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using fixed point above2.646 - it's easy to obtain by hand.And as a number?@"Compute [$]\sqrt{7}[$] by hand to two decimal positions accuracy. And make it snappy."
sqrt(64) - 1/2/sqrt(64) = 8 - 1/16 \approx 3*sqrt(7) => sqrt(7) \approx 127/16/3
This discrepancy can possibly be attributed to the fact that CF has convergence rate bounded by CF for Golden Ratio, i.e. 0.3819.. (Corollary 1.5).Sorry, the fraction 127/16/3 is precisely 2.6458(3). You required two decimal points and accidentally my method sufficed. It's not a robust method of estimating decimal values of square roots, obviously.
Which leads T4As to the proof that it's a Cauchy sequenceThis discrepancy can possibly be attributed to the fact that CF has convergence rate bounded by CF for Golden Ratio, i.e. 0.3819.. (Corollary 1.5).Sorry, the fraction 127/16/3 is precisely 2.6458(3). You required two decimal points and accidentally my method sufficed. It's not a robust method of estimating decimal values of square roots, obviously.
http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf
The fixed-point algo has (at most) linear convergence as we can see. Aitken acceleration gives 2nd order acceleration.
Which leads T4As to the proof that it's a Cauchy sequenceThis discrepancy can possibly be attributed to the fact that CF has convergence rate bounded by CF for Golden Ratio, i.e. 0.3819.. (Corollary 1.5).Sorry, the fraction 127/16/3 is precisely 2.6458(3). You required two decimal points and accidentally my method sufficed. It's not a robust method of estimating decimal values of square roots, obviously.
http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf
The fixed-point algo has (at most) linear convergence as we can see. Aitken acceleration gives 2nd order acceleration.
and when fate hands you demons, make a Demonade. Maxwell's Demonade! Is it vegan or not?Which leads T4As to the proof that it's a Cauchy sequenceThis discrepancy can possibly be attributed to the fact that CF has convergence rate bounded by CF for Golden Ratio, i.e. 0.3819.. (Corollary 1.5).
http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf
The fixed-point algo has (at most) linear convergence as we can see. Aitken acceleration gives 2nd order acceleration.
Who knows if it's vegan but at least the beverage always stays cold no matter what the conditions.and when fate hands you demons, make a Demonade. Maxwell's Demonade! Is it vegan or not?Which leads T4As to the proof that it's a Cauchy sequence
That sounds like Raynaud's Syndrome.Definitely vegan in that case. I'm always cold - except in the middle of the summer in Italy when temperatures go above 30C. My hands and feet are still cold though. Anyway, doesn't one also need some sugar for the lemonade? Or is this another (sugar) tax avoidance scheme?
yes! See Kevin Spacey' blackboard QEDAny conclusion yet? closing in on the light? or still in the darkness? any word from the jury yet? needs another over-painting or not?
What yer Norwegian artist (like Kevin Spacey) wanted to say was that every algebraic number such as [$]\sqrt n[$] for each [$]n \in \mathbb{Z} [$] has a continued fraction expansion (but there was not enough space on the margin). So, can we close this discussion by saying that each algebraic number can be characterised by Cauchy sequence of continued fractions?so he messed up that also?The [$]\sqrt 7[$] identity is wrong. The continued fraction should be the golden ratio with the only radical being [$]\sqrt 5[$]. So the two side cannot be equal.
well supposedly he forgot the ln in front of 2, so they had to add "ln" above the 2 later on, can be seen from the painting kind off. And now u say one more error, I need to tell them and ask if they can fix it. this is what happen when one hire painters/artists with no math skills?
or a math-smart painter putting in math errors on purpose? wanting people to think...?
"Grattan-Guinness 1970 has suggested that Cauchy "stole" this and other ideas from Bolzano's paper of 1817."yes! See Kevin Spacey' blackboard QEDAny conclusion yet? closing in on the light? or still in the darkness? any word from the jury yet? needs another over-painting or not?
