Horner's method

Given the polynomial

${\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n},}$

where ${\displaystyle a_{0},\ldots ,a_{n}}$ are constant coefficients, the problem is to evaluate the polynomial at a specific value ${\displaystyle x_{0}}$ of ${\displaystyle x.}$

For this, a new sequence of constants is defined recursively as follows:

${\displaystyle {\begin{aligned}b_{n}&:=a_{n}\\b_{n-1}&:=a_{n-1}+b_{n}x_{0}\\&~~~\vdots \\b_{1}&:=a_{1}+b_{2}x_{0}\\b_{0}&:=a_{0}+b_{1}x_{0}.\end{aligned}}}$

Then ${\displaystyle b_{0}}$ is the value of ${\displaystyle p(x_{0})}$.

To see why this works, the polynomial can be written in the form

${\displaystyle p(x)=a_{0}+x{\bigg (}a_{1}+x{\Big (}a_{2}+x{\big (}a_{3}+\cdots +x(a_{n-1}+x\,a_{n})\cdots {\big )}{\Big )}{\bigg )}\ .}$

Thus, by iteratively substituting the ${\displaystyle b_{i}}$ into the expression,

${\displaystyle {\begin{aligned}p(x_{0})&=a_{0}+x_{0}{\Big (}a_{1}+x_{0}{\big (}a_{2}+\cdots +x_{0}(a_{n-1}+b_{n}x_{0})\cdots {\big )}{\Big )}\\&=a_{0}+x_{0}{\Big (}a_{1}+x_{0}{\big (}a_{2}+\cdots +x_{0}b_{n-1}{\big )}{\Big )}\\&~~\vdots \\&=a_{0}+x_{0}b_{1}\\&=b_{0}.\end{aligned}}}$

Now, it can be proven that;

${\displaystyle p(x)=(b_{1}+b_{2}x+b_{3}x^{2}+b_{4}x^{3}+\cdots +b_{n-1}x^{n-2}+b_{n}x^{n-1})(x-x_{0})+b_{0}}$

This expression constitutes Horner's practical application, as it offers a very quick way of determining the outcome of;

${\displaystyle p(x)/(x-x_{0})}$

with b₀ (which is equal to p(x₀)) being the division's remainder, as is demonstrated by the examples below. if x₀ is a root of p(x), then b₀ = 0 (meaning the remainder is 0), which means you can factor p(x) with (x-x₀).
As to finding the consecutive b-values, you start with determining b_n, which is simply equal to a_n. You then work your way down to the other b's, using the formula;

${\displaystyle b_{n-1}=a_{n-1}+b_{n}x_{0}}$

till you arrive at b₀.

 x0│   x3    x2    x1    x0
 3 │   2    −6     2    −1
   │         6     0     6
   └────────────────────────
       2     0     2     5

 2 │   1    −6    11    −6
   │         2    −8     6
   └────────────────────────
       1    −4     3     0

  0.5 │ 4  -6   0   3  -5
      │     2  -2  -1   1
└───────────────────────
        2  -2  -1   1  -2

Using the long division algorithm in combination with Newton's method, it is possible to approximate the real roots of a polynomial. The algorithm works as follows. Given a polynomial ${\displaystyle p_{n}(x)}$ of degree ${\displaystyle n}$ with zeros ${\displaystyle z_{n}<z_{n-1}<\cdots <z_{1},}$ make some initial guess ${\displaystyle x_{0}}$ such that ${\displaystyle x_{0}>z_{1}}$. Now iterate the following two steps:

1. Using Newton's method, find the largest zero ${\displaystyle z_{1}}$ of ${\displaystyle p_{n}(x)}$ using the guess ${\displaystyle x_{0}}$.
2. Using Horner's method, divide out ${\displaystyle (x-z_{1})}$ to obtain ${\displaystyle p_{n-1}}$. Return to step 1 but use the polynomial ${\displaystyle p_{n-1}}$ and the initial guess ${\displaystyle z_{1}}$.

