4751:
4025:
4746:{\displaystyle {\begin{aligned}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}\\&=\left(a_{0}+a_{2}x^{2}+a_{4}x^{4}+\cdots \right)+\left(a_{1}x+a_{3}x^{3}+a_{5}x^{5}+\cdots \right)\\&=\left(a_{0}+a_{2}x^{2}+a_{4}x^{4}+\cdots \right)+x\left(a_{1}+a_{3}x^{2}+a_{5}x^{4}+\cdots \right)\\&=\sum _{i=0}^{\lfloor n/2\rfloor }a_{2i}x^{2i}+x\sum _{i=0}^{\lfloor n/2\rfloor }a_{2i+1}x^{2i}\\&=p_{0}(x^{2})+xp_{1}(x^{2}).\end{aligned}}}
1745:
8127:
8743:"... 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."
25:
1339:
7740:
417:
5428:
1031:
127:
7163:
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
5151:
1740:{\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}}}
8612:
for
September, 1821, concludes that Holdred was the first person to discover a direct and general practical solution of numerical equations. Fuller 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
4018:
on modern computers. In most applications where the efficiency of polynomial evaluation matters, many low-order polynomials are evaluated simultaneously (for each pixel or polygon in computer graphics, or for each grid square in a numerical simulation), so it is not necessary to find parallelism
8515:
8122:{\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}}}
5003:
6723:
5939:
1305:
1975:
8318:
6713:
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.
790:
6227:
The method is particularly fast on processors supporting a single-instruction shift-and-addition-accumulate. Compared to a C floating-point library, Horner's method sacrifices some accuracy, however it is nominally 13 times faster (16 times faster when the
5764:
7735:
678:
9333:
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
4762:
4030:
5773:
412:{\displaystyle {\begin{aligned}&a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n}\\={}&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 )}.\end{aligned}}}
3488:
1118:
6985:
2931:, the entries in the third row. So, synthetic division (which was actually invented and published by Ruffini 10 years before Horner's publication) is easier to use; it can be shown to be equivalent to Horner's method.
112:
by Horner himself, and can be traced back many hundreds of years to
Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials.
6989:
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
5423:{\displaystyle {\begin{aligned}(0.15625)m&=(0.00101_{b})m=\left(2^{-3}+2^{-5}\right)m=\left(2^{-3})m+(2^{-5}\right)m\\&=2^{-3}\left(m+\left(2^{-2}\right)m\right)=2^{-3}\left(m+2^{-2}(m)\right).\end{aligned}}}
1757:
8134:
6204:
5473:
If all the non-zero bits were counted, then the intermediate result register now holds the final result. Otherwise, add d to the intermediate result, and continue in step 2 with the next most significant bit in
7161:
9051:
Analysis Per
Quantitatum Series, Fluctiones ac Differentias : Cum Enumeratione Linearum Tertii Ordinis, Londini. Ex Officina Pearsoniana. Anno MDCCXI, p. 10, 4th
6064:
7436:
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
6857:
5766:
At this stage in the algorithm, it is required that terms with zero-valued coefficients are dropped, so that only binary coefficients equal to one are counted, thus the problem of multiplication or
7290:
5547:
6407:
9179:
8139:
7745:
5156:
1344:
795:
132:
2839:
8577:
2929:
7370:
3941:
of the representation is allowed, which makes sense if the polynomial is evaluated only once. However, if preconditioning is allowed and the polynomial is to be evaluated many times, then
7560:
503:
3216:
2433:
2544:
5005:
where the inner summations may be evaluated using separate parallel instances of Horner's method. This requires slightly more operations than the basic Horner's method, but allows
8627:
Although Horner is credited with making the method accessible and practical, it was known long before Horner. In reverse chronological order, Horner's method was already known to:
7434:
8754:. He said, Fibonacci probably learned of it from Arabs, who perhaps borrowed from the Chinese. The extraction of square and cube roots along similar lines is already discussed by
5542:
5464:
Count (to the left) the number of bit positions to the next most significant non-zero bit. If there are no more-significant bits, then take the value of the current bit position.
724:
7555:
3056:
8475:
1026:{\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}}}
2612:
in this example) with the third-row entry immediately to the left. The entries in the first row are the coefficients of the polynomial to be evaluated. Then the remainder of
6474:
2047:
6224:), and a (2) results in a left arithmetic shift. The multiplication product can now be quickly calculated using only arithmetic shift operations, addition and subtraction.
3267:
8727:; 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
3582:
8402:
3340:
3130:
6614:
7470:
6545:
6319:
3359:
3303:
2110:
1110:
6680:
6647:
5070:
2281:
2199:
2740:
7194:
7049:
6707:
6572:
6509:
6434:
3751:
3724:
2460:
2358:
2331:
2137:
2074:
1334:
1074:
751:
8504:
8376:
8347:
7017:
3631:
2961:
2705:
2639:
2248:
2166:
8428:
6862:
5134:
5096:
3082:
2987:
2665:
2570:
7292:
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
3897:
3677:
774:
6339:
3994:
3963:
3928:
3874:
3851:
3831:
3811:
3791:
3771:
3697:
3651:
3602:
3535:
3515:
2304:
2219:
467:
447:
6069:
8731:; 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.
2934:
As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of
4998:{\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}=\sum _{j=0}^{k-1}x^{j}\sum _{i=0}^{\lfloor n/k\rfloor }a_{ki+j}x^{ki}=\sum _{j=0}^{k-1}x^{j}p_{j}(x^{k})}
9157:
7054:
5946:
9243:
492:
made more efficient for hand calculation by application of Horner's rule. It was widely used until computers came into general use around 1970.
46:
9675:
3584:
multiplications, if powers are calculated by repeated multiplication and each monomial is evaluated individually. The cost can be reduced to
6733:
7199:
5934:{\displaystyle =d_{0}\left(m+2{\frac {d_{1}}{d_{0}}}\left(m+2{\frac {d_{2}}{d_{1}}}\left(m+2{\frac {d_{3}}{d_{2}}}(m)\right)\right)\right).}
9259:
1300:{\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 )}\ .}
3902:
Horner's method is optimal, in the sense that any algorithm to evaluate an arbitrary polynomial must use at least as many operations.
9519:
8698:
9331:
7295:
1970:{\displaystyle p(x)=\left(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}\right)\left(x-x_{0}\right)+b_{0}}
9034:
8313:{\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}}}
9454:
2363:
8723:
8666:
9718:
9758:
9727:
9632:
9513:
9466:
9425:
9320:
8620:. Horner is also known to have made a close reading of John Bonneycastle's book on algebra, though he neglected the work of
3656:
If numerical data are represented in terms of digits (or bits), then the naive algorithm also entails storing approximately
7375:
6344:
3942:
1993:
This expression constitutes Horner's practical application, as it offers a very quick way of determining the outcome of;
53:
9444:
Kripasagar, Venkat (March 2008). "Efficient Micro
Mathematics – Multiplication and Division Techniques for MCUs".
8592:
before the Royal
Society of London, at its meeting on July 1, 1819, with a sequel in 1823. Horner's paper in Part II of
488:
also refers to a method for approximating the roots of polynomials, described by Horner in 1819. It is a variant of the
9938:
5467:
Using that value, perform a left-shift operation by that number of bits on the register holding the intermediate result
5039:. One of the binary numbers to be multiplied is represented as a trivial polynomial, where (using the above notation)
2748:
2604:
The entries in the third row are the sum of those in the first two. Each entry in the second row is the product of the
8617:
8524:
9646:
9613:
8588:
Horner's paper, titled "A new method of solving numerical equations of all orders, by continuous approximation", was
2844:
71:
8852:
8680:
5013:
execution of most of them. Modern compilers generally evaluate polynomials this way when advantageous, although for
1996:
6285:, it is possible to approximate the real roots of a polynomial. The algorithm works as follows. Given a polynomial
4022:
If, however, one is evaluating a single polynomial of very high order, it may be useful to break it up as follows:
5026:
3138:
6270:
3931:
5759:{\displaystyle (d_{3}2^{3}+d_{2}2^{2}+d_{1}2^{1}+d_{0}2^{0})m=d_{3}2^{3}m+d_{2}2^{2}m+d_{1}2^{1}m+d_{0}2^{0}m.}
9710:
9358:
Holdred's method is in the supplement following page numbered 45 (which is the 52nd page of the pdf version).
8677:
8631:
8621:
4015:
3087:
2 β 1 β6 11 β6 β 2 β8 6 βββββββββββββββββββββββββ 1 β4 3 0
2601:
3 β 2 β6 2 β1 β 6 0 6 βββββββββββββββββββββββββ 2 0 2 5
2473:
9933:
9917:
9420:. Vol. 2: Seminumerical Algorithms (3rd ed.). Addison-Wesley. pp. 486β488 in section 4.6.4.
3345:
0.5 β 4 β6 0 3 β5 β 2 β2 β1 1 ββββββββββββββββββββββββ 2 β2 β1 1 β4
2675:
9265:(Report). PAM. University of California, Berkeley: Center for Pure and Applied Mathematics. Archived from
8766:
in the 7th century supposes his readers can solve cubics by an approximation method described in his book
5490:
9999:
9953:
9705:
9488:
8802:
7372:
which is shown in green and found to have a zero at −3. This polynomial is further reduced to
683:
7487:
5031:
Horner's method is a fast, code-efficient method for multiplication and division of binary numbers on a
3003:
9798:
9367:(July 1819). "A new method of solving numerical equations of all orders, by continuous approximation".
8597:
9751:
9569:
8589:
7730:{\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},}
673:{\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},}
40:
6439:
9943:
9700:
9039:
8752:
It is obvious that this procedure is a
Chinese invention ... the method was not known in India
3221:
2994:
8616:
Unlike his
English contemporaries, Horner drew on the Continental literature, notably the work of
9882:
8433:
3540:
9029:
8381:
3308:
3093:
9912:
9907:
9902:
9780:
9191:
Read before the
Southwestern Section of the American Mathematical Society on November 26, 1910.
8788:
6580:
6229:
3853:
8320:
This computation of the divided difference is subject to less round-off error than evaluating
7476:. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were found.
7439:
6514:
6288:
6269:, in which case the gain in computational efficiency is even greater. However, for such cases
3945:. They involve a transformation of the representation of the polynomial. In general, a degree-
3272:
2079:
1079:
9892:
9872:
9364:
9266:
9237:
6652:
6619:
5107:
5042:
2253:
2171:
109:
105:
101:
2710:
9994:
9958:
9877:
9744:
9199:
7167:
7022:
6685:
6550:
6487:
6412:
3729:
3702:
2438:
2336:
2309:
2115:
2052:
1312:
1052:
729:
9501:
8480:
8352:
8323:
6993:
3607:
2937:
2681:
2615:
2224:
2142:
8:
9771:
9657:
9403:
Directly available online via the link, but also reprinted with appraisal in D.E. Smith:
9341:
9061:
Newton's collected papers, the edition 1779, in a footnote, vol. I, p. 270-271
9025:
8658:
8648:
8407:
5113:
5075:
3903:
3061:
2966:
2644:
2549:
779:
726:
are constant coefficients, the problem is to evaluate the polynomial at a specific value
9124:
Berggren, J. L. (1990). "Innovation and
Tradition in Sharaf al-Din al-Tusi's Muadalat".
3879:
3659:
756:
9815:
9810:
9736:
9669:
9588:
9565:"Algorithm 337: calculation of a polynomial and its derivative values by Horner scheme"
9551:
9392:
9384:
9141:
8827:
8811:
8779:
6481:
6324:
6282:
4005:
3979:
3948:
3913:
3859:
3836:
3816:
3796:
3776:
3756:
3682:
3636:
3587:
3520:
3500:
2576:
2289:
2204:
489:
452:
432:
9194:
8683:
in the 12th century (the first to use that method in a general case of cubic equation)
9793:
9628:
9609:
9602:
9555:
9509:
9472:
9462:
9421:
9396:
9316:
9228:
9211:
8823:
8635:
5036:
35:
9592:
9547:
9174:
9989:
9846:
9839:
9834:
9578:
9543:
9376:
9297:
9223:
9195:
9169:
9133:
8817:
8747:
6221:
6217:
6213:
6209:
5767:
89:
9968:
9725:
9856:
9851:
9803:
9788:
9731:
9642:
9251:
8605:
7473:
5032:
4011:
3938:
3483:{\displaystyle {\frac {f_{1}(x)}{f_{2}(x)}}=2x^{3}-2x^{2}-x+1-{\frac {4}{2x-1}}.}
9827:
9822:
9207:
9153:
8783:
8674:
6241:
5014:
2112:) being the division's remainder, as is demonstrated by the examples below. If
8806:
9983:
9255:
8767:
8763:
8732:
6266:
6980:{\displaystyle p_{6}(x)=x^{6}+4x^{5}-72x^{4}-214x^{3}+1127x^{2}+1602x-5040.}
5020:
3910:
proved in 1966 that the number of multiplications is minimal. However, when
9963:
9413:
9380:
9302:
9285:
8792:
8691:
8641:
5943:
The denominators all equal one (or the term is absent), so this reduces to
9583:
9564:
8848:
8714:
8703:
8661:
8518:
85:
9767:
9203:
8651:
6199:{\displaystyle =d_{3}(m+2^{-1}{d_{2}}(m+2^{-1}{d_{1}}(m+{d_{0}}(m)))).}
3907:
422:
9388:
9286:"Horner versus Holdred: An Episode in the History of Root Computation"
9145:
3937:
This assumes that the polynomial is evaluated in monomial form and no
9897:
9625:
The Genius of China: 3,000 Years of Science, Discovery, and Invention
9534:
Pan, Y. Ja (1966). "On means of calculating values of polynomials".
