Horner scheme

Horner scheme
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In numerical analysis, the Horner scheme or Horner algorithm, named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. Horner's method describes a manual process by which one may approximate the roots of a polynomial equation. The Horner scheme can also be viewed as a fast algorithm for dividing a polynomial by a linear polynomial with Ruffini's rule.

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1 Description of the algorithm
2 Examples
- 2.1 Floating point multiplication and division
3 Application
4 Efficiency
5 History
6 See also
7 References
8 External links

Description of the algorithm

Given the polynomial

$p(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n,$

where $a_0, \ldots, a_n$ are real numbers, we wish to evaluate the polynomial at a specific value of $x\,\!$ , say $x_0\,\!$ .

To accomplish this, we define a new sequence of constants as follows:


$b_n\,\!$	$:=\,\!$	$a_n\,\!$
$b_{n-1}\,\!$	$:=\,\!$	$a_{n-1} + b_n x_0\,\!$
	$\vdots$
$b_0\,\!$	$:=\,\!$	$a_0 + b_1 x_0\,\!$

Then $b_0\,\!$ is the value of $p(x_0)\,\!$ .

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

$p(x) = a_0 + x(a_1 + x(a_2 + \cdots x(a_{n-1} + a_n x)\dots))$

Thus, by iteratively substituting the $b i$ into the expression,


$p(x_0)\,\!$	$=\,\!$	$a_0 + x_0(a_1 + x_0(a_2 + \cdots x_0(a_{n-1} + b_n x_0)\dots))$
	$=\,\!$	$a_0 + x_0(a_1 + x_0(a_2 + \cdots x_0(b_{n-1})\dots))$
	$\vdots$
	$=\,\!$	$a_0 + x_0(b_1)\,\!$
	$=\,\!$	$b_0\,\!$

Examples

Evaluate $f_1(x)=2x^3-6x^2+2x-1\,$ for $x=3\;$ . By repeatedly factoring out $x$ , $f 1$ may be rewritten as $x(x(2x-6)+2)-1\;$ . We use a synthetic diagram to organize these calculations and make the process faster.

 $x 0$  |    $x 3$      $x 2$       $x 1$     $x 0$

 3 |   2    -6     2    -1
   |         6     0     6    
   |----------------------
       2     0     2     5

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 x-value (3 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. The answer is 5.

As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial that is the quotient of f₁/(x-3). The remainder is 5. This makes Horner's method useful for polynomial long division.

Divide $x^3-6x^2+11x-6\,$ by $x - 2$ :

 2 |   1    -6    11    -6
   |         2    -8     6    
   |----------------------
       1    -4     3     0

The quotient is $x 2 - 4 x + 3$ .

Let $f_1(x)=4x^4-6x^3+3x-5\,$ and $f_2(x)=2x-1\,$ . Divide $f_1(x)\,$ by $f_2\,(x)$ using Horner's scheme.

  2 |  4    -6    0    3   |   -5
---------------------------|------
  1 |        2   -2   -1   |    1
    |                      |  
    |----------------------|-------
       2    -2    -1   1   |   -4

The third row is the sum of the first two rows, divided by 2. Each entry in the second row is the product of 1 with the third-row entry to the left. The answer is

$\frac{f_1(x)}{f_2(x)}=2x^3-2x^2-x+1-\frac{4}{(2x-1)}.$

Floating point multiplication and division

Horner's method is a fast, code-efficient method for multiplication and division of binary numbers on a microcontroller with no math coprocessor. One of the binary numbers to be multiplied is represented as a trivial polynomial, where, (using the above notation): a_i = 1, and x = 2. Then, x (or x to some power) is repeatedly factored out. In this binary numeral system (base 2), x = 2, so powers of 2 are repeatedly factored out.

Example

For example, to find the product of two numbers, (0.15625) and "m":

(0.15625) m = (0.00101 b) m = (2 - 3 + 2 - 5) m = (2 - 3) m + (2 - 5) m = 2 - 3 (m + (2 - 2) m) = 2 - 3 (m + 2 - 2 (m))

Method

To find the product of two binary numbers, "d" and "m".

1. A register holding the intermediate result is initialized to (d).
2. Begin in (m) with the least significant (rightmost) non-zero bit,
- 2b. 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.
- 2c. Using that value, perform a right-shift operation by that number of bits on the register holding the intermediate result
3. 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 (m).

