4762:
4036:
4757:{\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}}}
1756:
8138:
8754:"... 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."
36:
1350:
7751:
428:
5439:
1042:
138:
7174:
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
5162:
1751:{\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}}}
8623:
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
4029:
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
8526:
8133:{\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}}}
5014:
6734:
5950:
1316:
1986:
8329:
6724:
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.
801:
6238:
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
5775:
7746:
689:
9344:
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
4773:
4041:
5784:
423:{\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}}}
3499:
1129:
6996:
2942:, 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.
123:
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.
7000:
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
5434:{\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}}}
1768:
8145:
6215:
5484:
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
7172:
9062:
Analysis Per
Quantitatum Series, Fluctiones ac Differentias : Cum Enumeratione Linearum Tertii Ordinis, Londini. Ex Officina Pearsoniana. Anno MDCCXI, p. 10, 4th
6075:
7447:
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
6868:
5777:
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
7301:
5558:
6418:
9190:
8150:
7756:
5167:
1355:
806:
143:
2850:
8588:
2940:
7381:
3952:
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
7571:
514:
3227:
2444:
2555:
5016:
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
8638:
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:
7445:
8765:. 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
5553:
5475:
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.
735:
7566:
3067:
8486:
1037:{\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}}}
2623:
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
6485:
2058:
6235:), 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.
3278:
8738:; 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
3593:
8413:
3351:
3141:
6625:
7481:
6556:
6330:
3370:
3314:
2121:
1121:
6691:
6658:
5081:
2292:
2210:
2751:
7205:
7060:
6718:
6583:
6520:
6445:
3762:
3735:
2471:
2369:
2342:
2148:
2085:
1345:
1085:
762:
8515:
8387:
8358:
7028:
3642:
2972:
2716:
2650:
2259:
2177:
8439:
6873:
5145:
5107:
3093:
2998:
2676:
2581:
7303:
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
3908:
3688:
785:
6350:
4005:
3974:
3939:
3885:
3862:
3842:
3822:
3802:
3782:
3708:
3662:
3613:
3546:
3526:
2315:
2230:
478:
458:
6080:
8742:; 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.
2945:
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
5009:{\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})}
9168:
7065:
5957:
9254:
503:
made more efficient for hand calculation by application of Horner's rule. It was widely used until computers came into general use around 1970.
57:
9686:
3595:
multiplications, if powers are calculated by repeated multiplication and each monomial is evaluated individually. The cost can be reduced to
6744:
7210:
5945:{\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).}
9270:
1311:{\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 )}\ .}
3913:
Horner's method is optimal, in the sense that any algorithm to evaluate an arbitrary polynomial must use at least as many operations.
9530:
8709:
9342:
7306:
1981:{\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}}
9045:
8324:{\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}}}
9465:
2374:
8734:
8677:
9729:
9769:
9738:
9643:
9524:
9477:
9436:
9331:
8631:. Horner is also known to have made a close reading of John Bonneycastle's book on algebra, though he neglected the work of
3667:
If numerical data are represented in terms of digits (or bits), then the naive algorithm also entails storing approximately
7386:
6355:
3953:
2004:
This expression constitutes Horner's practical application, as it offers a very quick way of determining the outcome of;
64:
9455:
Kripasagar, Venkat (March 2008). "Efficient Micro
Mathematics – Multiplication and Division Techniques for MCUs".
8603:
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
499:
also refers to a method for approximating the roots of polynomials, described by Horner in 1819. It is a variant of the
9949:
5478:
Using that value, perform a left-shift operation by that number of bits on the register holding the intermediate result
5050:. One of the binary numbers to be multiplied is represented as a trivial polynomial, where (using the above notation)
2759:
2615:
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
8628:
8535:
9657:
9624:
8599:
Horner's paper, titled "A new method of solving numerical equations of all orders, by continuous approximation", was
2855:
82:
8863:
8691:
5024:
execution of most of them. Modern compilers generally evaluate polynomials this way when advantageous, although for
2007:
6296:, it is possible to approximate the real roots of a polynomial. The algorithm works as follows. Given a polynomial
4033:
If, however, one is evaluating a single polynomial of very high order, it may be useful to break it up as follows:
5037:
3149:
6281:
3942:
5770:{\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.}
9721:
9369:
Holdred's method is in the supplement following page numbered 45 (which is the 52nd page of the pdf version).
8688:
8642:
8632:
4026:
3098:
2 β 1 β6 11 β6 β 2 β8 6 βββββββββββββββββββββββββ 1 β4 3 0
2612:
3 β 2 β6 2 β1 β 6 0 6 βββββββββββββββββββββββββ 2 0 2 5
2484:
17:
9944:
9928:
9431:. Vol. 2: Seminumerical Algorithms (3rd ed.). Addison-Wesley. pp. 486β488 in section 4.6.4.
3356:
0.5 β 4 β6 0 3 β5 β 2 β2 β1 1 ββββββββββββββββββββββββ 2 β2 β1 1 β4
2686:
9276:(Report). PAM. University of California, Berkeley: Center for Pure and Applied Mathematics. Archived from
8777:
in the 7th century supposes his readers can solve cubics by an approximation method described in his book
5501:
10010:
9964:
9716:
9499:
8813:
7383:
which is shown in green and found to have a zero at −3. This polynomial is further reduced to
694:
7498:
5042:
Horner's method is a fast, code-efficient method for multiplication and division of binary numbers on a
3014:
9809:
9378:(July 1819). "A new method of solving numerical equations of all orders, by continuous approximation".
8608:
9762:
9580:
8600:
7741:{\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},}
684:{\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},}
51:
6450:
9954:
9711:
9050:
8763:
It is obvious that this procedure is a
Chinese invention ... the method was not known in India
3232:
3005:
8627:
Unlike his
English contemporaries, Horner drew on the Continental literature, notably the work of
9893:
8444:
3551:
9040:
8392:
3319:
3104:
9923:
9918:
9913:
9791:
9202:
Read before the
Southwestern Section of the American Mathematical Society on November 26, 1910.
8799:
6591:
6240:
3864:
8331:
This computation of the divided difference is subject to less round-off error than evaluating
7487:. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were found.
7450:
6525:
6299:
6280:, in which case the gain in computational efficiency is even greater. However, for such cases
3956:. They involve a transformation of the representation of the polynomial. In general, a degree-
3283:
2090:
1090:
9903:
9883:
9375:
9277:
9248:
6663:
6630:
5118:
5053:
2264:
2182:
120:
116:
112:
2721:
10005:
9969:
9888:
9755:
9210:
7178:
7033:
6696:
6561:
6498:
6423:
3740:
3713:
2449:
2347:
2320:
2126:
2063:
1323:
1063:
740:
9512:
8491:
8363:
8334:
7004:
3618:
2948:
2692:
2626:
2235:
2153:
8:
9782:
9668:
9414:
Directly available online via the link, but also reprinted with appraisal in D.E. Smith:
9352:
9072:
Newton's collected papers, the edition 1779, in a footnote, vol. I, p. 270-271
9036:
8669:
8659:
8418:
5124:
5086:
3914:
3072:
2977:
2655:
2560:
790:
737:
are constant coefficients, the problem is to evaluate the polynomial at a specific value
9135:
Berggren, J. L. (1990). "Innovation and
Tradition in Sharaf al-Din al-Tusi's Muadalat".
3890:
3670:
767:
9826:
9821:
9747:
9680:
9599:
9576:"Algorithm 337: calculation of a polynomial and its derivative values by Horner scheme"
9562:
9403:
9395:
9152:
8838:
8822:
8790:
6492:
6335:
6293:
4016:
3990:
3959:
3924:
3870:
3847:
3827:
3807:
3787:
3767:
3693:
3647:
3598:
3531:
3511:
2587:
2300:
2215:
500:
463:
443:
9205:
8694:
in the 12th century (the first to use that method in a general case of cubic equation)
9804:
9639:
9620:
9613:
9566:
9520:
9483:
9473:
9432:
9407:
9327:
9239:
9222:
8834:
8646:
5047:
46:
9603:
9558:
9185:
10000:
9857:
9850:
9845:
9589:
9554:
9387:
9308:
9234:
9206:
9180:
9144:
8828:
8758:
6232:
6228:
6224:
6220:
5778:
100:
9979:
9736:
9867:
9862:
9814:
9799:
9742:
9653:
9262:
8616:
7484:
5043:
4022:
3949:
3494:{\displaystyle {\frac {f_{1}(x)}{f_{2}(x)}}=2x^{3}-2x^{2}-x+1-{\frac {4}{2x-1}}.}
9838:
9833:
9218:
9164:
8794:
8685:
6252:
5025:
2123:) being the division's remainder, as is demonstrated by the examples below. If
8817:
9994:
9266:
8778:
8774:
8743:
6277:
6991:{\displaystyle p_{6}(x)=x^{6}+4x^{5}-72x^{4}-214x^{3}+1127x^{2}+1602x-5040.}
5031:
3921:
proved in 1966 that the number of multiplications is minimal. However, when
9974:
9424:
9391:
9313:
9296:
8803:
8702:
8652:
5954:
The denominators all equal one (or the term is absent), so this reduces to
9594:
9575:
8859:
8725:
8714:
8672:
8529:
96:
9778:
9214:
8662:
6210:{\displaystyle =d_{3}(m+2^{-1}{d_{2}}(m+2^{-1}{d_{1}}(m+{d_{0}}(m)))).}
3918:
433:
9399:
9297:"Horner versus Holdred: An Episode in the History of Root Computation"
9156:
3948:
This assumes that the polynomial is evaluated in monomial form and no
9908:
9636:
The Genius of China: 3,000 Years of Science, Discovery, and Invention
9545:
Pan, Y. Ja (1966). "On means of calculating values of polynomials".
6251:
Horner's method can be used to convert between different positional
6219:
In binary (base-2) math, multiplication by a power of 2 is merely a
5781:
is not an issue, despite this implication in the factored equation:
9898:
9517:
Studies in Mathematics and Mechanics presented to Richard von Mises
9148:
8807:
8739:
8698:
8525:
6231:, a (0) results in no operation (since 2 = 1 is the multiplicative
58:
Talk:Horner's method#This Article is about Two Different Algorithms
9513:"On two problems in abstract algebra connected with Horner's rule"
7495:
Horner's method can be modified to compute the divided difference
6733:
6077:
or equivalently (as consistent with the "method" described above)
4021:
A disadvantage of Horner's rule is that all of the operations are
480:
additions. This is optimal, since there are polynomials of degree
8766:
8718:
6223:
operation. Thus, multiplying by 2 is calculated in base-2 by an
5028:
calculations this requires enabling (unsafe) reassociative math.
3917:
proved in 1954 that the number of additions required is minimal.
1126:
To see why this works, the polynomial can be written in the form
8619:
is dismissed curtly in this review. The sequence of reviews in
7167:{\displaystyle p_{5}(x)=x^{5}+11x^{4}+5x^{3}-179x^{2}-126x+720}
9506:(1st ed.). Chelsea Publishing Co reprint. pp. 74β77.
8825:
to facilitate parallelization on modern computer architectures
6070:{\displaystyle =d_{0}(m+2{d_{1}}(m+2{d_{2}}(m+2{d_{3}}(m)))),}
5155:
For example, to find the product of two numbers (0.15625) and
3867:. Horner's method can also be extended to evaluate the first
5467:
Begin with the least significant (rightmost) non-zero bit in
5460:
A register holding the intermediate result is initialized to
6272:
representation of a given number β and can also be used if
6243:" (CSD) form is used) and uses only 20% of the code space.
5021:
3359:
The third row is the sum of the first two rows, divided by
484:
that cannot be evaluated with fewer arithmetic operations.
