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This paper concerns test matrices for numerical linear algebra using an error-free transformation of floating-point arithmetic. For specified eigenvalues given by a user, we propose methods of generating a matrix whose eigenvalues are exactly known based on, for example, Schur or Jordan normal form and a block diagonal form. It is also possible to produce a real matrix with specified complex eigenvalues. Such test matrices with exactly known eigenvalues are useful for numerical algorithms in checking the accuracy of computed results. In particular, exact errors of eigenvalues can be monitored. To generate test matrices, we first propose an error-free transformation for the product of three matrices YSX. We approximate S by ${S^{\prime }}$ S ′ to compute ${YS^{\prime }X}$ Y S ′ X without a rounding error. Next, the error-free transformation is applied to the generation of test matrices with exactly known eigenvalues. Note that the exactly known eigenvalues of the constructed matrix may differ from the anticipated given eigenvalues. Finally, numerical examples are introduced in checking the accuracy of numerical computations for symmetric and unsymmetric eigenvalue problems.

We propose fast numerical algorithms to improve the accuracy of singular vectors for a real matrix.Recently, Ogita and Aishima proposed an iterative refinement algorithm for singular value decomposition that is constructed with highly accurate matrix multiplications carried out six times per iteration.The algorithm runs for the problem that has no multiple and clustered singular values.In this study, we show that the same algorithm can be run with highly accurate matrix multiplications carried out five times.Also, we proposed four algorithms constructed with highly accurate matrix multiplications, two algorithms with the multiplications carried out four times, and the other two with the multiplications carried out five times. These algorithms adopt the idea of a mixed-precision iterative refinement method for linear systems.Numerical experiments demonstrate speed-up and quadratic convergence of the proposed algorithms.As a result, the fastest algorithm is 1.7 and 1.4 times faster than the Ogita-Aishima algorithm per iteration on a CPU and GPU, respectively.

This paper concerns accuracy-guaranteed numerical computations for linear systems. Due to the rapid progress of supercomputers, the treatable problem size is getting larger. The larger the problem size, the more rounding errors in floating-point arithmetic can accumulate in general, and the more inaccurate numerical solutions are obtained. Therefore, it is important to verify the accuracy of numerical solutions. Verified numerical computations are used to produce error bounds on numerical solutions. We report the implementation of a verification method for large-scale linear systems and some numerical results using the RIKEN K computer and the Fujitsu PRIMEHPC FX100, which show the high performance of the verified numerical computations.

Although IEEE 754-2008 binary128 (with a 15-bit exponent and 113-bit significand, i.e., quadruple-precision) is not currently implemented on x86 in hardware, software emulation is available on some compilers. However, the performance is significantly slower compared to the binary64 operation, which is supported natively in hardware. This study proposes a fast implementation of matrix multiplication on matrices stored in the binary128 format on x86 CPUs. The proposed implementation utilizes the Ozaki scheme, which is an accurate matrix multiplication algorithm proposed by Ozaki et al. in 2012. This scheme enables one to perform most computations using the binary64 matrix multiplication (the DGEMM routine in Basic Linear Algebra Subprograms (BLAS)); it can exploit the high-performance of highly-optimized vendor BLAS. Although the achievable performance depends on the input matrices (the inner-product dimension, the absolute range, and the significand bit length), the proposed implementation can achieve better performance and accuracy compared to naive matrix multiplication performed using the GCC’s binary128 emulation in many cases. In addition, we discuss GPU acceleration, performance on reduced precision inputs, an implementation based on binary32 matrix multiplication (SGEMM), application to memory-intensive operations, and the possibility of a distributed parallel implementation.

Lepidopteran insects are mostly monophagous or oligophagous. Female butterflies distinguish their host plants by detecting a combination of specific phytochemicals through the gustatory sensilla densely distributed on their foreleg tarsi, thereby ensuring oviposition on appropriate host plants. In this study, to gain insight into the molecular mechanism underlying host plant recognition by the gustatory sensilla, using Asian swallowtail, Papilio xuthus, we focused on a family of small soluble ligand-binding molecules, odorant-binding proteins (OBPs), and found that three OBP genes showed enriched expression in the foreleg tarsus. Multicolor fluorescence in situ hybridization analyses demonstrated the coexpression of these three OBP genes at the bases of the foreleg gustatory sensilla. Further analyses on other appendages revealed that PxutOBP3 was exclusively expressed in the tissues which could have direct contact with the leaf surface, suggesting that this OBP gene specifically plays an important role in phytochemicals perception.

