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Yahoo! Mail

''' Yahoo ! Mail (also written as Yahoo Mail''') is an email service offered by the American company Yahoo, Inc. The service is free for personal use, with an optional monthly fee for additional features. Business email was previously available with the Yahoo! Small Business brand, before it transitioned to Verizon Small Business Essentials in early 2022. Launched on October 8, 1997, as of January 2020, Yahoo! Mail has 225 million users.

Users are able to access and manage their mailboxes using webmail interface, accessible using a standard web browser. Some accounts also supported the use of standard mail protocols (POP3 and SMTP). Since 2015, users can also connect non-Yahoo e-mail accounts to the webmail client.

New Yahoo! Mail accounts, and most of the service's accounts, use  as the email suffix. Previously, users could choose  or   as a suffix, or one of several country-specific suffixes. Many countries were available, such as  in the United Kingdom,   in France (also used by francophones) and   in Italy. While these suffixes are discontinued for new accounts, they are preserved for existing accounts. Yahoo! Japan Mail, a separate service, offers both  and   as suffixes. Internet service providers using Yahoo! Mail offer their own suffixes for subscribers, with AT&T also offering free accounts to non-subscribers.

1997–2002
Yahoo! made a deal with the online communications company Four11 for co-branded white pages. Marvin Gavin, who worked at Four11 as the director of international business development, said "We always had a bias about being acquired by Yahoo. They were more entrepreneurial than Microsoft. We had a great cultural fit – it made a lot of sense." In the end, Yahoo! acquired Four11 for $96 million. The purpose behind acquiring Four11 was the company's RocketMail webmail service, which launched in 1997. Yahoo! announced the acquisition on October 8, 1997, close to the time that Yahoo! Mail was launched. Yahoo! chose acquisition rather than internal platform development, because, as Healy said, "Hotmail was growing at thousands and thousands users per week. We did an analysis. For us to build, it would have taken four to six months, and by then, so many users would have taken an email account. The speed of the market was critical."

The transition to Yahoo! Mail was not easy for many Rocketmail users. On March 21, 2002, Yahoo! eliminated free software client access and introduced the $29.99 per year Mail Forwarding Service. Mary Osako, a Yahoo! Spokeswoman, told CNET, "For-pay services on Yahoo!, originally launched in February 1999, have experienced great acceptance from our base of active registered users, and we expect this adoption to continue to grow."

2002–2010
During 2002, the Yahoo! network was gradually redesigned. On July 2, 2002, the company website was changed, and it was announced that Yahoo! Mail and other services would also be redesigned. Along with this new design, new features were to be implemented, including drop-down menus in DHTML, different category tabs, and a new user-customizable color scheme. Yahoo! Network Services senior vice president Geoff Ralston stated that

"The launch of Yahoo! Mail Plus is part of Yahoo's strategic initiative to offer premium services that deliver innovative, reliable and relevant services to consumers ... In just five years, Yahoo! Mail has grown from one million to tens of millions of users, illustrating how consumers have made email an essential part of their daily lives. Through Yahoo! Mail Plus, Yahoo! continues to demonstrate leadership and innovation by offering consumers the industry's most complete and powerful email solution."

On July 9, 2004, Yahoo! acquired Oddpost, a webmail service which simulated a desktop email client. Oddpost had features such as drag-and-drop support, right-click menus, RSS feeds, a preview pane, and increased speed using email caching to shorten response time. Many of the features were incorporated into an updated Yahoo! Mail service.

Competition
On April 1, 2004, Google announced its Gmail service with 1 GB of storage, however Gmail's invitation-only accounts kept the other webmail services at the forefront. Most major webmail providers, including Yahoo! Mail, increased their mailbox storage in response. Yahoo! first announced 100 MB of storage for basic accounts and 2 GB of storage for premium users. However, soon Yahoo! Mail increased its free storage quota to 1 GB, before eventually allowing unlimited storage. Yahoo! Mail had free unlimited storage from March 27, 2007, until October 8, 2013.

2011–present
On September 17, 2010, Yahoo! showed off a new Mail program to reporters. Codenamed "Minty", the 2011 release was announced on September 16, 2010. It included a new interface, enhanced performance, improved Facebook and Twitter integration, the ability to watch YouTube videos straight from email, and improved search. Public beta began on October 26, 2010. In May 2011, the new Yahoo! Mail became the default interface. As the new interface became mandatory for users, some users of Yahoo! Mail reported slow typing speeds, contradicting Yahoo's claims of faster performance. Yahoo! offered no resolution to the problem. Users also missed the ability to paste textual email addresses into the sender box. The new version disabled the use of the "secondary" addresses provided in the previous version. The new interface overrode the browser's right mouse button (making functions such as opening mails in new tab windows unavailable).

