Solving systems of symmetric Toeplitz tridiagonal equations: Rojo’s algorithm revisited

Vidal, Antonio M.; Alonso, Pedro

doi:10.1016/j.amc.2012.08.030

Cited by 4 publications

(4 citation statements)

References 10 publications

(15 reference statements)

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“…Tridiagonal Toeplitz systems of linear equations appear in many theoretical and practical applications. For example, numerical algorithms for solving boundary value problems for ordinary and partial differential equations reduce to such systems [ 13 , 15 ]. They also play an important role in piecewise cubic interpolation and spline algorithms [ 4 , 14 ].…”

Section: Introductionmentioning

confidence: 99%

High Performance Portable Solver for Tridiagonal Toeplitz Systems of Linear Equations

Dmitruk

Stpiczynski

2021

Euro-Par 2020: Parallel Processing Workshops

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We show that recently developed divide and conquer parallel algorithm for solving tridiagonal Toeplitz systems of linear equations can be easily and efficiently implemented for a variety of modern multicore and GPU architectures, as well as hybrid systems. Our new portable implementation that uses OpenACC can be executed on both CPU-based and GPU-accelerated systems. More sophisticated variants of the implementation are suitable for systems with multiple GPUs and it can use CPU and GPU cores. We consider the use of both column-wise and row-wise storage formats for two dimensional double precision arrays and show how to efficiently convert between these two formats using cache memory. Numerical experiments performed on Intel CPUs and Nvidia GPUs show that our new implementation achieves relatively good performance.

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Section: Introductionmentioning

confidence: 99%

High Performance Portable Solver for Tridiagonal Toeplitz Systems of Linear Equations

Dmitruk

Stpiczynski

2021

Euro-Par 2020: Parallel Processing Workshops

View full text Add to dashboard Buy / Rent full text

show abstract

“…Tridiagonal Toeplitz systems of linear equations appear in many theoretical and practical applications. For example, numerical methods for boundary value problems for ordinary and partial differential equations require the use of reliable and effective algorithms for solving such systems 1,2 . They also play an important role in piecewise cubic interpolation, spline algorithms, 3,4 signal processing, and control theory 5,6 .…”

Section: Introductionmentioning

confidence: 99%

“…For example, numerical methods for boundary value problems for ordinary and partial differential equations require the use of reliable and effective algorithms for solving such systems. 1,2 They also play an important role in piecewise cubic interpolation, spline algorithms, 3,4 signal processing, and control theory. 5,6 There are several methods for solving such systems.…”

Section: Introductionmentioning

confidence: 99%

“…In this article, we additionally consider a new, parallel and vectorizable algorithm based on a recently introduced sequential method for solving such systems. 16 Both presented algorithms are used for solving tridiagonal Toeplitz systems with matrices satisfying |t 2 | = |t 1 | + |t 3 |, t 2 2 − 4t 1 t 3 > 0 and for the second algorithm also |t 1 | > |t 3 | (Liu et al 16 presented two algorithms: for |t 1 | > |t 3 | and for |t 1 | < |t 3 |). We show how to improve the efficiency of the implementation utilizing column-wise storage by the use of cache memory.…”

Section: Introductionmentioning

confidence: 99%

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Solving tridiagonal Toeplitz systems of linear equations on GPU‐accelerated computers

Dmitruk

Stpiczynski

2021

Concurrency and Computation

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The aim of this article is to show that solvers for tridiagonal Toeplitz systems of linear equations can be efficiently implemented for a variety of modern GPU-accelerated and multicore architectures using OpenACC. We consider two parallel algorithms for solving such systems with special assumptions about coefficient matrices. As the first algorithm, we propose a new, faster implementation of the divide and conquer method.The next algorithm is a new, vectorizable algorithm based on a recently introduced sequential method. We consider the use of both column-wise and row-wise storage formats for two-dimensional arrays and show how to efficiently convert between these two formats using cache memory and improve the overall performance of our implementations. We also show how to tune the performance by predicting the best values of the methods' parameters. Numerical experiments performed on Intel CPUs and Nvidia GPUs show that our new implementations achieve relatively good performance and accuracy.

show abstract

Solving systems of symmetric Toeplitz tridiagonal equations: Rojo’s algorithm revisited

Cited by 4 publications

References 10 publications

High Performance Portable Solver for Tridiagonal Toeplitz Systems of Linear Equations

High Performance Portable Solver for Tridiagonal Toeplitz Systems of Linear Equations

Solving tridiagonal Toeplitz systems of linear equations on GPU‐accelerated computers

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