On Krylov subspace methods such as the Conjugate Gradient (CG), the number of iterations until convergence may increase due to the loss of computation accuracy caused by rounding errors in floating-point computations. Besides, as the order of operations is non-deterministic on parallel computations, the result and the behavior of the convergence may be non-identical in different environments, even for the same input. This paper presents a new approach for the CG method with high accuracy as well as bit-level reproducibility of computed solutions on many-core processors, including both x86 CPUs and NVIDIA GPUs. In our proposed approach, accurate and reproducible operations are installed into all the inner-product based operations such as matrix-vector multiplication and dot-product, which are the main sources that may disturb reproducibility in the CG method. The accurate and reproducible operations are performed using the Ozaki scheme, which is the error-free transformation for dot-product that can ensure the correct-rounding. As this method can be built upon vendor-provided linear algebra libraries such as Intel Math Kernel Library and NVIDIA cuBLAS/ cuSparse, it reduces the development cost. In this paper, showing some examples with the non-identical conver-gences and computed solutions on different platforms, we demonstrate the applicability and the effectiveness of the proposed approach as well as its performance on both CPUs and GPUs. Besides, we compare against an existing accurate and reproducible CG implementation based on the Exact BLAS (ExBLAS) on CPUs.

The Oriental fruit fly, Bactrocera dorsalis, is a highly destructive pest of various fruits. The reproductive and host-finding behaviors of this species are affected by several plant semiochemicals that are perceived through chemosensory receptors. However, the chemosensory mechanisms by which this perception occurs have not been fully elucidated. We conducted RNA sequencing analysis of the chemosensory organs of B. dorsalis to identify the genes coding for chemosensory receptors. We identified 60 olfactory receptors (ORs), 17 gustatory receptors and 23 ionotropic receptors-including their homologs and variants-from the transcriptome of male antennae and proboscises. We functionally analyzed ten ORs co-expressed with the obligatory co-receptor ORCO in Xenopus oocytes to identify their ligands. We tested 24 compounds including attractants for several Bactrocera species and volatiles from the host fruits of B. dorsalis. We found that BdorOR13a co-expressed with ORCO responded robustly to 1-octen-3-ol. BdorOR82a co-expressed with ORCO responded significantly to geranyl acetate, but responded weakly to farnesenes (a mixture of isomers) and linalyl acetate. These four compounds were subsequently subjected to behavioral bioassays. When each of the aforementioned compound was presented in combination with a sphere model as a visual cue to adult flies, 1-octen-3-ol, geranyl acetate, and farnesenes significantly enhanced landing behavior in mated females, but not in unmated females or males. These results suggest that the ORs characterized in the present study are involved in the perception of plant volatiles that affect host-finding behavior in B. dorsalis.

Generally, floating-point computations comprise rounding errors; the result may be inaccurate and not identical (non-reproducible). Particularly, heterogeneous computing has many factors that affect reproducibility. The loss of accuracy and reproducibility could be a crucial issue in debugging complex codes and the reliability of computations. In this paper, we propose high-performance implementations of reproducible basic linear algebra subprograms (BLAS) routines with tunable accuracy for many-core architectures. Our approach is based on an accurate matrix-multiplication method, Ozaki scheme, which can be constructed on level-3 BLAS that performs standard floating-point operations. We demonstrate the performance of three routines: inner product (DOT), matrix-vector multiplication (GEMV), and matrix-multiplication (GEMM) on NVIDIA’s Volta GPU by comparing these with the standard routines provided by the vendor. Furthermore, we demonstrate the reproducibility between CPU and GPU and its accuracy.

Dacini fruit flies (Tephritidae: Diptera), including destructive pest species, are strongly affected in their reproductive behaviors by semiochemicals. Notably, male lures have been developed for pest management e.g., aromatic compounds for the Oriental fruit fly Bactrocera dorsalis and the melon fruit fly Zeugodacus cucurbitae; terpenic α-ionone analogs for the solanaceous fruit fly, B. latifrons. Other than those specific male attractants, 1-nonanol analogs have been noticed as major aliphatic components in the male rectal gland, which is considered as a secretory organ of male sex pheromones. Although multiple semiochemicals associated with the life cycle of Dacini fruit flies have been identified, their behavioral role(s) and chemosensory mechanisms by which the perception occurs have not been fully elucidated. In this study, we conducted RNA sequencing analysis of the chemosensory organs of B. latifrons and Z. cucurbitae to identify the genes coding for chemosensory receptors. Because the skeletons of male attractants are different among Dacini fruit fly species, we analyzed phylogenetic relationships of candidate olfactory receptors (ORs) among the three species. We found that the OR phylogeny reflects the taxonomic relationships of the three species. We further characterized functional properties of OR74a in the three Dacini species to the 1-nonanol analogs related to components in the rectal glands. The three OR74a homologs responded to 1-nonanol, but their sensitivities differed from each other. The OR74a homologs identified from B. dorsalis and Z. cucurbitae responded significantly to 6-oxo-1-nonanol, but not to 1, 3-nonanediol and nonyl acetate, indicating similar binding properties of the homologous ORs.