These two steps are repeated until all real zeros are found for the polynomial. If the approximated zeros are not precise enough, the obtained values can be used as initial guesses for Newton's method but using the full polynomial rather than the reduced polynomials.[9]

Example

Polynomial root finding using Horner's method

Consider the polynomial

${\displaystyle p_{6}(x)=(x-3)(x+3)(x+5)(x+8)(x-2)(x-7)}$

which can be expanded to

${\displaystyle p_{6}(x)=x^{6}+4x^{5}-72x^{4}-214x^{3}+1127x^{2}+1602x-5040.}$

From the above we know that the largest root of this polynomial is 7 so we are able to make an initial guess of 8. Using Newton's method the first zero of 7 is found as shown in black in the figure to the right. Next ${\displaystyle p(x)}$ is divided by ${\displaystyle (x-7)}$ to obtain

${\displaystyle p_{5}(x)=x^{5}+11x^{4}+5x^{3}-179x^{2}-126x+720}$

which is drawn in red in the figure to the right. Newton's method is used to find the largest zero of this polynomial with an initial guess of 7. The largest zero of this polynomial which corresponds to the second largest zero of the original polynomial is found at 3 and is circled in red. The degree 5 polynomial is now divided by ${\displaystyle (x-3)}$ to obtain

${\displaystyle p_{4}(x)=x^{4}+14x^{3}+47x^{2}-38x-240}$

which is shown in yellow. The zero for this polynomial is found at 2 again using Newton's method and is circled in yellow. Horner's method is now used to obtain

${\displaystyle p_{3}(x)=x^{3}+16x^{2}+79x+120}$

which is shown in green and found to have a zero at −3. This polynomial is further reduced to

${\displaystyle p_{2}(x)=x^{2}+13x+40}$

which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing ${\displaystyle p_{2}(x)}$ and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were found.

Horner's method can be modified to compute the divided difference ${\displaystyle (p(y)-p(x))/(y-x).}$ Given the polynomial (as before)

${\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n},}$

proceed as follows[10]

${\displaystyle {\begin{aligned}b_{n}&=a_{n},&\quad d_{n}&=b_{n},\\b_{n-1}&=a_{n-1}+b_{n}x,&\quad d_{n-1}&=b_{n-1}+d_{n}y,\\&{}\ \ \vdots &\quad &{}\ \ \vdots \\b_{1}&=a_{1}+b_{2}x,&\quad d_{1}&=b_{1}+d_{2}y,\\b_{0}&=a_{0}+b_{1}x.\end{aligned}}}$

At completion, we have

${\displaystyle {\begin{aligned}p(x)&=b_{0},\\{\frac {p(y)-p(x)}{y-x}}&=d_{1},\\p(y)&=b_{0}+(y-x)d_{1}.\end{aligned}}}$

This computation of the divided difference is subject to less round-off error than evaluating ${\displaystyle p(x)}$ and ${\displaystyle p(y)}$ separately, particularly when ${\displaystyle x\approx y}$. Substituting ${\displaystyle y=x}$ in this method gives ${\displaystyle d_{1}=p'(x)}$, the derivative of ${\displaystyle p(x)}$.

Qin Jiushao's algorithm for solving the quadratic polynomial equation${\displaystyle -x^{4}+763200x^{2}-40642560000=0}$
result: x=840[11]

Horner's paper, titled "A new method of solving numerical equations of all orders, by continuous approximation",[12] was read before the Royal Society of London, at its meeting on July 1, 1819, with Davies Gilbert, Vice-President and Treasurer, in the chair; this was the final meeting of the session before the Society adjourned for its Summer recess. When a sequel was read before the Society in 1823, it was again at the final meeting of the session. On both occasions, papers by James Ivory, FRS, were also read. In 1819, it was Horner's paper that got through to publication in the "Philosophical Transactions".[12] later in the year, Ivory's paper falling by the way, despite Ivory being a Fellow; in 1823, when a total of ten papers were read, fortunes as regards publication, were reversed. Gilbert, who had strong connections with the West of England and may have had social contact with Horner, resident as Horner was in Bristol and Bath, published his own survey of Horner-type methods earlier in 1823.

