This paper presents the non-linear generalization of a previous work on matrix differential models. It focusses on the construction of approximate solutions of first-order matrix differential equations Y(x)=f(x,Y(x)) using matrix-cubic splines. An estimation of the approximation error, an algorithm for its implementation and illustrative examples for Sylvester and Riccati matrix differential equations are given.
Second order spiral splines are $C^2$ unit-speed planar curves that can be used to interpolate a list $Y$ of $n+1$ points in $R ^2$ at times specified in some list $T$, where $ngeq 2$. Asymptotic methods are used to develop a fast algorithm, based on a pair of tridiagonal linear systems and standard software. The algorithm constructs a second order spiral spline interpolant for data that is convex and sufficiently finely sampled.
We study a family of models for an $N_1 times N_2$ matrix worth of Ising spins $S_{aB}$. In the large $N_i$ limit we show that the spins soften, so that the partition function is described by a bosonic matrix integral with a single `spherical constraint. In this way we generalize the results of [1] to a wide class of Ising Hamiltonians with $O(N_1,mathbb{Z})times O(N_2,mathbb{Z})$ symmetry. The models can undergo topological large $N$ phase transitions in which the thermal expectation value of the distribution of singular values of the matrix $S_{aB}$ becomes disconnected. This topological transition competes with low temperature glassy and magnetically ordered phases.
We present a matrix-product state (MPS)-based quadratically convergent density-matrix renormalization group self-consistent-field (DMRG-SCF) approach. Following a proposal by Werner and Knowles (JCP 82, 5053, (1985)), our DMRG-SCF algorithm is based on a direct minimization of an energy expression which is correct to second-order with respect to changes in the molecular orbital basis. We exploit a simultaneous optimization of the MPS wave function and molecular orbitals in order to achieve quadratic convergence. In contrast to previously reported (augmented Hessian) Newton-Raphson and super-configuration-interaction algorithms for DMRG-SCF, energy convergence beyond a quadratic scaling is possible in our ansatz. Discarding the set of redundant active-active orbital rotations, the DMRG-SCF energy converges typically within two to four cycles of the self-consistent procedure
In this paper, we apply the hierarchical modeling technique and study some numerical linear algebra problems arising from the Brownian dynamics simulations of biomolecular systems where molecules are modeled as ensembles of rigid bodies. Given a rigid body $p$ consisting of $n$ beads, the $6 times 3n$ transformation matrix $Z$ that maps the force on each bead to $p$s translational and rotational forces (a $6times 1$ vector), and $V$ the row space of $Z$, we show how to explicitly construct the $(3n-6) times 3n$ matrix $tilde{Q}$ consisting of $(3n-6)$ orthonormal basis vectors of $V^{perp}$ (orthogonal complement of $V$) using only $mathcal{O}(n log n)$ operations and storage. For applications where only the matrix-vector multiplications $tilde{Q}{bf v}$ and $tilde{Q}^T {bf v}$ are needed, we introduce asymptotically optimal $mathcal{O}(n)$ hierarchical algorithms without explicitly forming $tilde{Q}$. Preliminary numerical results are presented to demonstrate the performance and accuracy of the numerical algorithms.
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