lsmr (least squares minimal residual) is an iterative method for the solution of the linear system of equations and leastsquares problems. this paper presents a block version of the lsmr algorithm for solving linear systems with multiple right-hand sides. the new algorithm is based on the block bidiagonalization and derived by minimizing the frobenius norm of the resid ual matrix of normal equa...

In the present paper, we investigate a connection between two fuzzy inner product one of which arises from Felbin's fuzzy norm and the other is based on Bag and Samanta's fuzzy norm. Also we show that, considering a fuzzy inner product space, how one can construct another kind of fuzzy inner product on this space.

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Let k be a field, T an algebraic k-torus, X smooth k-compactification of and PicX‾ the Picard group X‾=X×kk‾. Hoshi, Kanai Yamasaki [HKY22] determined H1(k,PicX‾) for norm one tori T=RK/k(1)(Gm) gave necessary sufficient condition Hasse principle extensions K/k number fields with [K:k]=n≤15 n≠12. In this paper, we determine 64 cases H1(k,PicX‾)≠0 give [K:k]=12.