In 2013, Yahoo! redesigned the site and removed several features, such as simultaneously opening multiple emails in tabs, sorting by sender name, and dragging mails to folders. The new email interface was geared to give an improved user-experience for mobile devices, but was criticized for having an inferior desktop interface. Many users objected to the unannounced nature of the changes through an online post asking Yahoo! to bring back mail tabs with one hundred thousand voting and nearly ten thousand commenting. The redesign produced a problem that caused an unknown number of users to lose access to their accounts for several weeks.

In December 2013, Yahoo! Mail suffered a major outage where approximately one million users, one percent of the site's total users, could not access their emails for several days. Then Yahoo! CEO Marissa Mayer publicly apologized to the site's users.

China Yahoo! Mail officially announced its shutdown on April 18, 2013. Users were warned that all emails, contacts, and account settings would be inaccessible, unless users migrated to the American version of Yahoo! Mail. Individuals who made China Yahoo! Mail accounts during the Alibaba takeover were required to create new accounts under new usernames.

In January 2014, an undisclosed number of usernames and passwords were released to hackers, following a security breach that Yahoo! believed had occurred through a third-party website. Yahoo! contacted affected users and requested that passwords be changed.

In October 2015, Yahoo! updated the mail service with a "more subtle" redesign, as well as improved mobile features. The same release introduced the Yahoo! Account Key, a smartphone-based replacement for password logins.

In 2017, Yahoo! again redesigned the web interface with a "more minimal" look, and introduced the option to customize it with different color themes and layouts.

In 2020, Yahoo! added a feature to view NFL matches.

User interface
As many as three web interfaces were available at any given time. The traditional "Yahoo! Mail Classic" preserved the availability of their original 1997 interface until July 2013 in North America. A 2005 version included a new Ajax interface, drag-and-drop, improved search, keyboard shortcuts, address auto-completion, and tabs. However, other features were removed, such as column widths and one click delete-move-to-next. In October 2010, Yahoo! released a beta version of Yahoo! Mail, which included improvements to performance, search, and Facebook integration. In May 2011, this became the default interface. Their current Webmail interface was introduced in 2017.

Spam policy
Yahoo! Mail is often used by spammers to provide a "remove me" email address. Often, these addresses are used to verify the recipient's address, thus opening the door for more spam.

Yahoo! does not tolerate this practice and terminates accounts connected with spam-related activities without warning, causing spammers to lose access to any other Yahoo! services connected with their ID under the Terms of Service. Additionally, Yahoo! stresses that its servers are based in California and any spam-related activity which uses its servers could potentially violate that state's anti-spam laws.

In February 2006, Yahoo! announced its decision (along with AOL) to give some organizations the option to "certify" mail by paying up to one cent for each outgoing message, allowing the mail in question to bypass inbound spam filters.

Filters
In order to prevent abuse, in 2002 Yahoo! Mail activated filters which changed certain words (that could trigger unwanted JavaScript events) and word fragments into other words. "mocha" was changed to "espresso", "expression" became "statement", and "eval" (short for "evaluation") became "review". This resulted in many unintended corrections, such as "prevent" (prevalent), "revalidation" (evaluation) and "media review" (medieval).

When asked about these changes, Yahoo! explained that the changed words were common terms used in their privacy dashboard and were blacklisted to prevent hackers from sending damaging commands via the program's HTML function. Starting before February 7, 2006, Yahoo! Mail ended the practice, and began to add an underscore as a prefix to certain suspicious words and word fragments.

Greylisting
Incoming mail to Yahoo! addresses can be subjected to deferred delivery as part of Yahoo's incoming spam controls. This can delay delivery of mail sent to Yahoo! addresses without the sender or recipients being aware of it. The deferral is typically of short duration, but may extend up to several hours. Yahoo! does not specifically document this policy in detail, although some information is available.

Shi Tao arrest
In 2004, Yahoo's Hong Kong office provided technical information to the Chinese authorities about the account of journalist Shi Tao, who was subsequently sentenced to ten years' imprisonment for "leaking state secrets". Yahoo! was criticized by Reporters Without Borders for acting as a "police informant" to increase its profits. In August 2007, the United States Congress began an investigation into Yahoo's handling of the case. Yahoo! founder Jerry Yang testified before Congress. On November 6, 2007, the congressional panel criticized Yahoo! for not giving full details to the House Foreign Affairs Committee the previous year, stating it had been "at best inexcusably negligent" and at worst "deceptive". Representative Tom Lantos described its executives as moral "pygmies". Yang stated that Yahoo! no longer controlled its Chinese operations, and was collaborating with human rights groups to formulate ethical code for technology companies.

In a February 2006 hearing, Yahoo! executives swore that they had received no information about the investigation. Several months later, it was discovered that the document provided to Yahoo! China on April 22, 2004, by the Beijing State Security Bureau stated that "Your office is in possession of the following items relating to a case of suspected illegal provision of state secrets to foreign entities."

