Fast solvers for tridiagonal Toeplitz linear systems

Liu, Zhongyun; Li, Shan; Yin, Yilong; Zhang, Y.

doi:10.1007/s40314-020-01369-3

Cited by 5 publications

(9 citation statements)

References 21 publications

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“…Let us also note that for the second algorithm, the problem size is equal to

n = 2^{k} + 1

. We consider problems that appear when solving systems of linear equations arising for one‐dimension convection‐diffusion equations: 20

\{\begin{array}{l} prefix- a u^{'^{'}} + b u^{'} = 0, \\ x \in false(0, 1 false), u false(0 false) = 0, u false(1 false) = 1, \end{array}

All presented results have been obtained for Example 2 presented by Liu et al, 16 namely,

t_{1} = - 1 - c

t_{2} = 2 + c

t_{3} = - 1

and

c = 9

. It ensures that the previously described conditions are satisfy, that is,

| t_{2} | = | t_{1} | + | t_{3} |

t_{2}^{2} - 4 t_{1} t_{3} > 0

and

| t_{1} | > | t_{3} |

.…”

Section: Results Of Experimentsmentioning

confidence: 99%

“…Recently, Liu et al 16 formulated the following sequential method for solving the problem. For the sake of simplicity, we assume that

n = 2^{k} + 1

k \in ℕ

.…”

Section: Parallel Algorithmsmentioning

confidence: 99%

“…For the sake of simplicity, we assume that

n = 2^{k} + 1

k \in ℕ

. Let:

P = [\begin{array}{ccccc} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ 1 & 0 & 0 & \dots & 0 \end{array}] .

The solution to () is the same as the solution to

P T x = P f

, which is of the form: and it can be rewritten in the following block from: It can be easily verified that

x_{0} = T_{11}^{- 1} f_{1} - x_{n - 1} T_{11}^{- 1} p

, thus the basic formula for solving (14) can be expressed as follows: 16

\{\begin{array}{l} {boldx}_{0} = boldv prefix- x_{n prefix- 1} boldu \\ x_{n prefix- 1} = false(t_{2} v_{0} + t_{3} v_{1} prefix- f_{0} false) false/ false(t_{2} u_{0} + t_{3} \end{array}

…”

Section: Parallel Algorithmsmentioning

confidence: 99%

“…It can be easily verified that x 0 = T −1 11 f 1 − x n−1 T −1 11 p, thus the basic formula for solving (14) can be expressed as follows: 16 { x 0 = v − x n−1 u…”

Section: Second Algorithmmentioning

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} - 4 t_{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

Stpiczyński

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

“…Let us also note that for the second algorithm, the problem size is equal to

n = 2^{k} + 1

. We consider problems that appear when solving systems of linear equations arising for one‐dimension convection‐diffusion equations: 20

\{\begin{array}{l} prefix- a u^{'^{'}} + b u^{'} = 0, \\ x \in false(0, 1 false), u false(0 false) = 0, u false(1 false) = 1, \end{array}

All presented results have been obtained for Example 2 presented by Liu et al, 16 namely,

t_{1} = - 1 - c

t_{2} = 2 + c

t_{3} = - 1

and

c = 9

. It ensures that the previously described conditions are satisfy, that is,

| t_{2} | = | t_{1} | + | t_{3} |

t_{2}^{2} - 4 t_{1} t_{3} > 0

and

| t_{1} | > | t_{3} |

.…”

Section: Results Of Experimentsmentioning

confidence: 99%

“…Recently, Liu et al 16 formulated the following sequential method for solving the problem. For the sake of simplicity, we assume that

n = 2^{k} + 1

k \in ℕ

.…”

Section: Parallel Algorithmsmentioning

confidence: 99%

“…For the sake of simplicity, we assume that

n = 2^{k} + 1

k \in ℕ

. Let:

P = [\begin{array}{ccccc} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ 1 & 0 & 0 & \dots & 0 \end{array}] .

The solution to () is the same as the solution to

P T x = P f

, which is of the form: and it can be rewritten in the following block from: It can be easily verified that

x_{0} = T_{11}^{- 1} f_{1} - x_{n - 1} T_{11}^{- 1} p

, thus the basic formula for solving (14) can be expressed as follows: 16

\{\begin{array}{l} {boldx}_{0} = boldv prefix- x_{n prefix- 1} boldu \\ x_{n prefix- 1} = false(t_{2} v_{0} + t_{3} v_{1} prefix- f_{0} false) false/ false(t_{2} u_{0} + t_{3} \end{array}

…”

Section: Parallel Algorithmsmentioning

confidence: 99%

“…It can be easily verified that x 0 = T −1 11 f 1 − x n−1 T −1 11 p, thus the basic formula for solving (14) can be expressed as follows: 16 { x 0 = v − x n−1 u…”

Section: Second Algorithmmentioning

confidence: 99%