6240:
Horner's method can be used to convert between different positional
6208:
In binary (base-2) math, multiplication by a power of 2 is merely a
5770:
is not an issue, despite this implication in the factored equation:
9887:
9506:
Studies in Mathematics and Mechanics presented to Richard von Mises
9137:
8796:
8728:
8687:
8514:
6220:, a (0) results in no operation (since 2 = 1 is the multiplicative
47:
Talk:Horner's method#This Article is about Two Different Algorithms
9502:"On two problems in abstract algebra connected with Horner's rule"
7484:
Horner's method can be modified to compute the divided difference
6722:
6066:
or equivalently (as consistent with the "method" described above)
4010:
A disadvantage of Horner's rule is that all of the operations are
469:
additions. This is optimal, since there are polynomials of degree
8755:
8707:
6212:
operation. Thus, multiplying by 2 is calculated in base-2 by an
5017:
calculations this requires enabling (unsafe) reassociative math.
3906:
proved in 1954 that the number of additions required is minimal.
1115:
To see why this works, the polynomial can be written in the form
8608:
is dismissed curtly in this review. The sequence of reviews in
7156:{\displaystyle p_{5}(x)=x^{5}+11x^{4}+5x^{3}-179x^{2}-126x+720}
9495:(1st ed.). Chelsea Publishing Co reprint. pp. 74β77.
8814:
to facilitate parallelization on modern computer architectures
6059:{\displaystyle =d_{0}(m+2{d_{1}}(m+2{d_{2}}(m+2{d_{3}}(m)))),}
5144:
For example, to find the product of two numbers (0.15625) and
3856:. Horner's method can also be extended to evaluate the first
5456:
Begin with the least significant (rightmost) non-zero bit in
5449:
A register holding the intermediate result is initialized to
6261:
representation of a given number β and can also be used if
6232:" (CSD) form is used) and uses only 20% of the code space.
5010:
3348:
The third row is the sum of the first two rows, divided by
473:
that cannot be evaluated with fewer arithmetic operations.
9604:
Schaum's Outline of Theory and Problems of College Algebra
8521:'s algorithm for solving the quadratic polynomial equation
495:
108:, this method is much older, as it has been attributed to
9158:"Horner's method of approximation anticipated by Ruffini"
8594:
Philosophical Transactions of the Royal Society of London
5021:
Application to floating-point multiplication and division
9766:
9260:
Improving exact integrals from symbolic algebra systems
6852:{\displaystyle p_{6}(x)=(x+8)(x+5)(x+3)(x-2)(x-3)(x-7)}
3793:
multiplications, and its storage requirements are only
8830:
to divide a polynomial by a binomial of the form x β r
7285:{\displaystyle p_{4}(x)=x^{4}+14x^{3}+47x^{2}-38x-240}
6281:
Using the long division algorithm in combination with
3833:. Alternatively, Horner's method can be computed with
8604:
for April, 1820; in comparison, a technical paper by
8527:
8483:
8436:
8410:
8384:
8355:
8326:
8137:
7743:
7563:
7490:
7479:
7442:
7378:
7298:
7202:
7170:
7057:
7025:
6996:
6865:
6736:
6688:
6655:
6622:
6583:
6553:
6517:
6490:
6442:
6415:
6402:{\displaystyle z_{n}<z_{n-1}<\cdots <z_{1},}
6347:
6327:
6291:
6072:
5949:
5776:
5550:
5493:
5154:
5116:
5078:
5045:
4765:
4028:
3982:
3951:
3916:
3882:
3862:
3839:
3819:
3799:
3779:
3759:
3732:
3705:
3699:: the evaluated polynomial has approximate magnitude
3685:
3662:
3639:
3610:
3590:
3543:
3523:
3503:
3362:
3311:
3275:
3224:
3141:
3096:
3064:
3006:
2969:
2940:
2847:
2751:
2713:
2684:
2647:
2618:
2552:
2476:
2441:
2366:
2360:. Then you then work recursively using the formula:
2339:
2312:
2292:
2256:
2227:
2207:
2174:
2145:
2118:
2082:
2055:
1999:
1760:
1342:
1315:
1121:
1082:
1055:
793:
759:
732:
686:
506:
455:
435:
130:
8851:
and 700 years earlier, by the Persian mathematician
5106:
to some power) is repeatedly factored out. In this
3753:
itself. By contrast, Horner's method requires only
3356:
with the third-row entry to the left. The answer is
8952:
9601:
8596:for 1819 was warmly and expansively welcomed by a
8571:
8498:
8469:
8422:
8396:
8370:
8341:
8312:
8121:
7729:
7549:
7464:
7428:
7364:
7284:
7188:
7155:
7043:
7011:
6979:
6851:
6701:
6674:
6641:
6608:
6566:
6539:
6503:
6468:
6428:
6401:
6333:
6313:
6198:
6058:
5933:
5758:
5536:
5422:
5128:
5090:
5064:
4997:
4745:
3988:
3957:
3922:
3891:
3868:
3845:
3825:
3805:
3785:
3765:
3745:
3718:
3691:
3671:
3645:
3625:
3596:
3576:
3529:
3509:
3482:
3334:
3297:
3261:
3210:
3124:
3076:
3050:
2981:
2955:
2923:
2833:
2734:
2699:
2659:
2633:
2564:
2538:
2454:
2427:
2352:
2325:
2298:
2275:
2242:
2213:
2193:
2160:
2131:
2104:
2068:
2041:
1969:
1739:
1328:
1299:
1104:
1068:
1025:
768:
745:
718:
672:
461:
441:
411:
9641:
9493:The Development of Mathematics in China and Japan
9407:, McGraw-Hill, 1929; Dover reprint, 2 vols, 1959.
9023:
5487:In general, for a binary number with bit values (
4755:More generally, the summation can be broken into
3352:. Each entry in the second row is the product of
2834:{\displaystyle a_{3}=2,a_{2}=-6,a_{1}=2,a_{0}=-1}
1648:
1559:
1519:
1398:
1286:
1279:
1178:
1155:
778:For this, a new sequence of constants is defined
397:
390:
289:
266:
9981:
8847:600 years earlier, by the Chinese mathematician
8572:{\displaystyle -x^{4}+763200x^{2}-40642560000=0}
9658:"Jottings on the Science of Chinese Arithmetic"
3497:Evaluation using the monomial form of a degree
2924:{\displaystyle b_{3}=2,b_{2}=0,b_{1}=2,b_{0}=5}
9789:Zero polynomial (degree undefined or β1 or ββ)
9313:Accuracy and Stability of Numerical Algorithms
7365:{\displaystyle p_{3}(x)=x^{3}+16x^{2}+79x+120}
5136:, so powers of 2 are repeatedly factored out.
9752:
9724:For more on the root-finding application see
9459:Chinese Mathematics in the Thirteenth Century
9162:Bulletin of the American Mathematical Society
9019:
9017:
8737:Development of Mathematics in China and Japan
6726:Polynomial root finding using Horner's method
5027:multiplication algorithm Β§ Shift and add
4014:, so it is not possible to take advantage of
1641:
1589:
1512:
1428:
1272:
1201:
383:
312:
9250:
8958:
8758:in connection with Problems IV.16 and 22 in
4892:
4878:
4631:
4617:
4564:
4550:
3633:multiplications by evaluating the powers of
9242:: CS1 maint: numeric names: authors list (
9208:Stein10.1016/0315-0860(81)90069-0, Clifford
3211:{\displaystyle f_{1}(x)=4x^{4}-6x^{3}+3x-5}
9759:
9745:
9674:: CS1 maint: location missing publisher (
9562:
9443:
9014:
8922:
8874:
6649:. Return to step 1 but use the polynomial
6276:
6248:is the base of the number system, and the
5437:To find the product of two binary numbers
2428:{\displaystyle b_{n-1}=a_{n-1}+b_{n}x_{0}}
9582:
9499:
9452:
9301:
9227:
9173:
9106:
8980:
8978:
8969:
8886:
8699:The Nine Chapters on the Mathematical Art
8613:this method should go to Holdred (1820).
3322:
1253:
364:
72:Learn how and when to remove this message
9222:(3) (3rd ed.). MIT Press: 277β318.
9126:Journal of the American Oriental Society
9123:
9070:
8602:The Monthly Review: or, Literary Journal
8513:
6721:
6257:coefficients are the digits of the base-
2993:. This makes Horner's method useful for
9599:
9329:
9035:MacTutor History of Mathematics Archive
6476:. Now iterate the following two steps:
4019:within a single polynomial evaluation.
3965:polynomial can be evaluated using only
2539:{\displaystyle f(x)=2x^{3}-6x^{2}+2x-1}
496:Polynomial evaluation and long division
9982:
9622:
9486:
9363:
9310:
9283:
9152:
9094:
9082:
9008:
9002:
8996:
8984:
8975:
8934:
8724:Mathematical Treatise in Nine Sections
8667:Mathematical Treatise in Nine Sections
3999:
1309:Thus, by iteratively substituting the
120:, in which a polynomial is written in
9740:
9655:
9434:
9412:
8946:
8910:
7429:{\displaystyle p_{2}(x)=x^{2}+13x+40}
6235:
9375:. Royal Society of London: 308β335.
5537:{\displaystyle d_{3}d_{2}d_{1}d_{0}}
2306:-values, you start with determining
1751:
784:
18:
9533:
8898:
8863:
3876:derivatives of the polynomial with
719:{\displaystyle a_{0},\ldots ,a_{n}}
16:Algorithm for polynomial evaluation
13:
9508:. Academic Press. pp. 40β48.
7550:{\displaystyle (p(y)-p(x))/(y-x).}
7480:Divided difference of a polynomial
6577:Using Horner's method, divide out
3051:{\displaystyle x^{3}-6x^{2}+11x-6}
14:
10011:
9693:
8706:(202 BC β 220 AD) edited by
7557:Given the polynomial (as before)
9500:Ostrowski, Alexander M. (1954).
8820:to approximate roots graphically
2678:, we know that the remainder is
421:This allows the evaluation of a
23:
9548:10.1070/rm1966v021n01abeh004147
9522:from the original on 2019-04-15
9418:The Art of Computer Programming
9340:. Richard Watts. Archived from
9182:from the original on 2017-09-04
9175:10.1090/s0002-9904-1911-02072-9
9100:
9088:
9076:
9064:
9055:
9045:
8990:
8963:
8018:
7952:
7875:
7780:
7164:5 polynomial is now divided by
3899:additions and multiplications.
8940:
8928:
8916:
8904:
8892:
8880:
8868:
8857:
8841:
8493:
8487:
8464:
8458:
8378:separately, particularly when
8365:
8359:
8336:
8330:
8290:
8278:
8255:
8249:
8205:
8199:
8190:
8184:
8151:
8145:
7573:
7567:
7541:
7529:
7521:
7518:
7512:
7503:
7497:
7491:
7459:
7453:
7395:
7389:
7315:
7309:
7219:
7213:
7183:
7171:
7074:
7068:
7038:
7026:
7006:
7000:
6882:
6876:
6846:
6834:
6831:
6819:
6816:
6804:
6801:
6789:
6786:
6774:
6771:
6759:
6753:
6747:
6603:
6584:
6534:
6528:
6469:{\displaystyle z_{1}<x_{0}}
6308:
6302:
6190:
6187:
6184:
6181:
6175:
6154:
6120:
6086:
6050:
6047:
6044:
6041:
6035:
6011:
5987:
5963:
5910:
5904:
5643:
5551:
5405:
5399:
5272:
5263:
5191:
5178:
5165:
5159:
4992:
4979:
4775:
4769:
4733:
4720:
4701:
4688:
4042:
4036:
3943:faster algorithms are possible
3932:Horner's method is not optimal
3563:
3544:
3403:
3397:
3382:
3376:
3329:
3323:
3292:
3286:
3241:
3235:
3158:
3152:
2950:
2944:
2723:
2717:
2694:
2688:
2628:
2622:
2486:
2480:
2237:
2231:
2221:), which means you can factor
2155:
2149:
2099:
2086:
2042:{\displaystyle p(x)/(x-x_{0})}
2036:
2017:
2009:
2003:
1770:
1764:
1504:
1462:
1363:
1350:
1264:
1228:
1131:
1125:
1099:
1086:
516:
510:
375:
339:
1:
9949:Horner's method of evaluation
9664:. Shanghai. pp. 159β194.
9489:"Chapter 11. Ch'in Chiu-Shao"
9116:
6216:. The factor (2) is a right
5482:
4016:instruction level parallelism
3492:
3262:{\displaystyle f_{2}(x)=2x-1}
9648:The Calculus of Observations
9405:A Source Book in Mathematics
9229:10.1016/0315-0860(81)90069-0
9212:"Introduction to Algorithms"
6717:
3813:times the number of bits of
3679:times the number of bits of
3517:polynomial requires at most
2676:polynomial remainder theorem
1749:Now, it can be proven that;
7:
9954:Polynomial identity testing
9706:Encyclopedia of Mathematics
9600:Spiegel, Murray R. (1956).