Horner's paper in Part II of Philosophical Transactions of the Royal Society of London for 1819 was warmly and expansively welcomed by a reviewer in the issue of The Monthly Review: or, Literary Journal for April, 1820; in comparison, a technical paper by Charles Babbage is dismissed curtly in this review. However, the reviewer noted that another, similar method had also recently been published by the architect and mathematical expositor, Peter Nicholson. This theme is developed in a further review of some of Nicholson's books in the issue of The Monthly Review for December, 1820, which in turn ends with notice of the appearance of a booklet by Theophilus Holdred, from whom Nicholson acknowledges he obtained the gist of his approach in the first place, although claiming to have improved upon it. The sequence of reviews is concluded in the issue of The Monthly Review for September, 1821, with the reviewer concluding that whereas Holdred was the first person to discover a direct and general practical solution of numerical equations, he had not reduced it to its simplest form by the time of Horner's publication, and saying that had Holdred published forty years earlier when he first discovered his method, his contribution could be more easily recognized. The reviewer is exceptionally well-informed, even having cited Horner's preparatory correspondence with Peter Barlow in 1818, seeking work of Budan. The Bodlean Library, Oxford has the Editor's annotated copy of The Monthly Review from which it is clear that the most active reviewer in mathematics in 1814 and 1815 (the last years for which this information has been published) was none other than Peter Barlow, one of the foremost specialists on approximation theory of the period, suggesting that it was Barlow, who wrote this sequence of reviews. As it also happened, Henry Atkinson, of Newcastle, devised a similar approximation scheme in 1809; he had consulted his fellow Geordie, Charles Hutton, another specialist and a senior colleague of Barlow at the Royal Military Academy, Woolwich, only to be advised that, while his work was publishable, it was unlikely to have much impact. J. R. Young, writing in the mid-1830s, concluded that Holdred's first method replicated Atkinson's while his improved method was only added to Holdred's booklet some months after its first appearance in 1820, when Horner's paper was already in circulation.

The feature of Horner's writing that most distinguishes it from his English contemporaries is the way he draws on the Continental literature, notably the work of Arbogast. The advocacy, as well as the detraction, of Horner's Method has this as an unspoken subtext. Quite how he gained that familiarity has not been determined. Horner is known to have made a close reading of John Bonneycastle's book on algebra. Bonneycastle recognizes that Arbogast has the general, combinatorial expression for the reversion of series, a project going back at least to Newton. But Bonneycastle's main purpose in mentioning Arbogast is not to praise him, but to observe that Arbogast's notation is incompatible with the approach he adopts. The gap in Horner's reading was the work of Paolo Ruffini, except that, as far as awareness of Ruffini goes, citations of Ruffini's work by authors, including medical authors, in Philosophical Transactions speak volumes, as there are none — Ruffini's name[13] only appears in 1814, recording a work he donated to the Royal Society. Ruffini might have done better if his work had appeared in French, as had Malfatti's Problem in the reformulation of Joseph Diez Gergonne, or had he written in French, as had Antonio Cagnoli, a source quoted by Bonneycastle on series reversion. (Today, Cagnoli is in the Italian Wikipedia, as shown, but has yet to make it into either French or English.)

Fuller[14] showed that the method in Horner's 1819 paper differs from what afterwards became known as "Horner's method" and that in consequence the priority for this method should go to Holdred (1920). This view may be compared with the remarks concerning the works of Horner and Holdred in the previous paragraph. Fuller also takes aim at Augustus De Morgan. Precocious though Augustus de Morgan was, he was not the reviewer for The Monthly Review, while several others — Thomas Stephens Davies, J. R. Young, Stephen Fenwick, T. T. Wilkinson — wrote Horner firmly into their records, not least Horner, as he published extensively up until the year of his death in 1837. His paper in 1819 was one that would have been difficult to miss. In contrast, the only other mathematical sighting of Holdred is a single named contribution to The Gentleman's Mathematical Companion, an answer to a problem.

It is questionable to what extent it was De Morgan's advocacy of Horner's priority in discovery[15][16] that led to "Horner's method" being so called in textbooks, but it is true that those suggesting this tend to know of Horner largely through intermediaries, of whom De Morgan made himself a prime example. However, this method qua method was known long before Horner. In reverse chronological order, Horner's method was already known to:

• Paolo Ruffini in 1809 (see Ruffini's rule)[15][16]
• Isaac Newton in 1669 (but precise reference needed)
• the Chinese mathematician Zhu Shijie in the 14th century[16]
• the Chinese mathematician Qin Jiushao in his Mathematical Treatise in Nine Sections in the 13th century
• the Persian mathematician Sharaf al-Dīn al-Ṭūsī in the 12th century (the first to use that method in a general case of cubic equation)[17]
• the Chinese mathematician Jia Xian in the 11th century (Song dynasty)
• The Nine Chapters on the Mathematical Art, a Chinese work of the Han dynasty (202 BC – 220 AD) edited by Liu Hui (fl. 3rd century).[18]

However, this observation on its own masks significant differences in conception and also, as noted with Ruffini's work, issues of accessibility.

