(redirected from Modified Gram-Schmidt)
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References in periodicals archive ?
Conventionally, three major algorithms have been used to compute the QR decomposition: the Householder QR algorithm, the classical Gram-Schmidt (CGS) algorithm, and the modified Gram-Schmidt (MGS) algorithm.
In this section we evaluate the numerical stability of CholeskyQR2 and compare it with the stability of other popular QR decomposition algorithms, namely, Householder QR, classical and modified Gram-Schmidt (CGS and MGS, we also run them twice, shown as CGS2 and MGS2), and Cholesky QR.
There are different algorithms and architectures proposed in the literature which includes Gram-Schmidt orthogonalization, modified Gram-Schmidt orthogonalization [2, 3], Givens rotation [4-11], householder transformations [12], and various other hybrid methods [13, 14].
Prasad, and P T Balsara, "VLSI architecture for matrix inversion using modified gram-schmidt based QR decomposition," Proceedings of the 20th International Conference on VLSI Design Held Jointly with 6th International Conference on Embedded Systems, pp.
This paper presents the implementation results on the 16bit-fixed-point DSP chip (TMS 320C6474 which is optimized for 16bits arithmetic operations), [1] of a linear system solving performed by Cholesky decomposition and modified Gram-Schmidt process.
Essentially it is shown that the QR factorization using the modified Gram-Schmidt process (MGS) is less sensitive to the round-off error than the classical Gram-Schmidt process (GS) when using the fixed-point calculation format [4, 5].
In some contexts (as in, say, [3]), we may assume that [xi] [less than or equal to] f (m, n)[[epsilon].sub.M] where [[epsilon].sub.M] is machine precision and f (m, n) is a modestly growing function, but, in our discussion, we simply assume that it is "small." It is possible that [xi] depends on the condition number of R, for example, when it is the result of a modified Gram-Schmidt QR factorization [6].
If we compute the QR factorization of X (1) with Householder transformations instead of the modified Gram-Schmidt method, the loss of orthogonality starts out at near machine precision and stays there.
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