9615:
Schaum's Outline of Theory and Problems of College Algebra
8532:'s algorithm for solving the quadratic polynomial equation
506:
119:, this method is much older, as it has been attributed to
9169:"Horner's method of approximation anticipated by Ruffini"
8605:
Philosophical Transactions of the Royal Society of London
5032:
Application to floating-point multiplication and division
9777:
9271:
Improving exact integrals from symbolic algebra systems
6863:{\displaystyle p_{6}(x)=(x+8)(x+5)(x+3)(x-2)(x-3)(x-7)}
3804:
multiplications, and its storage requirements are only
8841:
to divide a polynomial by a binomial of the form x β r
7296:{\displaystyle p_{4}(x)=x^{4}+14x^{3}+47x^{2}-38x-240}
6292:
Using the long division algorithm in combination with
3844:. Alternatively, Horner's method can be computed with
8615:
for April, 1820; in comparison, a technical paper by
8538:
8494:
8447:
8421:
8395:
8366:
8337:
8148:
7754:
7574:
7501:
7490:
7453:
7389:
7309:
7213:
7181:
7068:
7036:
7007:
6876:
6747:
6699:
6666:
6633:
6594:
6564:
6528:
6501:
6453:
6426:
6413:{\displaystyle z_{n}<z_{n-1}<\cdots <z_{1},}
6358:
6338:
6302:
6083:
5960:
5787:
5561:
5504:
5165:
5127:
5089:
5056:
4776:
4039:
3993:
3962:
3927:
3893:
3873:
3850:
3830:
3810:
3790:
3770:
3743:
3716:
3710:: the evaluated polynomial has approximate magnitude
3696:
3673:
3650:
3621:
3601:
3554:
3534:
3514:
3373:
3322:
3286:
3235:
3152:
3107:
3075:
3017:
2980:
2951:
2858:
2762:
2724:
2695:
2658:
2629:
2563:
2487:
2452:
2377:
2371:. Then you then work recursively using the formula:
2350:
2323:
2303:
2267:
2238:
2218:
2185:
2156:
2129:
2093:
2066:
2010:
1771:
1353:
1326:
1132:
1093:
1066:
804:
770:
743:
697:
517:
466:
446:
141:
8862:
and 700 years earlier, by the Persian mathematician
5117:
to some power) is repeatedly factored out. In this
3764:
itself. By contrast, Horner's method requires only
3367:
with the third-row entry to the left. The answer is
8963:
9612:
8607:for 1819 was warmly and expansively welcomed by a
8582:
8509:
8480:
8433:
8407:
8381:
8352:
8323:
8132:
7740:
7560:
7475:
7439:
7375:
7295:
7199:
7166:
7054:
7022:
6990:
6862:
6712:
6685:
6652:
6619:
6577:
6550:
6514:
6479:
6439:
6412:
6344:
6324:
6209:
6069:
5944:
5769:
5547:
5433:
5139:
5101:
5075:
5008:
4756:
3999:
3968:
3933:
3902:
3879:
3856:
3836:
3816:
3796:
3776:
3756:
3729:
3702:
3682:
3656:
3636:
3607:
3587:
3540:
3520:
3493:
3345:
3308:
3272:
3221:
3135:
3087:
3061:
2992:
2966:
2934:
2844:
2745:
2710:
2670:
2644:
2575:
2549:
2465:
2438:
2363:
2336:
2309:
2286:
2253:
2224:
2204:
2171:
2142:
2115:
2079:
2052:
1980:
1750:
1339:
1310:
1115:
1079:
1036:
779:
756:
729:
683:
472:
452:
422:
9652:
9504:The Development of Mathematics in China and Japan
9418:, McGraw-Hill, 1929; Dover reprint, 2 vols, 1959.
9034:
5498:In general, for a binary number with bit values (
4766:More generally, the summation can be broken into
3363:. Each entry in the second row is the product of
2845:{\displaystyle a_{3}=2,a_{2}=-6,a_{1}=2,a_{0}=-1}
1659:
1570:
1530:
1409:
1297:
1290:
1189:
1166:
789:For this, a new sequence of constants is defined
408:
401:
300:
277:
9992:
8858:600 years earlier, by the Chinese mathematician
8583:{\displaystyle -x^{4}+763200x^{2}-40642560000=0}
9669:"Jottings on the Science of Chinese Arithmetic"
3508:Evaluation using the monomial form of a degree
2935:{\displaystyle b_{3}=2,b_{2}=0,b_{1}=2,b_{0}=5}
9800:Zero polynomial (degree undefined or β1 or ββ)
9324:Accuracy and Stability of Numerical Algorithms
7376:{\displaystyle p_{3}(x)=x^{3}+16x^{2}+79x+120}
5147:, so powers of 2 are repeatedly factored out.
9763:
9735:For more on the root-finding application see
9470:Chinese Mathematics in the Thirteenth Century
9173:Bulletin of the American Mathematical Society
9030:
9028:
8748:Development of Mathematics in China and Japan
6737:Polynomial root finding using Horner's method
5038:multiplication algorithm Β§ Shift and add
4025:, so it is not possible to take advantage of
1652:
1600:
1523:
1439:
1283:
1212:
394:
323:
9261:
8969:
8769:in connection with Problems IV.16 and 22 in
4903:
4889:
4642:
4628:
4575:
4561:
3644:multiplications by evaluating the powers of
9253:: CS1 maint: numeric names: authors list (
9219:Stein10.1016/0315-0860(81)90069-0, Clifford
3222:{\displaystyle f_{1}(x)=4x^{4}-6x^{3}+3x-5}
9770:
9756:
9685:: CS1 maint: location missing publisher (
9573:
9454:
9025:
8933:
8885:
6660:. Return to step 1 but use the polynomial
6287:
6259:is the base of the number system, and the
5448:To find the product of two binary numbers
2439:{\displaystyle b_{n-1}=a_{n-1}+b_{n}x_{0}}
9593:
9510:
9463:
9312:
9238:
9184:
9117:
8991:
8989:
8980:
8897:
8710:The Nine Chapters on the Mathematical Art
8624:this method should go to Holdred (1820).
3333:
1264:
375:
83:Learn how and when to remove this message
9233:(3) (3rd ed.). MIT Press: 277β318.
9137:Journal of the American Oriental Society
9134:
9081:
8613:The Monthly Review: or, Literary Journal
8524:
6732:
6268:coefficients are the digits of the base-
3004:. This makes Horner's method useful for
9610:
9340:
9046:MacTutor History of Mathematics Archive
6487:. Now iterate the following two steps:
4030:within a single polynomial evaluation.
3976:polynomial can be evaluated using only
2550:{\displaystyle f(x)=2x^{3}-6x^{2}+2x-1}
507:Polynomial evaluation and long division
14:
9993:
9633:
9497:
9374:
9321:
9294:
9163:
9105:
9093:
9019:
9013:
9007:
8995:
8986:
8945:
8735:Mathematical Treatise in Nine Sections
8678:Mathematical Treatise in Nine Sections
4010:
1320:Thus, by iteratively substituting the
131:, in which a polynomial is written in
9751:
9666:
9445:
9423:
8957:
8921:
7440:{\displaystyle p_{2}(x)=x^{2}+13x+40}
6246:
9386:. Royal Society of London: 308β335.
5548:{\displaystyle d_{3}d_{2}d_{1}d_{0}}
2317:-values, you start with determining
1762:
795:
29:
9544:
8909:
8874:
3887:derivatives of the polynomial with
730:{\displaystyle a_{0},\ldots ,a_{n}}
27:Algorithm for polynomial evaluation
24:
9519:. Academic Press. pp. 40β48.
7561:{\displaystyle (p(y)-p(x))/(y-x).}
7491:Divided difference of a polynomial
6588:Using Horner's method, divide out
3062:{\displaystyle x^{3}-6x^{2}+11x-6}
25:
10022:
9704:
8717:(202 BC β 220 AD) edited by
7568:Given the polynomial (as before)
9511:Ostrowski, Alexander M. (1954).
8831:to approximate roots graphically
2689:, we know that the remainder is
432:This allows the evaluation of a
34:
9559:10.1070/rm1966v021n01abeh004147
9533:from the original on 2019-04-15
9429:The Art of Computer Programming
9351:. Richard Watts. Archived from
9193:from the original on 2017-09-04
9186:10.1090/s0002-9904-1911-02072-9
9111:
9099:
9087:
9075:
9066:
9056:
9001:
8974:
8029:
7963:
7886:
7791:
7175:5 polynomial is now divided by
3910:additions and multiplications.
8951:
8939:
8927:
8915:
8903:
8891:
8879:
8868:
8852:
8504:
8498:
8475:
8469:
8389:separately, particularly when
8376:
8370:
8347:
8341:
8301:
8289:
8266:
8260:
8216:
8210:
8201:
8195:
8162:
8156:
7584:
7578:
7552:
7540:
7532:
7529:
7523:
7514:
7508:
7502:
7470:
7464:
7406:
7400:
7326:
7320:
7230:
7224:
7194:
7182:
7085:
7079:
7049:
7037:
7017:
7011:
6893:
6887:
6857:
6845:
6842:
6830:
6827:
6815:
6812:
6800:
6797:
6785:
6782:
6770:
6764:
6758:
6614:
6595:
6545:
6539:
6480:{\displaystyle z_{1}<x_{0}}
6319:
6313:
6201:
6198:
6195:
6192:
6186:
6165:
6131:
6097:
6061:
6058:
6055:
6052:
6046:
6022:
5998:
5974:
5921:
5915:
5654:
5562:
5416:
5410:
5283:
5274:
5202:
5189:
5176:
5170:
5003:
4990:
4786:
4780:
4744:
4731:
4712:
4699:
4053:
4047:
3954:faster algorithms are possible
3943:Horner's method is not optimal
3574:
3555:
3414:
3408:
3393:
3387:
3340:
3334:
3303:
3297:
3252:
3246:
3169:
3163:
2961:
2955:
2734:
2728:
2705:
2699:
2639:
2633:
2497:
2491:
2248:
2242:
2232:), which means you can factor
2166:
2160:
2110:
2097:
2053:{\displaystyle p(x)/(x-x_{0})}
2047:
2028:
2020:
2014:
1781:
1775:
1515:
1473:
1374:
1361:
1275:
1239:
1142:
1136:
1110:
1097:
527:
521:
386:
350:
13:
1:
9960:Horner's method of evaluation
9675:. Shanghai. pp. 159β194.
9500:"Chapter 11. Ch'in Chiu-Shao"
9127:
6227:. The factor (2) is a right
5493:
4027:instruction level parallelism
3503:
3273:{\displaystyle f_{2}(x)=2x-1}
9659:The Calculus of Observations
9416:A Source Book in Mathematics
9240:10.1016/0315-0860(81)90069-0
9223:"Introduction to Algorithms"
6728:
3824:times the number of bits of
3690:times the number of bits of
3528:polynomial requires at most
2687:polynomial remainder theorem
1760:Now, it can be proven that;
7:
9965:Polynomial identity testing
9717:Encyclopedia of Mathematics
9611:Spiegel, Murray R. (1956).
8816:to evaluate polynomials in
8793:to evaluate polynomials in
8784:
8481:{\displaystyle d_{1}=p'(x)}
3588:{\displaystyle (n^{2}+n)/2}
2476:
2344:, which is simply equal to
2297:To finding the consecutive
1994:
1050:
54:. The specific problem is:
10:
10027:
9464:Libbrecht, Ulrich (2005).
9380:Philosophical Transactions
8697:the Chinese mathematician
8520:
8408:{\displaystyle x\approx y}
5150:
5035:
4014:
3737:, and one must also store
3346:{\displaystyle f_{2}\,(x)}
3136:{\displaystyle x^{2}-4x+3}
2212:(meaning the remainder is
127:The algorithm is based on
9937:
9876:
9789:
9693:Reprinted from issues of
9667:Wylie, Alexander (1897).
9581:Communications of the ACM
9322:Higham, Nicholas (2002).
6870:which can be expanded to
6620:{\displaystyle (x-z_{1})}
5443:
9051:University of St Andrews
8970:Fateman & Kahan 2000
8845:
8814:De Casteljau's algorithm
8713:, a Chinese work of the
7476:{\displaystyle p_{2}(x)}
6741:Consider the polynomial
6551:{\displaystyle p_{n}(x)}
6495:, find the largest zero
6420:make some initial guess
6325:{\displaystyle p_{n}(x)}
3309:{\displaystyle f_{1}(x)}
3006:polynomial long division
2116:{\displaystyle p(x_{0})}
1116:{\displaystyle p(x_{0})}
9950:Greatest common divisor
9732:(Cong Shu Ji Cheng ed.)
9656:; Robinson, G. (1924).
9634:Temple, Robert (1986).
9574:Pankiewicz, W. (1968).
9498:Mikami, Yoshio (1913).
9472:(2nd ed.). Dover.
9457:Circuit Cellar Magazine
8142:At completion, we have
6686:{\displaystyle p_{n-1}}
6653:{\displaystyle p_{n-1}}
6288:Polynomial root finding
5473:
5076:{\displaystyle a_{i}=1}
3987:+2 multiplications and
3353:using Horner's method.
2287:{\displaystyle x-x_{0}}
2205:{\displaystyle b_{0}=0}
115:. Although named after
9822:Quadratic function (2)
9695:The North China Herald
9638:. Simon and Schuster.
9446:Kress, Rainer (1991).
9392:10.1098/rstl.1819.0023
9376:Horner, William George
9314:10.1006/hmat.1998.2214
9295:Fuller, A. T. (1999).