Qin Jiushao, in his Shu Shu Jiu Zhang (Mathematical Treatise in Nine Sections; 1247), presents a portfolio of methods of Horner-type for solving polynomial equations, which was based on earlier works of the 11th century Song dynasty mathematician Jia Xian; for example, one method is specifically suited to bi-quintics, of which Qin gives an instance, in keeping with the then Chinese custom of case studies. The first person writing in English to note the connection with Horner's method was Alexander Wylie, writing in The North China Herald in 1852; perhaps conflating and misconstruing different Chinese phrases, Wylie calls the method Harmoniously Alternating Evolution (which does not agree with his Chinese, linglong kaifang, not that at that date he uses pinyin), working the case of one of Qin's quartics and giving, for comparison, the working with Horner's method. Yoshio Mikami in Development of Mathematics in China and Japan published in Leipzig in 1913, gave a detailed description of Qin's method, using the quartic illustrated to the above right in a worked example; he wrote:

"... who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe ... We of course don't intend in any way to ascribe Horner's invention to a Chinese origin, but the lapse of time sufficiently makes it not altogether impossible that the Europeans could have known of the Chinese method in a direct or indirect way."[19]

However, as Mikami is also aware, it was not altogether impossible that a related work, Si Yuan Yu Jian (Jade Mirror of the Four Unknowns; 1303) by Zhu Shijie might make the shorter journey across to Japan, but seemingly it never did, although another work of Zhu, Suan Xue Qi Meng, had a seminal influence on the development of traditional mathematics in the Edo period, starting in the mid-1600s. Ulrich Libbrecht (at the time teaching in school, but subsequently a professor of comparative philosophy) gave a detailed description in his doctoral thesis of Qin's method, he concluded: It is obvious that this procedure is a Chinese invention ... the method was not known in India. He said, Fibonacci probably learned of it from Arabs, who perhaps borrowed from the Chinese.[20] Here, the problems is that there is no more evidence for this speculation than there is of the method being known in India. Of course, the extraction of square and cube roots along similar lines is already discussed by Liu Hui in connection with Problems IV.16 and 22 in Jiu Zhang Suan Shu, while Wang Xiaotong in the 7th century supposes his readers can solve cubics by an approximation method described in his book Jigu Suanjing.

• Clenshaw algorithm to evaluate polynomials in Chebyshev form
• De Boor's algorithm to evaluate splines in B-spline form
• De Casteljau's algorithm to evaluate polynomials in Bézier form
• Estrin's scheme to facilitate parallelization on modern computer architectures
• Lill's method to approximate roots graphically
• Ruffini's rule and synthetic division to divide a polynomial by a binomial of the form x − r

• Berggren, J. L. (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat". Journal of the American Oriental Society. 110 (2): 304–309. doi:10.2307/604533. JSTOR 604533.CS1 maint: ref=harv (link)
• Cajori, Florian (1911). "Horner's method of approximation anticipated by Ruffini". Bulletin of the American Mathematical Society. 17 (8): 409–414. doi:10.1090/s0002-9904-1911-02072-9.CS1 maint: ref=harv (link) Read before the Southwestern Section of the American Mathematical Society on November 26, 1910.
• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein10.1016/0315-0860(81)90069-0, Clifford (2009). "Introduction to Algorithms". Historia Mathematica (3rd ed.). MIT Press. 8 (3): 277–318. doi:10.1016/0315-0860(81)90069-0.CS1 maint: ref=harv (link)
• Fateman, R. J.; Kahan, W. (2000). Improving exact integrals from symbolic algebra systems (PDF) (Report). PAM. University of California, Berkeley: Center for Pure and Applied Mathematics. Archived from the original (PDF) on 2017-08-14. Retrieved 2018-05-17.CS1 maint: ref=harv (link)
• Fuller, A. T. (1999). "Horner versus Holdred: An Episode in the History of Root Computation". Historia Mathematica. 26: 29–51. doi:10.1006/hmat.1998.2214.CS1 maint: ref=harv (link)
• Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms. SIAM. ISBN 978-0-89871-521-7.CS1 maint: ref=harv (link)
• Holdred, T. (1820). A New Method of Solving Equations with Ease and Expedition; by which the True Value of the Unknown Quantity is Found Without Previous Reduction. With a Supplement, Containing Two Other Methods of Solving Equations, Derived from the Same Principle (PDF). Richard Watts. Archived from the original (PDF) on 2014-01-06. Retrieved 2012-12-10.CS1 maint: ref=harv (link)

