Mobile platforms have matured to a point where they can provide the infrastructure needed by sophisticated optimization codes. This opens both the possibility to envisage new market interest for distributed application codes and the opportunity to intensify research on algorithms that require limited computational power, such as that which is oered by mobile platforms. This paper reports on some exploratory experiences in this area, with reference to requests and opportunities for real world applications and to opening computational validation using matheuristics codes. A specic use case requiring to address the feasibility version of the generalized assignment problem is discussed.

BACOLI is a Fortran software package for solving one-dimensional parabolic partial differential equations with separated boundary conditions by B-spline adaptive collocation methods. A distinguishing feature of BACOLI is its ability to estimate and control error as well as correspondingly adapt meshes in both space and time. However, many models of scientific interest can be formulated as multi-scale parabolic PDE systems, i.e., models that couple a system of parabolic partial differential equations describing dynamics on a global scale with a system of spatially uncoupled partial differential equations describing dynamics on a local scale. This article describes the Fortran software eBACOLI, the extension of \BACOLI\ to solve such multi-scale models. The performance of the extended code is demonstrated to be statistically equivalent to the original for purely parabolic PDE systems. Results from \eBACOLI\ are given for various multi-scale models from the extended problem class considered.Temporal work-precision behaviour for a difficult multi-scale example shows that it is generally advisable to choose the lowest-degree B-splines that yield a convergent solution.

In this remark, we point out a cause of loss-of-accuracy in calculating the Faddeyeva function, w(z) using Algorithm 680, near the real axis, and provide a simple remedy to recover the accuracy of the code as one of the important references for accuracy comparison.

We identify two faults in a published algorithm for efficient computation of multiplicative inverses modulo prime powers. We patch the algorithm and present machine assisted proofs of correctness of the repair. Our formal proofs also reveal that being prime is an unnecessary demand for the power base, thus attributing a wider scope of applications to the repaired algorithm.

Abstract. The paper begins with an axiomatic definition of rounded arithmetic. The concepts of rounding and of rounded arithmetic operations are defined in an axiomatic manner fully independent of special data formats and encodings. Basic properties of floating-point and interval arithmetic can directly be derived from this abstract model. Interval operations are defined as set operations for elements of the set IR of closed and connected sets of real numbers. As such they form an algebraically closed subset of the powerset of the real numbers. This property leads to explicit formulas for the arithmetic operations of floating-point intervals of IF, which are executable on the computer. Arithmetic for intervals of IF forms an exception free calculus, i.e., arithmetic operations for intervals of IF always lead to intervals of IF again. Later sections are concerned with programming support and hardware for interval arithmetic. Section 9 illustrates that interval arithmetic as developed in this paper has already a long tradition. Products based on these ideas have been available since 1980. Implementing what the paper advocates would have a profound effect on mathematical software. Modern processor architecture comes quite close to what is requested in this paper.

Interval arithmetic has emerged to solve problems with uncertain parameters which are represented by upper and lower bounds. In rectangular coordinate systems, the basic interval operations and improved interval algorithms have been developed and adopted in the numerical analysis. On the other hand, in polar coordinate systems, interval arithmetic still suffers from significant issues of complex computation and overestimation. This paper defines a polar affine quantity and develops a polar affine arithmetic (PAA) that extends affine arithmetic to the polar coordinate systems, which performs much better in many aspects than the corresponding polar interval arithmetic (PIA). Basic arithmetic operations are developed based on the complex affine arithmetic. The Chebyshev approximation theory and the min-range approximation theory are used to identify the best affine approximation of quantities. PAA can accurately keep track the interdependency among multiple variables throughout the calculation procedure, which prominently reduces the solution conservativeness. Numerical case studies in MATLAB programs show that, compared with benchmark results from the Monte Carlo (MC) method, the proposed PAA ensures the completeness of the exact solution, while presenting a much more compact solution region than PIA. PAA has a great potential in research fields including numerical analysis, computer graphics, and engineering optimization.