8805:to evaluate polynomials in
8782:to evaluate polynomials in
8773:
8470:{\displaystyle d_{1}=p'(x)}
3577:{\displaystyle (n^{2}+n)/2}
2465:
2333:, which is simply equal to
2286:To finding the consecutive
1983:
1039:
43:. The specific problem is:
10:
10016:
9453:Libbrecht, Ulrich (2005).
9369:Philosophical Transactions
8686:the Chinese mathematician
8509:
8397:{\displaystyle x\approx y}
5139:
5024:
4003:
3726:, and one must also store
3335:{\displaystyle f_{2}\,(x)}
3125:{\displaystyle x^{2}-4x+3}
2201:(meaning the remainder is
116:The algorithm is based on
9926:
9865:
9778:
9682:Reprinted from issues of
9656:Wylie, Alexander (1897).
9570:Communications of the ACM
9311:Higham, Nicholas (2002).
6859:which can be expanded to
6609:{\displaystyle (x-z_{1})}
5432:
9040:University of St Andrews
8959:Fateman & Kahan 2000
8834:
8803:De Casteljau's algorithm
8702:, a Chinese work of the
7465:{\displaystyle p_{2}(x)}
6730:Consider the polynomial
6540:{\displaystyle p_{n}(x)}
6484:, find the largest zero
6409:make some initial guess
6314:{\displaystyle p_{n}(x)}
3298:{\displaystyle f_{1}(x)}
2995:polynomial long division
2105:{\displaystyle p(x_{0})}
1105:{\displaystyle p(x_{0})}
9939:Greatest common divisor
9721:(Cong Shu Ji Cheng ed.)
9645:; Robinson, G. (1924).
9623:Temple, Robert (1986).
9563:Pankiewicz, W. (1968).
9487:Mikami, Yoshio (1913).
9461:(2nd ed.). Dover.
9446:Circuit Cellar Magazine
8131:At completion, we have
6675:{\displaystyle p_{n-1}}
6642:{\displaystyle p_{n-1}}
6277:Polynomial root finding
5462:
5065:{\displaystyle a_{i}=1}
3976:+2 multiplications and
3342:using Horner's method.
2276:{\displaystyle x-x_{0}}
2194:{\displaystyle b_{0}=0}
104:. Although named after
9811:Quadratic function (2)
9684:The North China Herald
9627:. Simon and Schuster.
9435:Kress, Rainer (1991).
9381:10.1098/rstl.1819.0023
9365:Horner, William George
9303:10.1006/hmat.1998.2214
9284:Fuller, A. T. (1999).
8745:
8585:
8573:
8500:
8471:
8424:
8398:
8372:
8343:
8314:
8123:
7731:
7599:
7551:
7466:
7430:
7366:
7286:
7190:
7157:
7045:
7013:
6981:
6853:
6727:
6703:
6682:and the initial guess
6676:
6643:
6610:
6568:
6541:
6505:
6470:
6430:
6403:
6335:
6315:
6230:canonical signed digit
6200:
6060:
5935:
5760:
5538:
5424:
5130:
5092:
5066:
4999:
4958:
4896:
4851:
4801:
4747:
4635:
4568:
4072:
4012:sequentially dependent
3990:
3959:
3924:
3893:
3870:
3847:
3827:
3807:
3787:
3767:
3747:
3720:
3693:
3673:
3647:
3627:
3598:
3578:
3531:
3511:
3484:
3336:
3299:
3263:
3212:
3126:
3078:
3052:
2983:
2957:
2925:
2835:
2736:
2735:{\displaystyle f(3)=5}
2701:
2661:
2635:
2566:
2540:
2456:
2429:
2354:
2327:
2300:
2277:
2244:
2215:
2195:
2162:
2133:
2106:
2070:
2043:
1971:
1741:
1330:
1301:
1106:
1070:
1027:
770:
747:
720:
674:
542:
463:
443:
413:
100:) is an algorithm for
9794:Constant function (0)
9584:10.1145/364063.364089
9536:Russian Math. Surveys
9200:Leiserson, Charles E.
9030:"Horner's method"
8853:Sharaf al-DΔ«n al-αΉ¬Ε«sΔ«
8741:
8739:(Leipzig 1913) wrote:
8690:in the 11th century (
8681:Sharaf al-DΔ«n al-αΉ¬Ε«sΔ«
8659:Chinese mathematician
8649:Chinese mathematician
8574:
8517:
8501:
8472:
8430:in this method gives
8425:
8399:
8373:
8344:
8315:
8124:
7732:
7579:
7552:
7467:
7431:
7367:
7287:
7191:
7189:{\displaystyle (x-3)}
7158:
7046:
7044:{\displaystyle (x-7)}
7014:
6982:
6854:
6725:
6704:
6702:{\displaystyle z_{1}}
6677:
6644:
6611:
6569:
6567:{\displaystyle x_{0}}
6542:
6506:
6504:{\displaystyle z_{1}}
6471:
6431:
6429:{\displaystyle x_{0}}
6404:
6336:
6316:
6201:
6061:
5936:
5761:
5539:
5425:
5131:
5108:binary numeral system
5093:
5067:
5000:
4932:
4862:
4825:
4781:
4748:
4601:
4534:
4052:
3991:
3960:
3925:
3894:
3871:
3848:
3828:
3808:
3788:
3768:
3748:
3746:{\displaystyle x^{n}}
3721:
3719:{\displaystyle x^{n}}
3694:
3674:
3648:
3628:
3599:
3579:
3532:
3512:
3485:
3337:
3300:
3264:
3213:
3127:
3079:
3053:
2984:
2958:
2926:
2836:
2737:
2702:
2662:
2636:
2567:
2541:
2457:
2455:{\displaystyle b_{0}}
2430:
2355:
2353:{\displaystyle a_{n}}
2328:
2326:{\displaystyle b_{n}}
2301:
2278:
2245:
2216:
2196:
2163:
2134:
2132:{\displaystyle x_{0}}
2107:
2071:
2069:{\displaystyle b_{0}}
2044:
1972:
1742:
1336:into the expression,
1331:
1329:{\displaystyle b_{i}}
1302:
1107:
1071:
1069:{\displaystyle b_{0}}
1028:
771:
748:
746:{\displaystyle x_{0}}
721:
675:
522:
500:Given the polynomial
490:NewtonβRaphson method
484:HornerβRuffini method
464:
444:
414:
110:Joseph-Louis Lagrange
106:William George Horner
102:polynomial evaluation
9927:Tools and algorithms
9847:Quintic function (5)
9835:Quartic function (4)
9772:polynomial functions
9330:Holdred, T. (1820).
9290:Historia Mathematica
9216:Historia Mathematica
9026:Robertson, Edmund F.
8525:
8499:{\displaystyle p(x)}
8481:
8477:, the derivative of
8434:
8408:
8382:
8371:{\displaystyle p(y)}
8353:
8342:{\displaystyle p(x)}
8324:
8135:
7741:
7561:
7488:
7440:
7376:
7296:
7200:
7168:
7055:
7023:
7012:{\displaystyle p(x)}
6994:
6863:
6734:
6686:
6653:
6620:
6581:
6551:
6515:
6488:
6440:
6413:
6345:
6325:
6289:
6070:
5947:
5774:
5548:
5491:
5152:
5114:
5076:
5043:
4763:
4026:
3980:
3949:
3914:
3880:
3860:
3837:
3817:
3797:
3777:
3757:
3730:
3703:
3683:
3660:
3637:
3626:{\displaystyle 2n-1}
3608:
3588:
3541:
3521:
3501:
3360:
3309:
3273:
3222:
3139:
3094:
3062:
3004:
2989:. The remainder is
2967:
2956:{\displaystyle f(x)}
2938:
2845:
2749:
2745:In this example, if
2711:
2700:{\displaystyle f(3)}
2682:
2645:
2634:{\displaystyle f(x)}
2616:
2550:
2474:
2439:
2364:
2337:
2310:
2290:
2254:
2243:{\displaystyle p(x)}
2225:
2205:
2172:
2161:{\displaystyle p(x)}
2143:
2116:
2080:
2053:
1997:
1758:
1340:
1313:
1119:
1080:
1053:
791:
757:
730:
684:
504:
453:
449:multiplications and
433:
128:
54:improve this article
39:to meet Knowledge's
9857:Septic equation (7)
9852:Sextic equation (6)
9799:Linear function (1)
9073:, pp. 304β309.
9024:O'Connor, John J.;
8972:, pp. 181β191.
8789:De Boor's algorithm
8670:in the 13th century
8654:in the 14th century
8423:{\displaystyle y=x}
7737:proceed as follows
5129:{\displaystyle x=2}
5091:{\displaystyle x=2}
5037:hardware multiplier
4000:Parallel evaluation
3904:Alexander Ostrowski
3854:fused multiplyβadds
3077:{\displaystyle x-2}
2982:{\displaystyle x-3}
2660:{\displaystyle x-3}
2565:{\displaystyle x=3}
2435:till you arrive at
2076:(which is equal to
10000:Numerical analysis
9823:Cubic function (3)
9816:Quadratic equation
9730:2018-09-28 at the
9662:Chinese Researches
9651:. London: Blackie.
9437:Numerical Analysis
8828:synthetic division
8780:Clenshaw algorithm
8760:Jiu Zhang Suan Shu
8710:(fl. 3rd century).
8610:The Monthly Review
8586:
8569:
8496:
8467:
8420:
8394:
8368:
8339:
8310:
8308:
8119:
8117:
7727:
7547:
7462:
7426:
7362:
7282:
7186:
7153:
7041:
7009:
6977:
6849:
6728:
6699:
6672:
6639:
6606:
6564:
6537:
6501:
6466:
6426:
6399:
6331:
6311:
6236:Other applications
6196:
6056:
5931:
5756:
5534:
5420:
5418:
5126:
5088:
5062:
4995:
4743:
4741:
3986:
3955:
3920:
3892:{\displaystyle kn}
3889:
3866:
3843:
3823:
3803:
3783:
3763:
3743:
3716:
3689:
3672:{\displaystyle 2n}
3669:
3643:
3623:
3594:
3574:
3527:
3507:
3480:
3332:
3295:
3259:
3208:
3122:
3074:
3048:
2979:
2953:
2921:
2831:
2732:
2697:
2657:
2631:
2577:synthetic division
2562:
2536:
2452:
2425:
2350:
2323:
2296:
2273:
2240:
2211:
2191:
2158:
2129:
2102:
2066:
2039:
1967:
1737:
1735:
1326:
1297:
1102:
1066:
1023:
1021:
769:{\displaystyle x.}
766:
743:
716:
670:
459:
439:
409:
407:
9977:
9976:
9918:Quasi-homogeneous
9719:Shu Shu Jiu Zhang
9634:978-0-671-62028-8
9515:978-1-4832-3272-0
9468:978-0-486-44619-6
9427:978-0-201-89684-8
9322:978-0-89871-521-7
9204:Rivest, Ronald L.
9196:Cormen, Thomas H.
8999:, pp. 29β51.