  Holdred's method is in the supplement following page numbered 45 (which is the 52nd page of the pdf version).

• Horner, William George (July 1819). "A new method of solving numerical equations of all orders, by continuous approximation" (PDF). Philosophical Transactions. Royal Society of London. 109: 308–335. doi:10.1098/rstl.1819.0023. JSTOR 107508. Archived from the original (PDF) on 2017-03-28. Retrieved 2017-03-28.CS1 maint: ref=harv (link)

  Directly available online via the link, but also reprinted with appraisal in D.E. Smith: A Source Book in Mathematics, McGraw-Hill, 1929; Dover reprint, 2 vols, 1959.

• Knuth, Donald (1997). The Art of Computer Programming. Vol. 2: Seminumerical Algorithms (3rd ed.). Addison-Wesley. pp. 486–488 in section 4.6.4. ISBN 978-0-201-89684-8.CS1 maint: ref=harv (link)
• Kress, Rainer (1991). Numerical Analysis. Springer.CS1 maint: ref=harv (link)
• Kripasagar, Venkat (March 2008). "Efficient Micro Mathematics – Multiplication and Division Techniques for MCUs". Circuit Cellar Magazine (212).CS1 maint: ref=harv (link)
• Libbrecht, Ulrich (2005). "Chapter 13". Chinese Mathematics in the Thirteenth Century (2nd ed.). Dover. ISBN 978-0-486-44619-6.CS1 maint: ref=harv (link)
• Mikami, Yoshio (1913). "Chapter 11. Ch'in Chiu-Shao". The Development of Mathematics in China and Japan (1st ed.). Chelsea Publishing Co reprint. pp. 74–77.CS1 maint: ref=harv (link)
• Ostrowski, Alexander M. (1954). "On two problems in abstract algebra connected with Horner's rule". Studies in Mathematics and Mechanics presented to Richard von Mises. Academic Press. pp. 40–48. ISBN 978-1-4832-3272-0.CS1 maint: ref=harv (link)
• Pan, Y. Ja (1966). "On means of calculating values of polynomials". Russian Math. Surveys. 21: 105–136. doi:10.1070/rm1966v021n01abeh004147.CS1 maint: ref=harv (link)
• Pankiewicz, W. (1968). "Algorithm 337: calculation of a polynomial and its derivative values by Horner scheme". Communications of the ACM. ACM. 11 (9): 633. doi:10.1145/364063.364089.CS1 maint: ref=harv (link)
• Spiegel, Murray R. (1956). Schaum's Outline of Theory and Problems of College Algebra. McGraw-Hill.CS1 maint: ref=harv (link)
• Temple, Robert (1986). The Genius of China: 3,000 Years of Science, Discovery, and Invention. Simon and Schuster. ISBN 978-0-671-62028-8.CS1 maint: ref=harv (link)
• Whittaker, E.T.; Robinson, G. (1924). The Calculus of Observations. London: Blackie.CS1 maint: ref=harv (link)
• Wylie, Alexander (1897). "Jottings on the Science of Chinese Arithmetic". Chinese Researches. Shanghai. pp. 159–194.CS1 maint: ref=harv (link)

  Reprinted from issues of The North China Herald (1852).

The Wikibook Algorithm Implementation has a page on the topic of: Polynomial evaluation

• Hazewinkel, Michiel, ed. (2001) [1994], "Horner scheme", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Qiu Jin-Shao, Shu Shu Jiu Zhang (Cong Shu Ji Cheng ed.)
• For more on the root-finding application see