A randomized algorithm for computing a so called UTV factorization efficiently is presented. Given a matrix $A$, the algorithm "randUTV" computes a factorization $A = UTV^{*}$, where $U$ and $V$ have orthonormal columns, and $T$ is triangular (either upper or lower, whichever is preferred). The algorithm randUTV is developed primarily to be a fast and easily parallelized alternative to algorithms for computing the Singular Value Decomposition (SVD). randUTV provides accuracy very close to that of the SVD for problems such as low-rank approximation, solving ill-conditioned linear systems, determining bases for various subspaces associated with the matrix, etc. Moreover, randUTV produces highly accurate approximations to the singular values of $A$. Unlike the SVD, the randomized algorithm proposed builds a UTV factorization in an incremental, single-stage, and non-iterative way, making it possible to halt the factorization process once a specified tolerance has been met. Numerical experiments comparing the accuracy and speed of randUTV to the SVD are presented. These experiments demonstrate that in comparison to column pivoted QR, which is another factorization that is often used as a relatively economic alternative to the SVD, randUTV compares favorably in terms of speed while providing far higher accuracy.

This paper presents a high performance software framework for computing a dense SVD on distributed-memory manycore systems. Originally introduced by Nakatsukasa et al., the SVD solver relies on the polar decomposition using the QR Dynamically-Weighted Halley algorithm (QDWH). Although the QDWH-based SVD algorithm performs a significant amount of extra floating-point operations compared to the traditional SVD with the one-stage bidiagonal reduction, the inherent high level of concurrency associated with Level 3 BLAS compute-bound kernels ultimately compensates for the arithmetic complexity overhead. Using the ScaLAPACK two-dimensional block cyclic data distribution with a rectangular processor topology, the resulting QDWH-SVD further reduces excessive communications during the panel factorization, while increasing the degree of parallelism during the update of the trailing submatrix, as opposed to relying to the default square processor grid. After detailing the algorithmic complexity and the memory footprint of the algorithm, we conduct a thorough performance analysis and study the impact of the grid topology on the performance by looking at the communication and computation profiling trade-offs. The QDWH-SVD framework achieves up to 3/8-fold on the Haswell/KNL-based platforms, respectively, against ScaLAPACK PDGESVD and turns out to be a competitive alternative for well and ill-conditioned matrices.

Besides tensor contractions, one of the most pronounced computational bottlenecks in the non-orthogonally spin-adapted forms of the quantum chemistry methods CCSDT and CCSDTQ, and their approximate formsincluding CCSD(T) and CCSDT(Q)are spin summations. At a first sight, spin summations are operations similar to tensor transpositions; a closer look instead reveals additional challenges to high- performance calculations, including temporal locality as well as scattered memory accesses. This publication explores a sequence of algorithmic solutions for spin summations, each exploiting individual properties of either the underlying hardware (e.g. caches, vectorization), or the problem itself (e.g. factorizability). The final algorithm combines the advantages of all the solutions, while avoiding their drawbacks; this algorithm, achieves high-performance through parallelization, vectorization, and by exploiting the temporal locality inherent to spin summations. Combined, these optimizations result in speedups between 2.4× and 5.5× over the NCC quantum chemistry software package. In addition to such a performance boost, our algorithm can perform the spin summations in-place, thus reducing the memory footprint by 2× over an out-of-place variant.

Matrices coming from elliptic Partial Differential Equations have been shown to have a low-rank property which can be efficiently exploited in multifrontal solvers to provide a substantial reduction of their complexity. Among the possible low-rank formats, the Block Low-Rank format (BLR) is easy to use in a general purpose multifrontal solver and its potential compared to standard (full-rank) solvers has been demonstrated. Recently, new variants have been introduced and it was proved that they can further reduce the complexity but their performance has never been analyzed. In this paper, we present a multithreaded BLR factorization, and analyze its efficiency and scalability in shared-memory multicore environments. We identify the challenges posed by the use of BLR approximations in multifrontal solvers and put forward several algorithmic variants of the BLR factorization that overcome these challenges by improving its efficiency and scalability. We illustrate the performance analysis of the BLR multifrontal factorization with numerical experiments on a large set of problems coming from a variety of real-life applications.

Braid monodromy is an important tool for computing invariants of curves and surfaces. In this paper, the rectangular braid diagram technique is proposed to compute the braid monodromy of a completely reducible $n$-gonal curve, i.e. the curves in the form $(y-y_1(x))...(y-y_n(x))=0$ where $n \in \mathbb{Z}^{+}$ and $y_i \in \mathbb{C}[x]$. Also, an algorithm is implemented to compute the Alexander polynomial of these curve complements using Burau representations of braid groups. Examples for each computation are provided.

