This is optimal, since there are polynomials of degree that cannot be evaluated with fewer arithmetic operations Alternatively, Horner's method also refers to a method for approximating the roots of polynomials, described by Horner in 1819.
Substituting y = x in this method gives d_1 = p'(x), the derivative of p(x). == History == Horner's paper, titled "A new method of solving numerical equations of all orders, by continuous approximation", was read 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 Philosophical Transactions of the Royal Society of London for 1819 was warmly and expansively welcomed by a reviewer in the issue of The Monthly Review: or, Literary Journal for April, 1820; in comparison, a technical paper by Charles Babbage is dismissed curtly in this review.
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 this method should go to Holdred (1820). Unlike his English contemporaries, Horner drew on the Continental literature, notably the work of Arbogast.
Horner's paper in Part II of Philosophical Transactions of the Royal Society of London for 1819 was warmly and expansively welcomed by a reviewer in the issue of The Monthly Review: or, Literary Journal for April, 1820; in comparison, a technical paper by Charles Babbage is dismissed curtly in this review.
The sequence of reviews in The Monthly Review for September, 1821, concludes that Holdred was the first person to discover a direct and general practical solution of numerical equations.
Substituting y = x in this method gives d_1 = p'(x), the derivative of p(x). == History == Horner's paper, titled "A new method of solving numerical equations of all orders, by continuous approximation", was read before the Royal Society of London, at its meeting on July 1, 1819, with a sequel in 1823.
Yoshio Mikami in Development of Mathematics in China and Japan (Leipzig 1913) wrote: Ulrich Libbrecht concluded: It is obvious that this procedure is a Chinese invention ...
Smith: A Source Book in Mathematics, McGraw-Hill, 1929; Dover reprint, 2 vols, 1959. Reprinted from issues of The North China Herald (1852). == External links == Qiu Jin-Shao, Shu Shu Jiu Zhang (Cong Shu Ji Cheng ed.) For more on the root-finding application see analysis] Articles with example Python (programming language) code Articles with example MATLAB/Octave code Articles with example C code
Alexander Ostrowski proved in 1954 that the number of additions required is minimal.
Smith: A Source Book in Mathematics, McGraw-Hill, 1929; Dover reprint, 2 vols, 1959. Reprinted from issues of The North China Herald (1852). == External links == Qiu Jin-Shao, Shu Shu Jiu Zhang (Cong Shu Ji Cheng ed.) For more on the root-finding application see analysis] Articles with example Python (programming language) code Articles with example MATLAB/Octave code Articles with example C code
Victor Pan proved in 1966 that the number of multiplications is minimal.
In fact, when x is a matrix, further acceleration is possible which exploits the structure of matrix multiplication, and only \sqrt{n} instead of n multiplies are needed (at the expense of requiring more storage) using the 1973 method of Paterson and Stockmeyer. == Polynomial root finding == Using the long division algorithm in combination with Newton's method, it is possible to approximate the real roots of a polynomial.
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