On November 13, 2007, Yahoo! settled with Shi for an undisclosed sum. Shi was released from prison in September 2013.

Username bans
On February 20, 2006, it was revealed that Yahoo! Mail was banning the word "Allah" in email usernames, both separately and as part of a user name such as. Shortly after the news of the ban, it was lifted on February 23, 2006. Along with this action, Yahoo! also made the following statement:

"We continuously evaluate abuse patterns in registration usernames to help prevent spam, fraud and other inappropriate behavior. A small number of people registered for IDs using specific terms with the sole purpose of promoting hate, and then used those IDs to post content that was harmful or threatening to others, thus violating Yahoo!'s Terms of Service.

'Allah' was one word being used for these purposes, with instances tied to defamatory language. We took steps to help protect our users by prohibiting use of the term in Yahoo! usernames. We recently re-evaluated the term 'Allah' and users can now register for IDs with this word because it is no longer a significant target for abuse. We regularly evaluate this type of activity and will continue to make adjustments to our registration process to help foster a positive customer experience."

Exploit
In November 2012, an exploit for Yahoo! Mail was sold for $700 by an Egyptian hacker, allowing hijackers to hack Yahoo! Mail user accounts and redirect users to a malicious website. The attack used cross-site scripting which let hackers steal cookies. In January 2013, hacker and security researcher Shahin Ramezany pointed out another DOM-based XSS loophole that placed 400 million users at risk.

Deletion of accounts
In 2013, Yahoo! deleted accounts that hadn't been logged into for more than 12 months and gave those names to other users. This included accounts that had set up email forwarding to other accounts, who didn't notice and act upon the message to preserve their account. Spillover from this action resulted in some users losing access to accounts on other services as their email resets were no longer forwarded to them, but to the new owner of their former account.

Phishing attack
From 2007, Yahoo! was the email service used by New Zealand Telecom, which came under criticism in early 2013 following a spam and phishing attack that was described as the biggest to have ever hit the country. Telecom and Yahoo! automatically reset "about 60,000" users' passwords. In April, Telecom announced that despite the issue, it would retain Yahoo! as an email provider.

Account theft
On October 3, 2017, Yahoo! reported that all Yahoo! user accounts, approximately 3 billion, were affected by the previously announced August 2013 theft of accounts. This information updates the December 14, 2016, announcement that more than 1 billion user accounts were hacked in a breach that had occurred back in 2013. Earlier that year in September, Yahoo! announced that an additional 500 million user accounts had been breached in 2014. The company was said to have discovered about the breach that affected hundreds of millions of accounts years before their initial announcement.

Automated scanning of email content
The contents of Yahoo! Mail messages are scanned for the purposes of targeted advertising, in contrast to its main competitors Gmail (which ended the practice in late 2017) and Outlook.com (which did not employ the practice).

Content sharing with the National Security Agency
In 2015, Reuters reported that Yahoo! has implemented a functionality to scan all incoming emails for specific keywords and share email content in real time with the US National Security Agency.

Platforms
The mobile app for Yahoo! Mail can be downloaded from the App Store for iOS, and the Google Play Store for Android. Both versions of the app perform the same function, but are tailored to the mobile device's operating system. Some characteristics include replying to a message by opening a notification and separately archiving, deleting and starring notifications. It is available in several languages.

title: Yahoo! Mail


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Plantaginaceae

Plantaginaceae, the plantain family, is a large, diverse family of flowering plants in the order Lamiales that includes common flowers such as snapdragon and foxglove. It is unrelated to the banana-like fruit also called "plantain." In older classifications, Plantaginaceae was the only family of the order Plantaginales, but numerous phylogenetic studies, summarized by the Angiosperm Phylogeny Group, have demonstrated that this taxon should be included within Lamiales.

Overview
The plantain family as traditionally circumscribed consisted of only three genera: Bougueria, Littorella, and Plantago. However phylogenetic research has indicated that Plantaginaceae sensu stricto (in the strict sense) were nested within Scrophulariaceae (but forming a group that did not include the type genus of that family, Scrophularia). Although Veronicaceae (1782) is the oldest family name for this group, Plantaginaceae (1789) is a conserved name under the International Code of Botanical Nomenclature (ICBN) and thus has priority over any earlier family name for a family including Plantago. Furthermore, the ICBN does not consider family names published before 1789 to be names eligible for conservation, thus ruling out Veronicaceae. The name Antirrhinaceae has been proposed for conservation over Plantaginaceae. In the meantime, the Angiosperm Phylogeny Group has accepted the name Plantaginaceae. However, Olmstead (2003) chose to use the name Veronicaceae, a later synonym.

A group of genera including Lindernia has now been segregated as the family Linderniaceae, which is recognized by Haston et al. 2007 (also known as LAPG II) as "Post-APG II family".