8719:Shu Shu Jiu Zhang
8220:
7962:
7959:
7946:
7943:
6334:{\displaystyle n}
5902:
5864:
5826:
5544:) the product is
3989:{\displaystyle n}
3958:{\displaystyle n}
3923:{\displaystyle x}
3869:{\displaystyle k}
3846:{\displaystyle n}
3826:{\displaystyle x}
3806:{\displaystyle n}
3786:{\displaystyle n}
3766:{\displaystyle n}
3692:{\displaystyle x}
3646:{\displaystyle x}
3597:{\displaystyle n}
3530:{\displaystyle n}
3510:{\displaystyle n}
3475:
3407:
2299:{\displaystyle b}
2214:{\displaystyle 0}
1991:
1990:
1663:
1660:
1293:
1047:
1046:
904:
901:
898:
462:{\displaystyle n}
442:{\displaystyle n}
82:
81:
74:
41:quality standards
32:This article may
10007:
9840:Quartic equation
9761:
9754:
9747:
9738:
9737:
9714:
9679:
9673:
9665:
9652:
9638:
9619:
9607:
9596:
9586:
9559:
9530:
9528:
9527:
9496:
9483:
9481:
9480:
9471:. Archived from
9449:
9440:
9431:
9400:
9355:
9353:
9352:
9346:
9339:
9326:
9307:
9305:
9280:
9278:
9277:
9271:
9264:
9247:
9241:
9233:
9231:
9190:
9188:
9187:
9177:
9149:
9110:
9104:
9098:
9092:
9086:
9080:
9074:
9068:
9062:
9059:
9053:
9049:
9043:
9042:
9021:
9012:
9006:
9000:
8994:
8988:
8982:
8973:
8967:
8961:
8956:
8950:
8944:
8938:
8932:
8926:
8920:
8914:
8908:
8902:
8896:
8890:
8884:
8878:
8872:
8866:
8861:
8855:
8845:
8748:Ulrich Libbrecht
8600:in the issue of
8578:
8576:
8575:
8570:
8556:
8555:
8540:
8539:
8505:
8503:
8502:
8497:
8476:
8474:
8473:
8468:
8457:
8446:
8445:
8429:
8427:
8426:
8421:
8404:. Substituting
8403:
8401:
8400:
8395:
8377:
8375:
8374:
8369:
8348:
8346:
8345:
8340:
8319:
8317:
8316:
8311:
8309:
8302:
8301:
8274:
8273:
8238:
8237:
8221:
8219:
8208:
8179:
8170:
8169:
8128:
8126:
8125:
8120:
8118:
8108:
8107:
8095:
8094:
8078:
8077:
8058:
8057:
8045:
8044:
8028:
8027:
8009:
8008:
7996:
7995:
7979:
7978:
7960:
7957:
7956:
7944:
7941:
7940:
7937:
7927:
7926:
7914:
7913:
7891:
7890:
7866:
7865:
7853:
7852:
7830:
7829:
7807:
7806:
7790:
7789:
7774:
7773:
7757:
7756:
7736:
7734:
7733:
7728:
7723:
7722:
7713:
7712:
7694:
7693:
7684:
7683:
7671:
7670:
7661:
7660:
7645:
7644:
7632:
7631:
7619:
7618:
7609:
7608:
7598:
7593:
7556:
7554:
7553:
7548:
7528:
7472:and solving the
7471:
7469:
7468:
7463:
7452:
7451:
7435:
7433:
7432:
7427:
7410:
7409:
7388:
7387:
7371:
7369:
7368:
7363:
7346:
7345:
7330:
7329:
7308:
7307:
7291:
7289:
7288:
7283:
7266:
7265:
7250:
7249:
7234:
7233:
7212:
7211:
7195:
7193:
7192:
7187:
7162:
7160:
7159:
7154:
7137:
7136:
7121:
7120:
7105:
7104:
7089:
7088:
7067:
7066:
7050:
7048:
7047:
7042:
7018:
7016:
7015:
7010:
6986:
6984:
6983:
6978:
6961:
6960:
6945:
6944:
6929:
6928:
6913:
6912:
6897:
6896:
6875:
6874:
6858:
6856:
6855:
6850:
6746:
6745:
6708:
6706:
6705:
6700:
6698:
6697:
6681:
6679:
6678:
6673:
6671:
6670:
6648:
6646:
6645:
6640:
6638:
6637:
6615:
6613:
6612:
6607:
6602:
6601:
6573:
6571:
6570:
6565:
6563:
6562:
6547:using the guess
6546:
6544:
6543:
6538:
6527:
6526:
6510:
6508:
6507:
6502:
6500:
6499:
6475:
6473:
6472:
6467:
6465:
6464:
6452:
6451:
6435:
6433:
6432:
6427:
6425:
6424:
6408:
6406:
6405:
6400:
6395:
6394:
6376:
6375:
6357:
6356:
6340:
6338:
6337:
6332:
6320:
6318:
6317:
6312:
6301:
6300:
6244:β in which case
6222:identity element
6218:arithmetic shift
6214:arithmetic shift
6205:
6203:
6202:
6197:
6174:
6173:
6172:
6153:
6152:
6151:
6141:
6140:
6119:
6118:
6117:
6107:
6106:
6085:
6084:
6065:
6063:
6062:
6057:
6034:
6033:
6032:
6010:
6009:
6008:
5986:
5985:
5984:
5962:
5961:
5940:
5938:
5937:
5932:
5927:
5923:
5922:
5918:
5917:
5913:
5903:
5901:
5900:
5891:
5890:
5881:
5865:
5863:
5862:
5853:
5852:
5843:
5827:
5825:
5824:
5815:
5814:
5805:
5789:
5788:
5768:division by zero
5765:
5763:
5762:
5757:
5749:
5748:
5739:
5738:
5723:
5722:
5713:
5712:
5697:
5696:
5687:
5686:
5671:
5670:
5661:
5660:
5642:
5641:
5632:
5631:
5619:
5618:
5609:
5608:
5596:
5595:
5586:
5585:
5573:
5572:
5563:
5562:
5543:
5541:
5540:
5535:
5533:
5532:
5523:
5522:
5513:
5512:
5503:
5502:
5429:
5427:
5426:
5421:
5419:
5412:
5408:
5398:
5397:
5374:
5373:
5358:
5354:
5350:
5346:
5345:
5318:
5317:
5299:
5292:
5288:
5287:
5286:
5262:
5261:
5238:
5234:
5233:
5232:
5217:
5216:
5190:
5189:
5135:
5133:
5132:
5127:
5097:
5095:
5094:
5089:
5071:
5069:
5068:
5063:
5055:
5054:
5004:
5002:
5001:
4996:
4991:
4990:
4978:
4977:
4968:
4967:
4957:
4946:
4928:
4927:
4915:
4914:
4895:
4888:
4876:
4861:
4860:
4850:
4839:
4821:
4820:
4811:
4810:
4800:
4795:
4752:
4750:
4749:
4744:
4742:
4732:
4731:
4719:
4718:
4700:
4699:
4687:
4686:
4671:
4667:
4666:
4654:
4653:
4634:
4627:
4615:
4594:
4593:
4581:
4580:
4567:
4560:
4548:
4527:
4523:
4519:
4512:
4511:
4502:
4501:
4489:
4488:
4479:
4478:
4466:
4465:
4445:
4441:
4434:
4433:
4424:
4423:
4411:
4410:
4401:
4400:
4388:
4387:
4367:
4363:
4359:
4352:
4351:
4342:
4341:
4329:
4328:
4319:
4318:
4303:
4302:
4285:
4281:
4274:
4273:
4264:
4263:
4251:
4250:
4241:
4240:
4228:
4227:
4207:
4203:
4202:
4193:
4192:
4174:
4173:
4164:
4163:
4151:
4150:
4141:
4140:
4125:
4124:
4112:
4111:
4096:
4092:
4091:
4082:
4081:
4071:
4066:
3995:
3993:
3992:
3987:
3975:
3968:
3964:
3962:
3961:
3956:
3929:
3927:
3926:
3921:
3898:
3896:
3895:
3890:
3875:
3873:
3872:
3867:
3852:
3850:
3849:
3844:
3832:
3830:
3829:
3824:
3812:
3810:
3809:
3804:
3792:
3790:
3789:
3784:
3772:
3770:
3769:
3764:
3752:
3750:
3749:
3744:
3742:
3741:
3725:
3723:
3722:
3717:
3715:
3714:
3698:
3696:
3695:
3690:
3678:
3676:
3675:
3670:
3653:by iteration.
3652:
3650:
3649:
3644:
3632:
3630:
3629:
3624:
3603:
3601:
3600:
3595:
3583:
3581:
3580:
3575:
3570:
3556:
3555:
3536:
3534:
3533:
3528:
3516:
3514:
3513:
3508:
3489:
3487:
3486:
3481:
3476:
3474:
3457:
3440:
3439:
3424:
3423:
3408:
3406:
3396:
3395:
3385:
3375:
3374:
3364:
3355:
3351:
3341:
3339:
3338:
3333:
3321:
3320:
3304:
3302:
3301:
3296:
3285:
3284:
3268:
3266:
3265:
3260:
3234:
3233:
3217:
3215:
3214:
3209:
3192:
3191:
3176:
3175:
3151:
3150:
3131:
3129:
3128:
3123:
3106:
3105:
3090:The quotient is
3083:
3081:
3080:
3075:
3057:
3055:
3054:
3049:
3032:
3031:
3016:
3015:
2992:
2988:
2986:
2985:
2980:
2962:
2960:
2959:
2954:
2930:
2928:
2927:
2922:
2914:
2913:
2895:
2894:
2876:
2875:
2857:
2856:
2841:we can see that
2840:
2838:
2837:
2832:
2821:
2820:
2802:
2801:
2780:
2779:
2761:
2760:
2741:
2739:
2738:
2733:
2706:
2704:
2703:
2698:
2670:
2666:
2664:
2663:
2658:
2640:
2638:
2637:
2632:
2611:
2607:
2571:
2569:
2568:
2563:
2545:
2543:
2542:
2537:
2520:
2519:
2504:
2503:
2461:
2459:
2458:
2453:
2451:
2450:
2434:
2432:
2431:
2426:
2424:
2423:
2414:
2413:
2401:
2400:
2382:
2381:
2359:
2357:
2356:
2351:
2349:
2348:
2332:
2330:
2329:
2324:
2322:
2321:
2305:
2303:
2302:
2297:
2282:
2280:
2279:
2274:
2272:
2271:
2249:
2247:
2246:
2241:
2220:
2218:
2217:
2212:
2200:
2198:
2197:
2192:
2184:
2183:
2167:
2165:
2164:
2159:
2138:
2136:
2135:
2130:
2128:
2127:
2111:
2109:
2108:
2103:
2098:
2097:
2075:
2073:
2072:
2067:
2065:
2064:
2048:
2046:
2045:
2040:
2035:
2034:
2016:
1985:
1976:
1974:
1973:
1968:
1966:
1965:
1953:
1949:
1948:
1947:
1927:
1923:
1922:
1921:
1906:
1905:
1893:
1892:
1877:
1876:
1852:
1851:
1842:
1841:
1829:
1828:
1819:
1818:
1803:
1802:
1790:
1789:
1752:
1746:
1744:
1743:
1738:
1736:
1729:
1728:
1713:
1709:
1708:
1699:
1698:
1686:
1685:
1670:
1661:
1658:
1656:
1652:
1651:
1645:
1644:
1638:
1637:
1622:
1621:
1603:
1602:
1593:
1592:
1586:
1585:
1573:
1572:
1563:
1562:
1556:
1555:
1543:
1542:
1527:
1523:
1522:
1516:
1515:
1503:
1502:
1493:
1492:
1480:
1479:
1461:
1460:
1442:
1441:
1432:
1431:
1425:
1424:
1412:
1411:
1402:
1401:
1395:
1394:
1382:
1381:
1362:
1361:
1335:
1333:
1332:
1327:
1325:
1324:
1306:
1304:
1303:
1298:
1291:
1290:
1289:
1283:
1282:
1276:
1275:
1263:
1262:
1246:
1245:
1215:
1214:
1205:
1204:
1192:
1191:
1182:
1181:
1169:
1168:
1159:
1158:
1146:
1145:
1111:
1109:
1108:
1103:
1098:
1097:
1076:is the value of
1075:
1073:
1072:
1067:
1065:
1064:
1041:
1032:
1030:
1029:
1024:
1022:
1015:
1014:
1005:
1004:
992:
991:
975:
974:
961:
960:
951:
950:
938:
937:
921:
920:
902:
899:
896:
894:
890:
889:
880:
879:
867:
866:
844:
843:
824:
823:
807:
806:
785:
775:
773:
772:
767:
752:
750:
749:
744:
742:
741:
725:
723:
722:
717:
715:
714:
696:
695:
679:
677:
676:
671:
666:
665:
656:
655:
637:
636:
627:
626:
614:
613:
604:
603:
588:
587:
575:
574:
562:
561:
552:
551:
541:
536:
486:
485:
472:
468:
466:
465:
460:
448:
446:
445:
440:
428:
418:
416:
415:
410:
408:
401:
400:
394:
393:
387:
386:
374:
373:
357:
356:
326:
325:
316:
315:
303:
302:
293:
292:
280:
279:
270:
269:
257:
256:
245:
236:
235:
226:
225:
207:
206:
197:
196:
184:
183:
174:
173:
158:
157:
145:
144:
134:
90:computer science
77:
70:
66:
63:
57:
27:
26:
19:
10015:
10014:
10010:
10009:
10008:
10006:
10005:
10004:
9980:
9979:
9978:
9973:
9922:
9861:
9804:Linear equation
9774:
9765:
9732:Wayback Machine
9701:"Horner scheme"
9699:
9696:
9691:
9667:
9666:
9643:Whittaker, E.T.
9635:
9616:
9608:. McGraw-Hill.
9577:(9). ACM: 633.
