In this paper, a fast and accurate algorithm for solving ill-conditioned linear systems is proposed. The proposed algorithm is based on a preconditioned technique using a result of an LU factorization, which requires less computational cost than a previous method using an approximate inverse. The algorithm can provide accurate numerical solutions for ill-conditioned problems beyond the limit of the working precision. Results of numerical experiments are presented for confirming the effectiveness of the proposed algorithm.
The alternating least squares (ALS) method is frequently used for the computation of the canonical polyadic decomposition (CPD) of tensors. It generally gives accurate solutions, but demands much time. An alternative is the alternating slice-wise diagonalization (ASD) method, which provides an efficient way for third-order tensors, utilizing compression based on matrix singular value decomposition. In this paper, we propose a new simple algorithm, Reduced ALS, which employs the same compression procedure as ASD, but applies it more directly to ALS. Numerical experiments show that Reduced ALS runs as fast as ASD, avoiding instability ASD sometimes exhibits.
We introduce an extended oqds algorithm for singular values of lower tridiagonal matrix which is a condensed form of inputted full matrix. Reduction to the lower tridiagonal matrix is able to be performed using cache-efficient block Householder method based on BLAS 2.5 routines. In this letter, we describe the implementation details of the latter algorithm such as the shift strategy and criteria for deflation and splitting. The effectiveness of our approach is demonstrated by numerical experiments.
The joint singular value decomposition of multiple rectangular matrices is formulated as a Riemannian optimization problem on the product of two Stiefel manifolds. In this paper, the geometry of the objective function and the Riemannian manifold for this problem are studied to develop a Riemannian trust-region algorithm. The proposed algorithm globally and locally quadratically converges, and our numerical experiments demonstrate that it performs much better than the steepest descent method.
We consider application of the discrete gradient method for the Webster equation, which models sound waves in tubes. Typically Hamilton equations are described by the use of gradients of the Hamiltonian and it is indispensable to introduce an inner product to define a gradient. We first apply the discrete gradient method to design an energy-preserving method by using a weighted inner product. Comparing with another scheme that is derived by a standard inner product, we show that the discrete gradient method has a geometric invariance, which implies that the method reflects the symplectic geometric aspect of mechanics.
A family of curves with monotone curvature called the log-aesthetic curves (LAC) has been investigated in the field of industrial shape design. LAC has a radius of curvature in proportional to the power of a linear function of an arc-length parameter. In the present article we show that LAC can be naturally formulated in the similarity geometry and the Riccati equations satisfied by the similarity curvatures of LAC can be derived. Moreover, we clarify that certain generalizations of LAC (GLAC) can also be described in a uniform way.
We introduce a model to evaluate credit value adjustment (CVA) for large derivative portfolios considering general wrong-way risk. First, we showed the empirical evidence that suggests the existence of wrong-way risks for interest rate swaps and foreign exchange forwards. Next, we formulate a model to calculate CVA considering the correlations between the probability of default of the counterparty, the interest rate curve, and the foreign exchange rate. Finally, we show numerical examples to estimate the effect of wrong-way risk, which can be related to the CVA stress testing.