Plantaginaceae sensu lato (in the broad sense) are a diverse, cosmopolitan family, occurring mostly in temperate zones. The group consists of herbs, shrubs and also a few aquatic plants with roots (such as the genus Callitriche). As the family is so diverse, its circumscription is difficult to establish.

The leaves are spiral to opposite and simple to compound. Unusual in Lamiales is the absence of vertical partitions in the heads of the glandular hairs.

The structure and form of the flowers is variable. Some genera are 4-merous (i.e., with 4 sepals and 4 petals), such as Aragoa (but this one has 5 sepals); others are 5- to 8-merous, such as Sibthorpia. The flowers of most genera are polysymmetric. The corolla is often two-lipped. In some taxa, the androecium is formed before the corolla.

The fruit is a loculicidal capsule, dehiscing through the partitions between the cells.

Genera
The enlarged Plantaginaceae consists of 107 genera and about 1,900 species. The largest genus is Veronica, with about 450 species. Veronica also includes the genera Hebe, Parahebe and Synthyris, formerly often treated as distinct. All genera of Plantaginaceae were formerly included in Scrophulariaceae except where otherwise stated.

Although GRIN includes Lafuentea Lag. in the tribe Antirrhineae, in the phylogenetic analysis of Fernández-Mazuecos et al. (2013) it was a sister to the Antirrhineae, as also noted by Albach (2005). For the time being it should be considered an outgroup.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Plantaginaceae


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Ninth Amendment

Ninth Amendment may refer to the:
 * Ninth Amendment to the United States Constitution, part of the Bill of Rights
 * Ninth Amendment of the Constitution of India, 1961 amendment allowing transfer of territory to Pakistan in Berubari, West Bengal following certain treaties between the countries including the Nehru-Noon Agreement relating to India-East Pakistan enclaves
 * Ninth Amendment of the Constitution of Ireland, extended the right to vote to certain non-citizens
 * Ninth Amendment of the Constitution of South Africa

title: Ninth Amendment


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Regula falsi

In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and error technique of using test ("false") values for the variable and then adjusting the test value according to the outcome. This is sometimes also referred to as "guess and check". Versions of the method predate the advent of algebra and the use of equations.

As an example, consider problem 26 in the Rhind papyrus, which asks for a solution of (written in modern notation) the equation $x + x⁄4 = 15$. This is solved by false position. First, guess that $x = 4$ to obtain, on the left, 4 + $4⁄4$ = 5. This guess is a good choice since it produces an integer value. However, 4 is not the solution of the original equation, as it gives a value which is three times too small. To compensate, multiply $x$ (currently set to 4) by 3 and substitute again to get 12 + $12⁄4$ = 15, verifying that the solution is $x = 12$.

Modern versions of the technique employ systematic ways of choosing new test values and are concerned with the questions of whether or not an approximation to a solution can be obtained, and if it can, how fast can the approximation be found.

Two historical types
Two basic types of false position method can be distinguished historically, simple false position and double false position.

Simple false position is aimed at solving problems involving direct proportion. Such problems can be written algebraically in the form: determine $x$ such that
 * $$ax = b ,$$

if $a$ and $b$ are known. The method begins by using a test input value $x′$, and finding the corresponding output value $b′$ by multiplication: $ax′ = b′$. The correct answer is then found by proportional adjustment, $x = b⁄b′ x′$.

Double false position is aimed at solving more difficult problems that can be written algebraically in the form: determine $x$ such that
 * $$f(x) = ax + c = 0 ,$$

if it is known that
 * $$f(x_1) = b_1, \qquad f(x_2) = b_2 .$$

Double false position is mathematically equivalent to linear interpolation. By using a pair of test inputs and the corresponding pair of outputs, the result of this algorithm given by,
 * $$ x = \frac{b_1 x_2 - b_2 x_1}{b_1 - b_2},$$

would be memorized and carried out by rote. Indeed, the rule as given by Robert Recorde in his Ground of Artes (c. 1542) is:

 * Gesse at this woorke as happe doth leade.
 * By chaunce to truthe you may procede.
 * And firste woorke by the question,
 * Although no truthe therein be don.
 * Suche falsehode is so good a grounde,
 * That truth by it will soone be founde.
 * From many bate to many mo,
 * From to fewe take to fewe also.
 * With to much ioyne to fewe againe,
 * To to fewe adde to manye plaine.
 * In crossewaies multiplye contrary kinde,
 * All truthe by falsehode for to fynde.

For an affine linear function,
 * $$f(x) = ax + c ,$$

double false position provides the exact solution, while for a nonlinear function $f$ it provides an approximation that can be successively improved by iteration.

History
The simple false position technique is found in cuneiform tablets from ancient Babylonian mathematics, and in papyri from ancient Egyptian mathematics.