9525:
9523:
9516:
9478:
9476:
9469:
9428:
9350:
9348:
9344:
9337:
9323:
9275:
9273:
9269:
9262:
9235:
9234:
9185:
9183:
9154:Cajori, Florian
9119:
9114:
9113:
9105:
9101:
9093:
9089:
9081:
9077:
9069:
9065:
9060:
9056:
9050:
9046:
9022:
9015:
9007:
9003:
8995:
8991:
8983:
8976:
8968:
8964:
8957:
8953:
8945:
8941:
8933:
8929:
8923:Kripasagar 2008
8921:
8917:
8909:
8905:
8897:
8893:
8885:
8881:
8875:Pankiewicz 1968
8873:
8869:
8862:
8858:
8846:
8842:
8837:
8812:Estrin's scheme
8776:
8606:Charles Babbage
8579:
8551:
8547:
8535:
8531:
8526:
8523:
8522:
8512:
8482:
8479:
8478:
8450:
8441:
8437:
8435:
8432:
8431:
8409:
8406:
8405:
8383:
8380:
8379:
8354:
8351:
8350:
8325:
8322:
8321:
8307:
8306:
8297:
8293:
8269:
8265:
8258:
8243:
8242:
8233:
8229:
8222:
8209:
8180:
8178:
8175:
8174:
8165:
8161:
8154:
8138:
8136:
8133:
8132:
8116:
8115:
8103:
8099:
8090:
8086:
8079:
8073:
8069:
8066:
8065:
8053:
8049:
8040:
8036:
8029:
8023:
8019:
8016:
8004:
8000:
7991:
7987:
7980:
7974:
7970:
7967:
7966:
7955:
7953:
7950:
7939:
7935:
7934:
7922:
7918:
7903:
7899:
7892:
7880:
7876:
7873:
7861:
7857:
7842:
7838:
7831:
7819:
7815:
7812:
7811:
7802:
7798:
7791:
7785:
7781:
7778:
7769:
7765:
7758:
7752:
7748:
7744:
7742:
7739:
7738:
7718:
7714:
7708:
7704:
7689:
7685:
7679:
7675:
7666:
7662:
7656:
7652:
7640:
7636:
7627:
7623:
7614:
7610:
7604:
7600:
7594:
7583:
7562:
7559:
7558:
7524:
7489:
7486:
7485:
7482:
7474:linear equation
7447:
7443:
7441:
7438:
7437:
7405:
7401:
7383:
7379:
7377:
7374:
7373:
7341:
7337:
7325:
7321:
7303:
7299:
7297:
7294:
7293:
7261:
7257:
7245:
7241:
7229:
7225:
7207:
7203:
7201:
7198:
7197:
7169:
7166:
7165:
7132:
7128:
7116:
7112:
7100:
7096:
7084:
7080:
7062:
7058:
7056:
7053:
7052:
7024:
7021:
7020:
6995:
6992:
6991:
6956:
6952:
6940:
6936:
6924:
6920:
6908:
6904:
6892:
6888:
6870:
6866:
6864:
6861:
6860:
6741:
6737:
6735:
6732:
6731:
6720:
6693:
6689:
6687:
6684:
6683:
6660:
6656:
6654:
6651:
6650:
6627:
6623:
6621:
6618:
6617:
6597:
6593:
6582:
6579:
6578:
6558:
6554:
6552:
6549:
6548:
6522:
6518:
6516:
6513:
6512:
6495:
6491:
6489:
6486:
6485:
6482:Newton's method
6460:
6456:
6447:
6443:
6441:
6438:
6437:
6420:
6416:
6414:
6411:
6410:
6390:
6386:
6365:
6361:
6352:
6348:
6346:
6343:
6342:
6326:
6323:
6322:
6296:
6292:
6290:
6287:
6286:
6283:Newton's method
6279:
6256:
6242:numeral systems
6238:
6168:
6164:
6163:
6147:
6143:
6142:
6133:
6129:
6113:
6109:
6108:
6099:
6095:
6080:
6076:
6071:
6068:
6067:
6028:
6024:
6023:
6004:
6000:
5999:
5980:
5976:
5975:
5957:
5953:
5948:
5945:
5944:
5896:
5892:
5886:
5882:
5880:
5870:
5866:
5858:
5854:
5848:
5844:
5842:
5832:
5828:
5820:
5816:
5810:
5806:
5804:
5794:
5790:
5784:
5780:
5775:
5772:
5771:
5744:
5740:
5734:
5730:
5718:
5714:
5708:
5704:
5692:
5688:
5682:
5678:
5666:
5662:
5656:
5652:
5637:
5633:
5627:
5623:
5614:
5610:
5604:
5600:
5591:
5587:
5581:
5577:
5568:
5564:
5558:
5554:
5549:
5546:
5545:
5528:
5524:
5518:
5514:
5508:
5504:
5498:
5494:
5492:
5489:
5488:
5485:
5470:
5435:
5417:
5416:
5390:
5386:
5379:
5375:
5366:
5362:
5338:
5334:
5330:
5323:
5319:
5310:
5306:
5297:
5296:
5279:
5275:
5254:
5250:
5249:
5245:
5225:
5221:
5209:
5205:
5204:
5200:
5185:
5181:
5171:
5155:
5153:
5150:
5149:
5142:
5115:
5112:
5111:
5077:
5074:
5073:
5050:
5046:
5044:
5041:
5040:
5033:microcontroller
5029:
5023:
4986:
4982:
4973:
4969:
4963:
4959:
4947:
4936:
4920:
4916:
4901:
4897:
4884:
4877:
4866:
4856:
4852:
4840:
4829:
4816:
4812:
4806:
4802:
4796:
4785:
4764:
4761:
4760:
4740:
4739:
4727:
4723:
4714:
4710:
4695:
4691:
4682:
4678:
4669:
4668:
4659:
4655:
4640:
4636:
4623:
4616:
4605:
4586:
4582:
4573:
4569:
4556:
4549:
4538:
4525:
4524:
4507:
4503:
4497:
4493:
4484:
4480:
4474:
4470:
4461:
4457:
4456:
4452:
4429:
4425:
4419:
4415:
4406:
4402:
4396:
4392:
4383:
4379:
4378:
4374:
4365:
4364:
4347:
4343:
4337:
4333:
4324:
4320:
4314:
4310:
4298:
4294:
4293:
4289:
4269:
4265:
4259:
4255:
4246:
4242:
4236:
4232:
4223:
4219:
4218:
4214:
4205:
4204:
4198:
4194:
4188:
4184:
4169:
4165:
4159:
4155:
4146:
4142:
4136:
4132:
4120:
4116:
4107:
4103:
4094:
4093:
4087:
4083:
4077:
4073:
4067:
4056:
4045:
4029:
4027:
4024:
4023:
4008:
4006:Estrin's scheme
4002:
3981:
3978:
3977:
3973:
3966:
3950:
3947:
3946:
3939:preconditioning
3915:
3912:
3911:
3881:
3878:
3877:
3861:
3858:
3857:
3838:
3835:
3834:
3818:
3815:
3814:
3798:
3795:
3794:
3778:
3775:
3774:
3758:
3755:
3754:
3737:
3733:
3731:
3728:
3727:
3710:
3706:
3704:
3701:
3700:
3684:
3681:
3680:
3661:
3658:
3657:
3638:
3635:
3634:
3609:
3606:
3605:
3589:
3586:
3585:
3566:
3551:
3547:
3542:
3539:
3538:
3522:
3519:
3518:
3502:
3499:
3498:
3495:
3461:
3456:
3435:
3431:
3419:
3415:
3391:
3387:
3386:
3370:
3366:
3365:
3363:
3361:
3358:
3357:
3353:
3349:
3346:
3316:
3312:
3310:
3307:
3306:
3280:
3276:
3274:
3271:
3270:
3229:
3225:
3223:
3220:
3219:
3187:
3183:
3171:
3167:
3146:
3142:
3140:
3137:
3136:
3101:
3097:
3095:
3092:
3091:
3088:
3063:
3060:
3059:
3027:
3023:
3011:
3007:
3005:
3002:
3001:
2990:
2968:
2965:
2964:
2963:on division by
2939:
2936:
2935:
2909:
2905:
2890:
2886:
2871:
2867:
2852:
2848:
2846:
2843:
2842:
2816:
2812:
2797:
2793:
2775:
2771:
2756:
2752:
2750:
2747:
2746:
2712:
2709:
2708:
2683:
2680:
2679:
2668:
2646:
2643:
2642:
2641:on division by
2617:
2614:
2613:
2609:
2605:
2602:
2587:
2551:
2548:
2547:
2515:
2511:
2499:
2495:
2475:
2472:
2471:
2468:
2446:
2442:
2440:
2437:
2436:
2419:
2415:
2409:
2405:
2390:
2386:
2371:
2367:
2365:
2362:
2361:
2344:
2340:
2338:
2335:
2334:
2317:
2313:
2311:
2308:
2307:
2291:
2288:
2287:
2267:
2263:
2255:
2252:
2251:
2226:
2223:
2222:
2206:
2203:
2202:
2179:
2175:
2173:
2170:
2169:
2144:
2141:
2140:
2123:
2119:
2117:
2114:
2113:
2093:
2089:
2081:
2078:
2077:
2060:
2056:
2054:
2051:
2050:
2030:
2026:
2012:
1998:
1995:
1994:
1961:
1957:
1943:
1939:
1932:
1928:
1911:
1907:
1901:
1897:
1882:
1878:
1866:
1862:
1847:
1843:
1837:
1833:
1824:
1820:
1814:
1810:
1798:
1794:
1785:
1781:
1780:
1776:
1759:
1756:
1755:
1734:
1733:
1724:
1720:
1711:
1710:
1704:
1700:
1694:
1690:
1681:
1677:
1668:
1667:
1654:
1653:
1647:
1646:
1640:
1639:
1627:
1623:
1617:
1613:
1598:
1594:
1588:
1587:
1581:
1577:
1568:
1564:
1558:
1557:
1551:
1547:
1538:
1534:
1525:
1524:
1518:
1517:
1511:
1510:
1498:
1494:
1488:
1484:
1469:
1465:
1456:
1452:
1437:
1433:
1427:
1426:
1420:
1416:
1407:
1403:
1397:
1396:
1390:
1386:
1377:
1373:
1366:
1357:
1353:
1343:
1341:
1338:
1337:
1320:
1316:
1314:
1311:
1310:
1285:
1284:
1278:
1277:
1271:
1270:
1258:
1254:
1235:
1231:
1210:
1206:
1200:
1199:
1187:
1183:
1177:
1176:
1164:
1160:
1154:
1153:
1141:
1137:
1120:
1117:
1116:
1093:
1089:
1081:
1078:
1077:
1060:
1056:
1054:
1051:
1050:
1020:
1019:
1010:
1006:
1000:
996:
987:
983:
976:
970:
966:
963:
962:
956:
952:
946:
942:
933:
929:
922:
916:
912:
909:
908:
892:
891:
885:
881:
875:
871:
856:
852:
845:
833:
829:
826:
825:
819:
815:
808:
802:
798:
794:
792:
789:
788:
758:
755:
754:
737:
733:
731:
728:
727:
710:
706:
691:
687:
685:
682:
681:
661:
657:
651:
647:
632:
628:
622:
618:
609:
605:
599:
595:
583:
579:
570:
566:
557:
553:
547:
543:
537:
526:
505:
502:
501:
498:
483:
482:
478:Horner's method
476:Alternatively,
470:
454:
451:
450:
434:
431:
430:
426:
406:
405:
396:
395:
389:
388:
382:
381:
369:
365:
346:
342:
321:
317:
311:
310:
298:
294:
288:
287:
275:
271:
265:
264:
252:
248:
246:
244:
238:
237:
231:
227:
221:
217:
202:
198:
192:
188:
179:
175:
169:
165:
153:
149:
140:
136:
131:
129:
126:
125:
98:Horner's scheme
94:Horner's method
78:
67:
61:
58:
51:
28:
24:
17:
12:
11:
5:
10013:
10003:
10002:
9997:
9992:
9975:
9974:
9972:
9971:
9966:
9961:
9956:
9951:
9946:
9941:
9936:
9930:
9928:
9924:
9923:
9921:
9920:
9915:
9910:
9905:
9900:
9895:
9890:
9885:
9880:
9875:
9869:
9867:
9863:
9862:
9860:
9859:
9854:
9849:
9844:
9843:
9842:
9832:
9831:
9830:
9828:Cubic equation
9820:
9819:
9818:
9808:
9807:
9806:
9796:
9791:
9785:
9783:
9776:
9775:
9764:
9763:
9756:
9749:
9741:
9735:
9734:
9722:
9717:Qiu Jin-Shao,
9715:
9695:
9694:External links
9692:
9690:
9689:
9688:
9687:
9653:
9639:
9633:
9620:
9614:
9597:
9560:
9531:
9514:
9497:
9484:
9467:
9450:
9441:
9432:
9426:
9410:
9409:
9408:
9361:
9360:
9359:
9327:
9321:
9308:
9281:
9252:Fateman, R. J.
9248:
9192:
9168:(8): 409β414.
9150:
9138:10.2307/604533
9132:(2): 304β309.
9120:
9118:
9115:
9112:
9111:
9109:, p. 208.
9107:Libbrecht 2005
9099:
9087:
9085:, p. 142.
9075:
9063:
9054:
9044:
9013:
9001:
8989:
8974:
8970:Libbrecht 2005
8962:
8951:
8949:, p. 112.
8939:
8937:, Section 5.4.