The presence of rounding errors is frequently inevitable when performing arithmetic operations in computers due to the use of floating-point number system whose elements have finite precision. Consequently, error-free transformation algorithms are proposed as solutions, and a particular instance of these is the error-free transformation algorithm for real matrix multiplication. Based on the previously mentioned algorithm, this study proposes two error-error free transformation algorithms for complex matrix multiplication. By conducting the numerical experiment, one finds that the combination of both proposed algorithms with an accurate summation algorithm generates the most accurate results, but it also suffers from a notable increase in the computing time ratio. On the other hand, the second proposed algorithm, which is simply another form of the first one with the change of order in summation, combined with the pure floating-point addition produces results that are nearly as accurate as those generated by the former combination. Since these results are achieved with significantly smaller computing time ratio, then the study concludes that the last combination is the best method for complex matrix multiplication in terms of accuracy and computing time efficiency.

Odorant-binding proteins (OBPs) and chemosensory proteins (CSPs) play important roles in insect chemosensory systems. In the Asian corn borer Ostrinia furnacalis (Guenee) (Lepidoptera: Crambidae), only a fraction of them have been examined for their expression patterns. In this study, we compared the expression levels of 24 OBPs and 19 CSPs between sexes using quantitative RT-PCR. Sexually biased expression was observed for 17 OBPs and 7 CSPs in virgin individuals. Expression levels of 18 OBPs and 10 CSPs changed after mating, resulting in sexually biased expression in 16 OBPs and 1 CSP. The effect of mating was sex-specific in 9 OBPs and 2 CSPs. These results suggest a sexually divergent molecular composition and may indicate a functional divide between OBP and CSP genes in the chemosensory system of O. furnacalis.

This paper discusses accurate numerical algorithms for matrix multiplication. Matrix multiplication is a basic and important problem in numerical linear algebra. Numerical computations using floating-point arithmetic can be quickly performed on existing computers. However, the accumulation of rounding errors due to finite precision arithmetic is a critical problem. An error-free transformation for matrix multiplication is reviewed in this paper. Such a transformation is extremely useful for developing accurate numerical algorithms for matrix multiplication. One advantage of the transformation is that it exploits Basic Linear Algebra Subprograms (BLAS). We provide a rounding error analysis of reproducible algorithms for matrix multiplication based on the error-free transformation. In addition, we propose an error-free transformation for matrix multiplication that can be utilized with the divide and conquer methods.

This paper concerns thin QR decomposition in an oblique inner product. Cholesky QR is known as a fast algorithm for thin QR decomposition. On the other hand, this algorithm is not applicable for ill-conditioned matrices. The preconditioned Cholesky QR algorithms named LU-Cholesky QR for thin QR decomposition are recently proposed. We apply these preconditioning techniques to thin QR decomposition in an oblique inner product.

Basic Linear Algebra Subprograms (BLAS) is a frequently used numerical library for linear algebra computations. However, it places little emphasis on computational accuracy, especially with respect to the accuracy assurance of the results. Although some algorithms for ensuring the computational accuracy of BLAS operations have been studied, there is a need for performance evaluation in advanced computer architectures. In this study, we parallelize high-precision matrix-matrix multiplication using thread-level parallelism. In addition, we conduct a performance evaluation from the viewpoints of execution speed and accuracy. We implement a method to convert dense matrices into sparse matrices by exploiting the nature of the target algorithm and adapting sparse-vector multiplication. Results obtained using the FX100 supercomputer system at Nagoya University indicate that (1) implementation with the ELL format achieves 1.43x speedup and (2) a maximum of 38x speedup compared to conventional implementation for dense matrix operations with dgemm.

This paper is concerned with floating-point filters for a two dimensional orientation problem which is a basic problem in the field of computational geometry. If this problem is only approximately solved by floating-point arithmetic, then an incorrect result may be obtained due to accumulation of rounding errors. A floating-point filter can quickly guarantee the correctness of the computed result if the problem is well-conditioned. In this paper, a simple semi-static floating-point filter which handles floating-point exceptions such as overflow and underflow by only one branch is developed. In addition, an improved fully-static filter is developed.