Double false position arose in late antiquity as a purely arithmetical algorithm. In the ancient Chinese mathematical text called The Nine Chapters on the Mathematical Art (九章算術), dated from 200 BC to AD 100, most of Chapter 7 was devoted to the algorithm. There, the procedure was justified by concrete arithmetical arguments, then applied creatively to a wide variety of story problems, including one involving what we would call secant lines on a conic section. A more typical example is this "joint purchase" problem involving an "excess and deficit" condition: Now an item is purchased jointly; everyone contributes 8 [coins], the excess is 3; everyone contributes 7, the deficit is 4. Tell: The number of people, the item price, what is each? Answer: 7 people, item price 53.

Between the 9th and 10th centuries, the Egyptian mathematician Abu Kamil wrote a now-lost treatise on the use of double false position, known as the Book of the Two Errors (Kitāb al-khaṭāʾayn). The oldest surviving writing on double false position from the Middle East is that of Qusta ibn Luqa (10th century), an Arab mathematician from Baalbek, Lebanon. He justified the technique by a formal, Euclidean-style geometric proof. Within the tradition of medieval Muslim mathematics, double false position was known as hisāb al-khaṭāʾayn ("reckoning by two errors"). It was used for centuries to solve practical problems such as commercial and juridical questions (estate partitions according to rules of Quranic inheritance), as well as purely recreational problems. The algorithm was often memorized with the aid of mnemonics, such as a verse attributed to Ibn al-Yasamin and balance-scale diagrams explained by al-Hassar and Ibn al-Banna, all three being mathematicians of Moroccan origin.

Leonardo of Pisa (Fibonacci) devoted Chapter 13 of his book Liber Abaci (AD 1202) to explaining and demonstrating the uses of double false position, terming the method regulis elchatayn after the al-khaṭāʾayn method that he had learned from Arab sources. In 1494, Pacioli used the term el cataym in his book Summa de arithmetica, probably taking the term from Fibonacci. Other European writers would follow Pacioli and sometimes provided a translation into Latin or the vernacular. For instance, Tartaglia translates the Latinized version of Pacioli's term into the vernacular "false positions" in 1556. Pacioli's term nearly disappeared in the 16th century European works and the technique went by various names such as "Rule of False", "Rule of Position" and "Rule of False Position". Regula Falsi appears as the Latinized version of Rule of False as early as 1690.

Several 16th century European authors felt the need to apologize for the name of the method in a science that seeks to find the truth. For instance, in 1568 Humphrey Baker says: "The Rule of falsehoode is so named not for that it teacheth anye deceyte or falsehoode, but that by fayned numbers taken at all aduentures, it teacheth to finde out the true number that is demaunded, and this of all the vulgar Rules which are in practise) is ye most excellence."

Numerical analysis
The method of false position provides an exact solution for linear functions, but more direct algebraic techniques have supplanted its use for these functions. However, in numerical analysis, double false position became a root-finding algorithm used in iterative numerical approximation techniques.

Many equations, including most of the more complicated ones, can be solved only by iterative numerical approximation. This consists of trial and error, in which various values of the unknown quantity are tried. That trial-and-error may be guided by calculating, at each step of the procedure, a new estimate for the solution. There are many ways to arrive at a calculated-estimate and regula falsi provides one of these.

Given an equation, move all of its terms to one side so that it has the form, $f (x) = 0$, where $f$ is some function of the unknown variable $x$. A value $c$ that satisfies this equation, that is, $f (c) = 0$, is called a root or zero of the function $f$ and is a solution of the original equation. If $f$ is a continuous function and there exist two points $a_{0}$ and $b_{0}$ such that $f (a_{0})$ and $f (b_{0})$ are of opposite signs, then, by the intermediate value theorem, the function $f$ has a root in the interval $(a_{0}, b_{0})$.

There are many root-finding algorithms that can be used to obtain approximations to such a root. One of the most common is Newton's method, but it can fail to find a root under certain circumstances and it may be computationally costly since it requires a computation of the function's derivative. Other methods are needed and one general class of methods are the two-point bracketing methods. These methods proceed by producing a sequence of shrinking intervals $[a_{k}, b_{k}]$, at the $k$th step, such that $(a_{k}, b_{k})$ contains a root of $f$.

Two-point bracketing methods
These methods start with two $x$-values, initially found by trial-and-error, at which $f (x)$ has opposite signs. Under the continuity assumption, a root of $f$ is guaranteed to lie between these two values, that is to say, these values "bracket" the root. A point strictly between these two values is then selected and used to create a smaller interval that still brackets a root. If $c$ is the point selected, then the smaller interval goes from $c$ to the endpoint where $f (x)$ has the sign opposite that of $f (c)$. In the improbable case that $f (c) = 0$, a root has been found and the algorithm stops. Otherwise, the procedure is repeated as often as necessary to obtain an approximation to the root to any desired accuracy.

