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Cuchulainn
Posts: 62604
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### Re: what did the painter do wrong?

@"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
And as a number?
2.646 - it's easy to obtain by hand.
using fixed point above
$n, a_{n}$
////////////////////
1, 4
2, 2.875
3, 2.6548
4, 2.6457
Step over the gap, not into it. Watch the space between platform and train.
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Cuchulainn
Posts: 62604
Joined: July 16th, 2004, 7:38 am
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### Re: what did the painter do wrong?

There is hope according to Vico

During the Renaissance, imagination prevailed and many wonderful things were accomplished. Since then, Western civilization has gradually devolved into what we see today. We are close to the next turning point.  All societies undergo similar periods of evolution and devolution, as well.
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

katastrofa
Posts: 9444
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what did the painter do wrong?

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.

Cuchulainn
Posts: 62604
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: what did the painter do wrong?

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.
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).
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.
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

Collector
Topic Author
Posts: 4685
Joined: August 21st, 2001, 12:37 pm

### Re: what did the painter do wrong?

Any conclusion yet? closing in on the light? or still in the darkness? any word from the jury yet?  needs another  over-painting or not?

few can paint light like that! one light on, another off...

katastrofa
Posts: 9444
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what did the painter do wrong?

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.
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).
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 sequence

Cuchulainn
Posts: 62604
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: what did the painter do wrong?

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.
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).
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 sequence
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

Collector
Topic Author
Posts: 4685
Joined: August 21st, 2001, 12:37 pm

### Re: what did the painter do wrong?

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).
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 sequence
and when fate hands you demons, make a Demonade. Maxwell's Demonade!  Is it vegan or not?

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Re: what did the painter do wrong?

Which leads T4As to the proof that it's a Cauchy sequence
and when fate hands you demons, make a Demonade. Maxwell's Demonade!  Is it vegan or not?
Who knows if it's vegan but at least the beverage always stays cold no matter what the conditions.

katastrofa
Posts: 9444
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: what did the painter do wrong?

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?

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Re: what did the painter do wrong?

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?
That sounds like Raynaud's Syndrome.

Cuchulainn
Posts: 62604
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: what did the painter do wrong?

Any conclusion yet? closing in on the light? or still in the darkness? any word from the jury yet?  needs another  over-painting or not?
yes! See Kevin Spacey' blackboard QED
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

Cuchulainn
Posts: 62604
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: what did the painter do wrong?

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.
so he messed up that also?
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...?
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?
Last edited by Cuchulainn on June 2nd, 2018, 3:45 pm, edited 1 time in total.
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

Cuchulainn
Posts: 62604
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: what did the painter do wrong?

The artist knew his Joyce

Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

Collector
Topic Author
Posts: 4685
Joined: August 21st, 2001, 12:37 pm

### Re: what did the painter do wrong?

Any conclusion yet? closing in on the light? or still in the darkness? any word from the jury yet?  needs another  over-painting or not?
yes! See Kevin Spacey' blackboard QED
"Grattan-Guinness 1970 has suggested that Cauchy "stole" this and other ideas from Bolzano's paper of 1817."