Definition:Horner's Method
Disambiguation
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Horner's Method may refer to:
Ruffini-Horner Method
The Ruffini-Horner method is a technique for finding the roots of polynomial equations in $1$ real variable.
Let $E_0$ be the polynomial equation in $x$:
- $\map p x = 0$
Suppose that a root $x_0$ being sought is the positive real number expressed as the decimal expansion:
- $x_0 = \sqbrk {abc.def}$
The process begins by finding $a$ by inspection.
We then form a new equation $E_1$ whose roots are $100 a$ less than those of $E_0$.
This will have a root $x_1$ in the form:
- $x_1 = \sqbrk {bc.def}$
Similarly, $b$ is found by inspection.
We then form a new equation $E_2$ whose roots are $10 b$ less than those of $E_1$.
The process continues for as many digits accuracy as required.
Horner's Rule
Let $\map p x$ be a polynomial.
- $\map p x = a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0$
Then $\map p x$ can be expressed in the following form:
- $\map p x = \paren {\cdots \paren {\paren {a_n x + a_{n - 1} } x + a_{n - 2} } x + \cdots + a_1} x + a_0$
Source of Name
This entry was named for William George Horner.