8927:
8915:
8903:
8891:
8887:Ostrowski 1954
8879:
8867:
8856:
8839:
8838:
8836:
8833:
8832:
8831:
8824:Ruffini's rule
8821:
8815:
8809:
8800:
8786:
8784:Chebyshev form
8775:
8772:
8712:
8711:
8695:
8684:
8671:
8655:
8645:
8639:
8636:Ruffini's rule
8568:
8565:
8562:
8559:
8554:
8550:
8546:
8543:
8538:
8534:
8530:
8511:
8508:
8495:
8492:
8489:
8486:
8466:
8463:
8460:
8456:
8453:
8449:
8444:
8440:
8419:
8416:
8413:
8393:
8390:
8387:
8367:
8364:
8361:
8358:
8338:
8335:
8332:
8329:
8305:
8300:
8296:
8292:
8289:
8286:
8283:
8280:
8277:
8272:
8268:
8264:
8261:
8259:
8257:
8254:
8251:
8248:
8245:
8244:
8241:
8236:
8232:
8228:
8225:
8223:
8218:
8215:
8212:
8207:
8204:
8201:
8198:
8195:
8192:
8189:
8186:
8183:
8177:
8176:
8173:
8168:
8164:
8160:
8157:
8155:
8153:
8150:
8147:
8144:
8141:
8140:
8114:
8111:
8106:
8102:
8098:
8093:
8089:
8085:
8082:
8080:
8076:
8072:
8068:
8067:
8064:
8061:
8056:
8052:
8048:
8043:
8039:
8035:
8032:
8030:
8026:
8022:
8017:
8015:
8012:
8007:
8003:
7999:
7994:
7990:
7986:
7983:
7981:
7977:
7973:
7969:
7968:
7965:
7954:
7951:
7949:
7938:
7936:
7933:
7930:
7925:
7921:
7917:
7912:
7909:
7906:
7902:
7898:
7895:
7893:
7889:
7886:
7883:
7879:
7874:
7872:
7869:
7864:
7860:
7856:
7851:
7848:
7845:
7841:
7837:
7834:
7832:
7828:
7825:
7822:
7818:
7814:
7813:
7810:
7805:
7801:
7797:
7794:
7792:
7788:
7784:
7779:
7777:
7772:
7768:
7764:
7761:
7759:
7755:
7751:
7747:
7746:
7726:
7721:
7717:
7711:
7707:
7703:
7700:
7697:
7692:
7688:
7682:
7678:
7674:
7669:
7665:
7659:
7655:
7651:
7648:
7643:
7639:
7635:
7630:
7626:
7622:
7617:
7613:
7607:
7603:
7597:
7592:
7589:
7586:
7582:
7578:
7575:
7572:
7569:
7566:
7546:
7543:
7540:
7537:
7534:
7531:
7527:
7523:
7520:
7517:
7514:
7511:
7508:
7505:
7502:
7499:
7496:
7493:
7481:
7478:
7461:
7458:
7455:
7450:
7446:
7425:
7422:
7419:
7416:
7413:
7408:
7404:
7400:
7397:
7394:
7391:
7386:
7382:
7361:
7358:
7355:
7352:
7349:
7344:
7340:
7336:
7333:
7328:
7324:
7320:
7317:
7314:
7311:
7306:
7302:
7281:
7278:
7275:
7272:
7269:
7264:
7260:
7256:
7253:
7248:
7244:
7240:
7237:
7232:
7228:
7224:
7221:
7218:
7215:
7210:
7206:
7185:
7182:
7179:
7176:
7173:
7152:
7149:
7146:
7143:
7140:
7135:
7131:
7127:
7124:
7119:
7115:
7111:
7108:
7103:
7099:
7095:
7092:
7087:
7083:
7079:
7076:
7073:
7070:
7065:
7061:
7040:
7037:
7034:
7031:
7028:
7019:is divided by
7008:
7005:
7002:
6999:
6976:
6973:
6970:
6967:
6964:
6959:
6955:
6951:
6948:
6943:
6939:
6935:
6932:
6927:
6923:
6919:
6916:
6911:
6907:
6903:
6900:
6895:
6891:
6887:
6884:
6881:
6878:
6873:
6869:
6848:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6824:
6821:
6818:
6815:
6812:
6809:
6806:
6803:
6800:
6797:
6794:
6791:
6788:
6785:
6782:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6758:
6755:
6752:
6749:
6744:
6740:
6719:
6716:
6711:
6710:
6696:
6692:
6669:
6666:
6663:
6659:
6636:
6633:
6630:
6626:
6605:
6600:
6596:
6592:
6589:
6586:
6575:
6561:
6557:
6536:
6533:
6530:
6525:
6521:
6498:
6494:
6463:
6459:
6455:
6450:
6446:
6423:
6419:
6398:
6393:
6389:
6385:
6382:
6379:
6374:
6371:
6368:
6364:
6360:
6355:
6351:
6330:
6310:
6307:
6304:
6299:
6295:
6278:
6275:
6271:faster methods
6252:
6237:
6234:
6210:register shift
6195:
6192:
6189:
6186:
6183:
6180:
6177:
6171:
6167:
6162:
6159:
6156:
6150:
6146:
6139:
6136:
6132:
6128:
6125:
6122:
6116:
6112:
6105:
6102:
6098:
6094:
6091:
6088:
6083:
6079:
6075:
6055:
6052:
6049:
6046:
6043:
6040:
6037:
6031:
6027:
6022:
6019:
6016:
6013:
6007:
6003:
5998:
5995:
5992:
5989:
5983:
5979:
5974:
5971:
5968:
5965:
5960:
5956:
5952:
5930:
5926:
5921:
5916:
5912:
5909:
5906:
5899:
5895:
5889:
5885:
5879:
5876:
5873:
5869:
5861:
5857:
5851:
5847:
5841:
5838:
5835:
5831:
5823:
5819:
5813:
5809:
5803:
5800:
5797:
5793:
5787:
5783:
5779:
5755:
5752:
5747:
5743:
5737:
5733:
5729:
5726:
5721:
5717:
5711:
5707:
5703:
5700:
5695:
5691:
5685:
5681:
5677:
5674:
5669:
5665:
5659:
5655:
5651:
5648:
5645:
5640:
5636:
5630:
5626:
5622:
5617:
5613:
5607:
5603:
5599:
5594:
5590:
5584:
5580:
5576:
5571:
5567:
5561:
5557:
5553:
5531:
5527:
5521:
5517:
5511:
5507:
5501:
5497:
5484:
5481:
5480:
5479:
5471:
5469:
5468:
5465:
5461:
5454:
5434:
5431:
5415:
5411:
5407:
5404:
5401:
5396:
5393:
5389:
5385:
5382:
5378:
5372:
5369:
5365:
5361:
5357:
5353:
5349:
5344:
5341:
5337:
5333:
5329:
5326:
5322:
5316:
5313:
5309:
5305:
5302:
5300:
5298:
5295:
5291:
5285:
5282:
5278:
5274:
5271:
5268:
5265:
5260:
5257:
5253:
5248:
5244:
5241:
5237:
5231:
5228:
5224:
5220:
5215:
5212:
5208:
5203:
5199:
5196:
5193:
5188:
5184:
5180:
5177:
5174:
5172:
5170:
5167:
5164:
5161:
5158:
5157:
5141:
5138:
5125:
5122:
5119:
5087:
5084:
5081:
5061:
5058:
5053:
5049:
5025:Main article:
5022:
5019:
5015:floating-point
4994:
4989:
4985:
4981:
4976:
4972:
4966:
4962:
4956:
4953:
4950:
4945:
4942:
4939:
4935:
4931:
4926:
4923:
4919:
4913:
4910:
4907:
4904:
4900:
4894:
4891:
4887:
4883:
4880:
4875:
4872:
4869:
4865:
4859:
4855:
4849:
4846:
4843:
4838:
4835:
4832:
4828:
4824:
4819:
4815:
4809:
4805:
4799:
4794:
4791:
4788:
4784:
4780:
4777:
4774:
4771:
4768:
4738:
4735:
4730:
4726:
4722:
4717:
4713:
4709:
4706:
4703:
4698:
4694:
4690:
4685:
4681:
4677:
4674:
4672:
4670:
4665:
4662:
4658:
4652:
4649:
4646:
4643:
4639:
4633:
4630:
4626:
4622:
4619:
4614:
4611:
4608:
4604:
4600:
4597:
4592:
4589:
4585:
4579:
4576:
4572:
4566:
4563:
4559:
4555:
4552:
4547:
4544:
4541:
4537:
4533:
4530:
4528:
4526:
4522:
4518:
4515:
4510:
4506:
4500:
4496:
4492:
4487:
4483:
4477:
4473:
4469:
4464:
4460:
4455:
4451:
4448:
4444:
4440:
4437:
4432:
4428:
4422:
4418:
4414:
4409:
4405:
4399:
4395:
4391:
4386:
4382:
4377:
4373:
4370:
4368:
4366:
4362:
4358:
4355:
4350:
4346:
4340:
4336:
4332:
4327:
4323:
4317:
4313:
4309:
4306:
4301:
4297:
4292:
4288:
4284:
4280:
4277:
4272:
4268:
4262:
4258:
4254:
4249:
4245:
4239:
4235:
4231:
4226:
4222:
4217:
4213:
4210:
4208:
4206:
4201:
4197:
4191:
4187:
4183:
4180:
4177:
4172:
4168:
4162:
4158:
4154:
4149:
4145:
4139:
4135:
4131:
4128:
4123:
4119:
4115:
4110:
4106:
4102:
4099:
4097:
4095:
4090:
4086:
4080:
4076:
4070:
4065:
4062:
4059:
4055:
4051:
4048:
4046:
4044:
4041:
4038:
4035:
4032:
4031:
4001:
3998:
3985:
3954:
3919:
3888:
3885:
3865:
3842:
3822:
3802:
3782:
3773:additions and
3762:
3740:
3736:
3713:
3709:
3688:
3668:
3665:
3642:
3622:
3619:
3616:
3613:
3604:additions and
3593:
3573:
3569:
3565:
3562:
3559:
3554:
3550:
3546:
3537:additions and
3526:
3506:
3494:
3491:
3479:
3473:
3470:
3467:
3464:
3460:
3455:
3452:
3449:
3446:
3443:
3438:
3434:
3430:
3427:
3422:
3418:
3414:
3411:
3405:
3402:
3399:
3394:
3390:
3384:
3381:
3378:
3373:
3369:
3344:
3331:
3328:
3325:
3319:
3315:
3294:
3291:
3288:
3283:
3279:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3232:
3228:
3207:
3204:
3201:
3198:
3195:
3190:
3186:
3182:
3179:
3174:
3170:
3166:
3163:
3160:
3157:
3154:
3149:
3145:
3121:
3118:
3115:
3112:
3109:
3104:
3100:
3086:
3073:
3070:
3067:
3047:
3044:
3041:
3038:
3035:
3030:
3026:
3022:
3019:
3014:
3010:
2978:
2975:
2972:
2952:
2949:
2946:
2943:
2920:
2917:
2912:
2908:
2904:
2901:
2898:
2893:
2889:
2885:
2882:
2879:
2874:
2870:
2866:
2863:
2860:
2855:
2851:
2830:
2827:
2824:
2819:
2815:
2811:
2808:
2805:
2800:
2796:
2792:
2789:
2786:
2783:
2778:
2774:
2770:
2767:
2764:
2759:
2755:
2731:
2728:
2725:
2722:
2719:
2716:
2696:
2693:
2690:
2687:
2656:
2653:
2650:
2630:
2627:
2624:
2621:
2585:
2581:
2561:
2558:
2555:
2535:
2532:
2529:
2526:
2523:
2518:
2514:
2510:
2507:
2502:
2498:
2494:
2491:
2488:
2485:
2482:
2479:
2467:
2464:
2449:
2445:
2422:
2418:
2412:
2408:
2404:
2399:
2396:
2393:
2389:
2385:
2380:
2377:
2374:
2370:
2347:
2343:
2320:
2316:
2295:
2270:
2266:
2262:
2259:
2239:
2236:
2233:
2230:
2210:
2190:
2187:
2182:
2178:
2157:
2154:
2151:
2148:
2126:
2122:
2101:
2096:
2092:
2088:
2085:
2063:
2059:
2038:
2033:
2029:
2025:
2022:
2019:
2015:
2011:
2008:
2005:
2002:
1989:
1988:
1979:
1977:
1964:
1960:
1956:
1952:
1946:
1942:
1938:
1935:
1931:
1926:
1920:
1917:
1914:
1910:
1904:
1900:
1896:
1891:
1888:
1885:
1881:
1875:
1872:
1869:
1865:
1861:
1858:
1855:
1850:
1846:
1840:
1836:
1832:
1827:
1823:
1817:
1813:
1809:
1806:
1801:
1797:
1793:
1788:
1784:
1779:
1775:
1772:
1769:
1766:
1763:
1732:
1727:
1723:
1719:
1716:
1714:
1712:
1707:
1703:
1697:
1693:
1689:
1684:
1680:
1676:
1673:
1671:
1669:
1666:
1657:
1655:
1650:
1643:
1636:
1633:
1630:
1626:
1620:
1616:
1612:
1609:
1606:
1601:
1597:
1591:
1584:
1580:
1576:
1571:
1567:
1561:
1554:
1550:
1546:
1541:
1537:
1533:
1530:
1528:
1526:
1521:
1514:
1509:
1506:
1501:
1497:
1491:
1487:
1483:
1478:
1475:
1472:
1468:
1464:
1459:
1455:
1451:
1448:
1445:
1440:
1436:
1430:
1423:
1419:
1415:
1410:
1406:
1400:
1393:
1389:
1385:
1380:
1376:
1372:
1369:
1367:
1365:
1360:
1356:
1352:
1349:
1346:
1345:
1323:
1319:
1296:
1288:
1281:
1274:
1269:
1266:
1261:
1257:
1252:
1249:
1244:
1241:
1238:
1234:
1230:
1227:
1224:
1221:
1218:
1213:
1209:
1203:
1198:
1195:
1190:
1186:
1180:
1175:
1172:
1167:
1163:
1157:
1152:
1149:
1144:
1140:
1136:
1133:
1130:
1127:
1124:
1101:
1096:
1092:
1088:
1085:
1063:
1059:
1045:
1044:
1035:
1033:
1018:
1013:
1009:
1003:
999:
995:
990:
986:
982:
979:
977:
973:
969:
965:
964:
959:
955:
949:
945:
941:
936:
932:
928:
925:
923:
919:
915:
911:
910:
907:
895:
893:
888:
884:
878:
874:
870:
865:
862:
859:
855:
851:
848:
846:
842:
839:
836:
832:
828:
827:
822:
818:
814:
811:
809:
805:
801:
797:
796:
765:
762:
740:
736:
713:
709:
705:
702:
699:
694:
690:
669:
664:
660:
654:
650:
646:
643:
640:
635:
631:
625:
621:
617:
612:
608:
602:
598:
594:
591:
586:
582:
578:
573:
569:
565:
560:
556:
550:
546:
540:
535:
532:
529:
525:
521:
518:
515:
512:
509:
497:
494:
458:
438:
404:
399:
392:
385:
380:
377:
372:
368:
363:
360:
355:
352:
349:
345:
341:
338:
335:
332:
329:
324:
320:
314:
309:
306:
301:
297:
291:
286:
283:
278:
274:
268:
263:
260:
255:
251:
247:
243:
240:
239:
234:
230:
224:
220:
216:
213:
210:
205:
201:
195:
191:
187:
182:
178:
172:
168:
164:
161:
156:
152:
148:
143:
139:
135:
133:
80:
79:
31:
29:
22:
15:
9:
6:
4:
3:
2:
10012:
10001:
9998:
9996:
9993:
9991:
9988:
9987:
9985:
9970:
9969:GrΓΆbner basis
9967:
9965:
9962:
9960:
9957:
9955:
9952:
9950:
9947:
9945:
9942:
9940:
9937:
9935:
9934:Factorization
9932:
9931:
9929:
9925:
9919:
9916:
9914:
9911:
9909:
9906:
9904:
9901:
9899:
9896:
9894:
9891:
9889:
9886:
9884:
9881:
9879:
9876:
9874:
9871:
9870:
9868:
9866:By properties
9864:
9858:
9855:
9853:
9850:
9848:
9845:
9841:
9838:
9837:
9836:
9833:
9829:
9826:
9825:
9824:
9821:
9817:
9814:
9813:
9812:
9809:
9805:
9802:
9801:
9800:
9797:
9795:
9792:
9790:
9787:
9786:
9784:
9782:
9777:
9773:
9769:
9762:
9757:
9755:
9750:
9748:
9743:
9742:
9739:
9733:
9729:
9726:
9723:
9720:
9716:
9712:
9708:
9707:
9702:
9698:
9697:
9685:
9681:
9680:
9677:
9671:
9663:
9659:
9654:
9650:
9649:
9644:
9640:
9636:
9630:
9626:
9621:
9617:
9615:9780070602267
9611:
9606:
9605:
9598:
9594:
9590:
9585:
9580:
9576:
9572:
9571:
9566:
9561:
9557:
9553:
9549:
9545:
9541:
9537:
9532:
9521:
9517:
9511:
9507:
9503:
9498:
9494:
9490:
9485:
9475:on 2017-06-06
9474:
9470:
9464:
9460:
9456:
9451:
9447:
9442:
9438:
9433:
9429:
9423:
9419:
9415:
9414:Knuth, Donald
9411:
9406:
9402:
9401:
9398:
9394:
9390:
9386:
9382:
9378:
9374:
9370:
9366:
9362:
9357:
9356:
9347:on 2014-01-06
9343:
9336:
9335:
9328:
9324:
9318:
9314:
9309:
9304:
9299:
9295:
9291:
9287:
9282:
9272:on 2017-08-14
9268:
9261:
9257:
9253:
9249:
9245:
9239:
9230:
9225:
9221:
9217:
9213:
9209:
9205:
9201:
9197:
9193:
9181:
9176:
9171:
9167:
9163:
9159:
9155:
9151:
9147:
9143:
9139:
9135:
9131:
9127:
9122:
9121:
9108:
9103:
9096:
9091:
9084:
9079:
9072:
9071:Berggren 1990
9067:
9058:
9048:
9041:
9037:
9036:
9031:
9027:
9020:
9018:
9010:
9005:
8998:
8993:
8986:
8981:
8979:
8971:
8966:
8960:
8955:
8948:
8943:
8936:
8931:
8925:, p. 62.
8924:
8919:
8912:
8907:
8900:
8895:
8888:
8883:
8876:
8871:
8865:
8860:
8854:
8850:
8844:
8840:
8829:
8825:
8822:
8819:
8818:Lill's method
8816:
8813:
8810:
8808:
8804:
8801:
8798:
8794:
8790:
8787:
8785:
8781:
8778:
8777:
8771:
8769:
8768:Jigu Suanjing
8765:
8764:Wang Xiaotong
8761:
8757:
8753:
8749:
8744:
8740:
8738:
8734:
8733:Yoshio Mikami
8730:
8726:
8725:
8720:
8716:
8709:
8705:
8701:
8700:
8696:
8693:
8689:
8685:
8682:
8679:
8678:mathematician
8676:
8672:
8669:
8668:
8663:
8660:
8656:
8653:
8650:
8646:
8643:
8640:
8637:
8634:in 1809 (see
8633:
8632:Paolo Ruffini
8630:
8629:
8628:
8625:
8623:
8622:Paolo Ruffini
8619:
8614:
8611:
8607:
8603:
8599:
8595:
8591:
8583:
8566:
8563:
8560:
8557:
8552:
8548:
8544:
8541:
8536:
8532:
8528:
8520:
8516:
8507:
8490:
8484:
8461:
8454:
8451:
8447:
8442:
8438:
8417:
8414:
8411:
8391:
8388:
8385:
8362:
8356:
8333:
8327:
8303:
8298:
8294:
8287:
8284:
8281:
8275:
8270:
8266:
8262:
8260:
8252:
8246:
8239:
8234:
8230:
8226:
8224:
8216:
8213:
8210:
8202:
8196:
8193:
8187:
8181:
8171:
8166:
8162:
8158:
8156:
8148:
8142:
8129:
8112:
8109:
8104:
8100:
8096:
8091:
8087:
8083:
8081:
8074:
8070:
8062:
8059:
8054:
8050:
8046:
8041:
8037:
8033:
8031:
8024:
8020:
8013:
8010:
8005:
8001:
7997:
7992:
7988:
7984:
7982:
7975:
7971:
7963:
7947:
7931:
7928:
7923:
7919:
7915:
7910:
7907:
7904:
7900:
7896:
7894:
7887:
7884:
7881:
7877:
7870:
7867:
7862:
7858:
7854:
7849:
7846:
7843:
7839:
7835:
7833:
7826:
7823:
7820:
7816:
7808:
7803:
7799:
7795:
7793:
7786:
7782:
7775:
7770:
7766:
7762:
7760:
7753:
7749:
7724:
7719:
7715:
7709:
7705:
7701:
7698:
7695:
7690:
7686:
7680:
7676:
7672:
7667:
7663:
7657:
7653:
7649:
7646:
7641:
7637:
7633:
7628:
7624:
7620:
7615:
7611:
7605:
7601:
7595:
7590:
7587:
7584:
7580:
7576:
7570:
7564:
7544:
7538:
7535:
7532:
7525:
7515:
7509:
7506:
7500:
7494:
7477:
7475:
7456:
7448:
7444:
7423:
7420:
7417:
7414:
7411:
7406:
7402:
7398:
7392:
7384:
7380:
7359:
7356:
7353:
7350:
7347:
7342:
7338:
7334:
7331:
7326:
7322:
7318:
7312:
7304:
7300:
7279:
7276:
7273:
7270:
7267:
7262:
7258:
7254:
7251:
7246:
7242:
7238:
7235:
7230:
7226:
7222:
7216:
7208:
7204:
7180:
7177:
7174:
7150:
7147:
7144:
7141:
7138:
7133:
7129:
7125:
7122:
7117:
7113:
7109:
7106:
7101:
7097:
7093:
7090:
7085:
7081:
7077:
7071:
7063:
7059:
7035:
7032:
7029:
7003:
6997:
6987:
6974:
6971:
6968:
6965:
6962:
6957:
6953:
6949:
6946:
6941:
6937:
6933:
6930:
6925:
6921:
6917:
6914:
6909:
6905:
6901:
6898:
6893:
6889:
6885:
6879:
6871:
6867:
6843:
6840:
6837:
6828:
6825:
6822:
6813:
6810:
6807:
6798:
6795:
6792:
6783:
6780:
6777:
6768:
6765:
6762:
6756:
6750:
6742:
6738:
6724:
6715:
6694:
6690:
6667:
6664:
6661:
6657:
6634:
6631:
6628:
6624:
6598:
6594:
6590:
6587:
6576:
6559:
6555:
6531:
6523:
6519:
6496:
6492:
6483:
6479:
6478:
6477:
6461:
6457:
6453:
6448:
6444:
6421:
6417:
6396:
6391:
6387:
6383:
6380:
6377:
6372:
6369:
6366:
6362:
6358:
6353:
6349:
6328:
6305:
6297:
6293:
6284:
6274:
6272:
6268:
6264:
6260:
6255:
6251:
6247:
6243:
6233:
6231:
6225:
6223:
6219:
6215:
6211:
6206:
6193:
6178:
6169:
6165:
6160:
6157:
6148:
6144:
6137:
6134:
6130:
6126:
6123:
6114:
6110:
6103:
6100:
6096:
6092:
6089:
6081:
6077:
6073:
6053:
6038:
6029:
6025:
6020:
6017:
6014:
6005:
6001:
5996:
5993:
5990:
5981:
5977:
5972:
5969:
5966:
5958:
5954:
5950:
5941:
5928:
5924:
5919:
5914:
5907:
5897:
5893:
5887:
5883:
5877:
5874:
5871:
5867:
5859:
5855:
5849:
5845:
5839:
5836:
5833:
5829:
5821:
5817:
5811:
5807:
5801:
5798:
5795:
5791:
5785:
5781:
5777:
5769:
5753:
5750:
5745:
5741:
5735:
5731:
5727:
5724:
5719:
5715:
5709:
5705:
5701:
5698:
5693:
5689:
5683:
5679:
5675:
5672:
5667:
5663:
5657:
5653:
5649:
5646:
5638:
5634:
5628:
5624:
5620:
5615:
5611:
5605:
5601:
5597:
5592:
5588:
5582:
5578:
5574:
5569:
5565:
5559:
5555:
5529:
5525:
5519:
5515:
5509:
5505:
5499:
5495:
5477:
5472:
5466:
5463:
5459:
5455:
5452:
5448:
5447:
5446:
5444:
5440:
5430:
5413:
5409:
5402:
5394:
5391:
5387:
5383:
5380:
5376:
5370:
5367:
5363:
5359:
5355:
5351:
5347:
5342:
5339:
5335:
5331:
5327:
5324:
5320:
5314:
5311:
5307:
5303:
5301:
5293:
5289:
5283:
5280:
5276:
5269:
5266:
5258:
5255:
5251:
5246:
5242:
5239:
5235:
5229:
5226:
5222:
5218:
5213:
5210:
5206:
5201:
5197:
5194:
5186:
5182:
5175:
5173:
5168:
5162:
5147:
5137:
5123:
5120:
5117:
5109:
5105:
5101:
5085:
5082:
5079:
5059:
5056:
5051:
5047:
5038:
5034:
5028:
5018:
5016:
5012:
5008:
4987:
4983:
4974:
4970:
4964:
4960:
4954:
4951:
4948:
4943:
4940:
4937:
4933:
4929:
4924:
4921:
4917:
4911:
4908:
4905:
4902:
4898:
4889:
4885:
4881:
4873:
4870:
4867:
4863:
4857:
4853:
4847:
4844:
4841:
4836:
4833:
4830:
4826:
4822:
4817:
4813:
4807:
4803:
4797:
4792:
4789:
4786:
4782:
4778:
4772:
4766:
4758:
4753:
4736:
4728:
4724:
4715:
4711:
4707:
4704:
4696:
4692:
4683:
4679:
4675:
4673:
4663:
4660:
4656:
4650:
4647:
4644:
4641:
4637:
4628:
4624:
4620:
4612:
4609:
4606:
4602:
4598:
4595:
4590:
4587:
4583:
4577:
4574:
4570:
4561:
4557:
4553:
4545:
4542:
4539:
4535:
4531:
4529:
4520:
4516:
4513:
4508:
4504:
4498:
4494:
4490:
4485:
4481:
4475:
4471:
4467:
4462:
4458:
4453:
4449:
4446:
4442:
4438:
4435:
4430:
4426:
4420:
4416:
4412:
4407:
4403:
4397:
4393:
4389:
4384:
4380:
4375:
4371:
4369:
4360:
4356:
4353:
4348:
4344:
4338:
4334:
4330:
4325:
4321:
4315:
4311:
4307:
4304:
4299:
4295:
4290:
4286:
4282:
4278:
4275:
4270:
4266:
4260:
4256:
4252:
4247:
4243:
4237:
4233:
4229:
4224:
4220:
4215:
4211:
4209:
4199:
4195:
4189:
4185:
4181:
4178:
4175:
4170:
4166:
4160:
4156:
4152:
4147:
4143:
4137:
4133:
4129:
4126:
4121:
4117:
4113:
4108:
4104:
4100:
4098:
4088:
4084:
4078:
4074:
4068:
4063:
4060:
4057:
4053:
4049:
4047:
4039:
4033:
4020:
4017:
4013:
4007:
3997:
3983:
3971:
3952:
3944:
3940:
3935:
3933:
3930:is a matrix,
3917:
3909:
3905:
3900:
3886:
3883:
3863:
3855:
3840:
3820:
3800:
3780:
3760:
3738:
3734:
3711:
3707:
3686:
3666:
3663:
3654:
3640:
3620:
3617:
3614:
3611:
3591:
3571:
3567:
3560:
3557:
3552:
3548:
3524:
3504:
3490:
3477:
3471:
3468:
3465:
3462:
3458:
3453:
3450:
3447:
3444:
3441:
3436:
3432:
3428:
3425:
3420:
3416:
3412:
3409:
3400:
3392:
3388:
3379:
3371:
3367:
3343:
3326:
3317:
3313:
3289:
3281:
3277:
3256:
3253:
3250:
3247:
3244:
3238:
3230:
3226:
3205:
3202:
3199:
3196:
3193:
3188:
3184:
3180:
3177:
3172:
3168:
3164:
3161:
3155:
3147:
3143:
3133:
3119:
3116:
3113:
3110:
3107:
3102:
3098:
3085:
3071:
3068:
3065:
3045:
3042:
3039:
3036:
3033:
3028:
3024:
3020:
3017:
3012:
3008:
2998:
2996:
2976:
2973:
2970:
2947:
2941:
2932:
2918:
2915:
2910:
2906:
2902:
2899:
2896:
2891:
2887:
2883:
2880:
2877:
2872:
2868:
2864:
2861:
2858:
2853:
2849:
2828:
2825:
2822:
2817:
2813:
2809:
2806:
2803:
2798:
2794:
2790:
2787:
2784:
2781:
2776:
2772:
2768:
2765:
2762:
2757:
2753:
2743:
2729:
2726:
2720:
2714:
2691:
2685:
2677:
2672:
2654:
2651:
2648:
2625:
2619:
2600:
2597:
2594:
2591:
2584:
2580:
2578:
2573:
2559:
2556:
2553:
2533:
2530:
2527:
2524:
2521:
2516:
2512:
2508:
2505:
2500:
2496:
2492:
2489:
2483:
2477:
2463:
2447:
2443:
2420:
2416:
2410:
2406:
2402:
2397:
2394:
2391:
2387:
2383:
2378:
2375:
2372:
2368:
2345:
2341:
2318:
2314:
2293:
2284:
2268:
2264:
2260:
2257:
2234:
2228:
2208:
2188:
2185:
2180:
2176:
2152:
2146:
2139:is a root of
2124:
2120:
2094:
2090:
2083:
2061:
2057:
2031:
2027:
2023:
2020:
2013:
2006:
2000:
1987:
1980:
1978:
1962:
1958:
1954:
1950:
1944:
1940:
1936:
1933:
1929:
1924:
1918:
1915:
1912:
1908:
1902:
1898:
1894:
1889:
1886:
1883:
1879:
1873:
1870:
1867:
1863:
1859:
1856:
1853:
1848:
1844:
1838:
1834:
1830:
1825:
1821:
1815:
1811:
1807:
1804:
1799:
1795:
1791:
1786:
1782:
1777:
1773:
1767:
1761:
1754:
1753:
1750:
1747:
1730:
1725:
1721:
1717:
1715:
1705:
1701:
1695:
1691:
1687:
1682:
1678:
1674:
1672:
1664:
1634:
1631:
1628:
1624:
1618:
1614:
1610:
1607:
1604:
1599:
1595:
1582:
1578:
1574:
1569:
1565:
1552:
1548:
1544:
1539:
1535:
1531:
1529:
1507:
1499:
1495:
1489:
1485:
1481:
1476:
1473:
1470:
1466:
1457:
1453:
1449:
1446:
1443:
1438:
1434:
1421:
1417:
1413:
1408:
1404:
1391:
1387:
1383:
1378:
1374:
1370:
1368:
1358:
1354:
1347:
1321:
1317:
1307:
1294:
1267:
1259:
1255:
1250:
1247:
1242:
1239:
1236:
1232:
1225:
1222:
1219:
1216:
1211:
1207:
1196:
1193:
1188:
1184:
1173:
1170:
1165:
1161:
1150:
1147:
1142:
1138:
1134:
1128:
1122:
1113:
1094:
1090:
1083:
1061:
1057:
1043:
1036:
1034:
1016:
1011:
1007:
1001:
997:
993:
988:
984:
980:
978:
971:
967:
957:
953:
947:
943:
939:
934:
930:
926:
924:
917:
913:
905:
886:
882:
876:
872:
868:
863:
860:
857:
853:
849:
847:
840:
837:
834:
830:
820:
816:
812:
810:
803:
799:
787:
786:
783:
781:
776:
763:
760:
738:
734:
711:
707:
703:
700:
697:
692:
688:
667:
662:
658:
652:
648:
644:
641:
638:
633:
629:
623:
619:
615:
610:
606:
600:
596:
592:
589:
584:
580:
576:
571:
567:
563:
558:
554:
548:
544:
538:
533:
530:
527:
523:
519:
513:
507:
493:
491:
487:
479:
474:
456:
436:
424:
419:
402:
378:
370:
366:
361:
358:
353:
350:
347:
343:
336:
333:
330:
327:
322:
318:
307:
304:
299:
295:
284:
281:
276:
272:
261:
258:
253:
249:
241:
232:
228:
222:
218:
214:
211:
208:
203:
199:
193:
189:
185:
180:
176:
170:
166:
162:
159:
154:
150:
146:
141:
137:
123:
119:
118:Horner's rule
114:
111:
107:
103:
99:
95:
91:
87:
76:
73:
65:
62:November 2018
55:
50:
48:
42:
38:
37:
30:
21:
20:
9964:Discriminant
9948:
9883:Multivariate
9704:
9683:
9661:
9647:
9624:
9603:
9574:
9568:
9539:
9535:
9524:. Retrieved
9505:
9492:
9477:. Retrieved
9473:the original
9458:
9455:"Chapter 13"
9445:
9436:
9417:
9404:
9372:
9368:
9349:. Retrieved
9342:the original
9332:
9312:
9293:
9289:
9274:. Retrieved
9267:the original
9238:cite journal
9219:
9215:
9184:. Retrieved
9165:
9161:
9129:
9125:
9102:
9097:, p. 77
9090:
9078:
9066:
9057:
9047:
9033:
9004:
8992:
8965:
8954:
8942:
8930:
8918:
8906:
8894:
8882:
8870:
8859:
8843:
8791:to evaluate
8759:
8751:
8746:
8742:
8736:
8722:
8718:
8713:
8697:
8692:Song dynasty
8665:
8642:Isaac Newton
8626:
8615:
8609:
8601:
8593:
8587:
8581:
8130:
7483:
6988:
6729:
6712:
6280:
6262:
6258:
6253:
6249:
6245:
6239:
6226:
6207:
5942:
5486:
5475:
5457:
5450:
5442:
5438:
5436:
5145:
5143:
5103:
5099:
5030:
5006:
4756:
4754:
4021:
4009:
3969:
3936:
3901:
3655:
3496:
3347:
3134:
3089:
2999:
2933:
2744:
2673:
2603:
2598:
2595:
2592:
2589:
2582:
2579:as follows:
2574:
2469:
2285:
1992:
1981:
1748:
1308:
1114:
1048:
1037:
782:as follows:
777:
499:
481:
477:
475:
420:
121:
117:
115:
97:
93:
83:
68:
59:
52:Please help
44:
33:
9995:Polynomials
9913:Homogeneous
9908:Square-free
9903:Irreducible
9768:Polynomials
9542:: 105β136.
9439:. Springer.
9095:Mikami 1913
9083:Temple 1986
9009:Cajori 1911
8997:Fuller 1999
8985:Horner 1819
8935:Higham 2002
8849:Qin Jiushao
8807:BΓ©zier form
8750:concluded:
8715:Qin Jiushao
8704:Han dynasty
8662:Qin Jiushao
8561:40642560000
8519:Qin Jiushao
6341:with zeros
6273:are known.
3996:additions.
2674:But by the
780:recursively
122:nested form
86:mathematics
56:if you can.
9984:Categories
9873:Univariate
9526:2016-08-23
9479:2016-08-23
9351:2012-12-10
9276:2018-05-17
9186:2012-03-04
9117:References
9052:paragraph.
8947:Kress 1991
8911:Knuth 1997
8652:Zhu Shijie
7196:to obtain
7051:to obtain
6616:to obtain
6436:such that
6321:of degree
5483:Derivation
5110:(base 2),
4004:See also:
3908:Victor Pan
3493:Efficiency
429:with only
425:of degree
423:polynomial
9959:Resultant
9898:Trinomial
9878:Bivariate
9711:EMS Press
9670:cite book
9556:250869179
9397:186210512
9334:Principle
9296:: 29β51.
9256:Kahan, W.
8717:, in his
8558:−
8529:−
8389:≈
8285:−
8214:−
8194:−
7964:⋮
7948:⋮
7908:−
7885:−
7847:−
7824:−
7699:⋯
7581:∑
7536:−
7507:−
7277:−
7268:−
7178:−
7139:−
7123:−
7033:−
6972:−
6931:−
6915:−
6841:−
6826:−
6811:−
6665:−
6632:−
6591:−
6381:⋯
6370:−
6135:−
6101:−
5392:−
5368:−
5340:−
5312:−
5281:−
5256:−
5227:−
5211:−
5098:. Then,
4952:−
4934:∑
4893:⌋
4879:⌊
4864:∑
4845:−
4827:∑
4783:∑
4632:⌋
4618:⌊
4603:∑
4565:⌋
4551:⌊
4536:∑
4517:⋯
4439:⋯
4357:⋯
4279:⋯
4179:⋯
4054:∑
3618:−
3469:−
3454:−
3442:−
3426:−
3269:. Divide
3254:−
3203:−
3178:−
3108:−
3069:−
3043:−
3018:−
2974:−
2826:−
2785:−
2652:−
2531:−
2506:−
2470:Evaluate
2395:−
2376:−
2261:−
2024:−
1937:−
1916:−
1887:−
1871:−
1857:⋯
1665:⋮
1632:−
1608:⋯
1508:⋯
1474:−
1447:⋯
1268:⋯
1240:−
1220:⋯
906:⋮
861:−
838:−
701:…
642:⋯
524:∑
379:⋯
351:−
331:⋯
212:⋯
9944:Division
9893:Binomial
9888:Monomial
9728:Archived
9593:52859619
9520:Archived
9416:(1997).
9315:. SIAM.
9258:(2000).
9210:(2009).
9180:Archived
9156:(1911).
8899:Pan 1966
8864:Pan 1966
8797:B-spline
8774:See also
8762:, while
8729:Jia Xian
8688:Jia Xian
8618:Arbogast
8598:reviewer
8580:result:
8455:′
5035:with no
2707:. Thus,
2608:-value (
2466:Examples
34:require
9990:Algebra
9713:, 2001
9686:(1852).
8793:splines
8756:Liu Hui
8708:Liu Hui
8675:Persian
8664:in his
8644:in 1669
8510:History
6718:Example
5183:0.00101
5163:0.15625
5140:Example
4759:parts:
3000:Divide
2575:We use
2168:, then
36:cleanup
9781:degree
9631:
9612:
9591:
9554:
9512:
9465:
9448:(212).
9424:
9395:
9389:107508
9387:
9319:
9146:604533
9144:
8545:763200
7961:
7958:
7945:
7942:
6480:Using
6267:matrix
5433:Method
5072:, and
1662:
1659:
1292:
903:
900:
897:
680:where
9589:S2CID
9552:S2CID
9393:S2CID
9385:JSTOR
9345:(PDF)
9338:(PDF)
9270:(PDF)
9263:(PDF)
9142:JSTOR
8835:Notes
6975:5040.
6265:is a
5009:-way
2049:with
1049:Then
9770:and
9676:link
9629:ISBN
9610:ISBN
9510:ISBN
9463:ISBN
9422:ISBN
9317:ISBN
9244:link
8826:and
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590:x
585:1
581:a
577:+
572:0
568:a
564:=
559:i
555:x
549:i
545:a
539:n
534:0
531:=
528:i
520:=
517:)
514:x
511:(
508:p
471:n
457:n
437:n
427:n
403:.
398:)
391:)
384:)
376:)
371:n
367:a
362:x
359:+
354:1
348:n
344:a
340:(
337:x
334:+
328:+
323:3
319:a
313:(
308:x
305:+
300:2
296:a
290:(
285:x
282:+
277:1
273:a
267:(
262:x
259:+
254:0
250:a
242:=
233:n
229:x
223:n
219:a
215:+
209:+
204:3
200:x
194:3
190:a
186:+
181:2
177:x
171:2
167:a
163:+
160:x
155:1
151:a
147:+
142:0
138:a
75:)
69:(
64:)
60:(
49:.
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