8756:
8596:
8584:
8511:
8482:
8435:
8409:
8383:
8354:
8325:
8134:
7742:
7610:
7562:
7477:
7441:
7377:
7297:
7201:
7168:
7056:
7024:
6992:
6864:
6738:
6714:
6693:and the initial guess
6687:
6654:
6621:
6579:
6552:
6516:
6481:
6441:
6414:
6346:
6326:
6241:canonical signed digit
6211:
6071:
5946:
5771:
5549:
5435:
5141:
5103:
5077:
5010:
4969:
4907:
4862:
4812:
4758:
4646:
4579:
4083:
4023:sequentially dependent
4001:
3970:
3935:
3904:
3881:
3858:
3838:
3818:
3798:
3778:
3758:
3731:
3704:
3684:
3658:
3638:
3609:
3589:
3542:
3522:
3495:
3347:
3310:
3274:
3223:
3137:
3089:
3063:
2994:
2968:
2936:
2846:
2747:
2746:{\displaystyle f(3)=5}
2712:
2672:
2646:
2577:
2551:
2467:
2440:
2365:
2338:
2311:
2288:
2255:
2226:
2206:
2173:
2144:
2117:
2081:
2054:
1982:
1752:
1341:
1312:
1117:
1081:
1038:
781:
758:
731:
685:
553:
474:
454:
424:
111:) is an algorithm for
9805:Constant function (0)
9595:10.1145/364063.364089
9547:Russian Math. Surveys
9211:Leiserson, Charles E.
9041:"Horner's method"
8864:Sharaf al-DΔ«n al-αΉ¬Ε«sΔ«
8752:
8750:(Leipzig 1913) wrote:
8701:in the 11th century (
8692:Sharaf al-DΔ«n al-αΉ¬Ε«sΔ«
8670:Chinese mathematician
8660:Chinese mathematician
8585:
8528:
8512:
8483:
8441:in this method gives
8436:
8410:
8384:
8355:
8326:
8135:
7743:
7590:
7563:
7478:
7442:
7378:
7298:
7202:
7200:{\displaystyle (x-3)}
7169:
7057:
7055:{\displaystyle (x-7)}
7025:
6993:
6865:
6736:
6715:
6713:{\displaystyle z_{1}}
6688:
6655:
6622:
6580:
6578:{\displaystyle x_{0}}
6553:
6517:
6515:{\displaystyle z_{1}}
6482:
6442:
6440:{\displaystyle x_{0}}
6415:
6347:
6327:
6212:
6072:
5947:
5772:
5550:
5436:
5142:
5119:binary numeral system
5104:
5078:
5011:
4943:
4873:
4836:
4792:
4759:
4612:
4545:
4063:
4002:
3971:
3936:
3905:
3882:
3859:
3839:
3819:
3799:
3779:
3759:
3757:{\displaystyle x^{n}}
3732:
3730:{\displaystyle x^{n}}
3705:
3685:
3659:
3639:
3610:
3590:
3543:
3523:
3496:
3348:
3311:
3275:
3224:
3138:
3090:
3064:
2995:
2969:
2937:
2847:
2748:
2713:
2673:
2647:
2578:
2552:
2468:
2466:{\displaystyle b_{0}}
2441:
2366:
2364:{\displaystyle a_{n}}
2339:
2337:{\displaystyle b_{n}}
2312:
2289:
2256:
2227:
2207:
2174:
2145:
2143:{\displaystyle x_{0}}
2118:
2082:
2080:{\displaystyle b_{0}}
2055:
1983:
1753:
1347:into the expression,
1342:
1340:{\displaystyle b_{i}}
1313:
1118:
1082:
1080:{\displaystyle b_{0}}
1039:
782:
759:
757:{\displaystyle x_{0}}
732:
686:
533:
511:Given the polynomial
501:NewtonβRaphson method
495:HornerβRuffini method
475:
455:
425:
121:Joseph-Louis Lagrange
117:William George Horner
113:polynomial evaluation
9938:Tools and algorithms
9858:Quintic function (5)
9846:Quartic function (4)
9783:polynomial functions
9341:Holdred, T. (1820).
9301:Historia Mathematica
9227:Historia Mathematica
9037:Robertson, Edmund F.
8536:
8510:{\displaystyle p(x)}
8492:
8488:, the derivative of
8445:
8419:
8393:
8382:{\displaystyle p(y)}
8364:
8353:{\displaystyle p(x)}
8335:
8146:
7752:
7572:
7499:
7451:
7387:
7307:
7211:
7179:
7066:
7034:
7023:{\displaystyle p(x)}
7005:
6874:
6745:
6697:
6664:
6631:
6592:
6562:
6526:
6499:
6451:
6424:
6356:
6336:
6300:
6081:
5958:
5785:
5559:
5502:
5163:
5125:
5087:
5054:
4774:
4037:
3991:
3960:
3925:
3891:
3871:
3848:
3828:
3808:
3788:
3768:
3741:
3714:
3694:
3671:
3648:
3637:{\displaystyle 2n-1}
3619:
3599:
3552:
3532:
3512:
3371:
3320:
3284:
3233:
3150:
3105:
3073:
3015:
3000:. The remainder is
2978:
2967:{\displaystyle f(x)}
2949:
2856:
2760:
2756:In this example, if
2722:
2711:{\displaystyle f(3)}
2693:
2656:
2645:{\displaystyle f(x)}
2627:
2561:
2485:
2450:
2375:
2348:
2321:
2301:
2265:
2254:{\displaystyle p(x)}
2236:
2216:
2183:
2172:{\displaystyle p(x)}
2154:
2127:
2091:
2064:
2008:
1769:
1351:
1324:
1130:
1091:
1064:
802:
768:
741:
695:
515:
464:
460:multiplications and
444:
139:
65:improve this article
50:to meet Knowledge's
9868:Septic equation (7)
9863:Sextic equation (6)
9810:Linear function (1)
9084:, pp. 304β309.
9035:O'Connor, John J.;
8983:, pp. 181β191.
8800:De Boor's algorithm
8681:in the 13th century
8665:in the 14th century
8434:{\displaystyle y=x}
7748:proceed as follows
5140:{\displaystyle x=2}
5102:{\displaystyle x=2}
5048:hardware multiplier
4011:Parallel evaluation
3915:Alexander Ostrowski
3865:fused multiplyβadds
3088:{\displaystyle x-2}
2993:{\displaystyle x-3}
2671:{\displaystyle x-3}
2576:{\displaystyle x=3}
2446:till you arrive at
2087:(which is equal to
10011:Numerical analysis
9834:Cubic function (3)
9827:Quadratic equation
9741:2018-09-28 at the
9673:Chinese Researches
9662:. London: Blackie.
9448:Numerical Analysis
8839:synthetic division
8791:Clenshaw algorithm
8771:Jiu Zhang Suan Shu
8721:(fl. 3rd century).
8621:The Monthly Review
8597:
8580:
8507:
8478:
8431:
8405:
8379:
8350:
8321:
8319:
8130:
8128:
7738:
7558:
7473:
7437:
7373:
7293:
7197:
7164:
7052:
7020:
6988:
6860:
6739:
6710:
6683:
6650:
6617:
6575:
6548:
6512:
6477:
6437:
6410:
6342:
6322:
6247:Other applications
6207:
6067:
5942:
5767:
5545:
5431:
5429:
5137:
5099:
5073:
5006:
4754:
4752:
3997:
3966:
3931:
3903:{\displaystyle kn}
3900:
3877:
3854:
3834:
3814:
3794:
3774:
3754:
3727:
3700:
3683:{\displaystyle 2n}
3680:
3654:
3634:
3605:
3585:
3538:
3518:
3491:
3343:
3306:
3270:
3219:
3133:
3085:
3059:
2990:
2964:
2932:
2842:
2743:
2708:
2668:
2642:
2588:synthetic division
2573:
2547:
2463:
2436:
2361:
2334:
2307:
2284:
2251:
2222:
2202:
2169:
2140:
2113:
2077:
2050:
1978:
1748:
1746:
1337:
1308:
1113:
1077:
1034:
1032:
780:{\displaystyle x.}
777:
754:
727:
681:
470:
450:
420:
418:
9988:
9987:
9929:Quasi-homogeneous
9730:Shu Shu Jiu Zhang
9645:978-0-671-62028-8
9526:978-1-4832-3272-0
9479:978-0-486-44619-6
9438:978-0-201-89684-8
9333:978-0-89871-521-7
9215:Rivest, Ronald L.
9207:Cormen, Thomas H.
9010:, pp. 29β51.
8730:Shu Shu Jiu Zhang
8231:
7973:
7970:
7957:
7954:
6345:{\displaystyle n}
5913:
5875:
5837:
5555:) the product is
4000:{\displaystyle n}
3969:{\displaystyle n}
3934:{\displaystyle x}
3880:{\displaystyle k}
3857:{\displaystyle n}
3837:{\displaystyle x}
3817:{\displaystyle n}
3797:{\displaystyle n}
3777:{\displaystyle n}
3703:{\displaystyle x}
3657:{\displaystyle x}
3608:{\displaystyle n}
3541:{\displaystyle n}
3521:{\displaystyle n}
3486:
3418:
2310:{\displaystyle b}
2225:{\displaystyle 0}
2002:
2001:
1674:
1671:
1304:
1058:
1057:
915:
912:
909:
473:{\displaystyle n}
453:{\displaystyle n}
93:
92:
85:
52:quality standards
43:This article may
16:(Redirected from
10018:
9851:Quartic equation
9772:
9765:
9758:
9749:
9748:
9725:
9690:
9684:
9676:
9663:
9649:
9630:
9618:
9607:
9597:
9570:
9541:
9539:
9538:
9507:
9494:
9492:
9491:
9482:. Archived from
9460:
9451:
9442:
9411:
9366:
9364:
9363:
9357:
9350:
9337:
9318:
9316:
9291:
9289:
9288:
9282:
9275:
9258:
9252:
9244:
9242:
9201:
9199:
9198:
9188:
9160:
9121:
9115:
9109:
9103:
9097:
9091:
9085:
9079:
9073:
9070:
9064:
9060:
9054:
9053:
9032:
9023:
9017:
9011:
9005:
8999:
8993:
8984:
8978:
8972:
8967:
8961:
8955:
8949:
8943:
8937:
8931:
8925:
8919:
8913:
8907:
8901:
8895:
8889:
8883:
8877:
8872:
8866:
8856:
8759:Ulrich Libbrecht
8611:in the issue of
8589:
8587:
8586:
8581:
8567:
8566:
8551:
8550:
8516:
8514:
8513:
8508:
8487:
8485:
8484:
8479:
8468:
8457:
8456:
8440:
8438:
8437:
8432:
8415:. Substituting
8414:
8412:
8411:
8406:
8388:
8386:
8385:
8380:
8359:
8357:
8356:
8351:
8330:
8328:
8327:
8322:
8320:
8313:
8312:
8285:
8284:
8249:
8248:
8232:
8230:
8219:
8190:
8181:
8180:
8139:
8137:
8136:
8131:
8129:
8119:
8118:
8106:
8105:
8089:
8088:
8069:
8068:
8056:
8055:
8039:
8038:
8020:
8019:
8007:
8006:
7990:
7989:
7971:
7968:
7967:
7955:
7952:
7951:
7948:
7938:
7937:
7925:
7924:
7902:
7901:
7877:
7876:
7864:
7863:
7841:
7840:
7818:
7817:
7801:
7800:
7785:
7784:
7768:
7767:
7747:
7745:
7744:
7739:
7734:
7733:
7724:
7723:
7705:
7704:
7695:
7694:
7682:
7681:
7672:
7671:
7656:
7655:
7643:
7642:
7630:
7629:
7620:
7619:
7609:
7604:
7567:
7565:
7564:
7559:
7539:
7483:and solving the
7482:
7480:
7479:
7474:
7463:
7462:
7446:
7444:
7443:
7438:
7421:
7420:
7399:
7398:
7382:
7380:
7379:
7374:
7357:
7356:
7341:
7340:
7319:
7318:
7302:
7300:
7299:
7294:
7277:
7276:
7261:
7260:
7245:
7244:
7223:
7222:
7206:
7204:
7203:
7198:
7173:
7171:
7170:
7165:
7148:
7147:
7132:
7131:
7116:
7115:
7100:
7099:
7078:
7077:
7061:
7059:
7058:
7053:
7029:
7027:
7026:
7021:
6997:
6995:
6994:
6989:
6972:
6971:
6956:
6955:
6940:
6939:
6924:
6923:
6908:
6907:
6886:
6885:
6869:
6867:
6866:
6861:
6757:
6756:
6719:
6717:
6716:
6711:
6709:
6708:
6692:
6690:
6689:
6684:
6682:
6681:
6659:
6657:
6656:
6651:
6649:
6648:
6626:
6624:
6623:
6618:
6613:
6612:
6584:
6582:
6581:
6576:
6574:
6573:
6558:using the guess
6557:
6555:
6554:
6549:
6538:
6537:
6521:
6519:
6518:
6513:
6511:
6510:
6486:
6484:
6483:
6478:
6476:
6475:
6463:
6462:
6446:
6444:
6443:
6438:
6436:
6435:
6419:
6417:
6416:
6411:
6406:
6405:
6387:
6386:
6368:
6367:
6351:
6349:
6348:
6343:
6331:
6329:
6328:
6323:
6312:
6311:
6255:β in which case
6233:identity element
6229:arithmetic shift
6225:arithmetic shift
6216:
6214:
6213:
6208:
6185:
6184:
6183:
6164:
6163:
6162:
6152:
6151:
6130:
6129:
6128:
6118:
6117:
6096:
6095:
6076:
6074:
6073:
6068:
6045:
6044:
6043:
6021:
6020:
6019:
5997:
5996:
5995:
5973:
5972:
5951:
5949:
5948:
5943:
5938:
5934:
5933:
5929:
5928:
5924:
5914:
5912:
5911:
5902:
5901:
5892:
5876:
5874:
5873:
5864:
5863:
5854:
5838:
5836:
5835:
5826:
5825:
5816:
5800:
5799:
5779:division by zero
5776:
5774:
5773:
5768:
5760:
5759:
5750:
5749:
5734:
5733:
5724:
5723:
5708:
5707:
5698:
5697:
5682:
5681:
5672:
5671:
5653:
5652:
5643:
5642:
5630:
5629:
5620:
5619:
5607:
5606:
5597:
5596:
5584:
5583:
5574:
5573:
5554:
5552:
5551:
5546:
5544:
5543:
5534:
5533:
5524:
5523:
5514:
5513:
5440:
5438:
5437:
5432:
5430:
5423:
5419:
5409:
5408:
5385:
5384:
5369:
5365:
5361:
5357:
5356:
5329:
5328:
5310:
5303:
5299:
5298:
5297:
5273:
5272:
5249:
5245:
5244:
5243:
5228:
5227:
5201:
5200:
5146:
5144:
5143:
5138:
5108:
5106:
5105:
5100:
5082:
5080:
5079:
5074:
5066:
5065:
5015:
5013:
5012:
5007:
5002:
5001:
4989:
4988:
4979:
4978:
4968:
4957:
4939:
4938:
4926:
4925:
4906:
4899:
4887:
4872:
4871:
4861:
4850:
4832:
4831:
4822:
4821:
4811:
4806:
4763:
4761:
4760:
4755:
4753:
4743:
4742:
4730:
4729:
4711:
4710:
4698:
4697:
4682:
4678:
4677:
4665:
4664:
4645:
4638:
4626:
4605:
4604:
4592:
4591:
4578:
4571:
4559:
4538:
4534:
4530:
4523:
4522:
4513:
4512:
4500:
4499:
4490:
4489:
4477:
4476:
4456:
4452:
4445:
4444:
4435:
4434:
4422:
4421:
4412:
4411:
4399:
4398:
4378:
4374:
4370:
4363:
4362:
4353:
4352:
4340:
4339:
4330:
4329:
4314:
4313:
4296:
4292:
4285:
4284:
4275:
4274:
4262:
4261:
4252:
4251:
4239:
4238:
4218:
4214:
4213:
4204:
4203:
4185:
4184:
4175:
4174:
4162:
4161:
4152:
4151:
4136:
4135:
4123:
4122:
4107:
4103:
4102:
4093:
4092:
4082:
4077:
4006:
4004:
4003:
3998:
3986:
3979:
3975:
3973:
3972:
3967:
3940:
3938:
3937:
3932:
3909:
3907:
3906:
3901:
3886:
3884:
3883:
3878:
3863:
3861:
3860:
3855:
3843:
3841:
3840:
3835:
3823:
3821:
3820:
3815:
3803:
3801:
3800:
3795:
3783:
3781:
3780:
3775:
3763:
3761:
3760:
3755:
3753:
3752:
3736:
3734:
3733:
3728:
3726:
3725:
3709:
3707:
3706:
3701:
3689:
3687:
3686:
3681:
3664:by iteration.
3663:
3661:
3660:
3655:
3643:
3641:
3640:
3635:
3614:
3612:
3611:
3606:
3594:
3592:
3591:
3586:
3581:
3567:
3566:
3547:
3545:
3544:
3539:
3527:
3525:
3524:
3519:
3500:
3498:
3497:
3492:
3487:
3485:
3468:
3451:
3450:
3435:
3434:
3419:
3417:
3407:
3406:
3396:
3386:
3385:
3375:
3366:
3362:
3352:
3350:
3349:
3344:
3332:
3331:
3315:
3313:
3312:
3307:
3296:
3295:
3279:
3277:
3276:
3271:
3245:
3244:
3228:
3226:
3225:
3220:
3203:
3202:
3187:
3186:
3162:
3161:
3142:
3140:
3139:
3134:
3117:
3116:
3101:The quotient is
3094:
3092:
3091:
3086:
3068:
3066:
3065:
3060:
3043:
3042:
3027:
3026:
3003:
2999:
2997:
2996:
2991:
2973:
2971:
2970:
2965:
2941:
2939:
2938:
2933:
2925:
2924:
2906:
2905:
2887:
2886:
2868:
2867:
2852:we can see that
2851:
2849:
2848:
2843:
2832:
2831:
2813:
2812:
2791:
2790:
2772:
2771:
2752:
2750:
2749:
2744:
2717:
2715:
2714:
2709:
2681:
2677:
2675:
2674:
2669:
2651:
2649:
2648:
2643:
2622:
2618:
2582:
2580:
2579:
2574:
2556:
2554:
2553:
2548:
2531:
2530:
2515:
2514:
2472:
2470:
2469:
2464:
2462:
2461:
2445:
2443:
2442:
2437:
2435:
2434:
2425:
2424:
2412:
2411:
2393:
2392:
2370:
2368:
2367:
2362:
2360:
2359:
2343:
2341:
2340:
2335:
2333:
2332:
2316:
2314:
2313:
2308:
2293:
2291:
2290:
2285:
2283:
2282:
2260:
2258:
2257:
2252:
2231:
2229:
2228:
2223:
2211:
2209:
2208:
2203:
2195:
2194:
2178:
2176:
2175:
2170:
2149:
2147:
2146:
2141:
2139:
2138:
2122:
2120:
2119:
2114:
2109:
2108:
2086:
2084:
2083:
2078:
2076:
2075:
2059:
2057:
2056:
2051:
2046:
2045:
2027:
1996:
1987:
1985:
1984:
1979:
1977:
1976:
1964:
1960:
1959:
1958:
1938:
1934:
1933:
1932:
1917:
1916:
1904:
1903:
1888:
1887:
1863:
1862:
1853:
1852:
1840:
1839:
1830:
1829:
1814:
1813:
1801:
1800:
1763:
1757:
1755:
1754:
1749:
1747:
1740:
1739:
1724:
1720:
1719:
1710:
1709:
1697:
1696:
1681:
1672:
1669:
1667:
1663:
1662:
1656:
1655:
1649:
1648:
1633:
1632:
1614:
1613:
1604:
1603:
1597:
1596:
1584:
1583:
1574:
1573:
1567:
1566:
1554:
1553:
1538:
1534:
1533:
1527:
1526:
1514:
1513:
1504:
1503:
1491:
1490:
1472:
1471:
1453:
1452:
1443:
1442:
1436:
1435:
1423:
1422:
1413:
1412:
1406:
1405:
1393:
1392:
1373:
1372:
1346:
1344:
1343:
1338:
1336:
1335:
1317:
1315:
1314:
1309:
1302:
1301:
1300:
1294:
1293:
1287:
1286:
1274:
1273:
1257:
1256:
1226:
1225:
1216:
1215:
1203:
1202:
1193:
1192:
1180:
1179:
1170:
1169:
1157:
1156:
1122:
1120:
1119:
1114:
1109:
1108:
1087:is the value of
1086:
1084:
1083:
1078:
1076:
1075:
1052:
1043:
1041:
1040:
1035:
1033:
1026:
1025:
1016:
1015:
1003:
1002:
986:
985:
972:
971:
962:
961:
949:
948:
932:
931:
913:
910:
907:
905:
901:
900:
891:
890:
878:
877:
855:
854:
835:
834:
818:
817:
796:
786:
784:
783:
778:
763:
761:
760:
755:
753:
752:
736:
734:
733:
728:
726:
725:
707:
706:
690:
688:
687:
682:
677:
676:
667:
666:
648:
647:
638:
637:
625:
624:
615:
614:
599:
598:
586:
585:
573:
572:
563:
562:
552:
547:
497:
496:
483:
479:
477:
476:
471:
459:
457:
456:
451:
439:
429:
427:
426:
421:
419:
412:
411:
405:
404:
398:
397:
385:
384:
368:
367:
337:
336:
327:
326:
314:
313:
304:
303:
291:
290:
281:
280:
268:
267:
256:
247:
246:
237:
236:
218:
217:
208:
207:
195:
194:
185:
184:
169:
168:
156:
155:
145:
101:computer science
88:
81:
77:
74:
68:
38:
37:
30:
21:
10026:
10025:
10021:
10020:
10019:
10017:
10016:
10015:
9991:
9990:
9989:
9984:
9933:
9872:
9815:Linear equation
9785:
9776:
9743:Wayback Machine
9712:"Horner scheme"
9710:
9707:
9702:
9678:
9677:
9654:Whittaker, E.T.
9646:
9627:
9619:. McGraw-Hill.
9588:(9). ACM: 633.
