A fast symmetric SVD algorithm for square Hankel matrices

SUMMARYProper range and precision analysis play an important role in the development of fixed-point algorithms for embedded system applications. Numerical linear algebra algorithms used to find singular value decomposition of symmetric matrices are suitable for signal and image-processing applications. These algorithms have not been attempted much in fixed-point arithmetic. The reason is wide dynamic range of data and vulnerability of the algorithms to round-off errors. For any real-time application, the range of the input matrix may change frequently. This poses difficulty for constant and variable fixed-point formats to decide on integer wordlengths during float-to-fixed conversion process because these formats involve determination of integer wordlengths before the compilation of the program. Thus, these formats may not guarantee to avoid overflow for all ranges of input matrices. To circumvent this problem, a novel dynamic fixed-point format has been proposed to compute integer wordlengths adaptively during runtime. Lanczos algorithm with partial orthogonalization, which is a tridiagonalization step in computation of singular value decomposition of symmetric matrices, has been taken up as a case study. The fixed-point Lanczos algorithm is tested for matrices with different dimensions and condition numbers along with image covariance matrix. The accuracy of fixed-point Lanczos algorithm in three different formats has been compared on the basis of signal-toquantization-noise-ratio, number of accurate fractional bits, orthogonality and factorization errors. Results show that dynamic fixed-point format either outperforms or performs on par with constant and variable formats. Determination of fractional wordlengths requires minimization of hardware cost subject to accuracy constraint. In this context, we propose an analytical framework for deriving mean-square-error or quantization noise power among Lanczos vectors, which can serve as an accuracy constraint for wordlength optimization. Error is found to propagate through different arithmetic operations and finally accumulate in the last Lanczos vector. It is observed that variable and dynamic fixed-point formats produce vectors with lesser round-off error than constant format. All the three fixed-point formats of Lanczos algorithm have been synthesized on Virtex 7 field-programmable gate array using Vivado high-level synthesis design tool. A comparative study of resource usage and power consumption is carried out.

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“…A random variable Ψ j given by Ψ j :¼ w j;jþ1 (13) is defined for orthogonality estimates. Another random variable θ k,j given by…”

Section: Lanczos Tridiagonalization With Partial Orthogonalizationmentioning

confidence: 99%

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

Pradhan

Kabi

Mohanty

et al. 2015

Circuit Theory & Apps

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“…Then, by applying the twisted factorization method, the symmetric SVD of the symmetric tridiagonal matrix can be efficiently obtained in O(n 2 ). Thus, it leads to an O(n 2 log n) algorithm for computing the symmetric SVD of a complex square Hankel matrix [9]. Since a Toeplitz matrix can be transformed into a Hankel matrix by reversing its rows or columns, this method can be straightforwardly modified into a fast SVD algorithm for square Toeplitz matrices.…”

Section: W Xu and S Qiaomentioning

confidence: 99%

A twisted factorization method for symmetric SVD of a complex symmetric tridiagonal matrix

Qiao

2009

Numerical Linear Algebra App

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This paper presents an O(n 2 ) method based on the twisted factorization for computing the Takagi vectors of an n-by-n complex symmetric tridiagonal matrix with known singular values. Since the singular values can be obtained in O(n 2 ) flops, the total cost of symmetric singular value decomposition or the Takagi factorization is O(n 2 ) flops. An analysis shows the accuracy and orthogonality of Takagi vectors. Also, techniques for a practical implementation of our method are proposed. Our preliminary numerical experiments have verified our analysis and demonstrated that the twisted factorization method is much more efficient than the implicit QR method, divide-and-conquer method and Matlab singular value decomposition subroutine with comparable accuracy. TWISTED FACTORIZATION METHOD FOR SSVD 1 = P 11 − ; % (1, 1)-entry l 1 = P 21 / 1 ; % (2, 1)-entry 2 = P 22 − −|l 1 | 2 1 ; % (2, 2)-entry for i = 1 : n −2 Similar to Theorem 3.1, we show an upper bound for | k |/ z k 2 , k = k 1 , k 2 , for the case of multiple eigenvalues i = i+1 .Theorem 5.2 Suppose that P − I is invertible and (P − I )z k = k e k for k = 1, 2, . . . , n

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“…A general SVD method, like ZGESDD, is too expensive for large size matrices occured in seismic data processing. By exploiting the Hankel structure, Xu and Qiao [27] proposed a fast SVD algorithm for (level-1) Hankel matrices. The fast SVD algorithm for Hankel matrices in [27] consists of two stages.…”

Section: Introductionmentioning

confidence: 99%

“…By exploiting the Hankel structure, Xu and Qiao [27] proposed a fast SVD algorithm for (level-1) Hankel matrices. The fast SVD algorithm for Hankel matrices in [27] consists of two stages. First, the Hankel matrix is bidiagonalized through the Lanczos method.…”

Section: Introductionmentioning

confidence: 99%

A fast SVD for multilevel block Hankel matrices with minimal memory storage

Qiao

2014

Numer Algor

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Motivated by the Cadzow filtering in seismic data processing, this paper presents a fast SVD method for multilevel block Hankel matrices. A seismic data presented as a multidimensional array is first transformed into a two dimensional multilevel block Hankel (MBH) matrix. Then the Lanczos process is applied to reduce the MBH matrix into a bidiagonal or tridiagonal matrix. Finally, the SVD of the reduced matrix is computed using the twisted factorization method. To achieve high efficiency, we propose a novel fast MBH matrix-vector multiplication method for the Lanczos process. In comparison with existing fast Hankel matrix-vector multiplication methods, our method applies 1-D, instead of multidimensional, FFT and requires minimum storage. Moreover, a partial SVD is performed on the reduced matrix, since complete SVD is not required by the Caszow filtering. Our numerical experiments show that our fast MBH matrix-vector multiplication method significantly improves both the computational cost and storage requirement. Our fast MBH SVD algorithm is particularly efficient for large size multilevel block Hankel matrices.

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A fast symmetric SVD algorithm for square Hankel matrices

Cited by 30 publications

References 16 publications

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

A twisted factorization method for symmetric SVD of a complex symmetric tridiagonal matrix

A fast SVD for multilevel block Hankel matrices with minimal memory storage

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