Real solutions of third and fourth degree equations

A few hours ago I "discovered" that if a third or fourth degree equation has distinct real solutions, it’s possible to calculate them avoiding complex numbers.

In particular, we have:

poly = (x - 1)(x - 2)(x - 3); {c, b, a} = CoefficientList[poly, x][[1 ;; 3]]; d = a^2/3 - b; e = 2 a^3/27 - a b/3 + c; f = ArcCos[-3 Sqrt[3] e/(2 d Sqrt[d])]; N[{-a/3 + 2 Sqrt[3 d] Cos[(f - 4 Pi)/3]/3,    -a/3 + 2 Sqrt[3 d] Cos[(f - 2 Pi)/3]/3,    -a/3 + 2 Sqrt[3 d] Cos[(f - 0 Pi)/3]/3}] 

{1., 2., 3.}

poly = (x - 1)(x - 2)(x - 3)(x - 4); {d, c, b, a} = CoefficientList[poly, x][[1 ;; 4]]; e = 3 a^2/4 - 2 b; f = 2 c - a b + a^3/4; g = b^2 + 12 d - 3 a c; h = 27 a^2 d - 9 a b c + 2 b^3 - 72 b d + 27 c^2; i = ArcCos[h/(2 g Sqrt[g])]; j = (e + 2 Sqrt[g] Cos[i/3])/3; N[{-a/4 - (Sqrt[j] + Sqrt[e - j + f/Sqrt[j]])/2,    -a/4 - (Sqrt[j] - Sqrt[e - j + f/Sqrt[j]])/2,    -a/4 + (Sqrt[j] - Sqrt[e - j - f/Sqrt[j]])/2,    -a/4 + (Sqrt[j] + Sqrt[e - j - f/Sqrt[j]])/2}] 

{1., 2., 3., 4.}

After a few moments of joy, however, I noticed that, for example, if I write (x - 10^-15) instead of (x - 1), I get respectively 6.66134*10^-16 and 8.88178*10^-16 instead of 10^-15 as the solution.

I intuitively believe that it’s a numerical problem and that the main cause is ArcCos, but I’m not too sure and also I’m not able to establish if something can be done to solve the issue, or if I have to give up and I must necessarily rely on it to the good Newton method.

Thank you!

Is there any real distinction between a ring and wondrous item?

In 3.5 edition (and Pathfinder), there was a distinction between rings and wondrous items, including in part that there were distinctions within wondrous items, including what applied to various body parts and slotless items. Though I see rings remain classified as "rings" and not "wondrous items" in 5th edition, I can’t locate any particular rule that identifies why there’s a distinction. (The only rule that guides magic items appears to be attunement and realism regarding body parts.) So, is there a reason that there remains a distinction between these types of magic items, or is this a remnant from older editions of the game?

Managed WordPress Hosting, Optimized for WP, NVMe Storage, Real Time Backup, 50% OFF

Raisinghost offers one of the best managed wordpress hosting in the hosting industry along with a fast, secure, and reliable cloud hosting platform that is equipped with the latest speed technology and NVme SSD storage. We have designed our wordpress hosting plans with the perfect resources and have single click script installer so can install/update your wp installation any time from your hosting cPanel. cPanel also allows to set free ssl and always offers you the easy graphical interface to manage the things.

Raisinghost also provides shared, reseller, VPS, cloud, dedicated services at guaranteed competitive prices. The features an in-house staff, wholly owned network and server hardware, and certified techs available 24/7/365 to ensure your online venture is an outstanding success also most of our services are covered with the 30 days money back guarantee!

Running with 50% OFF : SPECIAL50

Why Raisinghost? :

Easy and friendly control panel named cPanel
30 Day Money Back Guarantee
24/7/365 Technical Support
99.9% Uptime Guarantee
20x Times Faster Solid-State Drive Storage

Economy-WP : $0.5/M

– Single Site Hosting
– 2 Mysql Databases
– 10 Email Accounts
– 10 Subdomains
– Tier1 Technical Support

Value-WP : $1/mo

– Double Site Hosting
– 4 Mysql Databases
– 20GBSSD Web Space
– Unlimited Email Accounts
– Unlimited Subdomains
– Tier2 Technical Support

Deluxe-WP : $3/mo

– 15 Site Hosting
– 50 Mysql Databases
– 50GBSSD Web Space
– 14 Addon domains
– Unlimited Subdomains
– Tier2 Technical Support

Ultimate-WP : $5/mo

– Free Domain Name
– Unlimited Site Hosting
– Unlimited Mysql Databases
– Unlimited Web Space
– Unlimited Subdomains
– Tier3 Technical Support

Join Now :

Thank you.

Help me understand the REAL deal about SMM Panels

So I have been involved in online advertising to clients across the globe for almost 20 years off and on. My most recent return was about 6 weeks ago hoping to build a viable business again but it was after about large break. When I left; interstitial ads, keyword traffic,desktop ads were the hot items with mobile and social on the horizon. I come back to SMM and organic and API driven options. In research I see many options with SMM Panels to obtain deep discount prices almost lower than…

Help me understand the REAL deal about SMM Panels

Are any of the kingdoms/realms of the Forgotten Realms based on real world civilisations?

Have designers/writers for D&D ever cited explicit real world corolaries (current or historical) for any of the civilizations or peoples of the Forgotten Realms?

I’m looking at building a new character for an upcoming campaign and I think it would help me to role play better if I can contextualize it as a citizen of a real country or region.

Course teaching time complexities in real life systems

Having mis-read What course in CS deals with the study of RAM, CPU, Storage? I now wonder what course in CS deals with time complexities including GPUs, CPU caches in multiple levels, seek times on hard disk vs. SSDs, and bandwidth to disk and RAM.

I was taught the big O-notation but it never took into account that I might have a GPU with 100s of cores, or a limited amount of extremely fast cache, or a harddisk that is has a high bandwidth, but a high seek time.

Which class teaches this extended version of algorithm time complexities, which takes real world limitations into account?

Real value of MAC models in Linux

I have read about MAC vs. DAC in the Internet, but I still fail to understand, what kind of attack it is impossible to protect against if one only uses DAC+capabilities in comparison to MAC+DAC+capabilities. If a process does not run as root and lacks CAP_DAC_OVERRIDE, CAP_FOWNER and other dangerous capabilities, it cannot overcome the already assigned ownership and ACL’s of the resources it uses. On the other hand, if a process runs as root and has CAP_MAC_ADMIN, it can overwrite the security context enforced by MAC.

So is MAC "just additional layer of protection" without any real advantage on modern Linux system?