9536:
9534:
9527:
9489:
9487:
9480:
9439:
9361:
9359:
9355:
9348:
9334:
9286:
9284:
9280:
9273:
9246:
9245:
9196:
9194:
9165:Cajori, Florian
9130:
9125:
9124:
9116:
9112:
9104:
9100:
9092:
9088:
9080:
9076:
9071:
9067:
9061:
9057:
9033:
9026:
9018:
9014:
9006:
9002:
8994:
8987:
8979:
8975:
8968:
8964:
8956:
8952:
8944:
8940:
8934:Kripasagar 2008
8932:
8928:
8920:
8916:
8908:
8904:
8896:
8892:
8886:Pankiewicz 1968
8884:
8880:
8873:
8869:
8857:
8853:
8848:
8823:Estrin's scheme
8787:
8617:Charles Babbage
8590:
8562:
8558:
8546:
8542:
8537:
8534:
8533:
8523:
8493:
8490:
8489:
8461:
8452:
8448:
8446:
8443:
8442:
8420:
8417:
8416:
8394:
8391:
8390:
8365:
8362:
8361:
8336:
8333:
8332:
8318:
8317:
8308:
8304:
8280:
8276:
8269:
8254:
8253:
8244:
8240:
8233:
8220:
8191:
8189:
8186:
8185:
8176:
8172:
8165:
8149:
8147:
8144:
8143:
8127:
8126:
8114:
8110:
8101:
8097:
8090:
8084:
8080:
8077:
8076:
8064:
8060:
8051:
8047:
8040:
8034:
8030:
8027:
8015:
8011:
8002:
7998:
7991:
7985:
7981:
7978:
7977:
7966:
7964:
7961:
7950:
7946:
7945:
7933:
7929:
7914:
7910:
7903:
7891:
7887:
7884:
7872:
7868:
7853:
7849:
7842:
7830:
7826:
7823:
7822:
7813:
7809:
7802:
7796:
7792:
7789:
7780:
7776:
7769:
7763:
7759:
7755:
7753:
7750:
7749:
7729:
7725:
7719:
7715:
7700:
7696:
7690:
7686:
7677:
7673:
7667:
7663:
7651:
7647:
7638:
7634:
7625:
7621:
7615:
7611:
7605:
7594:
7573:
7570:
7569:
7535:
7500:
7497:
7496:
7493:
7485:linear equation
7458:
7454:
7452:
7449:
7448:
7416:
7412:
7394:
7390:
7388:
7385:
7384:
7352:
7348:
7336:
7332:
7314:
7310:
7308:
7305:
7304:
7272:
7268:
7256:
7252:
7240:
7236:
7218:
7214:
7212:
7209:
7208:
7180:
7177:
7176:
7143:
7139:
7127:
7123:
7111:
7107:
7095:
7091:
7073:
7069:
7067:
7064:
7063:
7035:
7032:
7031:
7006:
7003:
7002:
6967:
6963:
6951:
6947:
6935:
6931:
6919:
6915:
6903:
6899:
6881:
6877:
6875:
6872:
6871:
6752:
6748:
6746:
6743:
6742:
6731:
6704:
6700:
6698:
6695:
6694:
6671:
6667:
6665:
6662:
6661:
6638:
6634:
6632:
6629:
6628:
6608:
6604:
6593:
6590:
6589:
6569:
6565:
6563:
6560:
6559:
6533:
6529:
6527:
6524:
6523:
6506:
6502:
6500:
6497:
6496:
6493:Newton's method
6471:
6467:
6458:
6454:
6452:
6449:
6448:
6431:
6427:
6425:
6422:
6421:
6401:
6397:
6376:
6372:
6363:
6359:
6357:
6354:
6353:
6337:
6334:
6333:
6307:
6303:
6301:
6298:
6297:
6294:Newton's method
6290:
6267:
6253:numeral systems
6249:
6179:
6175:
6174:
6158:
6154:
6153:
6144:
6140:
6124:
6120:
6119:
6110:
6106:
6091:
6087:
6082:
6079:
6078:
6039:
6035:
6034:
6015:
6011:
6010:
5991:
5987:
5986:
5968:
5964:
5959:
5956:
5955:
5907:
5903:
5897:
5893:
5891:
5881:
5877:
5869:
5865:
5859:
5855:
5853:
5843:
5839:
5831:
5827:
5821:
5817:
5815:
5805:
5801:
5795:
5791:
5786:
5783:
5782:
5755:
5751:
5745:
5741:
5729:
5725:
5719:
5715:
5703:
5699:
5693:
5689:
5677:
5673:
5667:
5663:
5648:
5644:
5638:
5634:
5625:
5621:
5615:
5611:
5602:
5598:
5592:
5588:
5579:
5575:
5569:
5565:
5560:
5557:
5556:
5539:
5535:
5529:
5525:
5519:
5515:
5509:
5505:
5503:
5500:
5499:
5496:
5481:
5446:
5428:
5427:
5401:
5397:
5390:
5386:
5377:
5373:
5349:
5345:
5341:
5334:
5330:
5321:
5317:
5308:
5307:
5290:
5286:
5265:
5261:
5260:
5256:
5236:
5232:
5220:
5216:
5215:
5211:
5196:
5192:
5182:
5166:
5164:
5161:
5160:
5153:
5126:
5123:
5122:
5088:
5085:
5084:
5061:
5057:
5055:
5052:
5051:
5044:microcontroller
5040:
5034:
4997:
4993:
4984:
4980:
4974:
4970:
4958:
4947:
4931:
4927:
4912:
4908:
4895:
4888:
4877:
4867:
4863:
4851:
4840:
4827:
4823:
4817:
4813:
4807:
4796:
4775:
4772:
4771:
4751:
4750:
4738:
4734:
4725:
4721:
4706:
4702:
4693:
4689:
4680:
4679:
4670:
4666:
4651:
4647:
4634:
4627:
4616:
4597:
4593:
4584:
4580:
4567:
4560:
4549:
4536:
4535:
4518:
4514:
4508:
4504:
4495:
4491:
4485:
4481:
4472:
4468:
4467:
4463:
4440:
4436:
4430:
4426:
4417:
4413:
4407:
4403:
4394:
4390:
4389:
4385:
4376:
4375:
4358:
4354:
4348:
4344:
4335:
4331:
4325:
4321:
4309:
4305:
4304:
4300:
4280:
4276:
4270:
4266:
4257:
4253:
4247:
4243:
4234:
4230:
4229:
4225:
4216:
4215:
4209:
4205:
4199:
4195:
4180:
4176:
4170:
4166:
4157:
4153:
4147:
4143:
4131:
4127:
4118:
4114:
4105:
4104:
4098:
4094:
4088:
4084:
4078:
4067:
4056:
4040:
4038:
4035:
4034:
4019:
4017:Estrin's scheme
4013:
3992:
3989:
3988:
3984:
3977:
3961:
3958:
3957:
3950:preconditioning
3926:
3923:
3922:
3892:
3889:
3888:
3872:
3869:
3868:
3849:
3846:
3845:
3829:
3826:
3825:
3809:
3806:
3805:
3789:
3786:
3785:
3769:
3766:
3765:
3748:
3744:
3742:
3739:
3738:
3721:
3717:
3715:
3712:
3711:
3695:
3692:
3691:
3672:
3669:
3668:
3649:
3646:
3645:
3620:
3617:
3616:
3600:
3597:
3596:
3577:
3562:
3558:
3553:
3550:
3549:
3533:
3530:
3529:
3513:
3510:
3509:
3506:
3472:
3467:
3446:
3442:
3430:
3426:
3402:
3398:
3397:
3381:
3377:
3376:
3374:
3372:
3369:
3368:
3364:
3360:
3357:
3327:
3323:
3321:
3318:
3317:
3291:
3287:
3285:
3282:
3281:
3240:
3236:
3234:
3231:
3230:
3198:
3194:
3182:
3178:
3157:
3153:
3151:
3148:
3147:
3112:
3108:
3106:
3103:
3102:
3099:
3074:
3071:
3070:
3038:
3034:
3022:
3018:
3016:
3013:
3012:
3001:
2979:
2976:
2975:
2974:on division by
2950:
2947:
2946:
2920:
2916:
2901:
2897:
2882:
2878:
2863:
2859:
2857:
2854:
2853:
2827:
2823:
2808:
2804:
2786:
2782:
2767:
2763:
2761:
2758:
2757:
2723:
2720:
2719:
2694:
2691:
2690:
2679:
2657:
2654:
2653:
2652:on division by
2628:
2625:
2624:
2620:
2616:
2613:
2598:
2562:
2559:
2558:
2526:
2522:
2510:
2506:
2486:
2483:
2482:
2479:
2457:
2453:
2451:
2448:
2447:
2430:
2426:
2420:
2416:
2401:
2397:
2382:
2378:
2376:
2373:
2372:
2355:
2351:
2349:
2346:
2345:
2328:
2324:
2322:
2319:
2318:
2302:
2299:
2298:
2278:
2274:
2266:
2263:
2262:
2237:
2234:
2233:
2217:
2214:
2213:
2190:
2186:
2184:
2181:
2180:
2155:
2152:
2151:
2134:
2130:
2128:
2125:
2124:
2104:
2100:
2092:
2089:
2088:
2071:
2067:
2065:
2062:
2061:
2041:
2037:
2023:
2009:
2006:
2005:
1972:
1968:
1954:
1950:
1943:
1939:
1922:
1918:
1912:
1908:
1893:
1889:
1877:
1873:
1858:
1854:
1848:
1844:
1835:
1831:
1825:
1821:
1809:
1805:
1796:
1792:
1791:
1787:
1770:
1767:
1766:
1745:
1744:
1735:
1731:
1722:
1721:
1715:
1711:
1705:
1701:
1692:
1688:
1679:
1678:
1665:
1664:
1658:
1657:
1651:
1650:
1638:
1634:
1628:
1624:
1609:
1605:
1599:
1598:
1592:
1588:
1579:
1575:
1569:
1568:
1562:
1558:
1549:
1545:
1536:
1535:
1529:
1528:
1522:
1521:
1509:
1505:
1499:
1495:
1480:
1476:
1467:
1463:
1448:
1444:
1438:
1437:
1431:
1427:
1418:
1414:
1408:
1407:
1401:
1397:
1388:
1384:
1377:
1368:
1364:
1354:
1352:
1349:
1348:
1331:
1327:
1325:
1322:
1321:
1296:
1295:
1289:
1288:
1282:
1281:
1269:
1265:
1246:
1242:
1221:
1217:
1211:
1210:
1198:
1194:
1188:
1187:
1175:
1171:
1165:
1164:
1152:
1148:
1131:
1128:
1127:
1104:
1100:
1092:
1089:
1088:
1071:
1067:
1065:
1062:
1061:
1031:
1030:
1021:
1017:
1011:
1007:
998:
994:
987:
981:
977:
974:
973:
967:
963:
957:
953:
944:
940:
933:
927:
923:
920:
919:
903:
902:
896:
892:
886:
882:
867:
863:
856:
844:
840:
837:
836:
830:
826:
819:
813:
809:
805:
803:
800:
799:
769:
766:
765:
748:
744:
742:
739:
738:
721:
717:
702:
698:
696:
693:
692:
672:
668:
662:
658:
643:
639:
633:
629:
620:
616:
610:
606:
594:
590:
581:
577:
568:
564:
558:
554:
548:
537:
516:
513:
512:
509:
494:
493:
489:Horner's method
487:Alternatively,
481:
465:
462:
461:
445:
442:
441:
437:
417:
416:
407:
406:
400:
399:
393:
392:
380:
376:
357:
353:
332:
328:
322:
321:
309:
305:
299:
298:
286:
282:
276:
275:
263:
259:
257:
255:
249:
248:
242:
238:
232:
228:
213:
209:
203:
199:
190:
186:
180:
176:
164:
160:
151:
147:
142:
140:
137:
136:
109:Horner's scheme
105:Horner's method
89:
78:
72:
69:
62:
39:
35:
28:
23:
22:
15:
12:
11:
5:
10024:
10014:
10013:
10008:
10003:
9986:
9985:
9983:
9982:
9977:
9972:
9967:
9962:
9957:
9952:
9947:
9941:
9939:
9935:
9934:
9932:
9931:
9926:
9921:
9916:
9911:
9906:
9901:
9896:
9891:
9886:
9880:
9878:
9874:
9873:
9871:
9870:
9865:
9860:
9855:
9854:
9853:
9843:
9842:
9841:
9839:Cubic equation
9831:
9830:
9829:
9819:
9818:
9817:
9807:
9802:
9796:
9794:
9787:
9786:
9775:
9774:
9767:
9760:
9752:
9746:
9745:
9733:
9728:Qiu Jin-Shao,
9726:
9706:
9705:External links
9703:
9701:
9700:
9699:
9698:
9664:
9650:
9644:
9631:
9625:
9608:
9571:
9542:
9525:
9508:
9495:
9478:
9461:
9452:
9443:
9437:
9421:
9420:
9419:
9372:
9371:
9370:
9338:
9332:
9319:
9292:
9263:Fateman, R. J.
9259:
9203:
9179:(8): 409β414.
9161:
9149:10.2307/604533
9143:(2): 304β309.
9131:
9129:
9126:
9123:
9122:
9120:, p. 208.
9118:Libbrecht 2005
9110:
9098:
9096:, p. 142.
9086:
9074:
9065:
9055:
9024:
9012:
9000:
8985:
8981:Libbrecht 2005
8973:
8962:
8960:, p. 112.
8950:
8948:, Section 5.4.