Sparse tiling is a technique to fuse loops that access common data, thus increasing data locality. Unlike traditional loop fusion or blocking, the loops may have different iteration spaces and access shared datasets through indirect memory accesses, such as A[map[i]]. One notable example of such loops arises in discontinuous-Galerkin finite element methods, because of the computation of numerical integrals over different domains. The major challenge with sparse tiling is implementation not only is it cumbersome to understand and synthesize, but it is also onerous to maintain and generalize, as it requires a complete rewrite of the bulk of the numerical computation. In this article, we propose an approach to extend the applicability of sparse tiling based on raising the level of abstraction. Through a sequence of compiler passes, the mathematical specification of a problem is progressively lowered, and eventually sparse-tiled C for-loops are generated. Besides automation, we advance the state-of-the-art introducing: a more efficient sparse tiling algorithm; support for distributed-memory parallelism; implementation in a publicly-available library, SLOPE; and a study of the performance impact in Seigen, an elastic wave equation solver for seismological problems, which shows speed-ups up to 1.28× on a platform consisting of 896 Intel Broadwell cores.

A bottom-up approach to parallel anisotropic mesh generation is presented by building a mesh generator from principles. Applications focusing on high-lift design or dynamic stall, or numerical methods and modeling test cases still focus on two-dimensional domains. This automated parallel mesh generation approach can generate high-fidelity unstructured meshes with anisotropic boundary layers for use in the computational fluid dynamics field. The anisotropy requirement adds a level of complexity to a parallel meshing algorithm by making computation depend on the local alignment of elements, which in turn is dictated by geometric boundaries and the density functions. This approach yields computational savings in mesh generation and flow solution through well-shaped anisotropic triangles, instead of isotropic triangles. A 79% parallel weak scaling efficiency on 1024 distributed memory nodes, and a 72% parallel efficiency over the fastest sequential isotropic mesh generator on 512 distributed memory nodes is shown through numerical experiments.

#### Algorithm xxx: The 2D Tree Sliding Window Discrete Fourier Transform

Lee Richardson (Carnegie Mellon University); William Eddy (Carnegie Mellon University)The minimum distance of a code is an important concept in information theory. Hence, computing the minimum distance of a code with a minimum computational cost is a crucial process to many problems in this area. In this paper, we present and evaluate a family of algorithms and implementations to compute the minimum distance of a random linear code over F2 that are faster than current implementations, both commercial and public domain. In addition to the basic sequential implementations, we present parallel and vectorized implementations that render high performances on modern architectures. The attained performance results show the benefits of the developed optimized algorithms, which obtain remarkable performance improvements compared with state-of-the-art implementations widely used nowadays.

Applying original and existing theoretical results, we propose a platform-independent multi-threaded function library that provides data structures to generate, differentiate and render both the ordinary basis and the non-negative normalized B-basis of an arbitrary extended Chebyshev (EC) space that comprises the constants and can be identified with the solution space of a user-defined constant-coefficient homogeneous linear differential equation. Using the obtained non-negative normalized B-bases, our library can also generate, (partially) differentiate, modify and visualize a large family of so-called B-curves and tensor product B-surfaces. Moreover, the library also implements methods that can be used to perform general order elevation, to subdivide B-curves and B-surfaces by means of general de Casteljau-like B-algorithms, and to generate general basis transformations for the control point based exact description of arbitrary integral curves and surfaces that are described in traditional parametric form by means of the ordinary bases of the underlying EC spaces. Independently of the algebraic, exponential, trigonometric or mixed type of the applied EC space, the proposed library is numerically stable and efficient up to a reasonable dimension number and may be useful for academics and engineers in the fields of Approximation Theory, Computer Aided Geometric Design, Computer Graphics, Isogeometric and Numerical Analysis.

We discuss the design decisions, design alternatives and rationale behind the third generation of Peano, a framework for dynamically adaptive Cartesian meshes derived from spacetrees. Peano ties the mesh traversal to the mesh storage and supports only one element-wise traversal order resulting from space-filling curves. The user is not free to choose a traversal order herself. The traversal can exploit regular grid subregions and shared memory as well as distributed memory systems with almost no modifications to a serial application code. Relying on a formalism of the software design at hands of two interacting automata---one automaton for the multiscale grid traversal and one for the application-specific algorithmic steps---we discuss the chosen callback-based programming paradigm, supported application types and the two data storage schemes realised, before we detail high-performance computing aspects and lessons learned. Special emphasis is put on observations regarding the used programming and algorithmic concepts. This transforms our report from a ``one way to implement things'' code description into a generic alternatives and rationale summary for some design decisions to be made for any tree-based adaptive mesh refinement software.