The point selected in any current interval can be thought of as an estimate of the solution. The different variations of this method involve different ways of calculating this solution estimate.

Preserving the bracketing and ensuring that the solution estimates lie in the interior of the bracketing intervals guarantees that the solution estimates will converge toward the solution, a guarantee not available with other root finding methods such as Newton's method or the secant method.

The simplest variation, called the bisection method, calculates the solution estimate as the midpoint of the bracketing interval. That is, if at step $k$, the current bracketing interval is $[a_{k}, b_{k}]$, then the new solution estimate $c_{k}$ is obtained by,
 * $$c_k=\frac{a_k+b_k}{2}.$$

This ensures that $c_{k}$ is between $a_{k}$ and $b_{k}$, thereby guaranteeing convergence toward the solution.

Since the bracketing interval's length is halved at each step, the bisection method's error is, on average, halved with each iteration. Hence, every 3 iterations, the method gains approximately a factor of 23, i.e. roughly a decimal place, in accuracy.

The regula falsi (false position) method
The convergence rate of the bisection method could possibly be improved by using a different solution estimate.

The regula falsi method calculates the new solution estimate as the $f$-intercept of the line segment joining the endpoints of the function on the current bracketing interval. Essentially, the root is being approximated by replacing the actual function by a line segment on the bracketing interval and then using the classical double false position formula on that line segment.

More precisely, suppose that in the $x$-th iteration the bracketing interval is $(a_{k}, b_{k})$. Construct the line through the points $(a_{k}, f (a_{k}))$ and $(b_{k}, f (b_{k}))$, as illustrated. This line is a secant or chord of the graph of the function $f$. In point-slope form, its equation is given by

 * $$ y - f(b_k) = \frac{f(b_k)-f(a_k)}{b_k-a_k} (x-b_k). $$

Now choose $c_{k}$ to be the $k$-intercept of this line, that is, the value of $x$ for which $y = 0$, and substitute these values to obtain

 * $$ f(b_k) + \frac{f(b_k)-f(a_k)}{b_k-a_k} (c_k-b_k) = 0. $$

Solving this equation for ck gives:

c_k = b_k - f(b_k) \frac{b_k-a_k}{f(b_k)-f(a_k)} = \frac{a_k f(b_k) - b_k f(a_k)}{f(b_k) - f(a_k)}.$$

This last symmetrical form has a computational advantage:

As a solution is approached, $x$ and $a_{k}$ will be very close together, and nearly always of the same sign. Such a subtraction can lose significant digits. Because $f (b_{k})$ and $f (a_{k})$ are always of opposite sign the “subtraction” in the numerator of the improved formula is effectively an addition (as is the subtraction in the denominator too).

At iteration number $k$, the number $b_{k}$ is calculated as above and then, if $f (a_{k})$ and $f (c_{k})$ have the same sign, set $a_{k + 1} = c_{k}$ and $b_{k + 1} = b_{k}$, otherwise set $a_{k + 1} = a_{k}$ and $b_{k + 1} = c_{k}$. This process is repeated until the root is approximated sufficiently well.

The above formula is also used in the secant method, but the secant method always retains the last two computed points, and so, while it is slightly faster, it does not preserve bracketing and may not converge.

The fact that regula falsi always converges, and has versions that do well at avoiding slowdowns, makes it a good choice when speed is needed. However, its rate of convergence can drop below that of the bisection method.

Analysis
Since the initial end-points $a_{0}$ and $b_{0}$ are chosen such that $f (a_{0})$ and $f (b_{0})$ are of opposite signs, at each step, one of the end-points will get closer to a root of $f$. If the second derivative of $f$ is of constant sign (so there is no inflection point) in the interval, then one endpoint (the one where $f$ also has the same sign) will remain fixed for all subsequent iterations while the converging endpoint becomes updated. As a result, unlike the bisection method, the width of the bracket does not tend to zero (unless the zero is at an inflection point around which $sign(f ) = −sign(f)$). As a consequence, the linear approximation to $f (x)$, which is used to pick the false position, does not improve as rapidly as possible.

One example of this phenomenon is the function
 * $$ f(x) = 2x^3-4x^2+3x $$

on the initial bracket [−1,1]. The left end, −1, is never replaced (it does not change at first and after the first three iterations, $f$ is negative on the interval) and thus the width of the bracket never falls below 1. Hence, the right endpoint approaches 0 at a linear rate (the number of accurate digits grows linearly, with a rate of convergence of 2/3).