In Batesian mimicry, animals avoid predation by resembling distasteful models. In the swallowtail butterfly Papilio polytes, only mimetic-form females resemble the unpalatable butterfly Pachliopta aristolochiae. A recent report showed that a single gene, doublesex (dsx), controls this mimicry; however, the detailed molecular mechanisms remain unclear. Here we determined two whole-genome sequences of P. polytes and a related species, Papilio xuthus, identifying a single ∼130-kb autosomal inversion, including dsx, between mimetic (H-type) and non-mimetic (h-type) chromosomes in P. polytes. This inversion is associated with the mimicry-related locus H, as identified by linkage mapping. Knockdown experiments demonstrated that female-specific dsx isoforms expressed from the inverted H allele (dsx(H)) induce mimetic coloration patterns and simultaneously repress non-mimetic patterns. In contrast, dsx(h) does not alter mimetic patterns. We propose that dsx(H) switches the coloration of predetermined wing patterns and that female-limited polymorphism is tightly maintained by chromosomal inversion.

Although host-plant selection is a central topic in ecology, its general underpinnings are poorly understood. Here, we performed a case study focusing on the publicly available data on Japanese butterflies. A combined statistical analysis of plant–herbivore relationships and taxonomy revealed that some butterfly subfamilies in different families feed on the same plant families, and the occurrence of this phenomenon more than just by chance, thus indicating the independent acquisition of adaptive phenotypes to the same hosts. We consequently integrated plant–herbivore and plant–compound relationship data and conducted a statistical analysis to identify compounds unique to host plants of specific butterfly families. Some of the identified plant compounds are known to attract certain butterfly groups while repelling others. The additional incorporation of insect–compound relationship data revealed potential metabolic processes that are related to host plant selection. Our results demonstrate that data integration enables the computational detection of compounds putatively involved in particular interspecies interactions and that further data enrichment and integration of genomic and transcriptomic data facilitates the unveiling of the molecular mechanisms involved in host plant selection.

Although numerical computation is very fast, however, the results may not be accurate due to the accumulation of rounding errors. Consequently, much research has focussed on ways to verifying the accuracy of approximate solutions. Floating-point filters are one such technique. These can, for example, be used to guarantee the signs of computed results, such as those of the matrix determinants that are so important in the computational geometry field. In this paper, we extend floating-point filters to guarantee absolute and relative errors.

The numerical computations with guaranteed accuracy called verified numerical computations are studied, herein. We focus on an accurate numerical solution for dense linear systems of the form Ax = b and error bound of ‖ x ^ − x ‖ ∞ in supercomputing, where x ^ is an approximate solution. First, we implemented accurate algorithms for a matrix-vector product for parallel and distributed computing, and then, applied them to iterative refinements. Next, we implemented verified numerical computations using the Parallel Basic Linear Algebra Subprograms (PBLAS) and the Scalable Linear Algebra Package library (ScaLAPACK). Numerical examples of the verified numerical computations for linear systems are presented using the Fujitsu FX100 supercomputer. They clearly illustrate the efficiency of the verified numerical computations.

The swallowtail butterfly,Papilio xuthus, selectively uses a limited number of plants in the Rutaceae family. The butterfly detects oviposition stimulants in leaves through foreleg chemosensilla and requires a specific combination of multiple oviposition stimulants to lay eggs on the leaf of its host plants. In this study, we sought to elucidate the mechanism underlying the regulation of oviposition behavior by multiple oviposition stimulants. We classified chemosensilla on the tarsomere of the foreleg into three types (L1, L2, and S) according to their size and response to oviposition stimulants and general tastants. The L1 was more abundant in females than in males and responded preferentially to oviposition stimulants. Both L2 and S were common to both sexes and responded to general tastants. We found that five oviposition stimulants (synephrine, stachydrine, 5-hydroxy-Nω-methyltryptamine, narirutin, andchiro-inositol) elicited spikes from three specific gustatory receptor neurons (GRNs) within L1 sensilla. These three GRNs responded to a mixture of the five stimulants at concentrations equivalent to those found in the whole-leaf extract of citrus, and the mixture induced oviposition at levels comparable to whole-leaf extract. We propose that oviposition is triggered by the firing of three specific GRNs in L1 sensilla that encode the chemical signatures of multiple oviposition stimulants.