We present a pair arithmetic for the four basic operations and square root. It can be regarded as a simplified, more efficient double-double arithmetic. We prove rigorous error bounds for the computed result depending on the relative rounding error unit **u** according to base ², the size of the arithmetic expression, and possibly a condition measure. Under precisely specified assumptions, the result is proved to be faithfully rounded for up to 1/sqrt(²**u**)-2 operations. The assumptions are weak enough to apply to many algorithms. For example, our findings cover a number of previously published algorithms to compute faithfully rounded results, among them Horner's scheme, products, sums and dot products, or Euclidean norm. Beyond that, several other problems can be analyzed such as polynomial interpolation, orientation problems, Householder transformations, or the smallest singular value of Hilbert matrices of large size.

The recent version of the PLASMA (Parallel Linear Algebra Software for Multicore Architectures) library is based on tasks with dependencies from the OpenMP standard. The main functionality of the library is presented. Extensive benchmarks are targeted on three recent multicore and manycore architectures, namely an Intel Xeon, Intel Xeon Phi, and IBM POWER 8 processors.

Branch cuts in complex functions in combination with signed zero and signed infinity have important uses in fracture mechanics, jet flow and aerofoil analysis. We present benchmarks for validating Fortran 2008 complex functions - LOG, SQRT, ASIN, ACOS, ATAN, ASINH, ACOSH and ATANH - on branch cuts with arguments of all 3 IEEE floating point binary formats: binary32, binary64 and binary128. Results are reported with 8 Fortran 2008 compilers: GCC, Flang, Cray, Oracle, PGI, Intel, NAG and IBM. Multiple test failures were revealed, e.g. wrong signs of results or unexpected overflow, underflow, or NaN. We conclude that the quality of implementation of these Fortran 2008 intrinsics in many compilers is not yet sufficient to remove the need for special code for branch cuts. The test results are complemented by conformal maps of the branch cuts and detailed derivations of the values of these functions on branch cuts, to be used as a reference. The benchmarks are freely available from cmplx.sf.net. This work will be of interest to engineers who use complex functions, as well as to compiler and maths library developers.

An algorithm for multiplying a chain of Kronecker products by a matrix is described. The algorithm does not require that the Kronecker chain actually be computed and the main computational work is a series of matrix multiplications. Use of the algorithm can lead to substantial savings in both memory usage and computational speed. Although similar algorithms have been described before, this paper makes two novel contributions. First, it shows how shuffling of data can be (largely) avoided. Second, it provides a simple method to determine the optimal ordering of the workflow. A \matlab~implementation is provided in an appendix.

Large scale parameter estimation problems are some of the most computationally demanding problems. An academic researcher's domain-specific knowledge often precludes that of software design, which results in software frameworks for inversion that are technically correct, but not scalable to realistically-sized problems. On the other hand, the computational demands of the problem for realistic problems result in industrial codebases that are geared solely for performance, rather than comprehensibility or flexibility. We propose a new software design that bridges the gap between these two seemingly disparate worlds. A hierarchical and modular design allows a user to delve into as much detail as she desires, while using high performance primitives at the lower levels. Our code has the added benefit of actually reflecting the underlying mathematics of the problem, which lowers the cognitive load on user using it and reduces the initial startup period before a researcher can be fully productive. We also introduce a new preconditioner for the Helmholtz equation that is suitable for fault-tolerant distributed systems. Numerical experiments on a variety of 2D and 3D test problems demonstrate the effectiveness of this approach on scaling algorithms from small to large scale problems with minimal code changes.

Batched dense linear algebra kernels are becoming ubiquitous in scientific applications, ranging from tensor contractions in deep learning to data compression in hierarchical low rank matrix approximation. Within a single API call, these kernels are capable of simultaneously launching up to thousands of similar matrix computations, removing the expensive overhead of multiple API calls while increasing the occupancy of the underlying hardware. A challenge is that for the existing hardware landscape (x86, GPUs, etc.) only a subset of the required batched operations is implemented by the vendors, with limited support for very small problem sizes. We describe the design and performance of a new class of batched triangular dense linear algebra kernels on very small data sizes using single and multiple GPUs. By deploying recursive formulations, stressing the register usage, maintaining data locality, reducing threads synchronization and fusing successive kernel calls, the new batched kernels outperform existing state-of-the-art implementations.