For discontinuous functions, this method can only be expected to find a point where the function changes sign (for example at $x = 0$ for $1/x$ or the sign function). In addition to sign changes, it is also possible for the method to converge to a point where the limit of the function is zero, even if the function is undefined (or has another value) at that point (for example at $x = 0$ for the function given by $f (x) = abs(x) − x^{2}$ when $x ≠ 0$ and by $f (0) = 5$, starting with the interval [-0.5, 3.0]). It is mathematically possible with discontinuous functions for the method to fail to converge to a zero limit or sign change, but this is not a problem in practice since it would require an infinite sequence of coincidences for both endpoints to get stuck converging to discontinuities where the sign does not change, for example at $x = ±1$ in
 * $$f(x) = \frac{1}{(x-1)^2} + \frac{1}{(x+1)^2}.$$

The method of bisection avoids this hypothetical convergence problem.

Improvements in regula falsi
Though regula falsi always converges, usually considerably faster than bisection, there are situations that can slow its convergence – sometimes to a prohibitive degree. That problem isn't unique to regula falsi: Other than bisection, all of the numerical equation-solving methods can have a slow-convergence or no-convergence problem under some conditions. Sometimes, Newton's method and the secant method diverge instead of converging – and often do so under the same conditions that slow regula falsi's convergence.

But, though regula falsi is one of the best methods, and even in its original un-improved version would often be the best choice; for example, when Newton's isn't used because the derivative is prohibitively time-consuming to evaluate, or when Newton's and Successive-Substitutions have failed to converge.

Regula falsi's failure mode is easy to detect: The same end-point is retained twice in a row. The problem is easily remedied by picking instead a modified false position, chosen to avoid slowdowns due to those relatively unusual unfavorable situations. A number of such improvements to regula falsi have been proposed; two of them, the Illinois algorithm and the Anderson–Björk algorithm, are described below.

The Illinois algorithm
The Illinois algorithm halves the $c_{k}$-value of the retained end point in the next estimate computation when the new $y$-value (that is, $f (c_{k}$)) has the same sign as the previous one ($f (c_{k − 1}$)), meaning that the end point of the previous step will be retained. Hence:

 * $$ c_k = \frac{\frac{1}{2}f(b_k) a_k - f(a_k) b_k}{\frac{1}{2}f(b_k) - f(a_k)}$$

or
 * $$ c_k = \frac{f(b_k) a_k - \frac{1}{2}f(a_k) b_k}{f(b_k) - \frac{1}{2}f(a_k)},$$

down-weighting one of the endpoint values to force the next $c_{k}$ to occur on that side of the function. The factor $y$ used above looks arbitrary, but it guarantees superlinear convergence (asymptotically, the algorithm will perform two regular steps after any modified step, and has order of convergence 1.442). There are other ways to pick the rescaling which give even better superlinear convergence rates.

The above adjustment to regula falsi is called the Illinois algorithm by some scholars. Ford (1995) summarizes and analyzes this and other similar superlinear variants of the method of false position.

Anderson–Björck algorithm
Suppose that in the $1⁄2$-th iteration the bracketing interval is $[a_{k}, b_{k}]$ and that the functional value of the new calculated estimate $c_{k}$ has the same sign as $f (b_{k})$. In this case, the new bracketing interval $[a_{k + 1}, b_{k + 1}] = [a_{k}, c_{k}]$ and the left-hand endpoint has been retained. (So far, that's the same as ordinary Regula Falsi and the Illinois algorithm.)

But, whereas the Illinois algorithm would multiply $f (a_{k})$ by $k$, Anderson–Björck algorithm multiplies it by $m$, where $m$ has one of the two following values:

$$ \begin{align} m' &= 1 - \frac{f(c_k)}{f(b_k)},\\ m &= \begin{cases} m' & \text{if } m' > 0, \\ \frac{1}{2} & \text{otherwise.} \end{cases} \end{align} $$

For simple roots, Anderson–Björck performs very well in practice.

ITP method
Given $$\kappa_1\in (0,\infty), \kappa_2 \in \left[1,1+\phi\right) $$, $$n_{1/2} \equiv \lceil(b_0-a_0)/2\epsilon\rceil $$ and $$n_0\in[0,\infty) $$  where $$\phi $$ is the golden ration $$\tfrac{1}{2}(1+\sqrt{5}) $$, in each iteration $$j = 0,1,2... $$ the ITP method calculates the point $$x_{\text{ITP}} $$ following three steps:

 * 1) [Interpolation Step]  Calculate the bisection and the regula falsi points:   $$x_{1/2} \equiv \frac{a+b}{2} $$   and   $$x_f \equiv \frac{bf(a)-af(b)}{f(a)-f(b)} $$ ;
 * 2) [Truncation Step] Perturb the estimator towards the center:  $$x_t \equiv x_f+\sigma \delta $$  where  $$\sigma \equiv \text{sign}(x_{1/2}-x_f) $$ and $$\delta \equiv \min\{\kappa_1|b-a|^{\kappa_2},|x_{1/2}-x_f|\} $$ ;
 * 3) [Projection Step] Project the estimator to minmax interval: $$x_{\text{ITP}} \equiv x_{1/2} -\sigma \rho_k  $$ where $$\rho_k \equiv \min\left\{\epsilon 2^{n_{1/2}+n_0-j} - \frac{b-a}{2},|x_t-x_{1/2}|\right\} $$.

