. ال َسنت طيقنتلا تامالع ! The three most commonly used matrix norms in numerical analysis are particular cases of the Hőlder p-norm = max ≠0 , ∈ ℝ×, where ≥ 1 and = =1 1/ .
matrix A is an ill- conditioned matrix. If > 1, < 0. If > 1 ℎ < 0. Or For = , 2.2 holds with = 0. Or For = , inquality 2.2 holds with = 0. • صنلا نمض زومرلل بسانملا عضوملا رتخا
I have done an easy special case. • By straightforward computation = I lost my notes. • The details are left to the reader = I can't do it. • The following alternative proof of X's result may be of interest = I cannot understand X. • It will be observed that = I hope you hadn't noticed that. • Correct to within an order of magnitude = wrong. • ءاربخلا تاحلطصم مدختسا ..
لواح ! Fast Methods for Computing Up to the 1970s all calculations of were done using linearly on convergent methods based on series, products, or continued fractions. For these, doubling the number of digits in the result roughly doubles the work. Brent and Salamin independently discovered a quadratically convergent iteration for in 1976. For such an iteration, the errors satisfy +1 = 2 , so each iteration step approximately doubles the number of correct digits. Borwein and Borwein (1984) discovered another quadratically convergent iteration, and later found even higher order iterations. Their 1984 iteration is 0 = 2, 0 = 0, 0 = 2 + 2, +1 = 1 2 1/2 + −1/2 , +1 = 1/2 + 1 + , +1 = +1 1 + +1 1 + +1 , For which +1 − ≤ 1 102 .
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