8938:
8926:
8914:
8902:
8898:Ostrowski 1954
8890:
8878:
8867:
8850:
8849:
8847:
8844:
8843:
8842:
8835:Ruffini's rule
8832:
8826:
8820:
8811:
8797:
8795:Chebyshev form
8786:
8783:
8723:
8722:
8706:
8695:
8682:
8666:
8656:
8650:
8647:Ruffini's rule
8579:
8576:
8573:
8570:
8565:
8561:
8557:
8554:
8549:
8545:
8541:
8522:
8519:
8506:
8503:
8500:
8497:
8477:
8474:
8471:
8467:
8464:
8460:
8455:
8451:
8430:
8427:
8424:
8404:
8401:
8398:
8378:
8375:
8372:
8369:
8349:
8346:
8343:
8340:
8316:
8311:
8307:
8303:
8300:
8297:
8294:
8291:
8288:
8283:
8279:
8275:
8272:
8270:
8268:
8265:
8262:
8259:
8256:
8255:
8252:
8247:
8243:
8239:
8236:
8234:
8229:
8226:
8223:
8218:
8215:
8212:
8209:
8206:
8203:
8200:
8197:
8194:
8188:
8187:
8184:
8179:
8175:
8171:
8168:
8166:
8164:
8161:
8158:
8155:
8152:
8151:
8125:
8122:
8117:
8113:
8109:
8104:
8100:
8096:
8093:
8091:
8087:
8083:
8079:
8078:
8075:
8072:
8067:
8063:
8059:
8054:
8050:
8046:
8043:
8041:
8037:
8033:
8028:
8026:
8023:
8018:
8014:
8010:
8005:
8001:
7997:
7994:
7992:
7988:
7984:
7980:
7979:
7976:
7965:
7962:
7960:
7949:
7947:
7944:
7941:
7936:
7932:
7928:
7923:
7920:
7917:
7913:
7909:
7906:
7904:
7900:
7897:
7894:
7890:
7885:
7883:
7880:
7875:
7871:
7867:
7862:
7859:
7856:
7852:
7848:
7845:
7843:
7839:
7836:
7833:
7829:
7825:
7824:
7821:
7816:
7812:
7808:
7805:
7803:
7799:
7795:
7790:
7788:
7783:
7779:
7775:
7772:
7770:
7766:
7762:
7758:
7757:
7737:
7732:
7728:
7722:
7718:
7714:
7711:
7708:
7703:
7699:
7693:
7689:
7685:
7680:
7676:
7670:
7666:
7662:
7659:
7654:
7650:
7646:
7641:
7637:
7633:
7628:
7624:
7618:
7614:
7608:
7603:
7600:
7597:
7593:
7589:
7586:
7583:
7580:
7577:
7557:
7554:
7551:
7548:
7545:
7542:
7538:
7534:
7531:
7528:
7525:
7522:
7519:
7516:
7513:
7510:
7507:
7504:
7492:
7489:
7472:
7469:
7466:
7461:
7457:
7436:
7433:
7430:
7427:
7424:
7419:
7415:
7411:
7408:
7405:
7402:
7397:
7393:
7372:
7369:
7366:
7363:
7360:
7355:
7351:
7347:
7344:
7339:
7335:
7331:
7328:
7325:
7322:
7317:
7313:
7292:
7289:
7286:
7283:
7280:
7275:
7271:
7267:
7264:
7259:
7255:
7251:
7248:
7243:
7239:
7235:
7232:
7229:
7226:
7221:
7217:
7196:
7193:
7190:
7187:
7184:
7163:
7160:
7157:
7154:
7151:
7146:
7142:
7138:
7135:
7130:
7126:
7122:
7119:
7114:
7110:
7106:
7103:
7098:
7094:
7090:
7087:
7084:
7081:
7076:
7072:
7051:
7048:
7045:
7042:
7039:
7030:is divided by
7019:
7016:
7013:
7010:
6987:
6984:
6981:
6978:
6975:
6970:
6966:
6962:
6959:
6954:
6950:
6946:
6943:
6938:
6934:
6930:
6927:
6922:
6918:
6914:
6911:
6906:
6902:
6898:
6895:
6892:
6889:
6884:
6880:
6859:
6856:
6853:
6850:
6847:
6844:
6841:
6838:
6835:
6832:
6829:
6826:
6823:
6820:
6817:
6814:
6811:
6808:
6805:
6802:
6799:
6796:
6793:
6790:
6787:
6784:
6781:
6778:
6775:
6772:
6769:
6766:
6763:
6760:
6755:
6751:
6730:
6727:
6722:
6721:
6707:
6703:
6680:
6677:
6674:
6670:
6647:
6644:
6641:
6637:
6616:
6611:
6607:
6603:
6600:
6597:
6586:
6572:
6568:
6547:
6544:
6541:
6536:
6532:
6509:
6505:
6474:
6470:
6466:
6461:
6457:
6434:
6430:
6409:
6404:
6400:
6396:
6393:
6390:
6385:
6382:
6379:
6375:
6371:
6366:
6362:
6341:
6321:
6318:
6315:
6310:
6306:
6289:
6286:
6282:faster methods
6263:
6248:
6245:
6221:register shift
6206:
6203:
6200:
6197:
6194:
6191:
6188:
6182:
6178:
6173:
6170:
6167:
6161:
6157:
6150:
6147:
6143:
6139:
6136:
6133:
6127:
6123:
6116:
6113:
6109:
6105:
6102:
6099:
6094:
6090:
6086:
6066:
6063:
6060:
6057:
6054:
6051:
6048:
6042:
6038:
6033:
6030:
6027:
6024:
6018:
6014:
6009:
6006:
6003:
6000:
5994:
5990:
5985:
5982:
5979:
5976:
5971:
5967:
5963:
5941:
5937:
5932:
5927:
5923:
5920:
5917:
5910:
5906:
5900:
5896:
5890:
5887:
5884:
5880:
5872:
5868:
5862:
5858:
5852:
5849:
5846:
5842:
5834:
5830:
5824:
5820:
5814:
5811:
5808:
5804:
5798:
5794:
5790:
5766:
5763:
5758:
5754:
5748:
5744:
5740:
5737:
5732:
5728:
5722:
5718:
5714:
5711:
5706:
5702:
5696:
5692:
5688:
5685:
5680:
5676:
5670:
5666:
5662:
5659:
5656:
5651:
5647:
5641:
5637:
5633:
5628:
5624:
5618:
5614:
5610:
5605:
5601:
5595:
5591:
5587:
5582:
5578:
5572:
5568:
5564:
5542:
5538:
5532:
5528:
5522:
5518:
5512:
5508:
5495:
5492:
5491:
5490:
5482:
5480:
5479:
5476:
5472:
5465:
5445:
5442:
5426:
5422:
5418:
5415:
5412:
5407:
5404:
5400:
5396:
5393:
5389:
5383:
5380:
5376:
5372:
5368:
5364:
5360:
5355:
5352:
5348:
5344:
5340:
5337:
5333:
5327:
5324:
5320:
5316:
5313:
5311:
5309:
5306:
5302:
5296:
5293:
5289:
5285:
5282:
5279:
5276:
5271:
5268:
5264:
5259:
5255:
5252:
5248:
5242:
5239:
5235:
5231:
5226:
5223:
5219:
5214:
5210:
5207:
5204:
5199:
5195:
5191:
5188:
5185:
5183:
5181:
5178:
5175:
5172:
5169:
5168:
5152:
5149:
5136:
5133:
5130:
5098:
5095:
5092:
5072:
5069:
5064:
5060:
5036:Main article:
5033:
5030:
5026:floating-point
5005:
5000:
4996:
4992:
4987:
4983:
4977:
4973:
4967:
4964:
4961:
4956:
4953:
4950:
4946:
4942:
4937:
4934:
4930:
4924:
4921:
4918:
4915:
4911:
4905:
4902:
4898:
4894:
4891:
4886:
4883:
4880:
4876:
4870:
4866:
4860:
4857:
4854:
4849:
4846:
4843:
4839:
4835:
4830:
4826:
4820:
4816:
4810:
4805:
4802:
4799:
4795:
4791:
4788:
4785:
4782:
4779:
4749:
4746:
4741:
4737:
4733:
4728:
4724:
4720:
4717:
4714:
4709:
4705:
4701:
4696:
4692:
4688:
4685:
4683:
4681:
4676:
4673:
4669:
4663:
4660:
4657:
4654:
4650:
4644:
4641:
4637:
4633:
4630:
4625:
4622:
4619:
4615:
4611:
4608:
4603:
4600:
4596:
4590:
4587:
4583:
4577:
4574:
4570:
4566:
4563:
4558:
4555:
4552:
4548:
4544:
4541:
4539:
4537:
4533:
4529:
4526:
4521:
4517:
4511:
4507:
4503:
4498:
4494:
4488:
4484:
4480:
4475:
4471:
4466:
4462:
4459:
4455:
4451:
4448:
4443:
4439:
4433:
4429:
4425:
4420:
4416:
4410:
4406:
4402:
4397:
4393:
4388:
4384:
4381:
4379:
4377:
4373:
4369:
4366:
4361:
4357:
4351:
4347:
4343:
4338:
4334:
4328:
4324:
4320:
4317:
4312:
4308:
4303:
4299:
4295:
4291:
4288:
4283:
4279:
4273:
4269:
4265:
4260:
4256:
4250:
4246:
4242:
4237:
4233:
4228:
4224:
4221:
4219:
4217:
4212:
4208:
4202:
4198:
4194:
4191:
4188:
4183:
4179:
4173:
4169:
4165:
4160:
4156:
4150:
4146:
4142:
4139:
4134:
4130:
4126:
4121:
4117:
4113:
4110:
4108:
4106:
4101:
4097:
4091:
4087:
4081:
4076:
4073:
4070:
4066:
4062:
4059:
4057:
4055:
4052:
4049:
4046:
4043:
4042:
4012:
4009:
3996:
3965:
3930:
3899:
3896:
3876:
3853:
3833:
3813:
3793:
3784:additions and
3773:
3751:
3747:
3724:
3720:
3699:
3679:
3676:
3653:
3633:
3630:
3627:
3624:
3615:additions and
3604:
3584:
3580:
3576:
3573:
3570:
3565:
3561:
3557:
3548:additions and
3537:
3517:
3505:
3502:
3490:
3484:
3481:
3478:
3475:
3471:
3466:
3463:
3460:
3457:
3454:
3449:
3445:
3441:
3438:
3433:
3429:
3425:
3422:
3416:
3413:
3410:
3405:
3401:
3395:
3392:
3389:
3384:
3380:
3355:
3342:
3339:
3336:
3330:
3326:
3305:
3302:
3299:
3294:
3290:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3243:
3239:
3218:
3215:
3212:
3209:
3206:
3201:
3197:
3193:
3190:
3185:
3181:
3177:
3174:
3171:
3168:
3165:
3160:
3156:
3132:
3129:
3126:
3123:
3120:
3115:
3111:
3097:
3084:
3081:
3078:
3058:
3055:
3052:
3049:
3046:
3041:
3037:
3033:
3030:
3025:
3021:
2989:
2986:
2983:
2963:
2960:
2957:
2954:
2931:
2928:
2923:
2919:
2915:
2912:
2909:
2904:
2900:
2896:
2893:
2890:
2885:
2881:
2877:
2874:
2871:
2866:
2862:
2841:
2838:
2835:
2830:
2826:
2822:
2819:
2816:
2811:
2807:
2803:
2800:
2797:
2794:
2789:
2785:
2781:
2778:
2775:
2770:
2766:
2742:
2739:
2736:
2733:
2730:
2727:
2707:
2704:
2701:
2698:
2667:
2664:
2661:
2641:
2638:
2635:
2632:
2596:
2592:
2572:
2569:
2566:
2546:
2543:
2540:
2537:
2534:
2529:
2525:
2521:
2518:
2513:
2509:
2505:
2502:
2499:
2496:
2493:
2490:
2478:
2475:
2460:
2456:
2433:
2429:
2423:
2419:
2415:
2410:
2407:
2404:
2400:
2396:
2391:
2388:
2385:
2381:
2358:
2354:
2331:
2327:
2306:
2281:
2277:
2273:
2270:
2250:
2247:
2244:
2241:
2221:
2201:
2198:
2193:
2189:
2168:
2165:
2162:
2159:
2137:
2133:
2112:
2107:
2103:
2099:
2096:
2074:
2070:
2049:
2044:
2040:
2036:
2033:
2030:
2026:
2022:
2019:
2016:
2013:
2000:
1999:
1990:
1988:
1975:
1971:
1967:
1963:
1957:
1953:
1949:
1946:
1942:
1937:
1931:
1928:
1925:
1921:
1915:
1911:
1907:
1902:
1899:
1896:
1892:
1886:
1883:
1880:
1876:
1872:
1869:
1866:
1861:
1857:
1851:
1847:
1843:
1838:
1834:
1828:
1824:
1820:
1817:
1812:
1808:
1804:
1799:
1795:
1790:
1786:
1783:
1780:
1777:
1774:
1743:
1738:
1734:
1730:
1727:
1725:
1723:
1718:
1714:
1708:
1704:
1700:
1695:
1691:
1687:
1684:
1682:
1680:
1677:
1668:
1666:
1661:
1654:
1647:
1644:
1641:
1637:
1631:
1627:
1623:
1620:
1617:
1612:
1608:
1602:
1595:
1591:
1587:
1582:
1578:
1572:
1565:
1561:
1557:
1552:
1548:
1544:
1541:
1539:
1537:
1532:
1525:
1520:
1517:
1512:
1508:
1502:
1498:
1494:
1489:
1486:
1483:
1479:
1475:
1470:
1466:
1462:
1459:
1456:
1451:
1447:
1441:
1434:
1430:
1426:
1421:
1417:
1411:
1404:
1400:
1396:
1391:
1387:
1383:
1380:
1378:
1376:
1371:
1367:
1363:
1360:
1357:
1356:
1334:
1330:
1307:
1299:
1292:
1285:
1280:
1277:
1272:
1268:
1263:
1260:
1255:
1252:
1249:
1245:
1241:
1238:
1235:
1232:
1229:
1224:
1220:
1214:
1209:
1206:
1201:
1197:
1191:
1186:
1183:
1178:
1174:
1168:
1163:
1160:
1155:
1151:
1147:
1144:
1141:
1138:
1135:
1112:
1107:
1103:
1099:
1096:
1074:
1070:
1056:
1055:
1046:
1044:
1029:
1024:
1020:
1014:
1010:
1006:
1001:
997:
993:
990:
988:
984:
980:
976:
975:
970:
966:
960:
956:
952:
947:
943:
939:
936:
934:
930:
926:
922:
921:
918:
906:
904:
899:
895:
889:
885:
881:
876:
873:
870:
866:
862:
859:
857:
853:
850:
847:
843:
839:
838:
833:
829:
825:
822:
820:
816:
812:
808:
807:
776:
773:
751:
747:
724:
720:
716:
713:
710:
705:
701:
680:
675:
671:
665:
661:
657:
654:
651:
646:
642:
636:
632:
628:
623:
619:
613:
609:
605:
602:
597:
593:
589:
584:
580:
576:
571:
567:
561:
557:
551:
546:
543:
540:
536:
532:
529:
526:
523:
520:
508:
505:
469:
449:
415:
410:
403:
396:
391:
388:
383:
379:
374:
371:
366:
363:
360:
356:
352:
349:
346:
343:
340:
335:
331:
325:
320:
317:
312:
308:
302:
297:
294:
289:
285:
279:
274:
271:
266:
262:
258:
254:
251:
250:
245:
241:
235:
231:
227:
224:
221:
216:
212:
206:
202:
198:
193:
189:
183:
179:
175:
172:
167:
163:
159:
154:
150:
146:
144:
91:
90:
42:
40:
33:
26:
9:
6:
4:
3:
2:
10023:
10012:
10009:
10007:
10004:
10002:
9999:
9998:
9996:
9981:
9980:GrΓΆbner basis
9978:
9976:
9973:
9971:
9968:
9966:
9963:
9961:
9958:
9956:
9953:
9951:
9948:
9946:
9945:Factorization
9943:
9942:
9940:
9936:
9930:
9927:
9925:
9922:
9920:
9917:
9915:
9912:
9910:
9907:
9905:
9902:
9900:
9897:
9895:
9892:
9890:
9887:
9885:
9882:
9881:
9879:
9877:By properties
9875:
9869:
9866:
9864:
9861:
9859:
9856:
9852:
9849:
9848:
9847:
9844:
9840:
9837:
9836:
9835:
9832:
9828:
9825:
9824:
9823:
9820:
9816:
9813:
9812:
9811:
9808:
9806:
9803:
9801:
9798:
9797:
9795:
9793:
9788:
9784:
9780:
9773:
9768:
9766:
9761:
9759:
9754:
9753:
9750:
9744:
9740:
9737:
9734:
9731:
9727:
9723:
9719:
9718:
9713:
9709:
9708:
9696:
9692:
9691:
9688:
9682:
9674:
9670:
9665:
9661:
9660:
9655:
9651:
9647:
9641:
9637:
9632:
9628:
9626:9780070602267
9622:
9617:
9616:
9609:
9605:
9601:
9596:
9591:
9587:
9583:
9582:
9577:
9572:
9568:
9564:
9560:
9556:
9552:
9548:
9543:
9532:
9528:
9522:
9518:
9514:
9509:
9505:
9501:
9496:
9486:on 2017-06-06
9485:
9481:
9475:
9471:
9467:
9462:
9458:
9453:
9449:
9444:
9440:
9434:
9430:
9426:
9425:Knuth, Donald
9422:
9417:
9413:
9412:
9409:
9405:
9401:
9397:
9393:
9389:
9385:
9381:
9377:
9373:
9368:
9367:
9358:on 2014-01-06
9354:
9347:
9346:
9339:
9335:
9329:
9325:
9320:
9315:
9310:
9306:
9302:
9298:
9293:
9283:on 2017-08-14
9279:
9272:
9268:
9264:
9260:
9256:
9250:
9241:
9236:
9232:
9228:
9224:
9220:
9216:
9212:
9208:
9204:
9192:
9187:
9182:
9178:
9174:
9170:
9166:
9162:
9158:
9154:
9150:
9146:
9142:
9138:
9133:
9132:
9119:
9114:
9107:
9102:
9095:
9090:
9083:
9082:Berggren 1990
9078:
9069:
9059:
9052:
9048:
9047:
9042:
9038:
9031:
9029:
9021:
9016:
9009:
9004:
8997:
8992:
8990:
8982:
8977:
8971:
8966:
8959:
8954:
8947:
8942:
8936:, p. 62.