The value of the function $$f(x_{\text{ITP}}) $$ on this point is queried, and the interval is then reduced to bracket the root by keeping the sub-interval with function values of opposite sign on each end. This three step procedure guarantees that the minmax properties of the bisection method are enjoyed by the estimate as well as the superlinear convergence of the secant method. And, is observed to outperform both bisection and interpolation based methods under smooth and non-smooth functions.

Practical considerations
When solving one equation, or just a few, using a computer, the bisection method is an adequate choice. Although bisection isn't as fast as the other methods—when they're at their best and don't have a problem—bisection nevertheless is guaranteed to converge at a useful rate, roughly halving the error with each iteration – gaining roughly a decimal place of accuracy with every 3 iterations.

For manual calculation, by calculator, one tends to want to use faster methods, and they usually, but not always, converge faster than bisection. But a computer, even using bisection, will solve an equation, to the desired accuracy, so rapidly that there's no need to try to save time by using a less reliable method—and every method is less reliable than bisection.

An exception would be if the computer program had to solve equations very many times during its run. Then the time saved by the faster methods could be significant.

Then, a program could start with Newton's method, and, if Newton's isn't converging, switch to regula falsi, maybe in one of its improved versions, such as the Illinois or Anderson–Björck versions. Or, if even that isn't converging as well as bisection would, switch to bisection, which always converges at a useful, if not spectacular, rate.

When the change in $y$ has become very small, and $x$ is also changing very little, then Newton's method most likely will not run into trouble, and will converge. So, under those favorable conditions, one could switch to Newton's method if one wanted the error to be very small and wanted very fast convergence.

Example: Growth of a bulrush
In chapter 7 of The Nine Chapters, a root finding problem can be translated to modern language as follows: Excess And Deficit Problem #11: Answer: $(2 + \frac{6}{13})$ days; the height is $(4 + \frac{8}{10} + \frac{6}{130})$ units.
 * A bulrush grew 3 units on its first day. At the end of each day, the plant is observed to have grown by $1⁄2$ of the previous day's growth.
 * A club-rush grew 1 unit on its first day. At the end of each day, the plant has grown by 2 times as much as the previous day's growth.
 * Find the time [in fractional days] that the club-rush becomes as tall as the bulrush.

Explanation:
 * Suppose it is day 2. The club-rush is shorter than the bulrush by 1.5 units.
 * Suppose it is day 3. The club-rush is taller than the bulrush by 1.75 units. ∎

To understand this, we shall model the heights of the plants on day $1⁄2$ ($F$ = 1, 2, 3...) after a geometric series.

 * $$B(n) = \sum_{i=1}^n 3 \cdot \frac{1}{2^{i-1}} \quad$$Bulrush
 * $$C(n) = \sum_{i=1}^n 1 \cdot 2^{i-1} \quad$$Club-rush

For the sake of better notations, let $$\ k = i-1 ~.$$ Rewrite the plant height series $$\ B(n),\ C(n)\ $$ in terms of $K$ and invoke the sum formula.

 * $$\ B(n) = \sum_{k=0}^{n-1} 3 \cdot \frac{1}{2^k} = 3 \left( \frac{1 - (\tfrac{1}{2})^{n-1+1}}{1 - \tfrac{1}{2}} \right) = 6 \left( 1 - \frac{1}{2^n} \right)$$
 * $$\ C(n) = \sum_{k=0}^{n-1} 2^k = \frac{ 1 - 2^n}{\ 1 - 2\ } = 2^n - 1\ $$

Now, use regula falsi to find the root of $$\ (C(n) - B(n))\ $$

 * $$\ F(n) := C(n) - B(n) = \frac{6}{2^n} + 2^n - 7\ $$

Set $$\ x_1 = 2\ $$ and compute $$\ F(x_1) = F(2)\ $$ which equals $−1.5$ (the "deficit"). Set $$\ x_2 = 3\ $$ and compute $$\ F(x_2) = F(3)\ $$ which equals $1.75$ (the "excess").

Estimated root (1st iteration):

 * $$\ \hat{x} ~=~ \frac{~ x_1 F(x_2) - x_2 F(x_1) ~}{F(x_2) - F(x_1)} ~=~ \frac{~ 2 \times 1.75 + 3 \times 1.5 ~}{1.75 + 1.5} ~\approx~ 2.4615\ $$

Example code
This example program, written in the C programming language, is an example of the Illinois algorithm. To find the positive number $x$ where $cos(x) = x^{3}$, the equation is transformed into a root-finding form $f (x) = cos(x) -- x^{3} = 0$.

After running this code, the final answer is approximately 0.865474033101614.

title: Regula falsi