Derivation

In general, for a binary number with bit values: ( $d 3 d 2 d 1 d 0$ ) the product is:

(d 323 + d 222 + d 121 + d 020) m = d 323 m + d 222 m + d 121 m + d 020 m

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 division by zero is not an issue, despite this implication in the factored equation:

$= d_0(m + 2 \frac{d_1}{d_0} (m + 2 \frac{d_2}{d_1} (m + 2 \frac{d_3}{d_2} (m))))$

The denominators all equal one (or the term is absent), so this reduces to:

= d 0 (m + 2 d 1 (m + 2 d 2 (m + 2 d 3 (m))))

or equivalently (as consistent with the "method" described above):

= d 3 (m + 2 - 1 d 2 (m + 2 - 1 d 1 (m + d 0 (m))))

In binary (base 2) math, multiplication by a power of 2 is merely an register shift operation. Thus, multiplying by 2 is calculated in base-2 by a right arithmetic shift. The factor (2^-1) is a right arithmetic shift, a (0) results in no operation (since 2⁰ = 1, is the multiplicative identity element), and a (2¹) results in a left arithmetic shift. The multiplication product can now be quickly calculated using only arithmetic shift operations, addition and subtraction.

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 "canonical signed digit" (CSD) form is used), and uses only 20% of the code space (Kripasagar 62).

Application

The Horner scheme is often used to convert between different positional numeral systems — in which case x is the base of the number system, and the a_i coefficients are the digits of the base-x representation of a given number — and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater.

Efficiency

Evaluation using the monomial form of a degree-n polynomial requires at most n additions and (n² + n)/2 multiplications, if powers are calculated by repeated multiplication and each monomial is evaluated individually. (This can be reduced to n additions and 2n + 1 multiplications by evaluating the powers of x iteratively.) If numerical data are represented in terms of digits (or bits), then the naive algorithm also entails storing approximately 2n times the number of bits of x (the evaluated polynomial has approximate magnitude xⁿ, and one must also store xⁿ itself). By contrast, Horner's scheme requires only n additions and n multiplications, and its storage requirements are only n times the number of bits of x. Alternatively, Horner's scheme can be computed with n fused multiply-adds.

It has been shown that the Horner scheme is optimal, in the sense that any algorithm to evaluate an arbitrary polynomial must use at least as many operations. That the number of additions required is minimal was shown by Alexander Ostrowski in 1954; that the number of multiplications is minimal by Victor Pan, in 1966. When x is a matrix, the Horner scheme is not optimal.

This assumes that the polynomial is evaluated in monomial form and no preconditioning 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 faster algorithms are possible. They involve a transformation of the representation of the polynomial. In general, a degree-n polynomial can be evaluated using only ${\scriptstyle{\left\lfloor \frac{n}{2} \right\rfloor + 2}}$ multiplications and n additions (see Knuth: The Art of Computer Programming, Vol.2).

History

Even though the algorithm is named after William George Horner, who described it in 1819, the method was already known to Isaac Newton in 1669, the Chinese mathematician Qin Jiushao in his Mathematical Treatise in Nine Sections in the 13th century, and even earlier to the Persian Muslim mathematician Sharaf al-Dīn al-Tūsī in the 12th century.^[1] The earliest use of Horner's scheme was in The Nine Chapters on the Mathematical Art, a Chinese work of the Han Dynasty (202 BC – 220 AD) edited by Liu Hui (fl. 3rd century).^[2]

References

^ J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2), p. 304-309.
^ Temple, Robert. (1986). The Genius of China: 3,000 Years of Science, Discovery, and Invention. With a forward by Joseph Needham. New York: Simon and Schuster, Inc. ISBN 0671620282. Page 142.

William George Horner. A new method of solving numerical equations of all orders, by continuous approximation. In Philosophical Transactions of the Royal Society of London, pp. 308-335, July 1819.
Spiegel, Murray R. (1956). Schaum's Outline of Theory and Problems of College Algebra. McGraw-Hill Book Company.
Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Pages 486–488 in section 4.6.4.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Problem 2-3 (pg.39) and page 823 of section 30.1: Representation of polynomials.
Kripasagar, Venkat (March 2008). "Efficient Micro Mathematics - Multiplication and Division Techniques for MCUs". Circuit Cellar magazine (212): p. 60.

External links

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