8935:
8930:
8923:
8918:
8911:
8906:
8899:
8894:
8887:
8882:
8876:
8871:
8865:
8861:
8855:
8851:
8840:
8836:
8833:
8830:
8829:Lill's method
8827:
8824:
8821:
8819:
8815:
8812:
8809:
8805:
8801:
8798:
8796:
8792:
8789:
8788:
8782:
8780:
8779:Jigu Suanjing
8776:
8775:Wang Xiaotong
8772:
8768:
8764:
8760:
8755:
8751:
8749:
8745:
8744:Yoshio Mikami
8741:
8737:
8736:
8731:
8727:
8720:
8716:
8712:
8711:
8707:
8704:
8700:
8696:
8693:
8690:
8689:mathematician
8687:
8683:
8680:
8679:
8674:
8671:
8667:
8664:
8661:
8657:
8654:
8651:
8648:
8645:in 1809 (see
8644:
8643:Paolo Ruffini
8641:
8640:
8639:
8636:
8634:
8633:Paolo Ruffini
8630:
8625:
8622:
8618:
8614:
8610:
8606:
8602:
8594:
8577:
8574:
8571:
8568:
8563:
8559:
8555:
8552:
8547:
8543:
8539:
8531:
8527:
8518:
8501:
8495:
8472:
8465:
8462:
8458:
8453:
8449:
8428:
8425:
8422:
8402:
8399:
8396:
8373:
8367:
8344:
8338:
8314:
8309:
8305:
8298:
8295:
8292:
8286:
8281:
8277:
8273:
8271:
8263:
8257:
8250:
8245:
8241:
8237:
8235:
8227:
8224:
8221:
8213:
8207:
8204:
8198:
8192:
8182:
8177:
8173:
8169:
8167:
8159:
8153:
8140:
8123:
8120:
8115:
8111:
8107:
8102:
8098:
8094:
8092:
8085:
8081:
8073:
8070:
8065:
8061:
8057:
8052:
8048:
8044:
8042:
8035:
8031:
8024:
8021:
8016:
8012:
8008:
8003:
7999:
7995:
7993:
7986:
7982:
7974:
7958:
7942:
7939:
7934:
7930:
7926:
7921:
7918:
7915:
7911:
7907:
7905:
7898:
7895:
7892:
7888:
7881:
7878:
7873:
7869:
7865:
7860:
7857:
7854:
7850:
7846:
7844:
7837:
7834:
7831:
7827:
7819:
7814:
7810:
7806:
7804:
7797:
7793:
7786:
7781:
7777:
7773:
7771:
7764:
7760:
7735:
7730:
7726:
7720:
7716:
7712:
7709:
7706:
7701:
7697:
7691:
7687:
7683:
7678:
7674:
7668:
7664:
7660:
7657:
7652:
7648:
7644:
7639:
7635:
7631:
7626:
7622:
7616:
7612:
7606:
7601:
7598:
7595:
7591:
7587:
7581:
7575:
7555:
7549:
7546:
7543:
7536:
7526:
7520:
7517:
7511:
7505:
7488:
7486:
7467:
7459:
7455:
7434:
7431:
7428:
7425:
7422:
7417:
7413:
7409:
7403:
7395:
7391:
7370:
7367:
7364:
7361:
7358:
7353:
7349:
7345:
7342:
7337:
7333:
7329:
7323:
7315:
7311:
7290:
7287:
7284:
7281:
7278:
7273:
7269:
7265:
7262:
7257:
7253:
7249:
7246:
7241:
7237:
7233:
7227:
7219:
7215:
7191:
7188:
7185:
7161:
7158:
7155:
7152:
7149:
7144:
7140:
7136:
7133:
7128:
7124:
7120:
7117:
7112:
7108:
7104:
7101:
7096:
7092:
7088:
7082:
7074:
7070:
7046:
7043:
7040:
7014:
7008:
6998:
6985:
6982:
6979:
6976:
6973:
6968:
6964:
6960:
6957:
6952:
6948:
6944:
6941:
6936:
6932:
6928:
6925:
6920:
6916:
6912:
6909:
6904:
6900:
6896:
6890:
6882:
6878:
6854:
6851:
6848:
6839:
6836:
6833:
6824:
6821:
6818:
6809:
6806:
6803:
6794:
6791:
6788:
6779:
6776:
6773:
6767:
6761:
6753:
6749:
6735:
6726:
6705:
6701:
6678:
6675:
6672:
6668:
6645:
6642:
6639:
6635:
6609:
6605:
6601:
6598:
6587:
6570:
6566:
6542:
6534:
6530:
6507:
6503:
6494:
6490:
6489:
6488:
6472:
6468:
6464:
6459:
6455:
6432:
6428:
6407:
6402:
6398:
6394:
6391:
6388:
6383:
6380:
6377:
6373:
6369:
6364:
6360:
6339:
6316:
6308:
6304:
6295:
6285:
6283:
6279:
6275:
6271:
6266:
6262:
6258:
6254:
6244:
6242:
6236:
6234:
6230:
6226:
6222:
6217:
6204:
6189:
6180:
6176:
6171:
6168:
6159:
6155:
6148:
6145:
6141:
6137:
6134:
6125:
6121:
6114:
6111:
6107:
6103:
6100:
6092:
6088:
6084:
6064:
6049:
6040:
6036:
6031:
6028:
6025:
6016:
6012:
6007:
6004:
6001:
5992:
5988:
5983:
5980:
5977:
5969:
5965:
5961:
5952:
5939:
5935:
5930:
5925:
5918:
5908:
5904:
5898:
5894:
5888:
5885:
5882:
5878:
5870:
5866:
5860:
5856:
5850:
5847:
5844:
5840:
5832:
5828:
5822:
5818:
5812:
5809:
5806:
5802:
5796:
5792:
5788:
5780:
5764:
5761:
5756:
5752:
5746:
5742:
5738:
5735:
5730:
5726:
5720:
5716:
5712:
5709:
5704:
5700:
5694:
5690:
5686:
5683:
5678:
5674:
5668:
5664:
5660:
5657:
5649:
5645:
5639:
5635:
5631:
5626:
5622:
5616:
5612:
5608:
5603:
5599:
5593:
5589:
5585:
5580:
5576:
5570:
5566:
5540:
5536:
5530:
5526:
5520:
5516:
5510:
5506:
5488:
5483:
5477:
5474:
5470:
5466:
5463:
5459:
5458:
5457:
5455:
5451:
5441:
5424:
5420:
5413:
5405:
5402:
5398:
5394:
5391:
5387:
5381:
5378:
5374:
5370:
5366:
5362:
5358:
5353:
5350:
5346:
5342:
5338:
5335:
5331:
5325:
5322:
5318:
5314:
5312:
5304:
5300:
5294:
5291:
5287:
5280:
5277:
5269:
5266:
5262:
5257:
5253:
5250:
5246:
5240:
5237:
5233:
5229:
5224:
5221:
5217:
5212:
5208:
5205:
5197:
5193:
5186:
5184:
5179:
5173:
5158:
5148:
5134:
5131:
5128:
5120:
5116:
5112:
5096:
5093:
5090:
5070:
5067:
5062:
5058:
5049:
5045:
5039:
5029:
5027:
5023:
5019:
4998:
4994:
4985:
4981:
4975:
4971:
4965:
4962:
4959:
4954:
4951:
4948:
4944:
4940:
4935:
4932:
4928:
4922:
4919:
4916:
4913:
4909:
4900:
4896:
4892:
4884:
4881:
4878:
4874:
4868:
4864:
4858:
4855:
4852:
4847:
4844:
4841:
4837:
4833:
4828:
4824:
4818:
4814:
4808:
4803:
4800:
4797:
4793:
4789:
4783:
4777:
4769:
4764:
4747:
4739:
4735:
4726:
4722:
4718:
4715:
4707:
4703:
4694:
4690:
4686:
4684:
4674:
4671:
4667:
4661:
4658:
4655:
4652:
4648:
4639:
4635:
4631:
4623:
4620:
4617:
4613:
4609:
4606:
4601:
4598:
4594:
4588:
4585:
4581:
4572:
4568:
4564:
4556:
4553:
4550:
4546:
4542:
4540:
4531:
4527:
4524:
4519:
4515:
4509:
4505:
4501:
4496:
4492:
4486:
4482:
4478:
4473:
4469:
4464:
4460:
4457:
4453:
4449:
4446:
4441:
4437:
4431:
4427:
4423:
4418:
4414:
4408:
4404:
4400:
4395:
4391:
4386:
4382:
4380:
4371:
4367:
4364:
4359:
4355:
4349:
4345:
4341:
4336:
4332:
4326:
4322:
4318:
4315:
4310:
4306:
4301:
4297:
4293:
4289:
4286:
4281:
4277:
4271:
4267:
4263:
4258:
4254:
4248:
4244:
4240:
4235:
4231:
4226:
4222:
4220:
4210:
4206:
4200:
4196:
4192:
4189:
4186:
4181:
4177:
4171:
4167:
4163:
4158:
4154:
4148:
4144:
4140:
4137:
4132:
4128:
4124:
4119:
4115:
4111:
4109:
4099:
4095:
4089:
4085:
4079:
4074:
4071:
4068:
4064:
4060:
4058:
4050:
4044:
4031:
4028:
4024:
4018:
4008:
3994:
3982:
3963:
3955:
3951:
3946:
3944:
3941:is a matrix,
3928:
3920:
3916:
3911:
3897:
3894:
3874:
3866:
3851:
3831:
3811:
3791:
3771:
3749:
3745:
3722:
3718:
3697:
3677:
3674:
3665:
3651:
3631:
3628:
3625:
3622:
3602:
3582:
3578:
3571:
3568:
3563:
3559:
3535:
3515:
3501:
3488:
3482:
3479:
3476:
3473:
3469:
3464:
3461:
3458:
3455:
3452:
3447:
3443:
3439:
3436:
3431:
3427:
3423:
3420:
3411:
3403:
3399:
3390:
3382:
3378:
3354:
3337:
3328:
3324:
3300:
3292:
3288:
3267:
3264:
3261:
3258:
3255:
3249:
3241:
3237:
3216:
3213:
3210:
3207:
3204:
3199:
3195:
3191:
3188:
3183:
3179:
3175:
3172:
3166:
3158:
3154:
3144:
3130:
3127:
3124:
3121:
3118:
3113:
3109:
3096:
3082:
3079:
3076:
3056:
3053:
3050:
3047:
3044:
3039:
3035:
3031:
3028:
3023:
3019:
3009:
3007:
2987:
2984:
2981:
2958:
2952:
2943:
2929:
2926:
2921:
2917:
2913:
2910:
2907:
2902:
2898:
2894:
2891:
2888:
2883:
2879:
2875:
2872:
2869:
2864:
2860:
2839:
2836:
2833:
2828:
2824:
2820:
2817:
2814:
2809:
2805:
2801:
2798:
2795:
2792:
2787:
2783:
2779:
2776:
2773:
2768:
2764:
2754:
2740:
2737:
2731:
2725:
2702:
2696:
2688:
2683:
2665:
2662:
2659:
2636:
2630:
2611:
2608:
2605:
2602:
2595:
2591:
2589:
2584:
2570:
2567:
2564:
2544:
2541:
2538:
2535:
2532:
2527:
2523:
2519:
2516:
2511:
2507:
2503:
2500:
2494:
2488:
2474:
2458:
2454:
2431:
2427:
2421:
2417:
2413:
2408:
2405:
2402:
2398:
2394:
2389:
2386:
2383:
2379:
2356:
2352:
2329:
2325:
2304:
2295:
2279:
2275:
2271:
2268:
2245:
2239:
2219:
2199:
2196:
2191:
2187:
2163:
2157:
2150:is a root of
2135:
2131:
2105:
2101:
2094:
2072:
2068:
2042:
2038:
2034:
2031:
2024:
2017:
2011:
1998:
1991:
1989:
1973:
1969:
1965:
1961:
1955:
1951:
1947:
1944:
1940:
1935:
1929:
1926:
1923:
1919:
1913:
1909:
1905:
1900:
1897:
1894:
1890:
1884:
1881:
1878:
1874:
1870:
1867:
1864:
1859:
1855:
1849:
1845:
1841:
1836:
1832:
1826:
1822:
1818:
1815:
1810:
1806:
1802:
1797:
1793:
1788:
1784:
1778:
1772:
1765:
1764:
1761:
1758:
1741:
1736:
1732:
1728:
1726:
1716:
1712:
1706:
1702:
1698:
1693:
1689:
1685:
1683:
1675:
1645:
1642:
1639:
1635:
1629:
1625:
1621:
1618:
1615:
1610:
1606:
1593:
1589:
1585:
1580:
1576:
1563:
1559:
1555:
1550:
1546:
1542:
1540:
1518:
1510:
1506:
1500:
1496:
1492:
1487:
1484:
1481:
1477:
1468:
1464:
1460:
1457:
1454:
1449:
1445:
1432:
1428:
1424:
1419:
1415:
1402:
1398:
1394:
1389:
1385:
1381:
1379:
1369:
1365:
1358:
1332:
1328:
1318:
1305:
1278:
1270:
1266:
1261:
1258:
1253:
1250:
1247:
1243:
1236:
1233:
1230:
1227:
1222:
1218:
1207:
1204:
1199:
1195:
1184:
1181:
1176:
1172:
1161:
1158:
1153:
1149:
1145:
1139:
1133:
1124:
1105:
1101:
1094:
1072:
1068:
1054:
1047:
1045:
1027:
1022:
1018:
1012:
1008:
1004:
999:
995:
991:
989:
982:
978:
968:
964:
958:
954:
950:
945:
941:
937:
935:
928:
924:
916:
897:
893:
887:
883:
879:
874:
871:
868:
864:
860:
858:
851:
848:
845:
841:
831:
827:
823:
821:
814:
810:
798:
797:
794:
792:
787:
774:
771:
749:
745:
722:
718:
714:
711:
708:
703:
699:
678:
673:
669:
663:
659:
655:
652:
649:
644:
640:
634:
630:
626:
621:
617:
611:
607:
603:
600:
595:
591:
587:
582:
578:
574:
569:
565:
559:
555:
549:
544:
541:
538:
534:
530:
524:
518:
504:
502:
498:
490:
485:
467:
447:
435:
430:
413:
389:
381:
377:
372:
369:
364:
361:
358:
354:
347:
344:
341:
338:
333:
329:
318:
315:
310:
306:
295:
292:
287:
283:
272:
269:
264:
260:
252:
243:
239:
233:
229:
225:
222:
219:
214:
210:
204:
200:
196:
191:
187:
181:
177:
173:
170:
165:
161:
157:
152:
148:
134:
130:
129:Horner's rule
125:
122:
118:
114:
110:
106:
102:
98:
87:
84:
76:
73:November 2018
66:
61:
59:
53:
49:
48:
41:
32:
31:
19:
18:Horner scheme
9975:Discriminant
9959:
9894:Multivariate
9715:
9694:
9672:
9658:
9635:
9614:
9585:
9579:
9550:
9546:
9535:. Retrieved
9516:
9503:
9488:. Retrieved
9484:the original
9469:
9466:"Chapter 13"
9456:
9447:
9428:
9415:
9383:
9379:
9360:. Retrieved
9353:the original
9343:
9323:
9304:
9300:
9285:. Retrieved
9278:the original
9249:cite journal
9230:
9226:
9195:. Retrieved
9176:
9172:
9140:
9136:
9113:
9108:, p. 77
9101:
9089:
9077:
9068:
9058:
9044:
9015:
9003:
8976:
8965:
8953:
8941:
8929:
8917:
8905:
8893:
8881:
8870:
8854:
8802:to evaluate
8770:
8762:
8757:
8753:
8747:
8733:
8729:
8724:
8708:
8703:Song dynasty
8676:
8653:Isaac Newton
8637:
8626:
8620:
8612:
8604:
8598:
8592:
8141:
7494:
6999:
6740:
6723:
6291:
6273:
6269:
6264:
6260:
6256:
6250:
6237:
6218:
5953:
5497:
5486:
5468:
5461:
5453:
5449:
5447:
5156:
5154:
5114:
5110:
5041:
5017:
4767:
4765:
4032:
4020:
3980:
3947:
3912:
3666:
3507:
3358:
3145:
3100:
3010:
2944:
2755:
2684:
2614:
2609:
2606:
2603:
2600:
2593:
2590:as follows:
2585:
2480:
2296:
2003:
1992:
1759:
1319:
1125:
1059:
1048:
793:as follows:
788:
510:
492:
488:
486:
431:
132:
128:
126:
108:
104:
94:
79:
70:
63:Please help
55:
44:
10006:Polynomials
9924:Homogeneous
9919:Square-free
9914:Irreducible
9779:Polynomials
9553:: 105β136.
9450:. Springer.
9106:Mikami 1913
9094:Temple 1986
9020:Cajori 1911
9008:Fuller 1999
8996:Horner 1819
8946:Higham 2002
8860:Qin Jiushao
8818:BΓ©zier form
8761:concluded:
8726:Qin Jiushao
8715:Han dynasty
8673:Qin Jiushao
8572:40642560000
8530:Qin Jiushao
6352:with zeros
6284:are known.
4007:additions.
2685:But by the
791:recursively
133:nested form
97:mathematics
67:if you can.
9995:Categories
9884:Univariate
9537:2016-08-23
9490:2016-08-23
9362:2012-12-10
9287:2018-05-17
9197:2012-03-04
9128:References
9063:paragraph.
8958:Kress 1991
8922:Knuth 1997
8663:Zhu Shijie
7207:to obtain
7062:to obtain
6627:to obtain
6447:such that
6332:of degree
5494:Derivation
5121:(base 2),
4015:See also:
3919:Victor Pan
3504:Efficiency
440:with only
436:of degree
434:polynomial
9970:Resultant
9909:Trinomial
9889:Bivariate
9722:EMS Press
9681:cite book
9567:250869179
9408:186210512
9345:Principle
9307:: 29β51.
9267:Kahan, W.
8728:, in his
8569:−
8540:−
8400:≈
8296:−
8225:−
8205:−
7975:⋮
7959:⋮
7919:−
7896:−
7858:−
7835:−
7710:⋯
7592:∑
7547:−
7518:−
7288:−
7279:−
7189:−
7150:−
7134:−
7044:−
6983:−
6942:−
6926:−
6852:−
6837:−
6822:−
6676:−
6643:−
6602:−
6392:⋯
6381:−
6146:−
6112:−
5403:−
5379:−
5351:−
5323:−
5292:−
5267:−
5238:−
5222:−
5109:. Then,
4963:−
4945:∑
4904:⌋
4890:⌊
4875:∑
4856:−
4838:∑
4794:∑
4643:⌋
4629:⌊
4614:∑
4576:⌋
4562:⌊
4547:∑
4528:⋯
4450:⋯
4368:⋯
4290:⋯
4190:⋯
4065:∑
3629:−
3480:−
3465:−
3453:−
3437:−
3280:. Divide
3265:−
3214:−
3189:−
3119:−
3080:−
3054:−
3029:−
2985:−
2837:−
2796:−
2663:−
2542:−
2517:−
2481:Evaluate
2406:−
2387:−
2272:−
2035:−
1948:−
1927:−
1898:−
1882:−
1868:⋯
1676:⋮
1643:−
1619:⋯
1519:⋯
1485:−
1458:⋯
1279:⋯
1251:−
1231:⋯
917:⋮
872:−
849:−
712:…
653:⋯
535:∑
390:⋯
362:−
342:⋯
223:⋯
9955:Division
9904:Binomial
9899:Monomial
9739:Archived
9604:52859619
9531:Archived
9427:(1997).
9326:. SIAM.
9269:(2000).
9221:(2009).
9191:Archived
9167:(1911).
8910:Pan 1966
8875:Pan 1966
8808:B-spline
8785:See also
8773:, while
8740:Jia Xian
8699:Jia Xian
8629:Arbogast
8609:reviewer
8591:result:
8466:′
5046:with no
2718:. Thus,
2619:-value (
2477:Examples
45:require
10001:Algebra
9724:, 2001
9697:(1852).
8804:splines
8767:Liu Hui
8719:Liu Hui
8686:Persian
8675:in his
8655:in 1669
8521:History
6729:Example
5194:0.00101
5174:0.15625
5151:Example
4770:parts:
3011:Divide
2586:We use
2179:, then
47:cleanup
9792:degree
9642:
9623:
9602:
9565:
9523:
9476:
9459:(212).
9435:
9406:
9400:107508
9398:
9330:
9157:604533
9155:
8556:763200
7972:
7969:
7956:
7953:
6491:Using
6278:matrix
5444:Method
5083:, and
1673:
1670:
1303:
914:
911:
908:
691:where
9600:S2CID
9563:S2CID
9404:S2CID
9396:JSTOR
9356:(PDF)
9349:(PDF)
9281:(PDF)
9274:(PDF)
9153:JSTOR
8846:Notes
6986:5040.
6276:is a
5020:-way
2060:with
1060:Then
9781:and
9687:link
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635:3
631:a
627:+
622:2
618:x
612:2
608:a
604:+
601:x
596:1
592:a
588:+
583:0
579:a
575:=
570:i
566:x
560:i
556:a
550:n
545:0
542:=
539:i
531:=
528:)
525:x
522:(
519:p
482:n
468:n
448:n
438:n
414:.
409:)
402:)
395:)
387:)
382:n
378:a
373:x
370:+
365:1
359:n
355:a
351:(
348:x
345:+
339:+
334:3
330:a
324:(
319:x
316:+
311:2
307:a
301:(
296:x
293:+
288:1
284:a
278:(
273:x
270:+
265:0
261:a
253:=
244:n
240:x
234:n
230:a
226:+
220:+
215:3
211:x
205:3
201:a
197:+
192:2
188:x
182:2
178:a
174:+
171:x
166:1
162:a
158:+
153:0
149:a
86:)
80:(
75:)
71:(
60:.
20:)
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